Subgroups — #3565

Order 128.

465 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	2	yes	maximal, complemented
2	#562	C₂	1	2	yes	maximal, complemented
3	#557	C₂	1	1	yes	maximal, characteristic, complemented
4	#556	C₂	1	2	yes	maximal, complemented
5	#556	C₂	1	1	yes	maximal, characteristic, complemented
6	#555	C₂	1	1	yes	maximal, characteristic, complemented
7	#554	C₂	1	1	yes	maximal, characteristic, complemented
8	#553	C₂	1	1	yes	maximal, characteristic, complemented
9	#553	C₂	1	1	yes	maximal, characteristic, complemented
10	#552	C₂	1	2	yes	maximal, complemented
11	#552	C₂	1	1	yes	maximal, characteristic, complemented
12	#552	C₂	1	1	yes	maximal, characteristic, complemented
13	#544	C₂	1	1	yes	maximal, characteristic, complemented
14	#544	C₂	1	1	yes	maximal, characteristic, complemented
15	#542	C₂	1	1	yes	maximal, characteristic, complemented
16	#542	C₂	1	1	yes	maximal, characteristic, complemented
17	#542	C₂	1	2	yes	maximal, complemented
18	#542	C₂	1	2	yes	maximal, complemented
19	#541	C₂	1	2	yes	maximal, complemented
20	#539	C₂	1	2	yes	maximal, complemented
21	#533	C₂	1	1	yes	maximal, characteristic, complemented
22	#529	C₂	1	1	yes	maximal, characteristic, complemented
23	#529	C₂	1	1	yes	maximal, characteristic, complemented
24	C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₄⋊Q₈	C₂²	1	2	yes	—
27	C₄⋊Q₈	C₂²	1	2	yes	—
28	C₄⋊Q₈	C₂²	1	1	yes	characteristic
29	C₄⋊Q₈	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈	C₂²	1	2	yes	—
31	#126	C₂²	1	2	yes	—
32	#126	C₂²	1	1	yes	characteristic
33	#126	C₂²	1	1	yes	characteristic
34	#126	C₂²	1	2	yes	—
35	#126	C₂²	1	2	yes	—
36	#126	C₂²	1	2	yes	—
37	#126	C₂²	1	2	yes	—
38	#126	C₂²	1	2	yes	—
39	#126	C₂²	1	2	yes	—
40	#126	C₂²	1	2	yes	—
41	#126	C₂²	1	2	yes	—
42	#125	C₂²	1	1	yes	characteristic
43	#125	C₂²	1	1	yes	characteristic
44	#125	C₂²	1	1	yes	characteristic
45	#125	C₂²	1	1	yes	characteristic
46	#125	C₂²	1	2	yes	—
47	#125	C₂²	1	2	yes	—
48	#125	C₂²	1	2	yes	—
49	#125	C₂²	1	2	yes	—
50	#125	C₂²	1	2	yes	—
51	#125	C₂²	1	1	yes	characteristic
52	#125	C₂²	1	2	yes	—
53	#125	C₂²	1	1	yes	characteristic
54	#124	C₂²	1	1	yes	characteristic
55	#124	C₂²	1	1	yes	characteristic
56	#124	C₂²	1	2	yes	—
57	#124	C₂²	1	2	yes	—
58	#124	C₂²	1	2	yes	—
59	#124	C₂²	1	2	yes	—
60	#124	C₂²	1	2	yes	—
61	#123	C₂²	1	1	yes	characteristic
62	#123	C₂²	1	1	yes	characteristic
63	#123	C₂²	1	2	yes	—
64	#123	C₂²	1	2	yes	—
65	#123	C₂²	1	2	yes	—
66	#123	C₂²	1	2	yes	—
67	#123	C₂²	1	2	yes	—
68	#123	C₂²	1	2	yes	—
69	#123	C₂²	1	2	yes	—
70	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
71	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
72	C₂²⋊Q₈	C₂²	1	2	yes	—
73	C₂²⋊Q₈	C₂²	1	2	yes	—
74	C₂²⋊Q₈	C₂²	1	2	yes	—
75	C₂²⋊Q₈	C₂²	1	2	yes	—
76	C₂²⋊Q₈	C₂²	1	2	yes	—
77	C₂²⋊Q₈	C₂²	1	2	yes	—
78	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
79	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
80	C₂²⋊Q₈	C₂²	1	2	yes	—
81	C₂²⋊Q₈	C₂²	1	2	yes	—
82	C₂²⋊Q₈	C₂²	1	2	yes	—
83	C₂²⋊Q₈	C₂²	1	2	yes	—
84	C₂²⋊Q₈	C₂²	1	2	yes	—
85	#121	C₂²	1	1	yes	characteristic
86	#121	C₂²	1	1	yes	characteristic
87	#121	C₂²	1	2	yes	—
88	#121	C₂²	1	1	yes	characteristic
89	#121	C₂²	1	1	yes	characteristic
90	#121	C₂²	1	2	yes	—
91	#121	C₂²	1	2	yes	—
92	C₄×Q₈	C₂²	1	1	yes	characteristic
93	C₄×Q₈	C₂²	1	1	yes	characteristic
94	C₄×Q₈	C₂²	1	2	yes	—
95	C₄×Q₈	C₂²	1	2	yes	—
96	C₄×Q₈	C₂²	1	2	yes	—
97	C₄×Q₈	C₂²	1	1	yes	characteristic
98	C₄×Q₈	C₂²	1	2	yes	—
99	C₄×Q₈	C₂²	1	1	yes	characteristic
100	C₄×D₄	C₂²	1	2	yes	—
101	C₄×D₄	C₂²	1	1	yes	characteristic
102	C₄×D₄	C₂²	1	1	yes	characteristic
103	C₄×D₄	C₂²	1	1	yes	characteristic
104	C₄×D₄	C₂²	1	1	yes	characteristic
105	C₄×D₄	C₂²	1	2	yes	—
106	C₄×D₄	C₂²	1	2	yes	—
107	C₄×D₄	C₂²	1	1	yes	characteristic
108	C₄×D₄	C₂²	1	1	yes	characteristic
109	#117	C₂²	1	1	yes	characteristic
110	#117	C₂²	1	1	yes	characteristic
111	#117	C₂²	1	2	yes	—
112	#117	C₂²	1	1	yes	characteristic
113	#117	C₂²	1	1	yes	characteristic
114	#117	C₂²	1	2	yes	—
115	#117	C₂²	1	2	yes	—
116	#117	C₂²	1	2	yes	—
117	#117	C₂²	1	1	yes	characteristic
118	#117	C₂²	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
122	C₂×C₄²	C₂²	1	1	yes	characteristic
123	C₂×Q₈	C₂³	1	1	yes	characteristic
124	C₂×Q₈	C₂³	1	1	yes	characteristic
125	C₂×Q₈	C₂³	1	2	yes	—
126	C₂×Q₈	C₂³	1	2	yes	—
127	C₂×Q₈	C₂³	1	2	yes	—
128	C₂×Q₈	C₂³	1	2	yes	—
129	C₂×D₄	C₂³	1	1	yes	characteristic
130	C₂×D₄	C₂³	1	1	yes	characteristic
131	C₂×D₄	C₂³	1	2	yes	—
132	C₂×D₄	C₂³	1	1	yes	characteristic
133	C₂×D₄	C₂³	1	2	yes	—
134	C₂×D₄	C₂³	1	1	yes	characteristic
135	C₂×D₄	C₂³	1	2	yes	—
136	C₂²×C₄	C₂³	1	1	yes	characteristic
137	C₂²×C₄	C₂³	1	1	yes	characteristic
138	C₂²×C₄	C₂³	1	1	yes	characteristic
139	C₂²×C₄	C₂³	1	1	yes	characteristic
140	C₂²×C₄	C₂³	1	1	yes	characteristic
141	C₂²×C₄	C₂³	1	2	yes	—
142	C₂²×C₄	C₂³	1	2	yes	—
143	C₂²×C₄	C₂³	1	2	yes	—
144	C₂²×C₄	C₂³	1	2	yes	—
145	C₂²×C₄	C₂³	1	2	yes	—
146	C₄⋊C₄	C₂³	1	1	yes	characteristic
147	C₄⋊C₄	C₂³	1	1	yes	characteristic
148	C₄⋊C₄	C₂³	1	1	yes	characteristic
149	C₄⋊C₄	C₂³	1	1	yes	characteristic
150	C₄⋊C₄	C₂³	1	1	yes	characteristic
151	C₄⋊C₄	C₂³	1	2	yes	—
152	C₄⋊C₄	C₂³	1	2	yes	—
153	C₄⋊C₄	C₂³	1	2	yes	—
154	C₄⋊C₄	C₂³	1	2	yes	—
155	C₄⋊C₄	C₂³	1	2	yes	—
156	C₄⋊C₄	C₂³	1	1	yes	characteristic
157	C₄⋊C₄	C₂³	1	1	yes	characteristic
158	C₄⋊C₄	C₂³	1	1	yes	characteristic
159	C₄⋊C₄	C₂³	1	2	yes	—
160	C₄⋊C₄	C₂³	1	2	yes	—
161	C₄⋊C₄	C₂³	1	1	yes	characteristic
162	C₄⋊C₄	C₂³	1	2	yes	—
163	C₄⋊C₄	C₂³	1	1	yes	characteristic
164	C₄⋊C₄	C₂³	1	2	yes	—
165	C₄⋊C₄	C₂³	1	2	yes	—
166	C₄⋊C₄	C₂³	1	1	yes	characteristic
167	C₄⋊C₄	C₂³	1	1	yes	characteristic
168	C₄⋊C₄	C₂³	1	2	yes	—
169	C₄⋊C₄	C₂³	1	1	yes	characteristic
170	C₄⋊C₄	C₂³	1	2	yes	—
171	C₄⋊C₄	C₂³	1	2	yes	—
172	C₄⋊C₄	C₂³	1	1	yes	characteristic
173	C₄⋊C₄	C₂³	1	1	yes	characteristic
174	C₄⋊C₄	C₂³	1	1	yes	characteristic
175	C₄⋊C₄	C₂³	1	2	yes	—
176	C₄⋊C₄	C₂³	1	2	yes	—
177	C₄⋊C₄	C₂³	1	2	yes	—
178	C₄⋊C₄	C₂³	1	2	yes	—
179	C₄⋊C₄	C₂³	1	2	yes	—
180	C₄⋊C₄	C₂³	1	2	yes	—
181	C₄⋊C₄	C₂³	1	2	yes	—
182	C₄⋊C₄	C₂³	1	2	yes	—
183	C₄⋊C₄	C₂³	1	2	yes	—
184	C₂²⋊C₄	C₂³	1	2	yes	—
185	C₂²⋊C₄	C₂³	1	1	yes	characteristic
186	C₂²⋊C₄	C₂³	1	1	yes	characteristic
187	C₂²⋊C₄	C₂³	1	1	yes	characteristic
188	C₂²⋊C₄	C₂³	1	2	yes	—
189	C₂²⋊C₄	C₂³	1	2	yes	—
190	C₂²⋊C₄	C₂³	1	2	yes	—
191	C₂²⋊C₄	C₂³	1	1	yes	characteristic
192	C₂²⋊C₄	C₂³	1	2	yes	—
193	C₂²⋊C₄	C₂³	1	2	yes	—
194	C₂²⋊C₄	C₂³	1	1	yes	characteristic
195	C₂²⋊C₄	C₂³	1	1	yes	characteristic
196	C₂²⋊C₄	C₂³	1	2	yes	—
197	C₂²⋊C₄	C₂³	1	2	yes	—
198	C₂²⋊C₄	C₂³	1	2	yes	—
199	C₂²⋊C₄	C₂³	1	1	yes	characteristic
200	C₂²⋊C₄	C₂³	1	1	yes	characteristic
201	C₂²⋊C₄	C₂³	1	2	yes	—
202	C₂²⋊C₄	C₂³	1	2	yes	—
203	C₂²⋊C₄	C₂³	1	2	yes	—
204	C₂²⋊C₄	C₂³	1	2	yes	—
205	C₂²⋊C₄	C₂³	1	1	yes	characteristic
206	C₂²⋊C₄	C₂³	1	2	yes	—
207	C₂²⋊C₄	C₂³	1	2	yes	—
208	C₂²⋊C₄	C₂³	1	1	yes	characteristic
209	C₄²	C₂³	1	1	yes	characteristic
210	C₄²	C₂³	1	1	yes	characteristic
211	C₄²	C₂³	1	1	yes	characteristic
212	C₄²	C₂³	1	1	yes	characteristic
213	C₄²	C₂³	1	2	yes	—
214	C₄²	C₂³	1	2	yes	—
215	C₄²	C₂³	1	2	yes	—
216	C₄²	C₂³	1	2	yes	—
217	C₄²	C₂³	1	1	yes	characteristic
218	C₄²	C₂³	1	1	yes	characteristic
219	C₄²	C₂³	1	2	yes	—
220	C₄²	C₂³	1	2	yes	—
221	C₄²	C₂³	1	2	yes	—
222	C₂³	C₂⁴	1	1	yes	characteristic
223	C₂³	C₂⁴	1	1	yes	characteristic
224	C₂³	C₂⁴	1	1	yes	characteristic
225	C₂³	C₂⁴	1	2	yes	—
226	Q₈	—	2	2	no	—
227	Q₈	—	4	1	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂⁴	1	1	yes	characteristic
250	C₂×C₄	C₂⁴	1	1	yes	characteristic
251	C₂×C₄	C₂⁴	1	2	yes	—
252	C₂×C₄	C₂⁴	1	1	yes	characteristic
253	C₂×C₄	C₂⁴	1	2	yes	—
254	C₂×C₄	C₂⁴	1	1	yes	characteristic
255	C₂×C₄	C₂⁴	1	1	yes	characteristic
256	C₂×C₄	C₂⁴	1	1	yes	characteristic
257	C₂×C₄	C₂⁴	1	1	yes	characteristic
258	C₂×C₄	C₂⁴	1	1	yes	characteristic
259	C₂×C₄	C₂⁴	1	2	yes	—
260	C₂×C₄	C₂⁴	1	1	yes	characteristic
261	C₂×C₄	C₂⁴	1	2	yes	—
262	C₂×C₄	C₂⁴	1	2	yes	—
263	C₂×C₄	C₂⁴	1	2	yes	—
264	C₂×C₄	C₂⁴	1	1	yes	characteristic
265	C₂×C₄	C₂⁴	1	2	yes	—
266	C₂×C₄	C₂⁴	1	1	yes	characteristic
267	C₂×C₄	C₂⁴	1	1	yes	characteristic
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	C₂⁵	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	2₋¹⁺⁴	1	2	yes	—
300	C₂	—	4	1	no	—
301	C₂	—	4	1	no	—
302	C₂	—	4	1	no	—
303	C₂	—	4	2	no	—
304	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
305	C₂	#585	1	1	yes	characteristic
306	C₂	#585	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #563	C₂	semidirect
2. #562	C₂	semidirect
3. #557	C₂	semidirect
4. #556	C₂	semidirect
5. #556	C₂	semidirect
6. #555	C₂	semidirect
7. #554	C₂	semidirect
8. #553	C₂	semidirect
9. #553	C₂	semidirect
10. #552	C₂	semidirect
11. #552	C₂	semidirect
12. #552	C₂	semidirect
13. #544	C₂	semidirect
14. #544	C₂	semidirect
15. #542	C₂	semidirect
16. #542	C₂	semidirect
17. #542	C₂	semidirect
18. #542	C₂	semidirect
19. #541	C₂	semidirect
20. #539	C₂	semidirect
21. #533	C₂	semidirect
22. #529	C₂	semidirect
23. #529	C₂	semidirect

Maximal subgroups

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

subgroup	index
#563	2
#562	2
#557	2
#556	2
#555	2
#554	2
#553	2
#552	2
#544	2
#542	2
#541	2
#539	2
#533	2
#529	2