Subgroups — #72311363

Order 8192.

Only the 280 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 232 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310618	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310510	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310498	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310459	C₂	1	1	yes	maximal, characteristic, complemented
5	#72310162	C₂	1	1	yes	maximal, characteristic
6	#72309729	C₄	1	1	yes	characteristic, complemented
7	#72309472	C₄	1	1	yes	characteristic, complemented
8	#72266039	C₄	1	1	yes	characteristic, complemented
9	#72266038	C₄	1	1	yes	characteristic, complemented
10	#72259048	C₂	1	1	yes	maximal, characteristic, complemented
11	#72259029	C₂²	1	1	yes	characteristic
12	#72259029	C₂²	1	1	yes	characteristic
13	#72259021	D₄	1	2	yes	—
14	#72259008	C₂×C₄	1	1	yes	characteristic
15	#72259008	D₄	1	1	yes	characteristic, complemented
16	#72237630	C₂²	1	1	yes	characteristic
17	#72237628	C₂²	1	1	yes	characteristic
18	#72237618	D₄	1	1	yes	characteristic
19	#72235847	C₂×C₄	1	1	yes	characteristic
20	#72235847	D₄	1	1	yes	characteristic, complemented
21	#72234668	C₂×C₄	1	1	yes	characteristic
22	#72234625	C₂×C₄	1	1	yes	characteristic
23	#72232634	D₄	1	2	yes	complemented
24	#72232592	D₄	1	1	yes	characteristic
25	#72225838	C₂	1	1	yes	maximal, characteristic, complemented
26	#72225781	C₂²	1	1	yes	characteristic
27	#72225777	D₄	1	1	yes	characteristic, complemented
28	#72225776	D₄	1	1	yes	characteristic, complemented
29	#72225615	C₂²	1	1	yes	characteristic
30	#72219309	C₂²	1	1	yes	characteristic
31	#72219301	D₄	1	1	yes	characteristic, complemented
32	#72219299	D₄	1	1	yes	characteristic, complemented
33	#72219257	C₂×C₄	1	1	yes	characteristic
34	#72219254	C₂×C₄	1	1	yes	characteristic
35	#72219225	D₄	1	1	yes	characteristic
36	C₂²×#15947	D₄	1	1	yes	characteristic
37	C₂×#7670203	C₂³	1	1	yes	characteristic
38	C₂²×#62817	D₄	1	1	yes	characteristic
39	#72217875	D₄	1	1	yes	characteristic
40	#72217789	D₄	1	1	yes	characteristic
41	#72217511	D₄	1	1	yes	characteristic
42	C₂×#115690	D₄	1	2	yes	—
43	C₂×#62817	C₂²⋊C₄	1	1	yes	characteristic, complemented
44	C₂²×#3484	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
46	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
47	C₂⁴×#120	C₂²⋊C₄	1	1	yes	characteristic
48	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
49	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
50	C₂²×#2846	C₂×D₄	1	1	yes	characteristic
51	C₂×#35390	#41	1	1	yes	characteristic
52	C₂×#35348	#41	1	1	yes	characteristic
53	C₂×#35348	#41	1	1	yes	characteristic
54	C₂×#35346	C₂×D₄	1	1	yes	characteristic
55	C₂×#34981	C₂×D₄	1	1	yes	characteristic
56	C₂×#34981	C₂×D₄	1	1	yes	characteristic
57	C₂×#34980	C₂²×C₄	1	1	yes	characteristic
58	C₂×#34980	C₂×D₄	1	1	yes	characteristic
59	#7626202	C₂²⋊C₄	1	1	yes	characteristic, complemented
60	C₂×#34198	C₂×D₄	1	1	yes	characteristic
61	C₂×#34198	#41	1	1	yes	characteristic
62	C₂×#21746	C₂²⋊C₄	1	1	yes	characteristic, complemented
63	C₂×#21746	C₂²⋊C₄	1	1	yes	characteristic, complemented
64	#517411	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×#1438	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×#8101	C₂²⋊C₄	1	2	yes	complemented
67	C₂×#8101	C₂²⋊C₄	1	2	yes	complemented
68	C₂⁸	#99	1	1	yes	characteristic, complemented
69	C₂⁸	#120	1	1	yes	characteristic
70	C₂³×H(5,𝔽₂)	#99	1	1	yes	characteristic, complemented
71	C₂³×H(5,𝔽₂)	#120	1	1	yes	characteristic
72	C₂⁵×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂⁵×D₄	#120	1	1	yes	characteristic
74	C₂²×#534	C₄×D₄	1	1	yes	characteristic
75	C₂²×#534	C₄×D₄	1	1	yes	characteristic
76	C₂²×#534	#121	1	1	yes	characteristic
77	C₂²×#534	#121	1	1	yes	characteristic
78	C₂²×#534	#121	1	1	yes	characteristic
79	C₂³×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×#120	#120	1	1	yes	characteristic
81	C₂³×#120	#120	1	1	yes	characteristic
82	C₂³×#120	#120	1	1	yes	characteristic
83	C₂³×#120	#121	1	1	yes	characteristic
84	C₂³×#120	#121	1	1	yes	characteristic
85	C₂³×#120	#121	1	1	yes	characteristic
86	C₂³×#120	#121	1	1	yes	characteristic
87	C₂³×#120	#123	1	1	yes	characteristic
88	C₂³×#120	#123	1	1	yes	characteristic
89	C₂³×#120	#127	1	1	yes	characteristic
90	C₂²×#518	#99	1	1	yes	characteristic, complemented
91	C₂²×#518	#121	1	1	yes	characteristic
92	C₂²×#518	#123	1	1	yes	characteristic
93	C₂⁴×C₂²⋊C₄	#99	1	1	yes	characteristic, complemented
94	C₂⁴×C₂²⋊C₄	#120	1	1	yes	characteristic
95	C₂⁴×C₂²⋊C₄	#121	1	1	yes	characteristic
96	C₂×#2315	#121	1	1	yes	characteristic
97	C₂×#2315	#123	1	1	yes	characteristic
98	C₂×#2308	C₄×D₄	1	1	yes	characteristic
99	C₂×#2308	C₄×D₄	1	1	yes	characteristic
100	C₂×#2308	#120	1	1	yes	characteristic
101	C₂×#2308	#121	1	1	yes	characteristic
102	C₂×#2308	#127	1	1	yes	characteristic
103	C₂²×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×#379	#121	1	1	yes	characteristic
105	C₂²×#379	#124	1	1	yes	characteristic
106	C₂×#2273	#124	1	1	yes	characteristic
107	#14681	#99	1	2	yes	complemented
108	C₂×#1438	#99	1	2	yes	complemented
109	C₂⁷	#379	1	1	yes	characteristic
110	C₂⁷	#392	1	1	yes	characteristic
111	C₂⁷	#394	1	1	yes	characteristic
112	C₂⁷	C₂×#99	1	1	yes	characteristic
113	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
114	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
115	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
116	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
117	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
118	C₂⁴×D₄	#386	1	1	yes	characteristic
119	C₂⁴×D₄	#386	1	1	yes	characteristic
120	C₂⁴×D₄	#386	1	1	yes	characteristic
121	C₂⁴×D₄	#392	1	1	yes	characteristic
122	C₂⁴×D₄	#392	1	1	yes	characteristic
123	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
124	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
125	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
126	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
127	C₂⁵×C₄	C₂×#99	1	1	yes	characteristic
128	C₂⁵×C₄	B(4,𝔽₂)	1	1	yes	characteristic
129	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
130	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
131	C₂²×#120	#458	1	2	yes	—
132	C₂²×#120	#458	1	2	yes	—
133	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
134	C₂³×C₂²⋊C₄	#390	1	1	yes	characteristic
135	C₂³×C₂²⋊C₄	#390	1	1	yes	characteristic
136	C₂³×C₂²⋊C₄	#392	1	1	yes	characteristic
137	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
138	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
139	C₂³×C₂²⋊C₄	#394	1	1	yes	characteristic
140	C₂×#381	B(4,𝔽₂)	1	2	yes	—
141	C₂×#379	B(4,𝔽₂)	1	2	yes	—
142	C₂⁶	#1438	1	1	yes	characteristic
143	C₂⁶	#1781	1	1	yes	characteristic
144	C₂⁶	#1781	1	1	yes	characteristic
145	C₂⁶	#1790	1	2	yes	—
146	C₂⁶	#1790	1	2	yes	—
147	C₂⁶	#1889	1	1	yes	characteristic
148	C₂⁶	#1898	1	1	yes	characteristic
149	C₂⁶	#1898	1	1	yes	characteristic
150	C₂⁶	#2021	1	1	yes	characteristic
151	C₂⁶	#2021	1	1	yes	characteristic
152	C₂⁶	#2021	1	1	yes	characteristic
153	C₂⁶	#2021	1	1	yes	characteristic
154	C₂³×D₄	#1889	1	1	yes	characteristic
155	C₂³×D₄	#1889	1	1	yes	characteristic
156	C₂³×D₄	#1913	1	1	yes	characteristic
157	C₂³×D₄	#1913	1	1	yes	characteristic
158	C₂³×D₄	#2021	1	1	yes	characteristic
159	C₂³×D₄	#2021	1	1	yes	characteristic
160	C₂³×D₄	#2021	1	1	yes	characteristic
161	C₂³×D₄	#2021	1	1	yes	characteristic
162	C₂³×D₄	#2021	1	1	yes	characteristic
163	C₂³×D₄	#2021	1	1	yes	characteristic
164	C₂³×D₄	#2022	1	1	yes	characteristic
165	C₂³×D₄	#2022	1	1	yes	characteristic
166	C₂⁴×C₄	#1889	1	1	yes	characteristic
167	C₂⁴×C₄	#1898	1	1	yes	characteristic
168	C₂⁴×C₄	#1898	1	1	yes	characteristic
169	C₂⁴×C₄	#2021	1	1	yes	characteristic
170	C₂⁴×C₄	#2021	1	1	yes	characteristic
171	C₂×#120	#2196	1	4	yes	—
172	C₂×#120	#2199	1	4	yes	—
173	C₂²×C₂²⋊C₄	#1889	1	2	yes	—
174	C₂²×C₂²⋊C₄	#2021	1	2	yes	—
175	C₂²×C₂²⋊C₄	#2044	1	2	yes	—
176	C₂²×C₂²⋊C₄	#2044	1	2	yes	—
177	C₂⁵	#8534	1	2	yes	—
178	C₂⁵	#8534	1	2	yes	—
179	C₂⁵	#9495	1	1	yes	characteristic
180	C₂⁵	#9495	1	1	yes	characteristic
181	C₂⁵	#10336	1	1	yes	characteristic
182	C₂⁵	#10336	1	1	yes	characteristic
183	C₂⁵	#10337	1	1	yes	characteristic
184	C₂⁵	#10337	1	1	yes	characteristic
185	C₂⁵	#10965	1	1	yes	characteristic
186	C₂⁵	#10972	1	1	yes	characteristic
187	C₂⁵	#10991	1	1	yes	characteristic
188	C₂⁵	#10991	1	1	yes	characteristic
189	C₂⁵	#10991	1	1	yes	characteristic
190	C₂⁵	#10995	1	1	yes	characteristic
191	C₂⁵	#10995	1	1	yes	characteristic
192	C₂⁵	#12753	1	2	yes	—
193	C₂⁵	#12753	1	2	yes	—
194	C₂⁵	#12759	1	2	yes	—
195	C₂⁵	#12763	1	2	yes	—
196	C₂²×D₄	#13343	1	4	yes	—
197	C₂³×C₄	#9497	1	1	yes	characteristic
198	C₂³×C₄	#9497	1	1	yes	characteristic
199	C₂³×C₄	#10336	1	1	yes	characteristic
200	C₂³×C₄	#10336	1	1	yes	characteristic
201	C₂³×C₄	#10337	1	1	yes	characteristic
202	C₂³×C₄	#10337	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	#13419	1	4	yes	—
204	C₂⁴	#111458	1	1	yes	characteristic
205	C₂⁴	#111458	1	1	yes	characteristic
206	C₂⁴	#115133	1	1	yes	characteristic
207	C₂⁴	#115204	1	1	yes	characteristic
208	C₂⁴	#115204	1	1	yes	characteristic
209	C₂⁴	#115348	1	1	yes	characteristic
210	C₂⁴	#115690	1	1	yes	characteristic
211	C₂⁴	#133390	1	2	yes	—
212	C₂⁴	#133390	1	2	yes	—
213	C₂⁴	#133394	1	2	yes	—
214	C₂⁴	#133394	1	2	yes	—
215	C₂⁴	#139743	1	2	yes	—
216	C₂⁴	#144366	1	2	yes	—
217	C₂⁴	#147779	1	2	yes	—
218	C₂⁴	#147815	1	2	yes	—
219	C₂²×C₄	#144366	1	2	yes	—
220	C₂³	#72218387	1	1	yes	characteristic
221	C₂³	#72218840	1	1	yes	characteristic
222	C₂³	#72219160	1	1	yes	characteristic
223	C₂³	#72219162	1	1	yes	characteristic
224	C₂³	#72225799	1	2	yes	—
225	C₂³	#72226368	1	2	yes	—
226	C₂³	#72236928	1	1	yes	characteristic
227	C₂²	#72226371	1	1	yes	characteristic
228	C₂²	#72258218	1	1	yes	characteristic
229	C₂²	#72258323	1	1	yes	characteristic
230	C₂	#72266182	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266182
commutator	C₂×#34980	C₂²×C₄
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	#72310162	C₂
LCS	G ▷ C₂×#34980 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72310618	C₂	semidirect
2. #72310510	C₂	semidirect
3. #72310498	C₂	semidirect
4. #72310459	C₂	semidirect
6. #72309729	C₄	semidirect
7. #72309472	C₄	semidirect
8. #72266039	C₄	semidirect
9. #72266038	C₄	semidirect
10. #72259048	C₂	semidirect
15. #72259008	D₄	semidirect
20. #72235847	D₄	semidirect
23. #72232634	D₄	semidirect
25. #72225838	C₂	semidirect
27. #72225777	D₄	semidirect
28. #72225776	D₄	semidirect
31. #72219301	D₄	semidirect
32. #72219299	D₄	semidirect
43. C₂×#62817	C₂²⋊C₄	semidirect
59. #7626202	C₂²⋊C₄	semidirect
62. C₂×#21746	C₂²⋊C₄	semidirect
63. C₂×#21746	C₂²⋊C₄	semidirect
66. C₂×#8101	C₂²⋊C₄	semidirect
67. C₂×#8101	C₂²⋊C₄	semidirect
68. C₂⁸	#99	semidirect
70. C₂³×H(5,𝔽₂)	#99	semidirect
90. C₂²×#518	#99	semidirect
93. C₂⁴×C₂²⋊C₄	#99	semidirect
107. #14681	#99	semidirect
108. C₂×#1438	#99	semidirect

Maximal subgroups

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

subgroup	index
#72310618	2
#72310510	2
#72310498	2
#72310459	2
#72310162	2
#72259048	2
#72225838	2