Subgroups — #72266372

Order 2048.

At least 13676 conjugacy classes of subgroups; only the 860 normal subgroups are shown.

The subgroup lattice has 397 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72236440	C₂	1	1	yes	maximal, characteristic, complemented
2	#72235057	C₂	1	4	yes	maximal, complemented
3	#72225673	C₂	1	1	yes	maximal, characteristic, complemented
4	#72225649	C₂	1	1	yes	maximal, characteristic, complemented
5	#72225626	C₂	1	4	yes	maximal, complemented
6	#72225622	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219457	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219280	C₂	1	1	yes	maximal, characteristic
9	#72218467	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#62817	C₄	1	1	yes	characteristic, complemented
11	C₂²×#3484	C₂²	1	1	yes	characteristic
12	#7744685	C₄	1	4	yes	complemented
13	#7670203	C₂²	1	4	yes	—
14	#7642245	C₄	1	1	yes	characteristic, complemented
15	#7642244	C₄	1	1	yes	characteristic, complemented
16	#7630939	C₂²	1	1	yes	characteristic
17	C₂²×#2403	C₂²	1	1	yes	characteristic
18	#7626243	C₄	1	1	yes	characteristic, complemented
19	C₂²×#1898	C₂²	1	2	yes	complemented
20	C₂²×#1781	C₂²	1	1	yes	characteristic, complemented
21	C₂×#13867	C₂²	1	1	yes	characteristic, complemented
22	#392675	C₂²	1	2	yes	complemented
23	#392109	C₂²	1	1	yes	characteristic, complemented
24	#379723	C₂²	1	2	yes	complemented
25	#374022	C₂²	1	4	yes	complemented
26	#359428	C₂²	1	4	yes	complemented
27	#359428	C₂²	1	4	yes	complemented
28	C₂×#10965	C₂²	1	2	yes	complemented
29	C₂×#10965	C₂²	1	2	yes	complemented
30	#154195	C₂²	1	2	yes	complemented
31	#154094	C₂²	1	1	yes	characteristic, complemented
32	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic
33	C₂⁵×D₄	D₄	1	1	yes	characteristic
34	C₂⁵×D₄	C₂³	1	1	yes	characteristic
35	C₂×#3484	C₂×C₄	1	1	yes	characteristic
36	C₂×#3484	C₂×C₄	1	1	yes	characteristic
37	C₂×#3484	C₂×C₄	1	2	yes	—
38	C₂×#3484	C₂×C₄	1	1	yes	characteristic
39	C₂×#3484	C₂×C₄	1	4	yes	—
40	C₂×#3484	D₄	1	2	yes	—
41	C₂×#3484	D₄	1	1	yes	characteristic
42	C₂×#3484	D₄	1	1	yes	characteristic
43	C₂×#3484	D₄	1	4	yes	—
44	C₂×#3484	D₄	1	1	yes	characteristic
45	C₂×#3484	D₄	1	1	yes	characteristic
46	C₂²×#534	D₄	1	2	yes	—
47	C₂³×#120	C₂×C₄	1	1	yes	characteristic
48	C₂³×#120	D₄	1	1	yes	characteristic
49	C₂³×#120	C₂³	1	2	yes	—
50	C₂³×#120	C₂³	1	2	yes	—
51	C₂²×#518	D₄	1	1	yes	characteristic
52	C₂⁴×C₂²⋊C₄	C₂³	1	1	yes	characteristic, complemented
53	C₂×#2846	C₂³	1	2	yes	—
54	#35390	D₄	1	4	yes	—
55	#35348	D₄	1	4	yes	—
56	#35348	D₄	1	4	yes	—
57	#35346	D₄	1	4	yes	—
58	#34981	C₂×C₄	1	4	yes	—
59	#34981	D₄	1	4	yes	—
60	#34980	C₂×C₄	1	2	yes	—
61	#34980	C₂×C₄	1	4	yes	—
62	#34980	D₄	1	4	yes	—
63	#34980	D₄	1	2	yes	—
64	#34980	D₄	1	2	yes	—
65	C₂²×#392	C₂×C₄	1	2	yes	—
66	C₂²×#392	D₄	1	2	yes	—
67	#34198	C₂×C₄	1	4	yes	—
68	#34198	D₄	1	4	yes	—
69	#34171	C₂×C₄	1	1	yes	characteristic
70	#34171	D₄	1	1	yes	characteristic
71	#34171	D₄	1	1	yes	characteristic
72	C₂²×#386	C₂×C₄	1	1	yes	characteristic
73	C₂²×#386	D₄	1	1	yes	characteristic
74	C₂²×#386	D₄	1	2	yes	—
75	C₂²×#379	D₄	1	1	yes	characteristic
76	C₂²×#379	C₂³	1	2	yes	complemented
77	C₂²×#379	C₂³	1	2	yes	complemented
78	C₂²×#379	C₂³	1	1	yes	characteristic, complemented
79	#14679	C₂³	1	2	yes	complemented
80	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic, complemented
81	C₂⁷	C₂×D₄	1	1	yes	characteristic
82	C₂⁷	C₂⁴	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
86	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic, complemented
87	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	4	yes	complemented
88	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	complemented
89	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂²×C₄	1	2	yes	—
92	C₂⁴×D₄	C₂×D₄	1	2	yes	—
93	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
94	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂×D₄	1	2	yes	—
96	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	C₂×D₄	1	2	yes	—
99	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
100	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
103	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
104	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×D₄	1	4	yes	—
106	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
107	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×#534	C₂²⋊C₄	1	2	yes	—
109	C₂×#534	C₂²⋊C₄	1	2	yes	—
110	C₂×#534	C₂²⋊C₄	1	2	yes	—
111	C₂×#534	C₂²⋊C₄	1	2	yes	—
112	C₂×#534	C₂²⋊C₄	1	4	yes	complemented
113	C₂×#534	C₂²⋊C₄	1	4	yes	complemented
114	C₂×#534	C₂²⋊C₄	1	2	yes	complemented
115	C₂×#534	C₂²⋊C₄	1	4	yes	complemented
116	C₂×#534	C₂²⋊C₄	1	4	yes	complemented
117	C₂×#534	C₂²⋊C₄	1	2	yes	complemented
118	C₂×#534	C₂²⋊C₄	1	2	yes	complemented
119	C₂×#534	C₂²⋊C₄	1	4	yes	complemented
120	C₂²×#120	C₂²×C₄	1	2	yes	—
121	C₂²×#120	C₂²×C₄	1	2	yes	—
122	C₂²×#120	C₂²×C₄	1	2	yes	—
123	C₂²×#120	C₂×D₄	1	2	yes	—
124	C₂²×#120	C₂×D₄	1	4	yes	—
125	C₂²×#120	C₂×D₄	1	2	yes	—
126	C₂²×#120	C₂×D₄	1	2	yes	—
127	C₂²×#120	C₂×D₄	1	2	yes	—
128	C₂²×#120	C₂×D₄	1	4	yes	—
129	C₂²×#120	C₂×D₄	1	4	yes	—
130	C₂²×#120	C₂×D₄	1	4	yes	—
131	C₂²×#120	C₂×D₄	1	4	yes	—
132	C₂²×#120	C₂×D₄	1	4	yes	—
133	C₂²×#120	C₂×D₄	1	4	yes	—
134	C₂²×#120	C₂×D₄	1	4	yes	—
135	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
136	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×#518	C₂²⋊C₄	1	2	yes	—
138	C₂×#518	C₂²⋊C₄	1	4	yes	complemented
139	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
140	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
141	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
142	C₂×#518	C₂²⋊C₄	1	4	yes	complemented
143	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
144	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
145	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
146	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
147	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
148	C₂³×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
149	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
150	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
151	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
152	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
153	#2315	C₂²⋊C₄	1	4	yes	complemented
154	#2315	C₂²⋊C₄	1	4	yes	complemented
155	#2308	C₂²⋊C₄	1	4	yes	complemented
156	#2308	C₂²⋊C₄	1	4	yes	complemented
157	#2308	C₂²⋊C₄	1	4	yes	complemented
158	#2308	C₂²⋊C₄	1	4	yes	complemented
159	#2308	C₂²⋊C₄	1	4	yes	complemented
160	#2308	C₂²⋊C₄	1	2	yes	complemented
161	#2308	C₂²⋊C₄	1	2	yes	complemented
162	#2308	C₂²⋊C₄	1	2	yes	complemented
163	C₂×#379	C₂²×C₄	1	2	yes	—
164	C₂×#379	C₂×D₄	1	4	yes	—
165	C₂×#379	C₂×D₄	1	4	yes	—
166	C₂×#379	C₂×D₄	1	2	yes	—
167	C₂×#379	C₂×D₄	1	2	yes	—
168	#2273	C₂²⋊C₄	1	4	yes	complemented
169	C₂²×#95	C₂²⋊C₄	1	2	yes	complemented
170	C₂²×#95	C₂²⋊C₄	1	2	yes	complemented
171	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂⁶	#120	1	1	yes	characteristic
175	C₂⁶	#120	1	1	yes	characteristic
176	C₂⁶	#120	1	1	yes	characteristic
177	C₂⁶	#120	1	1	yes	characteristic
178	C₂⁶	#120	1	2	yes	—
179	C₂⁶	C₂³×C₄	1	1	yes	characteristic
180	C₂⁶	C₂²×D₄	1	1	yes	characteristic
181	C₂⁶	C₂²×D₄	1	1	yes	characteristic
182	C₂⁶	C₂²×D₄	1	4	yes	—
183	C₂⁶	C₂²×D₄	1	1	yes	characteristic
184	C₂⁶	C₂²×D₄	1	1	yes	characteristic
185	C₂⁶	C₂²×D₄	1	1	yes	characteristic
186	C₂⁶	C₂²×D₄	1	1	yes	characteristic
187	C₂⁶	C₂²×D₄	1	4	yes	—
188	C₂×H(5,𝔽₂)	#99	1	4	yes	complemented
189	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
195	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
196	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
200	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
201	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
202	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
203	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
204	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
205	C₂³×D₄	#120	1	1	yes	characteristic
206	C₂³×D₄	#120	1	1	yes	characteristic
207	C₂³×D₄	#120	1	1	yes	characteristic
208	C₂³×D₄	#120	1	1	yes	characteristic
209	C₂³×D₄	#120	1	1	yes	characteristic
210	C₂³×D₄	#120	1	2	yes	—
211	C₂³×D₄	#120	1	1	yes	characteristic
212	C₂³×D₄	#120	1	1	yes	characteristic
213	C₂³×D₄	#120	1	2	yes	—
214	C₂³×D₄	#120	1	2	yes	—
215	C₂³×D₄	#120	1	1	yes	characteristic
216	C₂³×D₄	#120	1	2	yes	—
217	C₂³×D₄	#120	1	2	yes	—
218	C₂³×D₄	#120	1	2	yes	—
219	C₂³×D₄	#120	1	4	yes	—
220	C₂³×D₄	#120	1	4	yes	—
221	C₂³×D₄	#120	1	4	yes	—
222	C₂³×D₄	#120	1	2	yes	—
223	C₂³×D₄	#120	1	4	yes	—
224	C₂³×D₄	#120	1	4	yes	—
225	C₂³×D₄	#120	1	2	yes	—
226	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂⁴×C₄	#120	1	1	yes	characteristic
232	C₂⁴×C₄	#120	1	1	yes	characteristic
233	C₂⁴×C₄	#120	1	1	yes	characteristic
234	C₂⁴×C₄	#120	1	1	yes	characteristic
235	C₂⁴×C₄	#120	1	2	yes	—
236	C₂⁴×C₄	#120	1	4	yes	—
237	#534	#99	1	8	yes	complemented
238	#518	#99	1	4	yes	complemented
239	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
240	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
241	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
242	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
244	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
245	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
246	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
247	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
249	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
250	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
251	C₂²×C₂²⋊C₄	#120	1	4	yes	—
252	C₂²×C₂²⋊C₄	#120	1	4	yes	—
253	C₂²×C₂²⋊C₄	#120	1	4	yes	—
254	C₂²×C₂²⋊C₄	#120	1	4	yes	—
255	C₂²×C₂²⋊C₄	#120	1	4	yes	—
256	C₂²×C₂²⋊C₄	#120	1	4	yes	—
257	C₂²×C₂²⋊C₄	#120	1	4	yes	—
258	C₂²×C₂²⋊C₄	#120	1	2	yes	—
259	C₂²×C₂²⋊C₄	#120	1	4	yes	—
260	C₂²×C₂²⋊C₄	#120	1	4	yes	—
261	C₂²×C₂²⋊C₄	#120	1	4	yes	—
262	C₂²×C₂²⋊C₄	#120	1	2	yes	—
263	C₂²×C₂²⋊C₄	#120	1	4	yes	—
264	C₂²×C₂²⋊C₄	#120	1	4	yes	—
265	C₂⁵	#379	1	1	yes	characteristic
266	C₂⁵	#379	1	1	yes	characteristic
267	C₂⁵	#379	1	1	yes	characteristic
268	C₂⁵	#379	1	1	yes	characteristic
269	C₂⁵	#379	1	2	yes	—
270	C₂⁵	#379	1	2	yes	—
271	C₂⁵	#379	1	2	yes	—
272	C₂⁵	#379	1	2	yes	—
273	C₂⁵	#379	1	2	yes	—
274	C₂⁵	B(4,𝔽₂)	1	2	yes	—
275	C₂⁵	#458	1	2	yes	—
276	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
277	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
278	C₂⁵	C₂²×C₂²⋊C₄	1	4	yes	—
279	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
280	C₂⁵	C₂×#120	1	1	yes	characteristic
281	C₂⁵	C₂×#120	1	2	yes	—
282	C₂⁵	C₂×#120	1	1	yes	characteristic
283	C₂⁵	C₂×#120	1	2	yes	—
284	C₂⁵	C₂×#120	1	1	yes	characteristic
285	C₂⁵	C₂×#120	1	2	yes	—
286	C₂⁵	C₂×#120	1	2	yes	—
287	C₂⁵	C₂×#120	1	4	yes	—
288	C₂⁵	C₂×#120	1	4	yes	—
289	C₂⁵	C₂×#120	1	4	yes	—
290	C₂⁵	C₂×#120	1	1	yes	characteristic
291	C₂⁵	C₂×#120	1	2	yes	—
292	C₂⁵	C₂×#120	1	2	yes	—
293	C₂²×D₄	C₂×#99	1	2	yes	—
294	C₂²×D₄	C₂×#99	1	2	yes	—
295	C₂²×D₄	C₂×#99	1	4	yes	—
296	C₂²×D₄	C₂×#99	1	4	yes	—
297	C₂²×D₄	C₂×#99	1	4	yes	—
298	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
299	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
300	C₂²×D₄	#458	1	2	yes	—
301	C₂²×D₄	#458	1	4	yes	—
302	C₂³×C₄	#379	1	1	yes	characteristic
303	C₂³×C₄	#379	1	1	yes	characteristic
304	C₂³×C₄	#379	1	1	yes	characteristic
305	C₂³×C₄	#379	1	1	yes	characteristic
306	C₂³×C₄	#379	1	4	yes	—
307	C₂³×C₄	#379	1	4	yes	—
308	C₂³×C₄	#379	1	4	yes	—
309	C₂³×C₄	#379	1	2	yes	—
310	C₂³×C₄	#379	1	2	yes	—
311	C₂³×C₄	#379	1	4	yes	—
312	C₂³×C₄	#379	1	2	yes	—
313	C₂³×C₄	#379	1	2	yes	—
314	C₂³×C₄	#379	1	2	yes	—
315	C₂³×C₄	#379	1	4	yes	—
316	C₂³×C₄	#379	1	2	yes	—
317	C₂³×C₄	#379	1	2	yes	—
318	C₂³×C₄	#379	1	4	yes	—
319	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
320	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
321	C₂⁴	#1781	1	2	yes	—
322	C₂⁴	#1781	1	2	yes	—
323	C₂⁴	#1781	1	4	yes	complemented
324	C₂⁴	#1790	1	2	yes	—
325	C₂⁴	#1790	1	2	yes	—
326	C₂⁴	#1790	1	2	yes	—
327	C₂⁴	#1790	1	4	yes	complemented
328	C₂⁴	C₂×#379	1	2	yes	—
329	C₂⁴	C₂×#379	1	2	yes	—
330	C₂⁴	C₂×#379	1	2	yes	—
331	C₂⁴	C₂×#379	1	1	yes	characteristic
332	C₂⁴	#2846	1	2	yes	—
333	C₂⁴	#2846	1	2	yes	—
334	C₂⁴	#2846	1	1	yes	characteristic
335	C₂⁴	#2846	1	2	yes	—
336	C₂⁴	#2846	1	1	yes	characteristic
337	C₂⁴	C₂²×#99	1	1	yes	characteristic
338	C₂⁴	C₂²×#99	1	2	yes	—
339	C₂⁴	C₂²×#99	1	1	yes	characteristic
340	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
341	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
342	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
343	C₂⁴	C₂×#458	1	1	yes	characteristic
344	C₂⁴	C₂×#458	1	2	yes	—
345	C₂⁴	C₂×#458	1	1	yes	characteristic
346	C₂⁴	#3026	1	1	yes	characteristic
347	C₂⁴	#3026	1	1	yes	characteristic
348	C₂⁴	#3026	1	2	yes	—
349	C₂⁴	#3026	1	2	yes	—
350	C₂⁴	#3026	1	2	yes	—
351	C₂²×C₄	#1781	1	2	yes	—
352	C₂²×C₄	#1781	1	2	yes	—
353	C₂²×C₄	#1781	1	4	yes	complemented
354	C₂²×C₄	#1781	1	8	yes	complemented
355	C₂²×C₄	#1790	1	2	yes	—
356	C₂²×C₄	#1790	1	4	yes	complemented
357	C₂³	#8534	1	4	yes	complemented
358	C₂³	#14679	1	1	yes	characteristic
359	C₂³	C₂×#1781	1	2	yes	—
360	C₂³	C₂×#1781	1	2	yes	—
361	C₂³	C₂×#1781	1	1	yes	characteristic
362	C₂³	C₂×#1781	1	1	yes	characteristic
363	C₂³	C₂×#1790	1	2	yes	—
364	C₂³	C₂×#1790	1	1	yes	characteristic
365	C₂³	C₂×#1790	1	1	yes	characteristic
366	C₂³	#20609	1	2	yes	—
367	C₂³	#20609	1	1	yes	characteristic
368	C₂³	#20609	1	1	yes	characteristic
369	C₂³	#32898	1	2	yes	—
370	C₂³	#32898	1	1	yes	characteristic
371	C₂³	#32898	1	2	yes	—
372	C₂³	#32898	1	2	yes	—
373	C₂³	#32898	1	1	yes	characteristic
374	C₂³	#32900	1	2	yes	—
375	C₂³	#32900	1	1	yes	characteristic
376	C₂³	#32900	1	2	yes	—
377	C₂³	#32900	1	2	yes	—
378	C₂³	#32900	1	1	yes	characteristic
379	C₂²	C₂×#8534	1	1	yes	characteristic
380	C₂²	#187928	1	1	yes	characteristic
381	C₂²	#187928	1	1	yes	characteristic
382	C₂²	#187944	1	1	yes	characteristic
383	C₂²	#347042	1	1	yes	characteristic
384	C₂²	#347048	1	1	yes	characteristic
385	C₂²	#347048	1	2	yes	—
386	C₂²	#392109	1	1	yes	characteristic
387	C₂²	#392125	1	1	yes	characteristic
388	C₂²	#392125	1	1	yes	characteristic
389	C₂²	#392125	1	2	yes	—
390	C₂²	#392135	1	2	yes	—
391	C₂	#72218467	1	1	yes	characteristic
392	C₂	#72219045	1	1	yes	characteristic
393	C₂	#72232860	1	2	yes	—
394	C₂	#72234164	1	2	yes	—
395	C₂	#72251779	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#14679
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁷	C₂⁴
prime radical	#72219280	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72236440	C₂	semidirect
2. #72235057	C₂	semidirect
3. #72225673	C₂	semidirect
4. #72225649	C₂	semidirect
5. #72225626	C₂	semidirect
6. #72225622	C₂	semidirect
7. #72219457	C₂	semidirect
9. #72218467	C₂	semidirect
10. C₂×#62817	C₄	semidirect
12. #7744685	C₄	semidirect
14. #7642245	C₄	semidirect
15. #7642244	C₄	semidirect
18. #7626243	C₄	semidirect
19. C₂²×#1898	C₂²	semidirect
20. C₂²×#1781	C₂²	semidirect
21. C₂×#13867	C₂²	semidirect
22. #392675	C₂²	semidirect
23. #392109	C₂²	semidirect
24. #379723	C₂²	semidirect
25. #374022	C₂²	semidirect
26. #359428	C₂²	semidirect
27. #359428	C₂²	semidirect
28. C₂×#10965	C₂²	semidirect
29. C₂×#10965	C₂²	semidirect
30. #154195	C₂²	semidirect
31. #154094	C₂²	semidirect
52. C₂⁴×C₂²⋊C₄	C₂³	semidirect
76. C₂²×#379	C₂³	semidirect
77. C₂²×#379	C₂³	semidirect
78. C₂²×#379	C₂³	semidirect
79. #14679	C₂³	semidirect
80. C₂⁷	C₂²⋊C₄	semidirect
86. C₂²×H(5,𝔽₂)	C₂²⋊C₄	semidirect
87. C₂²×H(5,𝔽₂)	C₂²⋊C₄	semidirect
88. C₂⁴×D₄	C₂²⋊C₄	semidirect
106. C₂⁵×C₄	C₂²⋊C₄	semidirect
112. C₂×#534	C₂²⋊C₄	semidirect
113. C₂×#534	C₂²⋊C₄	semidirect
114. C₂×#534	C₂²⋊C₄	semidirect
115. C₂×#534	C₂²⋊C₄	semidirect
116. C₂×#534	C₂²⋊C₄	semidirect
117. C₂×#534	C₂²⋊C₄	semidirect
118. C₂×#534	C₂²⋊C₄	semidirect
119. C₂×#534	C₂²⋊C₄	semidirect
138. C₂×#518	C₂²⋊C₄	semidirect
139. C₂×#518	C₂²⋊C₄	semidirect
140. C₂×#518	C₂²⋊C₄	semidirect
141. C₂×#518	C₂²⋊C₄	semidirect
142. C₂×#518	C₂²⋊C₄	semidirect
143. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
144. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
145. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
146. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
147. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
153. #2315	C₂²⋊C₄	semidirect
154. #2315	C₂²⋊C₄	semidirect
155. #2308	C₂²⋊C₄	semidirect
156. #2308	C₂²⋊C₄	semidirect
157. #2308	C₂²⋊C₄	semidirect
158. #2308	C₂²⋊C₄	semidirect
159. #2308	C₂²⋊C₄	semidirect
160. #2308	C₂²⋊C₄	semidirect
161. #2308	C₂²⋊C₄	semidirect
162. #2308	C₂²⋊C₄	semidirect
168. #2273	C₂²⋊C₄	semidirect
169. C₂²×#95	C₂²⋊C₄	semidirect
170. C₂²×#95	C₂²⋊C₄	semidirect
188. C₂×H(5,𝔽₂)	#99	semidirect
237. #534	#99	semidirect
238. #518	#99	semidirect
323. C₂⁴	#1781	semidirect
327. C₂⁴	#1790	semidirect
353. C₂²×C₄	#1781	semidirect
354. C₂²×C₄	#1781	semidirect
356. C₂²×C₄	#1790	semidirect
357. C₂³	#8534	semidirect

Maximal subgroups

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

subgroup	index
#72236440	2
#72235057	2
#72225673	2
#72225649	2
#72225626	2
#72225622	2
#72219457	2
#72219280	2
#72218467	2