Subgroups — #72249573

Order 4096.

At least 12310 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

The subgroup lattice has 237 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72238454	C₂	1	1	yes	maximal, characteristic, complemented
2	#72237630	C₂	1	1	yes	maximal, characteristic
3	#72237622	C₄	1	2	yes	complemented
4	#72237589	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237586	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237475	C₂	1	1	yes	maximal, characteristic
7	#72237474	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237457	C₄	1	1	yes	characteristic, complemented
9	#72237457	C₄	1	1	yes	characteristic, complemented
10	#72237400	C₄	1	1	yes	characteristic, complemented
11	#72237351	C₄	1	1	yes	characteristic, complemented
12	#72235718	C₂²	1	1	yes	characteristic
13	#72235718	C₂²	1	1	yes	characteristic
14	#72225873	C₄	1	2	yes	complemented
15	#72219397	C₂	1	1	yes	maximal, characteristic
16	#72219314	C₂²	1	1	yes	characteristic
17	#72219314	C₂²	1	1	yes	characteristic
18	#72219311	C₂²	1	1	yes	characteristic
19	#72219310	C₂²	1	1	yes	characteristic
20	C₂×#7670203	C₂²	1	1	yes	characteristic
21	#72218306	C₄	1	2	yes	complemented
22	#72215179	C₄	1	2	yes	complemented
23	C₂²×#3484	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3484	C₂×C₄	1	1	yes	characteristic
25	C₂⁴×#120	C₂×C₄	1	1	yes	characteristic
26	C₂⁴×#120	D₄	1	1	yes	characteristic
27	C₂²×#2846	Q₈	1	1	yes	characteristic
28	C₂×#35390	D₄	1	1	yes	characteristic
29	C₂×#35348	C₂×C₄	1	1	yes	characteristic
30	C₂×#35348	D₄	1	1	yes	characteristic
31	C₂×#35346	C₂×C₄	1	1	yes	characteristic
32	C₂×#34981	D₄	1	1	yes	characteristic
33	C₂×#34981	D₄	1	1	yes	characteristic
34	C₂×#34980	D₄	1	1	yes	characteristic
35	C₂×#34980	C₂³	1	1	yes	characteristic
36	C₂×#34198	C₂×C₄	1	1	yes	characteristic
37	C₂×#34198	Q₈	1	1	yes	characteristic
38	#480250	C₂×C₄	1	2	yes	complemented
39	#480250	C₂×C₄	1	2	yes	complemented
40	#392113	C₂×C₄	1	2	yes	complemented
41	#392109	C₂×C₄	1	2	yes	complemented
42	#188109	C₂×C₄	1	1	yes	characteristic
43	#188109	C₂×C₄	1	1	yes	characteristic
44	#188049	C₂×C₄	1	1	yes	characteristic
45	#188049	C₂×C₄	1	1	yes	characteristic
46	#133826	D₄	1	2	yes	—
47	#133825	D₄	1	2	yes	—
48	#133824	D₄	1	2	yes	—
49	#133822	D₄	1	2	yes	—
50	C₂⁸	C₄⋊C₄	1	1	yes	characteristic
51	C₂³×H(5,𝔽₂)	C₄²	1	1	yes	characteristic
52	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
54	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×#534	C₂²×C₄	1	1	yes	characteristic
58	C₂²×#534	C₂×D₄	1	1	yes	characteristic
59	C₂³×#120	C₄²	1	1	yes	characteristic
60	C₂³×#120	C₄²	1	1	yes	characteristic
61	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
63	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
64	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
65	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
66	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
67	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
68	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×#120	C₂×D₄	1	1	yes	characteristic
70	C₂²×#518	C₄²	1	1	yes	characteristic
71	C₂²×#518	C₄⋊C₄	1	1	yes	characteristic
72	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂⁴×C₂²⋊C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#2849	C₂×D₄	1	1	yes	characteristic
75	C₂×#2849	#41	1	1	yes	characteristic
76	C₂×#2846	C₂×D₄	1	1	yes	characteristic
77	C₂×#2846	#41	1	1	yes	characteristic
78	C₂×#2613	C₂×D₄	1	1	yes	characteristic
79	C₂×#2613	C₂×D₄	1	1	yes	characteristic
80	C₂×#2613	#41	1	1	yes	characteristic
81	C₂×#2613	#41	1	1	yes	characteristic
82	C₂×#2315	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#2315	C₄⋊C₄	1	1	yes	characteristic
84	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#2308	C₄⋊C₄	1	1	yes	characteristic
87	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
88	C₂×#2308	C₂×D₄	1	1	yes	characteristic
89	C₂²×#379	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×#379	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×#379	C₂×Q₈	1	1	yes	characteristic
92	C₂×#2273	C₄⋊C₄	1	1	yes	characteristic
93	#8548	C₂²⋊C₄	1	2	yes	—
94	#8547	C₂²⋊C₄	1	2	yes	—
95	#8543	C₂²⋊C₄	1	2	yes	—
96	#8541	C₂²⋊C₄	1	2	yes	—
97	C₂⁷	#95	1	1	yes	characteristic
98	C₂⁷	#95	1	1	yes	characteristic
99	C₂⁷	#99	1	1	yes	characteristic
100	C₂⁷	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂⁷	C₄×D₄	1	1	yes	characteristic
102	C₂⁴×D₄	#95	1	1	yes	characteristic
103	C₂⁴×D₄	#95	1	1	yes	characteristic
104	C₂⁴×D₄	#99	1	1	yes	characteristic
105	C₂⁴×D₄	#99	1	1	yes	characteristic
106	C₂⁴×D₄	C₂×C₄²	1	1	yes	characteristic
107	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
110	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
111	C₂⁵×C₄	#99	1	1	yes	characteristic
112	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
113	C₂×#534	#99	1	2	yes	—
114	C₂×#534	#99	1	2	yes	—
115	C₂²×#123	#99	1	1	yes	characteristic
116	C₂²×#123	#99	1	1	yes	characteristic
117	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×#123	#117	1	1	yes	characteristic
119	C₂²×#123	#120	1	1	yes	characteristic
120	C₂²×#123	#123	1	1	yes	characteristic
121	C₂²×#120	#99	1	4	yes	—
122	C₂²×#120	#99	1	1	yes	characteristic
123	C₂²×#120	#99	1	1	yes	characteristic
124	C₂²×#120	#99	1	2	yes	—
125	C₂²×#120	#99	1	2	yes	—
126	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×#120	#117	1	1	yes	characteristic
128	C₂²×#120	#120	1	1	yes	characteristic
129	C₂²×#120	#120	1	1	yes	characteristic
130	C₂²×#120	#120	1	1	yes	characteristic
131	C₂²×#120	#123	1	1	yes	characteristic
132	C₂²×#120	#123	1	1	yes	characteristic
133	C₂²×#120	#123	1	1	yes	characteristic
134	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
135	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
136	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
137	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
138	C₂³×C₂²⋊C₄	#99	1	4	yes	—
139	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂³×C₂²⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂³×C₂²⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×#381	C₄×D₄	1	1	yes	characteristic
143	C₂×#381	C₄×D₄	1	1	yes	characteristic
144	C₂×#381	#121	1	1	yes	characteristic
145	C₂×#381	#121	1	1	yes	characteristic
146	C₂×#379	C₄×D₄	1	1	yes	characteristic
147	C₂×#379	C₄×D₄	1	1	yes	characteristic
148	C₂×#379	#121	1	1	yes	characteristic
149	C₂×#379	#121	1	1	yes	characteristic
150	C₂×#379	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×#379	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×#379	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×#379	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂⁶	#342	1	1	yes	characteristic
155	C₂⁶	#342	1	1	yes	characteristic
156	C₂⁶	#342	1	2	yes	—
157	C₂⁶	#342	1	2	yes	—
158	C₂⁶	#342	1	2	yes	—
159	C₂⁶	#342	1	1	yes	characteristic
160	C₂⁶	#342	1	1	yes	characteristic
161	C₂⁶	C₂×#95	1	1	yes	characteristic
162	C₂⁶	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₂⁶	#385	1	1	yes	characteristic
164	C₂⁶	#385	1	1	yes	characteristic
165	C₂⁶	#386	1	1	yes	characteristic
166	C₂⁶	C₂×#99	1	1	yes	characteristic
167	C₂³×D₄	#342	1	1	yes	characteristic
168	C₂³×D₄	#342	1	1	yes	characteristic
169	C₂³×D₄	#342	1	1	yes	characteristic
170	C₂³×D₄	#342	1	1	yes	characteristic
171	C₂³×D₄	#379	1	1	yes	characteristic
172	C₂³×D₄	#381	1	1	yes	characteristic
173	C₂³×D₄	C₂×#99	1	1	yes	characteristic
174	C₂³×D₄	C₂×#99	1	2	yes	—
175	C₂³×D₄	C₂×#99	1	2	yes	—
176	C₂³×D₄	C₂×#99	1	1	yes	characteristic
177	C₂⁴×C₄	#342	1	1	yes	characteristic
178	C₂⁴×C₄	#342	1	1	yes	characteristic
179	C₂⁴×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₂⁴×C₄	#386	1	1	yes	characteristic
181	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
182	C₂²×C₂²⋊C₄	#342	1	2	yes	—
183	C₂²×C₂²⋊C₄	#342	1	2	yes	—
184	C₂²×C₂²⋊C₄	#342	1	2	yes	—
185	C₂²×C₂²⋊C₄	#342	1	1	yes	characteristic
186	C₂²×C₂²⋊C₄	#342	1	1	yes	characteristic
187	C₂²×C₂²⋊C₄	#380	1	1	yes	characteristic
188	C₂²×C₂²⋊C₄	#380	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	#385	1	1	yes	characteristic
190	C₂²×C₂²⋊C₄	#385	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
193	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
194	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
195	C₂⁵	#1304	1	2	yes	—
196	C₂⁵	#1304	1	1	yes	characteristic
197	C₂⁵	#1304	1	2	yes	—
198	C₂⁵	#1304	1	1	yes	characteristic
199	C₂⁵	C₂²⋊#95	1	1	yes	characteristic
200	C₂⁵	C₂×#342	1	1	yes	characteristic
201	C₂⁵	C₂×#342	1	1	yes	characteristic
202	C₂⁵	C₂×#342	1	1	yes	characteristic
203	C₂⁵	C₂×#342	1	1	yes	characteristic
204	C₂⁵	C₂×#342	1	1	yes	characteristic
205	C₂⁵	C₂×#342	1	1	yes	characteristic
206	C₂⁵	#1781	1	1	yes	characteristic
207	C₂⁵	#1819	1	1	yes	characteristic
208	C₂⁵	#1898	1	2	yes	—
209	C₂³×C₄	#1304	1	1	yes	characteristic
210	C₂³×C₄	#1304	1	1	yes	characteristic
211	C₂³×C₄	#1781	1	1	yes	characteristic
212	C₂³×C₄	#1819	1	1	yes	characteristic
213	C₂³×C₄	#1898	1	2	yes	—
214	C₂×C₂²⋊C₄	#1898	1	2	yes	—
215	C₂×C₂²⋊C₄	#1898	1	2	yes	—
216	C₂⁴	#7310	1	2	yes	—
217	C₂⁴	#7316	1	2	yes	—
218	C₂⁴	#7316	1	2	yes	—
219	C₂⁴	C₂×#1304	1	1	yes	characteristic
220	C₂⁴	C₂×#1304	1	1	yes	characteristic
221	C₂⁴	#8101	1	1	yes	characteristic
222	C₂⁴	#8101	1	1	yes	characteristic
223	C₂⁴	#10725	1	1	yes	characteristic
224	C₂⁴	#10735	1	1	yes	characteristic
225	C₂⁴	#10735	1	1	yes	characteristic
226	C₂²×C₄	#7310	1	2	yes	—
227	C₂³	C₂²⋊#1304	1	1	yes	characteristic
228	C₂³	C₂×#7310	1	1	yes	characteristic
229	C₂³	C₂×#7310	1	1	yes	characteristic
230	C₂³	C₂×#7316	1	2	yes	—
231	C₂³	#146578	1	2	yes	—
232	C₂²	#72218381	1	1	yes	characteristic
233	C₂²	#72219414	1	1	yes	characteristic
234	C₂²	#72240679	1	1	yes	characteristic
235	C₂	#72240684	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240684
commutator	C₂⁴×D₄	C₂×C₄²
Frattini, cosocle	C₂×#34980	C₂³
prime radical	C₂×#7670203	C₂²
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72238454	C₂	semidirect
3. #72237622	C₄	semidirect
4. #72237589	C₂	semidirect
5. #72237586	C₂	semidirect
7. #72237474	C₂	semidirect
8. #72237457	C₄	semidirect
9. #72237457	C₄	semidirect
10. #72237400	C₄	semidirect
11. #72237351	C₄	semidirect
14. #72225873	C₄	semidirect
21. #72218306	C₄	semidirect
22. #72215179	C₄	semidirect
38. #480250	C₂×C₄	semidirect
39. #480250	C₂×C₄	semidirect
40. #392113	C₂×C₄	semidirect
41. #392109	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#72238454	2
#72237630	2
#72237589	2
#72237586	2
#72237475	2
#72237474	2
#72219397	2