Subgroups — C₂×#3561 #62731

Order 256.

3629 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3561	C₂	1	32	yes	maximal, direct product factor
2	C₂×#563	C₂	1	2	yes	maximal, complemented
3	C₂×#562	C₂	1	2	yes	maximal, complemented
4	C₂×#556	C₂	1	2	yes	maximal, complemented
5	C₂×#551	C₂	1	4	yes	maximal, complemented
6	C₂×D₄×Q₈	C₂	1	2	yes	maximal, complemented
7	C₂×#548	C₂	1	2	yes	maximal, complemented
8	C₂×#546	C₂	1	2	yes	maximal, complemented
9	C₂×#543	C₂	1	2	yes	maximal, complemented
10	C₂×#542	C₂	1	2	yes	maximal, complemented
11	C₂×#542	C₂	1	4	yes	maximal, complemented
12	C₂×#538	C₂	1	2	yes	maximal, complemented
13	C₂×#536	C₂	1	2	yes	maximal, complemented
14	C₂×#533	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#529	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#124	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂²×#41	C₂²	1	2	yes	—
18	C₂³×Q₈	C₂²	1	1	yes	characteristic
19	C₂³×Q₈	C₂²	1	2	yes	complemented
20	C₂³×D₄	C₂²	1	1	yes	characteristic
21	#563	C₂²	1	32	yes	complemented
22	#562	C₂²	1	32	yes	complemented
23	#556	C₂²	1	32	yes	complemented
24	#551	C₂²	1	64	yes	complemented
25	D₄×Q₈	C₂²	1	32	yes	complemented
26	#548	C₂²	1	32	yes	complemented
27	#546	C₂²	1	32	yes	complemented
28	#543	C₂²	1	32	yes	complemented
29	#542	C₂²	1	32	yes	complemented
30	#542	C₂²	1	64	yes	complemented
31	#538	C₂²	1	32	yes	complemented
32	#536	C₂²	1	32	yes	complemented
33	#533	C₂²	1	16	yes	complemented
34	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄⋊Q₈	C₂²	1	2	yes	complemented
37	C₂×C₄⋊Q₈	C₂²	1	4	yes	complemented
38	#529	C₂²	1	16	yes	complemented
39	C₂×#126	C₂²	1	2	yes	complemented
40	C₂×#126	C₂²	1	4	yes	complemented
41	C₂×#126	C₂²	1	4	yes	complemented
42	C₂×#126	C₂²	1	4	yes	complemented
43	C₂×#125	C₂²	1	1	yes	characteristic, complemented
44	C₂×#125	C₂²	1	2	yes	complemented
45	C₂×#124	C₂²	1	2	yes	—
46	C₂×#124	C₂²	1	2	yes	—
47	C₂×#124	C₂²	1	2	yes	—
48	C₂×#124	C₂²	1	2	yes	—
49	C₂×#124	C₂²	1	2	yes	—
50	C₂×#124	C₂²	1	16	yes	complemented
51	C₂×#124	C₂²	1	1	yes	characteristic, complemented
52	C₂×#124	C₂²	1	4	yes	complemented
53	C₂×#124	C₂²	1	4	yes	complemented
54	C₂×#123	C₂²	1	2	yes	—
55	C₂×#123	C₂²	1	4	yes	—
56	C₂×#123	C₂²	1	4	yes	complemented
57	C₂×#123	C₂²	1	4	yes	complemented
58	C₂×#123	C₂²	1	4	yes	complemented
59	C₂×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
60	C₂×C₂²⋊Q₈	C₂²	1	2	yes	—
61	C₂×C₂²⋊Q₈	C₂²	1	2	yes	—
62	C₂×C₂²⋊Q₈	C₂²	1	4	yes	—
63	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
64	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
65	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
66	C₂×C₂²⋊Q₈	C₂²	1	2	yes	complemented
67	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
68	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
69	C₂×C₂²⋊Q₈	C₂²	1	4	yes	complemented
70	C₂×#121	C₂²	1	1	yes	characteristic
71	C₂×#121	C₂²	1	2	yes	—
72	C₂×#121	C₂²	1	4	yes	—
73	C₂×#121	C₂²	1	4	yes	complemented
74	C₂×#120	C₂²	1	4	yes	—
75	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
76	C₂×C₄×Q₈	C₂²	1	2	yes	complemented
77	C₂×C₄×Q₈	C₂²	1	2	yes	complemented
78	C₂×C₄×Q₈	C₂²	1	2	yes	complemented
79	C₂×C₄×D₄	C₂²	1	2	yes	—
80	C₂×C₄×D₄	C₂²	1	4	yes	—
81	C₂×C₄×D₄	C₂²	1	2	yes	—
82	C₂×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
83	C₂×C₄×D₄	C₂²	1	4	yes	complemented
84	C₂×#117	C₂²	1	2	yes	—
85	C₂×#117	C₂²	1	2	yes	complemented
86	C₂×#117	C₂²	1	4	yes	complemented
87	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
88	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
89	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
90	C₂²×C₄²	C₂²	1	1	yes	characteristic
91	C₂⁵	C₂³	1	2	yes	—
92	C₂×#41	—	4	8	no	—
93	C₂×#41	C₂³	1	16	yes	—
94	C₂²×Q₈	—	4	4	no	—
95	C₂²×Q₈	—	4	8	no	—
96	C₂²×Q₈	C₂³	1	2	yes	—
97	C₂²×Q₈	C₂³	1	2	yes	—
98	C₂²×Q₈	C₂³	1	1	yes	characteristic
99	C₂²×Q₈	C₂³	1	2	yes	—
100	C₂²×Q₈	C₂³	1	4	yes	—
101	C₂²×Q₈	C₂³	1	2	yes	—
102	C₂²×Q₈	C₂³	1	8	yes	—
103	C₂²×Q₈	C₂³	1	16	yes	complemented
104	C₂²×D₄	—	4	4	no	—
105	C₂²×D₄	C₂³	1	2	yes	—
106	C₂²×D₄	C₂³	1	2	yes	—
107	C₂²×D₄	C₂³	1	4	yes	—
108	C₂²×D₄	C₂³	1	1	yes	characteristic
109	C₂²×D₄	C₂³	1	4	yes	—
110	C₂²×D₄	C₂³	1	2	yes	—
111	C₂²×D₄	C₂³	1	8	yes	—
112	C₂³×C₄	C₂³	1	1	yes	characteristic
113	C₂³×C₄	C₂³	1	2	yes	—
114	C₂³×C₄	C₂³	1	1	yes	characteristic
115	C₂³×C₄	C₂³	1	2	yes	—
116	C₂³×C₄	C₂³	1	1	yes	characteristic
117	C₂³×C₄	C₂³	1	4	yes	—
118	C₂³×C₄	C₂³	1	4	yes	—
119	C₂³×C₄	C₂³	1	4	yes	—
120	C₄⋊Q₈	C₂³	1	8	yes	complemented
121	C₄⋊Q₈	C₂³	1	8	yes	complemented
122	C₄⋊Q₈	C₂³	1	16	yes	complemented
123	C₄⋊Q₈	C₂³	1	32	yes	complemented
124	#126	C₂³	1	16	yes	complemented
125	#126	C₂³	1	32	yes	complemented
126	#126	C₂³	1	32	yes	complemented
127	#126	C₂³	1	32	yes	complemented
128	#125	C₂³	1	8	yes	complemented
129	#125	C₂³	1	16	yes	complemented
130	#124	C₂³	1	16	yes	—
131	#124	C₂³	1	16	yes	—
132	#124	C₂³	1	16	yes	—
133	#124	C₂³	1	16	yes	—
134	#124	C₂³	1	16	yes	—
135	#124	C₂³	1	8	yes	complemented
136	#124	C₂³	1	32	yes	complemented
137	#124	C₂³	1	32	yes	complemented
138	#123	C₂³	1	16	yes	—
139	#123	C₂³	1	32	yes	—
140	#123	C₂³	1	32	yes	complemented
141	#123	C₂³	1	32	yes	complemented
142	#123	C₂³	1	32	yes	complemented
143	C₂²⋊Q₈	C₂³	1	8	yes	—
144	C₂²⋊Q₈	C₂³	1	16	yes	—
145	C₂²⋊Q₈	C₂³	1	16	yes	—
146	C₂²⋊Q₈	C₂³	1	32	yes	—
147	C₂²⋊Q₈	C₂³	1	32	yes	complemented
148	C₂²⋊Q₈	C₂³	1	32	yes	complemented
149	C₂²⋊Q₈	C₂³	1	32	yes	complemented
150	C₂²⋊Q₈	C₂³	1	16	yes	complemented
151	C₂²⋊Q₈	C₂³	1	32	yes	complemented
152	C₂²⋊Q₈	C₂³	1	32	yes	complemented
153	C₂²⋊Q₈	C₂³	1	32	yes	complemented
154	#121	C₂³	1	8	yes	—
155	#121	C₂³	1	16	yes	—
156	#121	C₂³	1	32	yes	—
157	#121	C₂³	1	32	yes	complemented
158	#120	C₂³	1	32	yes	—
159	C₄×Q₈	C₂³	1	8	yes	complemented
160	C₄×Q₈	C₂³	1	16	yes	complemented
161	C₄×Q₈	C₂³	1	16	yes	complemented
162	C₄×Q₈	C₂³	1	16	yes	complemented
163	C₄×D₄	C₂³	1	16	yes	—
164	C₄×D₄	C₂³	1	32	yes	—
165	C₄×D₄	C₂³	1	16	yes	—
166	C₄×D₄	C₂³	1	8	yes	complemented
167	C₄×D₄	C₂³	1	32	yes	complemented
168	#117	C₂³	1	16	yes	—
169	#117	C₂³	1	16	yes	complemented
170	#117	C₂³	1	32	yes	complemented
171	C₂×C₄⋊C₄	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³	1	4	yes	—
176	C₂×C₄⋊C₄	C₂³	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³	1	4	yes	—
179	C₂×C₄⋊C₄	C₂³	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂³	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂³	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
189	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
190	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
191	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
192	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂³	1	16	yes	—
196	C₂×C₂²⋊C₄	C₂³	1	16	yes	—
197	C₂×C₂²⋊C₄	C₂³	1	32	yes	—
198	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
201	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
202	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
204	C₂×C₂²⋊C₄	C₂³	1	2	yes	—
205	C₂×C₄²	C₂³	1	4	yes	—
206	C₂×C₄²	C₂³	1	1	yes	characteristic
207	C₂×C₄²	C₂³	1	2	yes	—
208	C₂×C₄²	C₂³	1	4	yes	—
209	C₂×C₄²	C₂³	1	1	yes	characteristic
210	C₂×C₄²	C₂³	1	1	yes	characteristic
211	C₂×C₄²	C₂³	1	1	yes	characteristic
212	C₂×C₄²	C₂³	1	8	yes	—
213	C₂⁴	—	2	8	no	—
214	C₂⁴	—	4	4	no	—
215	C₂⁴	—	4	4	no	—
216	C₂⁴	C₂⁴	1	4	yes	—
217	C₂⁴	C₂⁴	1	2	yes	—
218	C₂⁴	C₂⁴	1	1	yes	characteristic
219	C₂⁴	C₂⁴	1	8	yes	—
220	#41	—	4	32	no	—
221	C₂×Q₈	—	2	8	no	—
222	C₂×Q₈	—	2	8	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	16	no	—
228	C₂×Q₈	—	4	32	no	—
229	C₂×Q₈	C₂⁴	1	8	yes	—
230	C₂×Q₈	C₂⁴	1	8	yes	—
231	C₂×Q₈	C₂⁴	1	4	yes	—
232	C₂×Q₈	C₂⁴	1	8	yes	—
233	C₂×Q₈	C₂⁴	1	16	yes	—
234	C₂×Q₈	C₂⁴	1	8	yes	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	8	no	—
237	C₂×D₄	—	4	16	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	8	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	8	no	—
242	C₂×D₄	C₂⁴	1	8	yes	—
243	C₂×D₄	C₂⁴	1	8	yes	—
244	C₂×D₄	C₂⁴	1	16	yes	—
245	C₂×D₄	C₂⁴	1	4	yes	—
246	C₂×D₄	C₂⁴	1	16	yes	—
247	C₂×D₄	C₂⁴	1	8	yes	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	8	no	—
254	C₂²×C₄	—	4	4	no	—
255	C₂²×C₄	—	4	8	no	—
256	C₂²×C₄	C₂⁴	1	4	yes	—
257	C₂²×C₄	C₂⁴	1	4	yes	—
258	C₂²×C₄	C₂⁴	1	8	yes	—
259	C₂²×C₄	C₂⁴	1	1	yes	characteristic
260	C₂²×C₄	C₂⁴	1	2	yes	—
261	C₂²×C₄	C₂⁴	1	1	yes	characteristic
262	C₂²×C₄	C₂⁴	1	1	yes	characteristic
263	C₂²×C₄	C₂⁴	1	4	yes	—
264	C₂²×C₄	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄	C₂⁴	1	2	yes	—
267	C₂²×C₄	C₂⁴	1	4	yes	—
268	C₂²×C₄	C₂⁴	1	4	yes	—
269	C₂²×C₄	C₂⁴	1	2	yes	—
270	C₂²×C₄	C₂⁴	1	4	yes	—
271	C₂²×C₄	C₂⁴	1	8	yes	—
272	C₂²×C₄	C₂⁴	1	16	yes	—
273	C₂²×C₄	C₂⁴	1	16	yes	—
274	C₂²×C₄	C₂⁴	1	16	yes	—
275	C₂²×C₄	C₂⁴	1	1	yes	characteristic
276	C₄⋊C₄	C₂⁴	1	8	yes	—
277	C₄⋊C₄	C₂⁴	1	8	yes	—
278	C₄⋊C₄	C₂⁴	1	8	yes	—
279	C₄⋊C₄	C₂⁴	1	8	yes	—
280	C₄⋊C₄	C₂⁴	1	16	yes	—
281	C₄⋊C₄	C₂⁴	1	8	yes	—
282	C₄⋊C₄	C₂⁴	1	8	yes	—
283	C₄⋊C₄	C₂⁴	1	16	yes	—
284	C₄⋊C₄	C₂⁴	1	8	yes	—
285	C₄⋊C₄	C₂⁴	1	4	yes	—
286	C₄⋊C₄	C₂⁴	1	16	yes	—
287	C₄⋊C₄	C₂⁴	1	16	yes	—
288	C₄⋊C₄	C₂⁴	1	4	yes	—
289	C₄⋊C₄	C₂⁴	1	16	yes	—
290	C₄⋊C₄	C₂⁴	1	16	yes	—
291	C₄⋊C₄	C₂⁴	1	8	yes	—
292	C₂²⋊C₄	C₂⁴	1	8	yes	—
293	C₂²⋊C₄	C₂⁴	1	16	yes	—
294	C₂²⋊C₄	C₂⁴	1	16	yes	—
295	C₂²⋊C₄	C₂⁴	1	16	yes	—
296	C₂²⋊C₄	C₂⁴	1	16	yes	—
297	C₂²⋊C₄	C₂⁴	1	16	yes	—
298	C₂²⋊C₄	C₂⁴	1	16	yes	—
299	C₂²⋊C₄	C₂⁴	1	16	yes	—
300	C₂²⋊C₄	C₂⁴	1	16	yes	—
301	C₂²⋊C₄	C₂⁴	1	8	yes	—
302	C₂²⋊C₄	C₂⁴	1	8	yes	—
303	C₂²⋊C₄	C₂⁴	1	16	yes	—
304	C₂²⋊C₄	C₂⁴	1	8	yes	—
305	C₂²⋊C₄	C₂⁴	1	16	yes	—
306	C₂²⋊C₄	C₂⁴	1	8	yes	—
307	C₄²	C₂⁴	1	4	yes	—
308	C₄²	C₂⁴	1	4	yes	—
309	C₄²	C₂⁴	1	4	yes	—
310	C₄²	C₂⁴	1	4	yes	—
311	C₄²	C₂⁴	1	16	yes	—
312	C₄²	C₂⁴	1	8	yes	—
313	C₄²	C₂⁴	1	16	yes	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	4	no	—
317	C₂³	—	2	8	no	—
318	C₂³	—	2	4	no	—
319	C₂³	—	2	8	no	—
320	C₂³	—	2	8	no	—
321	C₂³	—	2	4	no	—
322	C₂³	—	2	16	no	—
323	C₂³	—	4	8	no	—
324	C₂³	—	4	8	no	—
325	C₂³	—	8	16	no	—
326	C₂³	2₋¹⁺⁴	1	4	yes	—
327	C₂³	C₂⁵	1	8	yes	—
328	C₂³	C₂⁵	1	4	yes	—
329	C₂³	C₂⁵	1	2	yes	—
330	C₂³	C₂⁵	1	1	yes	characteristic
331	Q₈	—	2	16	no	—
332	Q₈	—	2	16	no	—
333	Q₈	—	4	16	no	—
334	Q₈	—	4	8	no	—
335	Q₈	—	4	4	no	—
336	Q₈	—	4	8	no	—
337	D₄	—	2	16	no	—
338	D₄	—	2	16	no	—
339	D₄	—	4	8	no	—
340	D₄	—	4	16	no	—
341	D₄	—	4	4	no	—
342	D₄	—	4	16	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	16	no	—
357	C₂×C₄	—	4	16	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	16	no	—
360	C₂×C₄	—	4	8	no	—
361	C₂×C₄	—	4	16	no	—
362	C₂×C₄	H(5,𝔽₂)	1	4	yes	—
363	C₂×C₄	C₂⁵	1	2	yes	—
364	C₂×C₄	C₂⁵	1	4	yes	—
365	C₂×C₄	C₂⁵	1	2	yes	—
366	C₂×C₄	C₂⁵	1	2	yes	—
367	C₂×C₄	C₂⁵	1	2	yes	—
368	C₂×C₄	C₂⁵	1	8	yes	—
369	C₂×C₄	C₂⁵	1	2	yes	—
370	C₂×C₄	C₂⁵	1	4	yes	—
371	C₂×C₄	C₂⁵	1	8	yes	—
372	C₂×C₄	C₂⁵	1	8	yes	—
373	C₂×C₄	C₂⁵	1	4	yes	—
374	C₂×C₄	C₂⁵	1	2	yes	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	8	no	—
378	C₂²	—	2	4	no	—
379	C₂²	—	2	8	no	—
380	C₂²	—	2	4	no	—
381	C₂²	—	2	8	no	—
382	C₂²	—	2	8	no	—
383	C₂²	—	2	4	no	—
384	C₂²	—	4	8	no	—
385	C₂²	—	4	16	no	—
386	C₂²	—	8	16	no	—
387	C₂²	C₂×H(5,𝔽₂)	1	2	yes	—
388	C₂²	C₂×2₋¹⁺⁴	1	4	yes	—
389	C₂²	C₂×2₋¹⁺⁴	1	2	yes	—
390	C₂²	#585	1	2	yes	—
391	C₂²	C₂⁶	1	1	yes	characteristic
392	C₄	—	2	4	no	—
393	C₄	—	2	2	no	—
394	C₄	—	2	8	no	—
395	C₄	—	2	8	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	2	4	no	—
399	C₄	—	2	4	no	—
400	C₄	—	2	8	no	—
401	C₄	—	2	2	no	—
402	C₄	—	2	2	no	—
403	C₄	C₂×H(5,𝔽₂)	1	4	yes	—
404	C₂	—	2	4	no	—
405	C₂	—	4	4	no	—
406	C₂	—	4	8	no	—
407	C₂	#3561	1	4	yes	direct product factor
408	C₂	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
409	C₂	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
410	C₂	C₂×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3561	C₂	direct (N × H)
2. C₂×#563	C₂	semidirect
3. C₂×#562	C₂	semidirect
4. C₂×#556	C₂	semidirect
5. C₂×#551	C₂	semidirect
6. C₂×D₄×Q₈	C₂	semidirect
7. C₂×#548	C₂	semidirect
8. C₂×#546	C₂	semidirect
9. C₂×#543	C₂	semidirect
10. C₂×#542	C₂	semidirect
11. C₂×#542	C₂	semidirect
12. C₂×#538	C₂	semidirect
13. C₂×#536	C₂	semidirect
14. C₂×#533	C₂	semidirect
15. C₂×#529	C₂	semidirect
16. C₂²×#124	C₂	semidirect
19. C₂³×Q₈	C₂²	semidirect
21. #563	C₂²	semidirect
22. #562	C₂²	semidirect
23. #556	C₂²	semidirect
24. #551	C₂²	semidirect
25. D₄×Q₈	C₂²	semidirect
26. #548	C₂²	semidirect
27. #546	C₂²	semidirect
28. #543	C₂²	semidirect
29. #542	C₂²	semidirect
30. #542	C₂²	semidirect
31. #538	C₂²	semidirect
32. #536	C₂²	semidirect
33. #533	C₂²	semidirect
34. C₂×C₄⋊Q₈	C₂²	semidirect
35. C₂×C₄⋊Q₈	C₂²	semidirect
36. C₂×C₄⋊Q₈	C₂²	semidirect
37. C₂×C₄⋊Q₈	C₂²	semidirect
38. #529	C₂²	semidirect
39. C₂×#126	C₂²	semidirect
40. C₂×#126	C₂²	semidirect
41. C₂×#126	C₂²	semidirect
42. C₂×#126	C₂²	semidirect
43. C₂×#125	C₂²	semidirect
44. C₂×#125	C₂²	semidirect
50. C₂×#124	C₂²	semidirect
51. C₂×#124	C₂²	semidirect
52. C₂×#124	C₂²	semidirect
53. C₂×#124	C₂²	semidirect
56. C₂×#123	C₂²	semidirect
57. C₂×#123	C₂²	semidirect
58. C₂×#123	C₂²	semidirect
63. C₂×C₂²⋊Q₈	C₂²	semidirect
64. C₂×C₂²⋊Q₈	C₂²	semidirect
65. C₂×C₂²⋊Q₈	C₂²	semidirect
66. C₂×C₂²⋊Q₈	C₂²	semidirect
67. C₂×C₂²⋊Q₈	C₂²	semidirect
68. C₂×C₂²⋊Q₈	C₂²	semidirect
69. C₂×C₂²⋊Q₈	C₂²	semidirect
73. C₂×#121	C₂²	semidirect
75. C₂×C₄×Q₈	C₂²	semidirect
76. C₂×C₄×Q₈	C₂²	semidirect
77. C₂×C₄×Q₈	C₂²	semidirect
78. C₂×C₄×Q₈	C₂²	semidirect
82. C₂×C₄×D₄	C₂²	semidirect
83. C₂×C₄×D₄	C₂²	semidirect
85. C₂×#117	C₂²	semidirect
86. C₂×#117	C₂²	semidirect
103. C₂²×Q₈	C₂³	semidirect
120. C₄⋊Q₈	C₂³	semidirect
121. C₄⋊Q₈	C₂³	semidirect
122. C₄⋊Q₈	C₂³	semidirect
123. C₄⋊Q₈	C₂³	semidirect
124. #126	C₂³	semidirect
125. #126	C₂³	semidirect
126. #126	C₂³	semidirect
127. #126	C₂³	semidirect
128. #125	C₂³	semidirect
129. #125	C₂³	semidirect
135. #124	C₂³	semidirect
136. #124	C₂³	semidirect
137. #124	C₂³	semidirect
140. #123	C₂³	semidirect
141. #123	C₂³	semidirect
142. #123	C₂³	semidirect
147. C₂²⋊Q₈	C₂³	semidirect
148. C₂²⋊Q₈	C₂³	semidirect
149. C₂²⋊Q₈	C₂³	semidirect
150. C₂²⋊Q₈	C₂³	semidirect
151. C₂²⋊Q₈	C₂³	semidirect
152. C₂²⋊Q₈	C₂³	semidirect
153. C₂²⋊Q₈	C₂³	semidirect
157. #121	C₂³	semidirect
159. C₄×Q₈	C₂³	semidirect
160. C₄×Q₈	C₂³	semidirect
161. C₄×Q₈	C₂³	semidirect
162. C₄×Q₈	C₂³	semidirect
166. C₄×D₄	C₂³	semidirect
167. C₄×D₄	C₂³	semidirect
169. #117	C₂³	semidirect
170. #117	C₂³	semidirect
407. C₂	#3561	direct (N × H)

Maximal subgroups

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

subgroup	index
#3561	2
C₂×#563	2
C₂×#562	2
C₂×#556	2
C₂×#551	2
C₂×D₄×Q₈	2
C₂×#548	2
C₂×#546	2
C₂×#543	2
C₂×#542	2
C₂×#538	2
C₂×#536	2
C₂×#533	2
C₂×#529	2
C₂²×#124	2

Subgroups — C2×#3561 #62731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3561 #62731