Subgroups — #72311250

Order 2048.

Only the 1758 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 578 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72311180	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#147767	C₂	1	4	yes	maximal, complemented
3	C₂×#146577	C₂	1	1	yes	maximal, characteristic, complemented
4	#72306279	C₂	1	1	yes	maximal, characteristic, complemented
5	#72285407	C₂	1	8	yes	maximal, complemented
6	C₂×#10574841	C₂	1	1	yes	maximal, characteristic, complemented
7	#72256607	C₂	1	4	yes	maximal, complemented
8	C₂×#493397	C₂	1	1	yes	maximal, characteristic, complemented
9	#72240597	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#392135	C₂	1	1	yes	maximal, characteristic, complemented
11	#72213504	C₂	1	8	yes	maximal, complemented
12	C₂²×H(7,𝔽₂)	C₂²	1	1	yes	characteristic
13	C₂×#62852	C₂²	1	1	yes	characteristic
14	C₂²×#3026	C₂²	1	2	yes	complemented
15	#10574841	C₂²	1	8	yes	—
16	C₂²×#3019	C₂²	1	4	yes	—
17	C₂²×#2888	C₂²	1	1	yes	characteristic
18	#7642008	C₂²	1	2	yes	complemented
19	#7630442	C₂²	1	1	yes	characteristic
20	C₂²×#2127	C₂²	1	4	yes	complemented
21	C₂²×#2121	C₂²	1	2	yes	complemented
22	#6628897	C₂²	1	4	yes	complemented
23	#6628896	C₂²	1	4	yes	complemented
24	#6470235	C₂²	1	4	yes	complemented
25	#6470234	C₂²	1	4	yes	complemented
26	C₂×#21750	C₂²	1	2	yes	complemented
27	C₂×#20609	C₂²	1	2	yes	complemented
28	C₂×#20609	C₂²	1	4	yes	complemented
29	C₂×#20585	C₂²	1	1	yes	characteristic
30	C₂×#20585	C₂²	1	1	yes	characteristic
31	C₂×#20585	C₂²	1	1	yes	characteristic
32	#498928	C₂²	1	16	yes	complemented
33	#498927	C₂²	1	16	yes	complemented
34	#494682	C₂²	1	4	yes	complemented
35	#494681	C₂²	1	4	yes	complemented
36	#493397	C₂²	1	8	yes	complemented
37	C₂×#12574	C₂²	1	2	yes	complemented
38	C₂×#12574	C₂²	1	4	yes	complemented
39	#392135	C₂²	1	8	yes	complemented
40	#147767	C₂²	1	32	yes	complemented
41	#146577	C₂²	1	8	yes	complemented
42	C₂×H(7,𝔽₂)	D₄	1	4	yes	complemented
43	C₂×H(7,𝔽₂)	C₂³	1	4	yes	—
44	C₂²×#585	D₄	1	1	yes	characteristic, complemented
45	C₂²×#585	D₄	1	2	yes	complemented
46	C₂³×H(5,𝔽₂)	D₄	1	2	yes	complemented
47	C₂³×H(5,𝔽₂)	C₂³	1	2	yes	—
48	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
49	#62852	D₄	1	4	yes	complemented
50	#62852	C₂³	1	4	yes	—
51	C₂×#3449	D₄	1	2	yes	complemented
52	C₂²×#535	C₂³	1	2	yes	—
53	C₂²×#534	D₄	1	1	yes	characteristic
54	C₂²×#534	D₄	1	1	yes	characteristic
55	C₂²×#534	C₂³	1	1	yes	characteristic
56	C₂×#3428	D₄	1	1	yes	characteristic, complemented
57	C₂²×#458	C₂³	1	4	yes	complemented
58	C₂²×B(4,𝔽₂)	C₂³	1	4	yes	complemented
59	C₂×#3019	D₄	1	16	yes	complemented
60	C₂×#3019	C₂³	1	16	yes	—
61	C₂²×#455	D₄	1	8	yes	complemented
62	C₂²×#453	D₄	1	8	yes	complemented
63	C₂×#2888	D₄	1	4	yes	complemented
64	C₂×#2888	C₂³	1	4	yes	—
65	C₂²×#413	D₄	1	2	yes	complemented
66	C₂³×#100	D₄	1	2	yes	complemented
67	C₂³×#100	C₂³	1	2	yes	—
68	C₂³×#99	D₄	1	2	yes	complemented
69	C₂³×#99	D₄	1	2	yes	complemented
70	C₂³×#99	C₂³	1	2	yes	—
71	C₂³×#99	C₂³	1	2	yes	—
72	#36955	C₂³	1	4	yes	complemented
73	#36702	C₂³	1	4	yes	complemented
74	#35446	D₄	1	2	yes	—
75	#35446	C₂³	1	2	yes	—
76	#35346	D₄	1	2	yes	—
77	#35346	C₂³	1	2	yes	—
78	C₂×#2308	D₄	1	1	yes	characteristic
79	C₂×#2308	D₄	1	1	yes	characteristic
80	C₂×#2308	C₂³	1	1	yes	characteristic
81	C₂×#2308	C₂³	1	2	yes	—
82	C₂×#2127	C₂³	1	16	yes	complemented
83	C₂²×#352	C₂³	1	4	yes	complemented
84	C₂²×AΣL(1,𝔽₁₆)	C₂³	1	4	yes	complemented
85	#21750	C₂³	1	8	yes	complemented
86	#20609	C₂³	1	8	yes	complemented
87	#20609	C₂³	1	16	yes	complemented
88	#20585	D₄	1	4	yes	complemented
89	#20585	D₄	1	4	yes	complemented
90	#20585	C₂³	1	4	yes	—
91	#20585	C₂³	1	4	yes	—
92	#20585	C₂³	1	4	yes	—
93	C₂×#1782	D₄	1	2	yes	complemented
94	C₂×#1781	D₄	1	2	yes	complemented
95	#12574	C₂³	1	8	yes	complemented
96	#12574	C₂³	1	16	yes	complemented
97	C₂×#585	C₂×D₄	1	4	yes	—
98	C₂×#585	C₂×D₄	1	8	yes	—
99	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
100	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
101	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
102	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
103	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
104	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
105	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
106	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
107	C₂²×H(5,𝔽₂)	C₂×D₄	1	8	yes	—
108	C₂²×H(5,𝔽₂)	C₂×D₄	1	8	yes	—
109	C₂²×H(5,𝔽₂)	C₂×D₄	1	8	yes	—
110	C₂²×H(5,𝔽₂)	#41	1	2	yes	—
111	C₂²×H(5,𝔽₂)	C₂⁴	1	2	yes	—
112	C₂²×H(5,𝔽₂)	C₂⁴	1	4	yes	—
113	C₂³×#41	C₂×D₄	1	1	yes	characteristic
114	C₂³×#41	C₂×D₄	1	1	yes	characteristic
115	C₂³×#41	C₂×D₄	1	1	yes	characteristic
116	C₂³×#41	C₂×D₄	1	1	yes	characteristic
117	C₂³×#41	C₂×D₄	1	1	yes	characteristic
118	C₂³×#41	C₂×D₄	1	2	yes	—
119	C₂³×#41	#41	1	1	yes	characteristic
120	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
121	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
123	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
124	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
125	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
126	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
127	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
128	C₂⁴×D₄	#41	1	1	yes	characteristic
129	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
130	#3449	C₂×D₄	1	8	yes	—
131	C₂×#535	C₂×D₄	1	8	yes	—
132	C₂×#535	C₂×D₄	1	4	yes	—
133	C₂×#535	C₂⁴	1	4	yes	—
134	C₂²×#127	C₂×D₄	1	4	yes	—
135	C₂²×#124	C₂×D₄	1	4	yes	—
136	C₂²×#123	C₂×D₄	1	1	yes	characteristic
137	C₂²×#123	C₂×D₄	1	2	yes	—
138	C₂²×#123	C₂×D₄	1	2	yes	—
139	C₂²×#123	C₂⁴	1	1	yes	characteristic
140	C₂²×#120	C₂×D₄	1	1	yes	characteristic
141	C₂²×#120	C₂×D₄	1	2	yes	—
142	C₂²×#120	C₂×D₄	1	2	yes	—
143	C₂²×#120	C₂⁴	1	1	yes	characteristic
144	#3428	C₂×D₄	1	4	yes	—
145	C₂²×#117	C₂×D₄	1	2	yes	—
146	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
147	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
148	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
149	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
150	C₂×#455	C₂×D₄	1	32	yes	—
151	C₂×#453	C₂×D₄	1	32	yes	—
152	C₂×#413	C₂×D₄	1	8	yes	—
153	C₂²×#100	C₂×D₄	1	4	yes	—
154	C₂²×#100	C₂×D₄	1	4	yes	—
155	C₂²×#100	C₂×D₄	1	8	yes	—
156	C₂²×#100	C₂×D₄	1	8	yes	—
157	C₂²×#100	C₂×D₄	1	8	yes	—
158	C₂²×#100	C₂⁴	1	4	yes	—
159	C₂²×#99	C₂×D₄	1	8	yes	—
160	C₂²×#99	C₂×D₄	1	8	yes	—
161	C₂²×#99	C₂×D₄	1	4	yes	—
162	C₂²×#99	C₂×D₄	1	4	yes	—
163	C₂²×#99	C₂⁴	1	4	yes	—
164	C₂²×#99	C₂⁴	1	4	yes	—
165	#2308	C₂×D₄	1	2	yes	—
166	#2308	C₂×D₄	1	2	yes	—
167	#2308	C₂×D₄	1	2	yes	—
168	#2308	C₂×D₄	1	2	yes	—
169	#2308	C₂×D₄	1	2	yes	—
170	#2308	C₂×D₄	1	4	yes	—
171	#2308	C₂⁴	1	2	yes	—
172	#2308	C₂⁴	1	4	yes	—
173	#1782	C₂×D₄	1	8	yes	—
174	#1781	C₂×D₄	1	8	yes	—
175	C₂⁶	#120	1	1	yes	characteristic
176	C₂⁶	#121	1	1	yes	characteristic
177	C₂⁶	#121	1	1	yes	characteristic
178	C₂⁶	#127	1	1	yes	characteristic
179	C₂⁶	C₂²×D₄	1	1	yes	characteristic
180	C₂⁶	C₂²×D₄	1	1	yes	characteristic
181	C₂⁶	C₂²×D₄	1	1	yes	characteristic
182	C₂×H(5,𝔽₂)	#120	1	8	yes	—
183	C₂×H(5,𝔽₂)	#120	1	8	yes	—
184	C₂×H(5,𝔽₂)	#120	1	16	yes	—
185	C₂×H(5,𝔽₂)	C₂²×D₄	1	8	yes	—
186	C₂×H(5,𝔽₂)	C₂²×D₄	1	16	yes	—
187	C₂²×#41	#120	1	2	yes	—
188	C₂²×#41	#120	1	2	yes	—
189	C₂²×#41	#120	1	2	yes	—
190	C₂²×#41	#120	1	2	yes	—
191	C₂²×#41	#120	1	2	yes	—
192	C₂²×#41	#120	1	2	yes	—
193	C₂²×#41	#120	1	2	yes	—
194	C₂²×#41	#120	1	2	yes	—
195	C₂²×#41	#120	1	4	yes	complemented
196	C₂²×#41	#121	1	2	yes	—
197	C₂²×#41	#121	1	2	yes	—
198	C₂²×#41	#121	1	2	yes	—
199	C₂²×#41	#121	1	2	yes	—
200	C₂²×#41	#121	1	2	yes	—
201	C₂²×#41	#121	1	2	yes	—
202	C₂²×#41	#121	1	2	yes	—
203	C₂²×#41	#121	1	2	yes	—
204	C₂²×#41	#127	1	2	yes	—
205	C₂²×#41	#127	1	2	yes	—
206	C₂²×#41	C₂²×D₄	1	2	yes	—
207	C₂²×#41	C₂²×D₄	1	2	yes	—
208	C₂²×#41	C₂²×D₄	1	2	yes	—
209	C₂²×#41	C₂²×D₄	1	2	yes	—
210	C₂²×#41	C₂²×D₄	1	2	yes	—
211	C₂²×#41	C₂²×D₄	1	4	yes	—
212	C₂²×#41	C₂×#41	1	2	yes	—
213	C₂³×Q₈	#120	1	1	yes	characteristic
214	C₂³×Q₈	#120	1	2	yes	complemented
215	C₂³×Q₈	#121	1	1	yes	characteristic
216	C₂³×D₄	#120	1	1	yes	characteristic
217	C₂³×D₄	#120	1	2	yes	—
218	C₂³×D₄	#120	1	2	yes	—
219	C₂³×D₄	#120	1	2	yes	—
220	C₂³×D₄	#120	1	1	yes	characteristic
221	C₂³×D₄	#120	1	1	yes	characteristic
222	C₂³×D₄	#120	1	1	yes	characteristic
223	C₂³×D₄	#120	1	1	yes	characteristic
224	C₂³×D₄	#120	1	2	yes	—
225	C₂³×D₄	#120	1	2	yes	—
226	C₂³×D₄	#120	1	2	yes	—
227	C₂³×D₄	#120	1	2	yes	—
228	C₂³×D₄	#120	1	2	yes	—
229	C₂³×D₄	#120	1	2	yes	—
230	C₂³×D₄	#120	1	2	yes	—
231	C₂³×D₄	#120	1	1	yes	characteristic
232	C₂³×D₄	#120	1	1	yes	characteristic
233	C₂³×D₄	#120	1	2	yes	complemented
234	C₂³×D₄	#121	1	1	yes	characteristic
235	C₂³×D₄	#121	1	2	yes	—
236	C₂³×D₄	#121	1	2	yes	—
237	C₂³×D₄	#121	1	2	yes	—
238	C₂³×D₄	#121	1	2	yes	—
239	C₂³×D₄	#121	1	1	yes	characteristic
240	C₂³×D₄	#121	1	1	yes	characteristic
241	C₂³×D₄	#121	1	1	yes	characteristic
242	C₂³×D₄	#121	1	1	yes	characteristic
243	C₂³×D₄	#121	1	2	yes	—
244	C₂³×D₄	#121	1	2	yes	—
245	C₂³×D₄	#121	1	2	yes	—
246	C₂³×D₄	#121	1	2	yes	—
247	C₂³×D₄	#121	1	2	yes	—
248	C₂³×D₄	#121	1	2	yes	—
249	C₂³×D₄	#127	1	2	yes	—
250	C₂³×D₄	#127	1	1	yes	characteristic
251	C₂³×D₄	#127	1	1	yes	characteristic
252	C₂³×D₄	#127	1	2	yes	—
253	C₂³×D₄	#127	1	2	yes	—
254	C₂³×D₄	#127	1	2	yes	—
255	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
256	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
257	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
258	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
259	C₂³×D₄	C₂²×D₄	1	4	yes	—
260	C₂³×D₄	C₂²×D₄	1	4	yes	—
261	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
262	C₂³×D₄	C₂²×D₄	1	2	yes	—
263	C₂³×D₄	C₂²×D₄	1	2	yes	—
264	C₂³×D₄	C₂²×D₄	1	2	yes	—
265	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
266	C₂³×D₄	C₂²×D₄	1	2	yes	—
267	C₂³×D₄	C₂²×D₄	1	2	yes	—
268	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
269	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
270	C₂³×D₄	C₂²×D₄	1	2	yes	—
271	C₂³×D₄	C₂²×D₄	1	2	yes	—
272	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
273	C₂³×D₄	C₂×#41	1	1	yes	characteristic
274	C₂³×D₄	C₂×#41	1	1	yes	characteristic
275	C₂³×D₄	C₂×#41	1	2	yes	—
276	C₂³×D₄	H(5,𝔽₂)	1	4	yes	—
277	C₂³×D₄	H(5,𝔽₂)	1	4	yes	—
278	C₂³×D₄	C₂⁵	1	1	yes	characteristic
279	C₂⁴×C₄	#120	1	1	yes	characteristic
280	C₂⁴×C₄	#120	1	1	yes	characteristic
281	C₂⁴×C₄	#120	1	1	yes	characteristic
282	C₂⁴×C₄	#121	1	1	yes	characteristic
283	C₂⁴×C₄	#121	1	1	yes	characteristic
284	C₂⁴×C₄	#121	1	1	yes	characteristic
285	C₂⁴×C₄	#121	1	1	yes	characteristic
286	C₂⁴×C₄	#127	1	1	yes	characteristic
287	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
288	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
289	C₂×#127	#120	1	8	yes	complemented
290	C₂×#127	C₂²×D₄	1	8	yes	—
291	C₂×#124	#120	1	8	yes	complemented
292	C₂×#124	C₂²×D₄	1	8	yes	—
293	C₂×#117	#120	1	4	yes	complemented
294	C₂×#117	C₂²×D₄	1	4	yes	—
295	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
296	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
297	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
298	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
299	C₂²×C₂²⋊C₄	C₂²×D₄	1	8	yes	—
300	C₂²×C₄²	#120	1	4	yes	complemented
301	C₂×#100	#120	1	8	yes	complemented
302	C₂×#100	#120	1	8	yes	complemented
303	C₂×#100	#120	1	16	yes	complemented
304	C₂×#100	C₂²×D₄	1	8	yes	—
305	C₂×#100	C₂²×D₄	1	16	yes	—
306	C₂×#99	C₂²×D₄	1	8	yes	—
307	C₂⁵	#392	1	1	yes	characteristic
308	C₂⁵	#392	1	1	yes	characteristic
309	C₂⁵	#392	1	2	yes	—
310	C₂⁵	#392	1	2	yes	—
311	C₂⁵	#392	1	2	yes	—
312	C₂⁵	#392	1	2	yes	—
313	C₂⁵	#392	1	1	yes	characteristic
314	C₂⁵	#392	1	1	yes	characteristic
315	C₂⁵	C₂×#120	1	1	yes	characteristic
316	C₂⁵	C₂×#120	1	1	yes	characteristic
317	C₂⁵	C₂×#120	1	1	yes	characteristic
318	C₂⁵	C₂×#120	1	1	yes	characteristic
319	C₂⁵	C₂×#120	1	2	yes	—
320	C₂⁵	C₂×#120	1	1	yes	characteristic
321	C₂⁵	C₂×#121	1	1	yes	characteristic
322	C₂⁵	C₂×#121	1	1	yes	characteristic
323	C₂⁵	C₂×#121	1	2	yes	—
324	C₂⁵	C₂×#121	1	2	yes	—
325	C₂⁵	C₂×#121	1	1	yes	characteristic
326	C₂⁵	C₂×#121	1	1	yes	characteristic
327	C₂⁵	C₂×#127	1	2	yes	—
328	C₂⁵	C₂×#127	1	1	yes	characteristic
329	C₂⁵	C₂×#127	1	1	yes	characteristic
330	C₂⁵	D₄²	1	4	yes	—
331	C₂⁵	D₄²	1	4	yes	—
332	C₂⁵	D₄²	1	4	yes	—
333	C₂⁵	D₄²	1	4	yes	—
334	C₂⁵	C₂³×D₄	1	1	yes	characteristic
335	C₂⁵	C₂³×D₄	1	1	yes	characteristic
336	H(5,𝔽₂)	C₂×#120	1	16	yes	—
337	C₂²×Q₈	#392	1	2	yes	—
338	C₂²×Q₈	#392	1	2	yes	—
339	C₂²×Q₈	C₂×#120	1	2	yes	—
340	C₂²×Q₈	C₂×#120	1	4	yes	—
341	C₂²×Q₈	C₂×#121	1	2	yes	—
342	C₂²×D₄	#392	1	2	yes	—
343	C₂²×D₄	#392	1	2	yes	—
344	C₂²×D₄	#392	1	2	yes	—
345	C₂²×D₄	#392	1	2	yes	—
346	C₂²×D₄	#392	1	2	yes	—
347	C₂²×D₄	#392	1	2	yes	—
348	C₂²×D₄	#392	1	2	yes	—
349	C₂²×D₄	#392	1	2	yes	—
350	C₂²×D₄	#392	1	2	yes	—
351	C₂²×D₄	#392	1	2	yes	—
352	C₂²×D₄	#392	1	2	yes	—
353	C₂²×D₄	#392	1	2	yes	—
354	C₂²×D₄	#392	1	2	yes	—
355	C₂²×D₄	#392	1	2	yes	—
356	C₂²×D₄	B(4,𝔽₂)	1	8	yes	—
357	C₂²×D₄	B(4,𝔽₂)	1	8	yes	—
358	C₂²×D₄	C₂×#120	1	2	yes	—
359	C₂²×D₄	C₂×#120	1	2	yes	—
360	C₂²×D₄	C₂×#120	1	4	yes	—
361	C₂²×D₄	C₂×#120	1	2	yes	—
362	C₂²×D₄	C₂×#120	1	2	yes	—
363	C₂²×D₄	C₂×#120	1	4	yes	—
364	C₂²×D₄	C₂×#120	1	1	yes	characteristic
365	C₂²×D₄	C₂×#120	1	1	yes	characteristic
366	C₂²×D₄	C₂×#120	1	2	yes	—
367	C₂²×D₄	C₂×#120	1	2	yes	—
368	C₂²×D₄	C₂×#120	1	2	yes	—
369	C₂²×D₄	C₂×#120	1	2	yes	—
370	C₂²×D₄	C₂×#120	1	2	yes	—
371	C₂²×D₄	C₂×#120	1	2	yes	—
372	C₂²×D₄	C₂×#120	1	4	yes	—
373	C₂²×D₄	C₂×#120	1	1	yes	characteristic
374	C₂²×D₄	C₂×#120	1	1	yes	characteristic
375	C₂²×D₄	C₂×#120	1	4	yes	—
376	C₂²×D₄	C₂×#120	1	2	yes	—
377	C₂²×D₄	C₂×#121	1	1	yes	characteristic
378	C₂²×D₄	C₂×#121	1	1	yes	characteristic
379	C₂²×D₄	C₂×#121	1	1	yes	characteristic
380	C₂²×D₄	C₂×#121	1	1	yes	characteristic
381	C₂²×D₄	C₂×#121	1	2	yes	—
382	C₂²×D₄	C₂×#121	1	2	yes	—
383	C₂²×D₄	C₂×#121	1	2	yes	—
384	C₂²×D₄	C₂×#121	1	2	yes	—
385	C₂²×D₄	C₂×#121	1	1	yes	characteristic
386	C₂²×D₄	C₂×#121	1	1	yes	characteristic
387	C₂²×D₄	C₂×#121	1	2	yes	—
388	C₂²×D₄	C₂×#127	1	1	yes	characteristic
389	C₂²×D₄	C₂×#127	1	1	yes	characteristic
390	C₂²×D₄	C₂×#127	1	2	yes	—
391	C₂²×D₄	C₂×#127	1	2	yes	—
392	C₂²×D₄	#534	1	2	yes	—
393	C₂²×D₄	#534	1	2	yes	—
394	C₂²×D₄	#534	1	4	yes	—
395	C₂²×D₄	#534	1	4	yes	—
396	C₂²×D₄	#535	1	2	yes	—
397	C₂²×D₄	#535	1	2	yes	—
398	C₂²×D₄	#535	1	4	yes	—
399	C₂²×D₄	D₄²	1	4	yes	—
400	C₂²×D₄	D₄²	1	4	yes	—
401	C₂²×D₄	D₄²	1	4	yes	—
402	C₂²×D₄	D₄²	1	4	yes	—
403	C₂²×D₄	D₄²	1	4	yes	—
404	C₂²×D₄	D₄²	1	4	yes	—
405	C₂²×D₄	D₄²	1	4	yes	—
406	C₂²×D₄	D₄²	1	4	yes	—
407	C₂²×D₄	#546	1	4	yes	—
408	C₂²×D₄	#546	1	4	yes	—
409	C₂²×D₄	#546	1	4	yes	—
410	C₂²×D₄	#546	1	4	yes	—
411	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
412	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
413	C₂²×D₄	C₂³×D₄	1	4	yes	—
414	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
415	C₂²×D₄	C₂²×#41	1	1	yes	characteristic
416	C₂²×D₄	C₂×H(5,𝔽₂)	1	4	yes	—
417	C₂³×C₄	#392	1	1	yes	characteristic
418	C₂³×C₄	#392	1	1	yes	characteristic
419	C₂³×C₄	#392	1	1	yes	characteristic
420	C₂³×C₄	#392	1	2	yes	—
421	C₂³×C₄	#392	1	2	yes	—
422	C₂³×C₄	#392	1	2	yes	—
423	C₂³×C₄	#392	1	1	yes	characteristic
424	C₂³×C₄	#392	1	2	yes	—
425	C₂³×C₄	#392	1	2	yes	—
426	C₂³×C₄	#392	1	2	yes	—
427	C₂³×C₄	#392	1	2	yes	—
428	C₂³×C₄	#392	1	2	yes	—
429	C₂³×C₄	C₂×#120	1	1	yes	characteristic
430	C₂³×C₄	C₂×#120	1	1	yes	characteristic
431	C₂³×C₄	C₂×#120	1	2	yes	—
432	C₂³×C₄	C₂×#120	1	2	yes	—
433	C₂³×C₄	C₂×#120	1	1	yes	characteristic
434	C₂³×C₄	C₂×#120	1	2	yes	—
435	C₂³×C₄	C₂×#120	1	2	yes	—
436	C₂³×C₄	C₂×#120	1	2	yes	—
437	C₂³×C₄	C₂×#121	1	2	yes	—
438	C₂³×C₄	C₂×#121	1	2	yes	—
439	C₂³×C₄	C₂×#121	1	1	yes	characteristic
440	C₂³×C₄	C₂×#121	1	1	yes	characteristic
441	C₂³×C₄	C₂×#121	1	2	yes	—
442	C₂³×C₄	C₂×#121	1	2	yes	—
443	C₂³×C₄	C₂×#127	1	2	yes	—
444	C₂³×C₄	#534	1	2	yes	—
445	C₂³×C₄	#534	1	2	yes	—
446	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
447	C₂×C₄²	C₂×#120	1	8	yes	—
448	#100	C₂×#120	1	16	yes	—
449	C₂⁴	#2021	1	4	yes	—
450	C₂⁴	#2021	1	4	yes	—
451	C₂⁴	C₂×#392	1	1	yes	characteristic
452	C₂⁴	C₂×#392	1	1	yes	characteristic
453	C₂⁴	C₂×#392	1	1	yes	characteristic
454	C₂⁴	C₂×#392	1	1	yes	characteristic
455	C₂⁴	C₂×#392	1	1	yes	characteristic
456	C₂⁴	C₂×#392	1	1	yes	characteristic
457	C₂⁴	C₂×#392	1	2	yes	—
458	C₂⁴	C₂×#392	1	2	yes	—
459	C₂⁴	C₂×#392	1	2	yes	—
460	C₂⁴	C₂×#392	1	2	yes	—
461	C₂⁴	#2403	1	2	yes	—
462	C₂⁴	#2403	1	2	yes	—
463	C₂⁴	#2403	1	2	yes	—
464	C₂⁴	#2403	1	2	yes	—
465	C₂⁴	#2403	1	4	yes	—
466	C₂⁴	#2403	1	4	yes	—
467	C₂⁴	#2403	1	4	yes	—
468	C₂⁴	#2408	1	2	yes	—
469	C₂⁴	#2408	1	2	yes	—
470	C₂⁴	#2408	1	4	yes	—
471	C₂⁴	#2410	1	2	yes	—
472	C₂⁴	#2410	1	2	yes	—
473	C₂⁴	#2410	1	4	yes	—
474	C₂⁴	#2433	1	2	yes	—
475	C₂⁴	#2433	1	2	yes	—
476	C₂⁴	#2433	1	4	yes	—
477	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
478	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
479	C₂⁴	C₂²×#120	1	1	yes	characteristic
480	C₂⁴	C₂²×#120	1	1	yes	characteristic
481	C₂⁴	C₂²×#121	1	1	yes	characteristic
482	C₂⁴	C₂²×#121	1	1	yes	characteristic
483	C₂⁴	C₂²×#127	1	1	yes	characteristic
484	C₂⁴	C₂×D₄²	1	4	yes	—
485	C₂⁴	C₂×D₄²	1	4	yes	—
486	C₂×D₄	#2021	1	8	yes	—
487	C₂×D₄	#2021	1	8	yes	—
488	C₂×D₄	#2403	1	2	yes	—
489	C₂×D₄	#2403	1	2	yes	—
490	C₂×D₄	#2403	1	4	yes	—
491	C₂×D₄	#2408	1	2	yes	—
492	C₂×D₄	#2408	1	2	yes	—
493	C₂×D₄	#2408	1	4	yes	—
494	C₂×D₄	#2410	1	4	yes	—
495	C₂×D₄	#2410	1	4	yes	—
496	C₂×D₄	#2410	1	2	yes	—
497	C₂×D₄	#2410	1	2	yes	—
498	C₂×D₄	#2410	1	4	yes	—
499	C₂×D₄	#2433	1	2	yes	—
500	C₂×D₄	#2433	1	2	yes	—
501	C₂×D₄	#2433	1	4	yes	—
502	C₂×D₄	#2613	1	2	yes	—
503	C₂×D₄	#2613	1	2	yes	—
504	C₂×D₄	#2846	1	2	yes	—
505	C₂×D₄	#2846	1	2	yes	—
506	C₂×D₄	C₂×B(4,𝔽₂)	1	8	yes	—
507	C₂×D₄	C₂×B(4,𝔽₂)	1	8	yes	—
508	C₂×D₄	C₂²×#120	1	2	yes	—
509	C₂×D₄	C₂²×#120	1	2	yes	—
510	C₂×D₄	C₂×#534	1	2	yes	—
511	C₂×D₄	C₂×#535	1	2	yes	—
512	C₂×D₄	C₂×D₄²	1	4	yes	—
513	C₂×D₄	C₂×#546	1	4	yes	—
514	C₂²×C₄	#2021	1	4	yes	—
515	C₂²×C₄	#2021	1	4	yes	—
516	C₂²×C₄	C₂×#392	1	1	yes	characteristic
517	C₂²×C₄	C₂×#392	1	1	yes	characteristic
518	C₂²×C₄	C₂×#392	1	2	yes	—
519	C₂²×C₄	C₂×#392	1	2	yes	—
520	C₂²×C₄	C₂×#392	1	2	yes	—
521	C₂²×C₄	C₂×#392	1	2	yes	—
522	C₂²×C₄	#2403	1	2	yes	—
523	C₂²×C₄	#2403	1	2	yes	—
524	C₂²×C₄	#2410	1	2	yes	—
525	C₂²×C₄	#2410	1	2	yes	—
526	C₂²×C₄	#2410	1	2	yes	—
527	C₂²×C₄	#2410	1	2	yes	—
528	C₂²×C₄	#2613	1	1	yes	characteristic
529	C₂²×C₄	#2613	1	1	yes	characteristic
530	C₂²×C₄	#2613	1	2	yes	—
531	C₂²×C₄	#2846	1	1	yes	characteristic
532	C₂²×C₄	#2846	1	1	yes	characteristic
533	C₂²×C₄	#2846	1	2	yes	—
534	C₂²×C₄	C₂²×#120	1	2	yes	—
535	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
536	C₂²×C₄	C₂²×#121	1	1	yes	characteristic
537	C₂²×C₄	C₂×#534	1	2	yes	—
538	C₂³	#14715	1	1	yes	characteristic
539	C₂³	#14715	1	1	yes	characteristic
540	C₂³	#14715	1	2	yes	—
541	C₂³	#14715	1	2	yes	—
542	C₂³	#14715	1	2	yes	—
543	C₂³	#14715	1	2	yes	—
544	C₂³	#14715	1	2	yes	—
545	C₂³	C₂×#2021	1	4	yes	—
546	C₂³	C₂×#2021	1	4	yes	—
547	C₂³	C₂×#2021	1	2	yes	—
548	C₂³	C₂×#2021	1	2	yes	—
549	C₂³	C₂×#2021	1	2	yes	—
550	C₂³	C₂×#2021	1	2	yes	—
551	C₂³	#23900	1	8	yes	—
552	C₂³	#23900	1	8	yes	—
553	C₂³	C₂²×#392	1	1	yes	characteristic
554	C₂³	C₂×#2403	1	2	yes	—
555	C₂³	C₂×#2403	1	2	yes	—
556	C₂³	C₂×#2410	1	2	yes	—
557	C₂³	C₂×#2408	1	2	yes	—
558	C₂³	C₂×#2433	1	2	yes	—
559	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
560	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
561	C₂×C₄	#14715	1	1	yes	characteristic
562	C₂×C₄	#14715	1	1	yes	characteristic
563	C₂×C₄	#14715	1	2	yes	—
564	C₂×C₄	C₂×#2021	1	4	yes	—
565	C₂×C₄	C₂×#2021	1	4	yes	—
566	C₂×C₄	C₂×#2613	1	1	yes	characteristic
567	C₂×C₄	C₂×#2410	1	2	yes	—
568	C₂×C₄	C₂×#2846	1	1	yes	characteristic
569	C₂²	#394058	1	2	yes	—
570	C₂²	#394058	1	2	yes	—
571	C₂²	#394058	1	4	yes	—
572	C₂²	C₂×#14715	1	1	yes	characteristic
573	C₂²	C₂²×#2021	1	1	yes	characteristic
574	C₂²	C₂²×#2021	1	1	yes	characteristic
575	C₂²	C₂×#23900	1	4	yes	—
576	C₂	C₂×#394058	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#394058
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #72311180	C₂	semidirect
2. C₂×#147767	C₂	semidirect
3. C₂×#146577	C₂	semidirect
4. #72306279	C₂	semidirect
5. #72285407	C₂	semidirect
6. C₂×#10574841	C₂	semidirect
7. #72256607	C₂	semidirect
8. C₂×#493397	C₂	semidirect
9. #72240597	C₂	semidirect
10. C₂×#392135	C₂	semidirect
11. #72213504	C₂	semidirect
14. C₂²×#3026	C₂²	semidirect
18. #7642008	C₂²	semidirect
20. C₂²×#2127	C₂²	semidirect
21. C₂²×#2121	C₂²	semidirect
22. #6628897	C₂²	semidirect
23. #6628896	C₂²	semidirect
24. #6470235	C₂²	semidirect
25. #6470234	C₂²	semidirect
26. C₂×#21750	C₂²	semidirect
27. C₂×#20609	C₂²	semidirect
28. C₂×#20609	C₂²	semidirect
32. #498928	C₂²	semidirect
33. #498927	C₂²	semidirect
34. #494682	C₂²	semidirect
35. #494681	C₂²	semidirect
36. #493397	C₂²	semidirect
37. C₂×#12574	C₂²	semidirect
38. C₂×#12574	C₂²	semidirect
39. #392135	C₂²	semidirect
40. #147767	C₂²	semidirect
41. #146577	C₂²	semidirect
42. C₂×H(7,𝔽₂)	D₄	semidirect
44. C₂²×#585	D₄	semidirect
45. C₂²×#585	D₄	semidirect
46. C₂³×H(5,𝔽₂)	D₄	semidirect
49. #62852	D₄	semidirect
51. C₂×#3449	D₄	semidirect
56. C₂×#3428	D₄	semidirect
57. C₂²×#458	C₂³	semidirect
58. C₂²×B(4,𝔽₂)	C₂³	semidirect
59. C₂×#3019	D₄	semidirect
61. C₂²×#455	D₄	semidirect
62. C₂²×#453	D₄	semidirect
63. C₂×#2888	D₄	semidirect
65. C₂²×#413	D₄	semidirect
66. C₂³×#100	D₄	semidirect
68. C₂³×#99	D₄	semidirect
69. C₂³×#99	D₄	semidirect
72. #36955	C₂³	semidirect
73. #36702	C₂³	semidirect
82. C₂×#2127	C₂³	semidirect
83. C₂²×#352	C₂³	semidirect
84. C₂²×AΣL(1,𝔽₁₆)	C₂³	semidirect
85. #21750	C₂³	semidirect
86. #20609	C₂³	semidirect
87. #20609	C₂³	semidirect
88. #20585	D₄	semidirect
89. #20585	D₄	semidirect
93. C₂×#1782	D₄	semidirect
94. C₂×#1781	D₄	semidirect
95. #12574	C₂³	semidirect
96. #12574	C₂³	semidirect
195. C₂²×#41	#120	semidirect
214. C₂³×Q₈	#120	semidirect
233. C₂³×D₄	#120	semidirect
289. C₂×#127	#120	semidirect
291. C₂×#124	#120	semidirect
293. C₂×#117	#120	semidirect
300. C₂²×C₄²	#120	semidirect
301. C₂×#100	#120	semidirect
302. C₂×#100	#120	semidirect
303. C₂×#100	#120	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#72311180	2
C₂×#147767	2
C₂×#146577	2
#72306279	2
#72285407	2
C₂×#10574841	2
#72256607	2
C₂×#493397	2
#72240597	2
C₂×#392135	2
#72213504	2