Subgroups — #10586085

Order 512.

32601 conjugacy classes of subgroups.

The subgroup lattice has 259 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63004	C₂	1	16	yes	maximal, complemented
2	#63004	C₂	1	32	yes	maximal, complemented
3	Q₈×2₋¹⁺⁴	C₂	1	16	yes	maximal, complemented
4	#62854	C₂	1	20	yes	maximal, complemented
5	#62849	C₂	1	10	yes	maximal, complemented
6	#62813	C₂	1	20	yes	maximal, complemented
7	C₂×#3502	C₂	1	10	yes	maximal, complemented
8	#62653	C₂	1	2	yes	maximal
9	C₂×C₄×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
10	C₂²×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
11	#3504	C₂²	1	160	yes	complemented
12	#3504	C₂²	1	320	yes	complemented
13	#3503	C₂²	1	80	yes	complemented
14	#3503	C₂²	1	320	yes	complemented
15	#3503	C₂²	1	80	yes	complemented
16	#3502	C₂²	1	40	yes	—
17	#3502	C₂²	1	80	yes	—
18	#3502	C₂²	1	40	yes	—
19	#3502	C₂²	1	80	yes	complemented
20	#3502	C₂²	1	160	yes	complemented
21	#3502	C₂²	1	80	yes	complemented
22	#3502	C₂²	1	160	yes	complemented
23	#3501	C₂²	1	40	yes	—
24	#3501	C₂²	1	120	yes	—
25	Q₈×#41	C₂²	1	160	yes	complemented
26	Q₈×#41	C₂²	1	160	yes	complemented
27	C₂×Q₈²	C₂²	1	40	yes	complemented
28	C₂×#557	C₂²	1	80	yes	complemented
29	C₂×#557	C₂²	1	40	yes	complemented
30	#3458	C₂²	1	120	yes	—
31	C₂×#544	C₂²	1	60	yes	—
32	C₂×#533	C₂²	1	60	yes	—
33	C₄×2₋¹⁺⁴	C₂²	1	16	yes	—
34	C₄×2₋¹⁺⁴	C₂²	1	32	yes	—
35	C₄×2₋¹⁺⁴	C₂²	1	16	yes	—
36	C₄×2₋¹⁺⁴	C₂²	1	32	yes	—
37	#3428	C₂²	1	40	yes	—
38	C₂×#520	C₂²	1	10	yes	—
39	C₂×#519	C₂²	1	20	yes	—
40	C₂×#519	C₂²	1	5	yes	—
41	C₂×C₄×#41	C₂²	1	10	yes	—
42	C₂²×C₄×Q₈	C₂²	1	5	yes	—
43	C₂×2₋¹⁺⁴	Q₈	1	32	yes	complemented
44	C₂×2₋¹⁺⁴	C₂³	1	16	yes	—
45	C₂²×#41	C₂³	1	10	yes	—
46	C₂³×Q₈	C₂³	1	5	yes	—
47	Q₈²	C₂³	1	160	yes	—
48	Q₈²	C₂³	1	320	yes	—
49	Q₈²	C₂³	1	160	yes	—
50	#557	C₂³	1	160	yes	—
51	#557	C₂³	1	320	yes	—
52	#557	C₂³	1	640	yes	—
53	#557	C₂³	1	320	yes	—
54	#557	C₂³	1	160	yes	—
55	#557	C₂³	1	320	yes	—
56	#554	C₂³	1	160	yes	—
57	#554	C₂³	1	320	yes	—
58	#554	C₂³	1	960	yes	—
59	#554	C₂³	1	480	yes	—
60	D₄×Q₈	C₂³	1	160	yes	—
61	D₄×Q₈	C₂³	1	480	yes	—
62	#544	C₂³	1	240	yes	—
63	#544	C₂³	1	480	yes	—
64	#544	C₂³	1	80	yes	—
65	#544	C₂³	1	480	yes	—
66	#544	C₂³	1	480	yes	—
67	#543	C₂³	1	160	yes	—
68	#533	C₂³	1	80	yes	—
69	#533	C₂³	1	240	yes	—
70	#533	C₂³	1	240	yes	—
71	#533	C₂³	1	240	yes	—
72	#533	C₂³	1	960	yes	—
73	C₂×C₄⋊Q₈	C₂³	1	60	yes	—
74	C₂×C₄⋊Q₈	C₂³	1	60	yes	—
75	C₂×C₄⋊Q₈	C₂³	1	240	yes	—
76	C₂×#125	C₂³	1	120	yes	—
77	C₂×#125	C₂³	1	240	yes	—
78	C₂×C₂²⋊Q₈	C₂³	1	40	yes	—
79	C₂×C₂²⋊Q₈	C₂³	1	120	yes	—
80	#520	C₂³	1	160	yes	—
81	#520	C₂³	1	160	yes	—
82	#520	C₂³	1	160	yes	—
83	#519	C₂³	1	80	yes	—
84	#519	C₂³	1	80	yes	—
85	#519	C₂³	1	160	yes	—
86	#518	C₂³	1	80	yes	—
87	C₄×#41	C₂³	1	80	yes	—
88	C₄×#41	C₂³	1	80	yes	—
89	C₄×#41	C₂³	1	320	yes	—
90	C₂×C₄×Q₈	C₂³	1	10	yes	—
91	C₂×C₄×Q₈	C₂³	1	20	yes	—
92	C₂×C₄×Q₈	C₂³	1	40	yes	—
93	C₂×C₄×Q₈	C₂³	1	40	yes	—
94	C₂×C₄×Q₈	C₂³	1	10	yes	—
95	C₂×C₄×Q₈	C₂³	1	80	yes	—
96	C₂×C₄×Q₈	C₂³	1	80	yes	—
97	C₂×C₄×Q₈	C₂³	1	120	yes	—
98	C₂×C₄×Q₈	C₂³	1	120	yes	—
99	C₂×C₄×D₄	C₂³	1	10	yes	—
100	C₂×C₄×D₄	C₂³	1	30	yes	—
101	C₂×#117	C₂³	1	30	yes	—
102	C₂×#117	C₂³	1	120	yes	—
103	C₂²×C₄⋊C₄	C₂³	1	15	yes	—
104	C₂²×C₄²	C₂³	1	15	yes	—
105	2₋¹⁺⁴	C₂×Q₈	1	256	yes	complemented
106	C₂×#41	C₂×Q₈	1	160	yes	complemented
107	C₂×#41	C₂⁴	1	80	yes	—
108	C₂²×Q₈	C₂×Q₈	1	80	yes	complemented
109	C₂²×Q₈	C₂⁴	1	40	yes	—
110	C₂²×Q₈	C₂⁴	1	120	yes	—
111	C₂²×Q₈	C₂⁴	1	40	yes	—
112	C₂²×Q₈	C₂⁴	1	10	yes	—
113	C₂²×D₄	C₂⁴	1	10	yes	—
114	C₂³×C₄	C₂⁴	1	5	yes	—
115	C₂³×C₄	C₂⁴	1	15	yes	—
116	C₂³×C₄	C₂⁴	1	15	yes	—
117	C₄⋊Q₈	C₂⁴	1	80	yes	—
118	C₄⋊Q₈	C₂⁴	1	80	yes	—
119	C₄⋊Q₈	C₂⁴	1	960	yes	—
120	C₄⋊Q₈	C₂⁴	1	240	yes	—
121	C₄⋊Q₈	C₂⁴	1	240	yes	—
122	C₄⋊Q₈	C₂⁴	1	320	yes	—
123	#125	C₂⁴	1	160	yes	—
124	#125	C₂⁴	1	960	yes	—
125	#125	C₂⁴	1	480	yes	—
126	#125	C₂⁴	1	320	yes	—
127	C₂²⋊Q₈	C₂⁴	1	16	yes	—
128	C₂²⋊Q₈	C₂⁴	1	160	yes	—
129	C₂²⋊Q₈	C₂⁴	1	80	yes	—
130	C₂²⋊Q₈	C₂⁴	1	80	yes	—
131	C₂²⋊Q₈	C₂⁴	1	80	yes	—
132	C₂²⋊Q₈	C₂⁴	1	480	yes	—
133	C₂²⋊Q₈	C₂⁴	1	640	yes	—
134	C₂²⋊Q₈	C₂⁴	1	640	yes	—
135	C₄×Q₈	C₂⁴	1	40	yes	—
136	C₄×Q₈	C₂⁴	1	40	yes	—
137	C₄×Q₈	C₂⁴	1	160	yes	—
138	C₄×Q₈	C₂⁴	1	160	yes	—
139	C₄×Q₈	C₂⁴	1	160	yes	—
140	C₄×Q₈	C₂⁴	1	320	yes	—
141	C₄×Q₈	C₂⁴	1	320	yes	—
142	C₄×Q₈	C₂⁴	1	80	yes	—
143	C₄×Q₈	C₂⁴	1	480	yes	—
144	C₄×Q₈	C₂⁴	1	480	yes	—
145	C₄×D₄	C₂⁴	1	40	yes	—
146	C₄×D₄	C₂⁴	1	40	yes	—
147	C₄×D₄	C₂⁴	1	240	yes	—
148	C₄×D₄	C₂⁴	1	320	yes	—
149	C₄×D₄	C₂⁴	1	320	yes	—
150	#117	C₂⁴	1	160	yes	—
151	#117	C₂⁴	1	240	yes	—
152	#117	C₂⁴	1	480	yes	—
153	C₂×C₄⋊C₄	C₂⁴	1	30	yes	—
154	C₂×C₄⋊C₄	C₂⁴	1	120	yes	—
155	C₂×C₄⋊C₄	C₂⁴	1	10	yes	—
156	C₂×C₄⋊C₄	C₂⁴	1	120	yes	—
157	C₂×C₄⋊C₄	C₂⁴	1	120	yes	—
158	C₂×C₄⋊C₄	C₂⁴	1	80	yes	—
159	C₂×C₄⋊C₄	C₂⁴	1	40	yes	—
160	C₂×C₄⋊C₄	C₂⁴	1	120	yes	—
161	C₂×C₄⋊C₄	C₂⁴	1	240	yes	—
162	C₂×C₂²⋊C₄	C₂⁴	1	20	yes	—
163	C₂×C₂²⋊C₄	C₂⁴	1	80	yes	—
164	C₂×C₄²	C₂⁴	1	30	yes	—
165	C₂×C₄²	C₂⁴	1	240	yes	—
166	C₂×C₄²	C₂⁴	1	10	yes	—
167	C₂×C₄²	C₂⁴	1	60	yes	—
168	C₂×C₄²	C₂⁴	1	60	yes	—
169	C₂⁴	C₂⁵	1	5	yes	—
170	#41	C₂²×Q₈	1	640	yes	—
171	C₂×Q₈	—	4	80	no	—
172	C₂×Q₈	—	8	120	no	—
173	C₂×Q₈	C₂²×Q₈	1	80	yes	—
174	C₂×Q₈	C₂²×Q₈	1	320	yes	—
175	C₂×Q₈	C₂⁵	1	16	yes	—
176	C₂×Q₈	C₂⁵	1	80	yes	—
177	C₂×Q₈	C₂⁵	1	160	yes	—
178	C₂×Q₈	C₂⁵	1	40	yes	—
179	C₂×D₄	C₂²×Q₈	1	80	yes	—
180	C₂×D₄	C₂⁵	1	40	yes	—
181	C₂²×C₄	—	4	10	no	—
182	C₂²×C₄	—	8	15	no	—
183	C₂²×C₄	C₂²×Q₈	1	120	yes	—
184	C₂²×C₄	C₂⁵	1	1	yes	characteristic
185	C₂²×C₄	C₂⁵	1	10	yes	—
186	C₂²×C₄	C₂⁵	1	5	yes	—
187	C₂²×C₄	C₂⁵	1	10	yes	—
188	C₂²×C₄	C₂⁵	1	10	yes	—
189	C₂²×C₄	C₂⁵	1	80	yes	—
190	C₂²×C₄	C₂⁵	1	80	yes	—
191	C₂²×C₄	C₂⁵	1	60	yes	—
192	C₂²×C₄	C₂⁵	1	60	yes	—
193	C₂²×C₄	C₂⁵	1	10	yes	—
194	C₄⋊C₄	C₂⁵	1	120	yes	—
195	C₄⋊C₄	C₂⁵	1	32	yes	—
196	C₄⋊C₄	C₂⁵	1	16	yes	—
197	C₄⋊C₄	C₂⁵	1	160	yes	—
198	C₄⋊C₄	C₂⁵	1	160	yes	—
199	C₄⋊C₄	C₂⁵	1	160	yes	—
200	C₄⋊C₄	C₂⁵	1	320	yes	—
201	C₄⋊C₄	C₂⁵	1	40	yes	—
202	C₄⋊C₄	C₂⁵	1	160	yes	—
203	C₄⋊C₄	C₂⁵	1	80	yes	—
204	C₂²⋊C₄	C₂⁵	1	32	yes	—
205	C₂²⋊C₄	C₂⁵	1	80	yes	—
206	C₂²⋊C₄	C₂⁵	1	160	yes	—
207	C₄²	C₂⁵	1	20	yes	—
208	C₄²	C₂⁵	1	20	yes	—
209	C₄²	C₂⁵	1	120	yes	—
210	C₄²	C₂⁵	1	320	yes	—
211	C₂³	—	4	5	no	—
212	C₂³	—	4	5	no	—
213	C₂³	C₂³×Q₈	1	20	yes	—
214	C₂³	C₂⁶	1	1	yes	characteristic
215	C₂³	C₂⁶	1	10	yes	—
216	Q₈	—	2	32	no	—
217	Q₈	—	4	80	no	—
218	Q₈	—	4	160	no	—
219	Q₈	C₂³×Q₈	1	160	yes	—
220	D₄	C₂³×Q₈	1	160	yes	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	20	no	—
223	C₂×C₄	—	4	10	no	—
224	C₂×C₄	—	4	5	no	—
225	C₂×C₄	—	4	10	no	—
226	C₂×C₄	—	4	60	no	—
227	C₂×C₄	—	4	80	no	—
228	C₂×C₄	—	4	80	no	—
229	C₂×C₄	C₂³×Q₈	1	40	yes	—
230	C₂×C₄	C₂³×Q₈	1	240	yes	—
231	C₂×C₄	C₂⁶	1	1	yes	characteristic
232	C₂×C₄	C₂⁶	1	10	yes	—
233	C₂×C₄	C₂⁶	1	32	yes	—
234	C₂×C₄	C₂⁶	1	32	yes	—
235	C₂×C₄	C₂⁶	1	20	yes	—
236	C₂×C₄	C₂⁶	1	20	yes	—
237	C₂×C₄	C₂⁶	1	1	yes	characteristic
238	C₂²	—	2	10	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	10	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	20	no	—
243	C₂²	C₂⁴×Q₈	1	2	yes	—
244	C₂²	C₂⁴×Q₈	1	20	yes	—
245	C₂²	C₂⁷	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	10	no	—
248	C₄	—	2	32	no	—
249	C₄	—	2	20	no	—
250	C₄	—	2	32	no	—
251	C₄	C₂⁴×Q₈	1	40	yes	—
252	C₄	2₋¹⁺⁶	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	20	no	—
255	C₂	C₂⁵×Q₈	1	1	yes	characteristic
256	C₂	C₂×2₋¹⁺⁶	1	1	yes	characteristic
257	C₂	#63103	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁷
prime radical	C₂²×2₋¹⁺⁴	C₂²
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. #63004	C₂	semidirect
2. #63004	C₂	semidirect
3. Q₈×2₋¹⁺⁴	C₂	semidirect
4. #62854	C₂	semidirect
5. #62849	C₂	semidirect
6. #62813	C₂	semidirect
7. C₂×#3502	C₂	semidirect
11. #3504	C₂²	semidirect
12. #3504	C₂²	semidirect
13. #3503	C₂²	semidirect
14. #3503	C₂²	semidirect
15. #3503	C₂²	semidirect
19. #3502	C₂²	semidirect
20. #3502	C₂²	semidirect
21. #3502	C₂²	semidirect
22. #3502	C₂²	semidirect
25. Q₈×#41	C₂²	semidirect
26. Q₈×#41	C₂²	semidirect
27. C₂×Q₈²	C₂²	semidirect
28. C₂×#557	C₂²	semidirect
29. C₂×#557	C₂²	semidirect
43. C₂×2₋¹⁺⁴	Q₈	semidirect
105. 2₋¹⁺⁴	C₂×Q₈	semidirect
106. C₂×#41	C₂×Q₈	semidirect
108. C₂²×Q₈	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
#63004	2
Q₈×2₋¹⁺⁴	2
#62854	2
#62849	2
#62813	2
C₂×#3502	2
#62653	2
C₂×C₄×2₋¹⁺⁴	2