Subgroups — #61475

Order 256.

950 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3514	C₂	1	2	yes	maximal, complemented
2	#3396	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3373	C₂	1	2	yes	maximal, complemented
5	#3342	C₂	1	1	yes	maximal, characteristic, complemented
6	#3341	C₂	1	2	yes	maximal, complemented
7	#3324	C₂	1	2	yes	maximal, complemented
8	#3319	C₂	1	4	yes	maximal, complemented
9	#3280	C₂	1	2	yes	maximal, complemented
10	D₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#3175	C₂	1	2	yes	maximal, complemented
12	#3169	C₂	1	1	yes	maximal, characteristic, complemented
13	#3118	C₂	1	4	yes	maximal, complemented
14	#3115	C₂	1	2	yes	maximal, complemented
15	#2966	C₂	1	1	yes	maximal, characteristic, complemented
16	#2948	C₂	1	2	yes	maximal, complemented
17	C₄×#135	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#135	C₂²	1	1	yes	characteristic
19	C₂²×D₈	C₂²	1	2	yes	—
20	#559	C₂²	1	2	yes	complemented
21	#556	C₂²	1	2	yes	—
22	#555	C₂²	1	4	yes	complemented
23	#551	—	2	4	no	—
24	D₄×Q₈	C₂²	1	2	yes	complemented
25	#548	—	2	2	no	—
26	#546	—	2	2	no	—
27	#546	C₂²	1	4	yes	complemented
28	D₄²	C₂²	1	2	yes	—
29	#542	C₂²	1	4	yes	complemented
30	#538	—	2	4	no	—
31	#532	C₂²	1	2	yes	complemented
32	#529	—	2	2	no	—
33	#529	C₂²	1	1	yes	characteristic
34	C₂×#124	—	2	2	no	—
35	#518	C₂²	1	4	yes	complemented
36	C₄×#41	C₂²	1	2	yes	complemented
37	#500	C₂²	1	2	yes	complemented
38	#500	C₂²	1	1	yes	characteristic, complemented
39	#495	C₂²	1	2	yes	complemented
40	#493	C₂²	1	1	yes	characteristic
41	#489	C₂²	1	4	yes	complemented
42	#486	C₂²	1	2	yes	complemented
43	#485	C₂²	1	4	yes	complemented
44	#482	C₂²	1	4	yes	complemented
45	#482	C₂²	1	4	yes	complemented
46	#476	C₂²	1	4	yes	complemented
47	#475	C₂²	1	2	yes	complemented
48	#474	C₂²	1	2	yes	complemented
49	#474	C₂²	1	4	yes	complemented
50	#468	C₂²	1	4	yes	complemented
51	#467	C₂²	1	2	yes	complemented
52	#466	C₂²	1	2	yes	—
53	#466	C₂²	1	2	yes	complemented
54	#466	C₂²	1	2	yes	complemented
55	#463	C₂²	1	4	yes	complemented
56	#463	C₂²	1	4	yes	complemented
57	#460	C₂²	1	2	yes	complemented
58	#459	C₂²	1	2	yes	—
59	#450	C₂²	1	4	yes	complemented
60	#449	C₂²	1	4	yes	complemented
61	#447	C₂²	1	4	yes	—
62	#446	C₂²	1	1	yes	characteristic, complemented
63	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
64	#442	C₂²	1	4	yes	complemented
65	#442	C₂²	1	4	yes	complemented
66	#440	C₂²	1	4	yes	complemented
67	C₄×Q₁₆	C₂²	1	1	yes	characteristic, complemented
68	C₄×Q₁₆	C₂²	1	1	yes	characteristic, complemented
69	C₄×SD₁₆	C₂²	1	2	yes	complemented
70	C₄×SD₁₆	C₂²	1	2	yes	complemented
71	C₄×D₈	C₂²	1	1	yes	characteristic
72	C₄×D₈	C₂²	1	2	yes	—
73	C₄×D₈	C₂²	1	1	yes	characteristic
74	C₄×D₈	C₂²	1	2	yes	complemented
75	C₄×D₈	C₂²	1	2	yes	complemented
76	C₄×D₈	C₂²	1	2	yes	complemented
77	#436	C₂²	1	1	yes	characteristic
78	#435	C₂²	1	2	yes	complemented
79	C₈×D₄	C₂²	1	1	yes	characteristic
80	C₈×D₄	C₂²	1	2	yes	complemented
81	#433	C₂²	1	2	yes	complemented
82	#432	C₂²	1	1	yes	characteristic, complemented
83	#427	C₂²	1	1	yes	characteristic, complemented
84	#427	C₂²	1	2	yes	complemented
85	#418	C₂²	1	4	yes	complemented
86	#416	C₂²	1	2	yes	complemented
87	#405	C₂²	1	2	yes	complemented
88	C₂×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
89	C₂×#41	—	2	2	no	—
90	C₂×#41	C₂³	1	2	yes	—
91	C₂²×Q₈	—	2	2	no	—
92	C₂²×D₄	—	2	2	no	—
93	C₂²×D₄	C₂³	1	4	yes	—
94	#135	—	2	4	no	—
95	C₂×Q₁₆	C₂³	1	1	yes	characteristic
96	C₂×SD₁₆	C₂³	1	2	yes	—
97	C₂×SD₁₆	C₂³	1	4	yes	—
98	C₂×D₈	—	2	8	no	—
99	C₂×D₈	C₂³	1	2	yes	—
100	C₂×D₈	C₂³	1	4	yes	—
101	C₂×D₈	C₂³	1	2	yes	—
102	C₂×D₈	C₂³	1	1	yes	characteristic
103	C₂×M₁₆	C₂³	1	2	yes	—
104	C₂²×C₈	C₂³	1	2	yes	—
105	C₂²×C₈	C₂³	1	1	yes	characteristic
106	C₂²×C₈	C₂³	1	1	yes	characteristic
107	C₄⋊Q₈	C₂³	1	2	yes	—
108	C₄⋊Q₈	C₂³	1	4	yes	—
109	#127	C₂³	1	2	yes	—
110	#126	—	2	4	no	—
111	#126	—	2	4	no	—
112	#126	D₄	1	2	yes	—
113	#125	D₄	1	1	yes	characteristic
114	#124	—	2	2	no	—
115	#124	—	2	2	no	—
116	#124	—	2	4	no	—
117	#124	—	2	4	no	—
118	#124	—	2	4	no	—
119	#124	—	2	2	no	—
120	#124	D₄	1	1	yes	characteristic
121	#124	C₂³	1	4	yes	—
122	#123	—	2	4	no	—
123	#123	—	2	4	no	—
124	#123	—	2	4	no	—
125	#123	D₄	1	2	yes	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₂²⋊Q₈	—	2	4	no	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₂²⋊Q₈	D₄	1	1	yes	characteristic
130	C₂²⋊Q₈	C₂³	1	4	yes	—
131	#121	—	2	4	no	—
132	#121	—	2	2	no	—
133	#121	—	2	2	no	—
134	#121	—	2	4	no	—
135	#121	D₄	1	1	yes	characteristic
136	#121	C₂³	1	4	yes	—
137	#121	C₂³	1	4	yes	—
138	#120	—	2	4	no	—
139	#120	—	2	4	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	C₂³	1	2	yes	—
142	C₄×Q₈	C₂³	1	2	yes	—
143	C₄×Q₈	C₂³	1	1	yes	characteristic
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	—	2	4	no	—
147	C₄×D₄	—	2	4	no	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	C₂³	1	2	yes	—
151	C₄×D₄	C₂³	1	2	yes	—
152	C₄×D₄	C₂³	1	4	yes	—
153	C₄×D₄	C₂³	1	4	yes	—
154	C₄×D₄	C₂³	1	4	yes	—
155	C₄×D₄	C₂³	1	1	yes	characteristic
156	C₄×D₄	C₂³	1	2	yes	—
157	#117	—	2	4	no	—
158	#117	—	2	2	no	—
159	#117	C₂³	1	2	yes	—
160	#117	C₂³	1	4	yes	—
161	#117	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂³	1	1	yes	characteristic
167	#107	C₂³	1	1	yes	characteristic
168	#107	C₂³	1	2	yes	—
169	#107	C₂³	1	1	yes	characteristic
170	#107	C₂³	1	2	yes	—
171	#107	C₂³	1	4	yes	—
172	#106	C₂³	1	2	yes	—
173	#106	C₂³	1	4	yes	—
174	C₄⋊C₈	C₂³	1	1	yes	characteristic
175	C₄⋊C₈	C₂³	1	1	yes	characteristic
176	C₄⋊C₈	C₂³	1	2	yes	—
177	C₄⋊C₈	C₂³	1	2	yes	—
178	#103	C₂³	1	2	yes	—
179	#103	C₂³	1	2	yes	—
180	#103	C₂³	1	4	yes	—
181	#102	C₂³	1	2	yes	—
182	#102	C₂³	1	2	yes	—
183	#102	C₂³	1	4	yes	—
184	#102	C₂³	1	4	yes	—
185	#102	C₂³	1	4	yes	—
186	#102	C₂³	1	4	yes	—
187	#102	C₂³	1	4	yes	—
188	C₂²⋊C₈	C₂³	1	2	yes	—
189	C₂²⋊C₈	C₂³	1	2	yes	—
190	C₂²⋊C₈	C₂³	1	2	yes	—
191	#97	C₂³	1	2	yes	—
192	#97	C₂³	1	2	yes	—
193	C₄×C₈	C₂³	1	1	yes	characteristic
194	C₄×C₈	C₂³	1	1	yes	characteristic
195	C₄×C₈	C₂³	1	1	yes	characteristic
196	C₄×C₈	C₂³	1	1	yes	characteristic
197	C₄×C₈	C₂³	1	2	yes	—
198	C₂⁴	—	2	4	no	—
199	#41	—	2	4	no	—
200	#41	—	4	2	no	—
201	#41	—	8	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	8	4	no	—
205	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×Q₈	C₂⁴	1	2	yes	—
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	8	4	no	—
216	C₂×D₄	C₂×D₄	1	1	yes	characteristic
217	C₂×D₄	C₂×D₄	1	2	yes	—
218	C₂×D₄	C₂⁴	1	4	yes	—
219	C₂×D₄	C₂⁴	1	2	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴	1	2	yes	—
230	Q₁₆	—	2	2	no	—
231	SD₁₆	—	2	4	no	—
232	SD₁₆	—	4	4	no	—
233	D₈	—	2	8	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	D₈	#41	1	4	yes	complemented
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	C₂⁴	1	1	yes	characteristic
242	C₂×C₈	C₂⁴	1	1	yes	characteristic
243	C₂×C₈	C₂⁴	1	1	yes	characteristic
244	C₂×C₈	C₂⁴	1	2	yes	—
245	C₂×C₈	C₂⁴	1	2	yes	—
246	C₂×C₈	C₂⁴	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂×D₄	1	2	yes	—
253	C₄⋊C₄	C₂×D₄	1	2	yes	—
254	C₄⋊C₄	C₂×D₄	1	2	yes	—
255	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
257	C₄⋊C₄	C₂×D₄	1	2	yes	—
258	C₄⋊C₄	C₂⁴	1	2	yes	—
259	C₄⋊C₄	C₂⁴	1	4	yes	—
260	C₄⋊C₄	C₂⁴	1	2	yes	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	4	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	C₂×D₄	1	2	yes	—
274	C₂²⋊C₄	C₂×D₄	1	2	yes	—
275	C₂²⋊C₄	C₂×D₄	1	2	yes	—
276	C₂²⋊C₄	C₂×D₄	1	2	yes	—
277	C₂²⋊C₄	C₂×D₄	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	4	no	—
282	C₄²	C₂×D₄	1	1	yes	characteristic
283	C₄²	C₂×D₄	1	1	yes	characteristic
284	C₄²	C₂⁴	1	1	yes	characteristic
285	C₄²	C₂⁴	1	1	yes	characteristic
286	C₄²	C₂⁴	1	2	yes	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	4	no	—
289	C₂³	—	4	8	no	—
290	C₂³	—	8	4	no	—
291	C₂³	—	8	4	no	—
292	C₂³	C₂²×D₄	1	2	yes	—
293	C₂³	C₂²×D₄	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	2	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	8	4	no	—
299	D₄	—	2	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	8	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	4	no	—
310	D₄	C₂×#41	1	4	yes	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	—	8	4	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₄	1	2	yes	—
330	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄	1	2	yes	—
333	C₂×C₄	C₂²×D₄	1	2	yes	—
334	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄	C₂⁵	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	2	no	—
341	C₈	C₂×#41	1	2	yes	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	4	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	4	2	no	—
351	C₂²	—	4	4	no	—
352	C₂²	—	4	4	no	—
353	C₂²	—	4	4	no	—
354	C₂²	—	16	8	no	—
355	C₂²	C₂³×D₄	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	4	no	—
367	C₄	#576	1	2	yes	—
368	C₄	C₂²×#41	1	1	yes	characteristic
369	C₄	#585	1	1	yes	characteristic
370	C₂	—	4	1	no	—
371	C₂	—	4	2	no	—
372	C₂	—	4	4	no	—
373	C₂	—	8	4	no	—
374	C₂	D₄×#41	1	1	yes	characteristic
375	C₂	C₂×#576	1	1	yes	characteristic
376	C₂	#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3514	C₂	semidirect
2. #3396	C₂	semidirect
3. Q₈×D₈	C₂	semidirect
4. #3373	C₂	semidirect
5. #3342	C₂	semidirect
6. #3341	C₂	semidirect
7. #3324	C₂	semidirect
8. #3319	C₂	semidirect
9. #3280	C₂	semidirect
10. D₄×D₈	C₂	semidirect
11. #3175	C₂	semidirect
12. #3169	C₂	semidirect
13. #3118	C₂	semidirect
14. #3115	C₂	semidirect
15. #2966	C₂	semidirect
16. #2948	C₂	semidirect
17. C₄×#135	C₂	semidirect
20. #559	C₂²	semidirect
22. #555	C₂²	semidirect
24. D₄×Q₈	C₂²	semidirect
27. #546	C₂²	semidirect
29. #542	C₂²	semidirect
31. #532	C₂²	semidirect
35. #518	C₂²	semidirect
36. C₄×#41	C₂²	semidirect
37. #500	C₂²	semidirect
38. #500	C₂²	semidirect
39. #495	C₂²	semidirect
41. #489	C₂²	semidirect
42. #486	C₂²	semidirect
43. #485	C₂²	semidirect
44. #482	C₂²	semidirect
45. #482	C₂²	semidirect
46. #476	C₂²	semidirect
47. #475	C₂²	semidirect
48. #474	C₂²	semidirect
49. #474	C₂²	semidirect
50. #468	C₂²	semidirect
51. #467	C₂²	semidirect
53. #466	C₂²	semidirect
54. #466	C₂²	semidirect
55. #463	C₂²	semidirect
56. #463	C₂²	semidirect
57. #460	C₂²	semidirect
59. #450	C₂²	semidirect
60. #449	C₂²	semidirect
62. #446	C₂²	semidirect
63. C₈×Q₈	C₂²	semidirect
64. #442	C₂²	semidirect
65. #442	C₂²	semidirect
66. #440	C₂²	semidirect
67. C₄×Q₁₆	C₂²	semidirect
68. C₄×Q₁₆	C₂²	semidirect
69. C₄×SD₁₆	C₂²	semidirect
70. C₄×SD₁₆	C₂²	semidirect
74. C₄×D₈	C₂²	semidirect
75. C₄×D₈	C₂²	semidirect
76. C₄×D₈	C₂²	semidirect
78. #435	C₂²	semidirect
80. C₈×D₄	C₂²	semidirect
81. #433	C₂²	semidirect
82. #432	C₂²	semidirect
83. #427	C₂²	semidirect
84. #427	C₂²	semidirect
85. #418	C₂²	semidirect
86. #416	C₂²	semidirect
87. #405	C₂²	semidirect
88. C₂×C₄×C₈	C₂²	semidirect
236. D₈	#41	semidirect

Maximal subgroups

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

subgroup	index
#3514	2
#3396	2
Q₈×D₈	2
#3373	2
#3342	2
#3341	2
#3324	2
#3319	2
#3280	2
D₄×D₈	2
#3175	2
#3169	2
#3118	2
#3115	2
#2966	2
#2948	2
C₄×#135	2