Subgroups — #72323280

Order 8192.

Only the 398 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 262 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323244	C₂	1	2	yes	maximal, complemented
2	#72323242	C₂	1	2	yes	maximal, complemented
3	#72323241	C₂	1	2	yes	maximal, complemented
4	#72323238	C₂	1	2	yes	maximal, complemented
5	#72323229	C₂	1	1	yes	maximal, characteristic, complemented
6	#72323228	C₂²	1	2	yes	complemented
7	#72323226	C₂²	1	2	yes	complemented
8	#72323220	C₂	1	1	yes	maximal, characteristic, complemented
9	#72323219	C₂²	1	2	yes	complemented
10	#72323217	C₂²	1	2	yes	complemented
11	#72317088	C₂	1	1	yes	maximal, characteristic, complemented
12	#72313692	C₂	1	1	yes	maximal, characteristic, complemented
13	#72313608	C₂²	1	2	yes	complemented
14	#72313556	C₂²	1	2	yes	complemented
15	#72310246	C₂	1	1	yes	maximal, characteristic, complemented
16	#72309737	C₂²	1	1	yes	characteristic, complemented
17	#72309645	C₂²	1	1	yes	characteristic, complemented
18	#72309490	C₂²	1	1	yes	characteristic, complemented
19	#72309421	C₂²	1	1	yes	characteristic, complemented
20	#72266165	C₂²	1	2	yes	complemented
21	#72266163	C₂²	1	2	yes	complemented
22	#72265785	C₂²	1	1	yes	characteristic
23	#72265766	C₂³	1	2	yes	—
24	#72259029	C₂²	1	1	yes	characteristic
25	#72259008	C₂³	1	1	yes	characteristic
26	#72256435	C₂	1	1	yes	maximal, characteristic, complemented
27	#72237886	C₂²	1	2	yes	complemented
28	#72237608	D₄	1	2	yes	complemented
29	#72235847	C₂³	1	1	yes	characteristic
30	#72234789	C₂²	1	2	yes	complemented
31	#72234751	C₂	1	1	yes	maximal, characteristic, complemented
32	#72234688	C₂²	1	1	yes	characteristic
33	#72234683	C₂²	1	1	yes	characteristic
34	#72234670	D₄	1	1	yes	characteristic, complemented
35	#72234665	C₂²	1	1	yes	characteristic
36	#72234663	D₄	1	1	yes	characteristic
37	#72234639	C₂²	1	2	yes	—
38	#72234639	C₂²	1	2	yes	—
39	#72234638	D₄	1	4	yes	complemented
40	#72234637	D₄	1	4	yes	complemented
41	#72234636	C₂³	1	2	yes	—
42	#72234619	D₄	1	2	yes	—
43	#72234588	D₄	1	2	yes	complemented
44	#72232884	D₄	1	1	yes	characteristic
45	C₂×#8932885	D₄	1	1	yes	characteristic, complemented
46	#72225883	C₂²	1	1	yes	characteristic
47	#72225870	C₂³	1	2	yes	—
48	#72225868	D₄	1	2	yes	complemented
49	#72225738	C₂²	1	1	yes	characteristic
50	#72225736	D₄	1	1	yes	characteristic, complemented
51	#72225735	D₄	1	1	yes	characteristic, complemented
52	#72225713	D₄	1	1	yes	characteristic, complemented
53	#72225713	C₂³	1	1	yes	characteristic
54	#72224136	D₄	1	1	yes	characteristic, complemented
55	#72224136	C₂³	1	2	yes	—
56	#72224136	C₂³	1	1	yes	characteristic
57	#72219311	C₂³	1	1	yes	characteristic
58	#72219311	C₂³	1	1	yes	characteristic
59	#72219301	D₄	1	2	yes	complemented
60	C₂×#7670203	C₂³	1	1	yes	characteristic
61	#72218518	D₄	1	1	yes	characteristic, complemented
62	#72218518	D₄	1	2	yes	complemented
63	#72218291	D₄	1	1	yes	characteristic, complemented
64	C₂×#134033	D₄	1	2	yes	complemented
65	#72211186	D₄	1	4	yes	complemented
66	#72210578	D₄	1	4	yes	complemented
67	#10586404	C₂×D₄	1	2	yes	—
68	#10586404	C₂×D₄	1	1	yes	characteristic, complemented
69	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
70	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
71	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
72	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
73	C₂×#46996	C₂×D₄	1	2	yes	—
74	C₂²×#2846	C₂×D₄	1	1	yes	characteristic
75	C₂×#35390	C₂×D₄	1	1	yes	characteristic
76	C₂×#35348	C₂×D₄	1	1	yes	characteristic
77	C₂×#35348	#41	1	1	yes	characteristic
78	C₂×#35346	C₂×D₄	1	1	yes	characteristic
79	C₂×#34981	C₂×D₄	1	1	yes	characteristic
80	C₂×#34981	#41	1	1	yes	characteristic
81	C₂×#34980	C₂×D₄	1	1	yes	characteristic
82	C₂×#34980	C₂⁴	1	1	yes	characteristic
83	C₂×#34198	C₂×D₄	1	1	yes	characteristic
84	C₂×#34198	C₂×D₄	1	1	yes	characteristic
85	#6499870	C₂×D₄	1	4	yes	—
86	#6499867	C₂×D₄	1	4	yes	—
87	#6374251	C₂×D₄	1	1	yes	characteristic, complemented
88	#6374128	C₂×D₄	1	2	yes	—
89	#6374128	C₂×D₄	1	1	yes	characteristic, complemented
90	#6374128	C₂×D₄	1	1	yes	characteristic, complemented
91	#5024691	C₂×D₄	1	2	yes	—
92	#5020751	C₂×D₄	1	2	yes	—
93	#4430405	C₂×D₄	1	2	yes	—
94	#648360	C₂×D₄	1	2	yes	—
95	#576160	C₂×D₄	1	2	yes	—
96	C₂×#11499	C₂×D₄	1	4	yes	complemented
97	#392109	C₂×D₄	1	2	yes	—
98	#392109	C₂×D₄	1	2	yes	—
99	C₂×#8547	C₂×D₄	1	2	yes	complemented
100	C₂×#8541	C₂×D₄	1	2	yes	complemented
101	#115690	C₂×D₄	1	4	yes	—
102	#115631	C₂×D₄	1	4	yes	—
103	C₂⁸	#127	1	1	yes	characteristic
104	C₂³×H(5,𝔽₂)	#120	1	1	yes	characteristic
105	C₂⁵×D₄	#120	1	1	yes	characteristic
106	C₂⁵×D₄	C₂²×D₄	1	1	yes	characteristic
107	#62817	#120	1	4	yes	—
108	C₂²×#534	#120	1	1	yes	characteristic
109	C₂²×#534	#121	1	1	yes	characteristic
110	C₂²×#534	#121	1	1	yes	characteristic
111	C₂²×#534	C₂²×D₄	1	1	yes	characteristic
112	C₂²×#534	C₂²×D₄	1	1	yes	characteristic
113	C₂³×#120	#120	1	1	yes	characteristic
114	C₂³×#120	#120	1	1	yes	characteristic
115	C₂³×#120	#120	1	1	yes	characteristic
116	C₂³×#120	#120	1	1	yes	characteristic
117	C₂³×#120	#121	1	1	yes	characteristic
118	C₂³×#120	#121	1	1	yes	characteristic
119	C₂³×#120	#121	1	1	yes	characteristic
120	C₂³×#120	#121	1	1	yes	characteristic
121	C₂³×#120	#127	1	1	yes	characteristic
122	C₂³×#120	#127	1	1	yes	characteristic
123	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
124	C₂²×#518	#120	1	1	yes	characteristic
125	C₂²×#518	#121	1	1	yes	characteristic
126	C₂⁴×C₂²⋊C₄	#121	1	1	yes	characteristic
127	C₂⁴×C₂²⋊C₄	#127	1	1	yes	characteristic
128	C₂×#2840	#120	1	2	yes	complemented
129	C₂×#2849	H(5,𝔽₂)	1	2	yes	—
130	C₂×#2846	C₂²×D₄	1	2	yes	—
131	C₂×#2403	#120	1	2	yes	complemented
132	C₂×#2613	C₂²×D₄	1	2	yes	—
133	C₂×#2613	H(5,𝔽₂)	1	2	yes	—
134	C₂×#2315	#121	1	1	yes	characteristic
135	C₂×#2315	#121	1	1	yes	characteristic
136	C₂×#2308	#120	1	1	yes	characteristic
137	C₂×#2308	#120	1	1	yes	characteristic
138	C₂×#2308	#121	1	1	yes	characteristic
139	C₂×#2308	C₂²×D₄	1	1	yes	characteristic
140	C₂×#2308	C₂×#41	1	1	yes	characteristic
141	C₂²×#379	#120	1	1	yes	characteristic
142	C₂²×#379	#121	1	1	yes	characteristic
143	C₂²×#379	C₂²×D₄	1	1	yes	characteristic
144	C₂×#2273	#120	1	1	yes	characteristic
145	#21746	#120	1	4	yes	—
146	C₂×#1781	#120	1	4	yes	complemented
147	#14679	#120	1	4	yes	complemented
148	#13827	#120	1	2	yes	complemented
149	C₂²⋊C₄²	#120	1	2	yes	complemented
150	C₂⁷	#392	1	1	yes	characteristic
151	C₂⁷	#392	1	1	yes	characteristic
152	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
153	C₂⁷	C₂×#120	1	1	yes	characteristic
154	C₂⁷	C₂×#127	1	1	yes	characteristic
155	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	2	yes	—
156	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	2	yes	—
157	C₂⁴×D₄	#392	1	1	yes	characteristic
158	C₂⁴×D₄	#392	1	1	yes	characteristic
159	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
160	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
161	C₂⁴×D₄	C₂×#120	1	1	yes	characteristic
162	C₂⁴×D₄	C₂×#120	1	1	yes	characteristic
163	C₂⁴×D₄	C₂×#120	1	1	yes	characteristic
164	C₂⁴×D₄	C₂×#121	1	1	yes	characteristic
165	C₂⁴×D₄	#535	1	1	yes	characteristic
166	C₂⁵×C₄	B(4,𝔽₂)	1	1	yes	characteristic
167	C₂⁵×C₄	#535	1	1	yes	characteristic
168	C₂²×#123	B(4,𝔽₂)	1	2	yes	—
169	C₂²×#123	#534	1	2	yes	—
170	C₂²×#123	D₄²	1	2	yes	—
171	C₂²×#120	B(4,𝔽₂)	1	4	yes	—
172	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
173	C₂²×#120	C₂×#120	1	2	yes	—
174	C₂²×#120	D₄²	1	2	yes	—
175	C₂²×#120	D₄²	1	2	yes	—
176	C₂²×#120	D₄²	1	2	yes	—
177	C₂³×C₂²⋊C₄	#392	1	1	yes	characteristic
178	C₂³×C₂²⋊C₄	#392	1	1	yes	characteristic
179	C₂³×C₂²⋊C₄	#392	1	1	yes	characteristic
180	C₂³×C₂²⋊C₄	#392	1	1	yes	characteristic
181	C₂³×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
182	C₂³×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
183	C₂³×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
184	#2308	B(4,𝔽₂)	1	2	yes	—
185	#2308	B(4,𝔽₂)	1	2	yes	—
186	C₂×#381	#534	1	2	yes	—
187	C₂×#381	#546	1	2	yes	—
188	C₂×#379	C₂×#120	1	2	yes	—
189	C₂×#379	D₄²	1	2	yes	—
190	C₂×#379	D₄²	1	2	yes	—
191	C₂×#379	#546	1	2	yes	—
192	C₂⁶	#2021	1	1	yes	characteristic
193	C₂⁶	#2021	1	1	yes	characteristic
194	C₂⁶	#2021	1	1	yes	characteristic
195	C₂⁶	#2021	1	1	yes	characteristic
196	C₂⁶	#2021	1	2	yes	—
197	C₂⁶	C₂×#392	1	1	yes	characteristic
198	C₂⁶	#2408	1	1	yes	characteristic
199	C₂⁶	#2433	1	1	yes	characteristic
200	C₂⁶	#2433	1	1	yes	characteristic
201	C₂⁶	#2846	1	1	yes	characteristic
202	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
203	C₂³×D₄	#2021	1	1	yes	characteristic
204	C₂³×D₄	#2021	1	1	yes	characteristic
205	C₂³×D₄	#2021	1	1	yes	characteristic
206	C₂³×D₄	#2021	1	2	yes	—
207	C₂³×D₄	#2021	1	1	yes	characteristic
208	C₂³×D₄	#2021	1	2	yes	—
209	C₂³×D₄	#2403	1	2	yes	—
210	C₂³×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
211	C₂³×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
212	C₂³×D₄	#3026	1	2	yes	—
213	C₂⁴×C₄	#2021	1	1	yes	characteristic
214	C₂⁴×C₄	#2021	1	1	yes	characteristic
215	C₂⁴×C₄	#2408	1	1	yes	characteristic
216	C₂⁴×C₄	#2613	1	1	yes	characteristic
217	C₂⁴×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
218	C₂²×C₂²⋊C₄	#2021	1	2	yes	—
219	C₂²×C₂²⋊C₄	#2021	1	2	yes	—
220	C₂²×C₂²⋊C₄	#2403	1	2	yes	—
221	C₂²×C₂²⋊C₄	#2410	1	2	yes	—
222	C₂²×C₂²⋊C₄	#2410	1	2	yes	—
223	C₂²×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
224	C₂²×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
225	C₂⁵	#10336	1	2	yes	—
226	C₂⁵	#10336	1	1	yes	characteristic
227	C₂⁵	#10336	1	1	yes	characteristic
228	C₂⁵	#14715	1	1	yes	characteristic
229	C₂⁵	C₂×#2021	1	1	yes	characteristic
230	C₂⁵	C₂×#2021	1	1	yes	characteristic
231	C₂⁵	C₂×#2021	1	1	yes	characteristic
232	C₂⁵	C₂×#2021	1	1	yes	characteristic
233	C₂⁵	C₂×#2021	1	1	yes	characteristic
234	C₂⁵	#23444	1	1	yes	characteristic
235	C₂⁵	#23900	1	2	yes	—
236	C₂⁵	#32898	1	2	yes	—
237	C₂³×C₄	#10336	1	2	yes	—
238	C₂³×C₄	#10336	1	1	yes	characteristic
239	C₂³×C₄	#10336	1	1	yes	characteristic
240	C₂³×C₄	#23900	1	2	yes	—
241	C₂³×C₄	#29634	1	2	yes	—
242	C₂×C₂²⋊C₄	#23900	1	4	yes	—
243	C₂⁴	#144364	1	2	yes	—
244	C₂⁴	C₂×#10336	1	1	yes	characteristic
245	C₂⁴	C₂×#10336	1	1	yes	characteristic
246	C₂⁴	#233541	1	1	yes	characteristic
247	C₂⁴	#233541	1	1	yes	characteristic
248	C₂⁴	#393929	1	1	yes	characteristic
249	C₂⁴	#394058	1	1	yes	characteristic
250	C₂⁴	#394058	1	1	yes	characteristic
251	C₂⁴	#511790	1	4	yes	—
252	C₂³	#72210453	1	2	yes	—
253	C₂³	#72214549	1	1	yes	characteristic
254	C₂³	C₂×#144364	1	1	yes	characteristic
255	C₂³	#72218114	1	2	yes	—
256	C₂³	#72236631	1	1	yes	characteristic
257	C₂²	#72218222	1	1	yes	characteristic
258	C₂²	#72258111	1	1	yes	characteristic
259	C₂²	#72258538	1	1	yes	characteristic
260	C₂	#72266303	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266303
commutator, Frattini, cosocle	C₂×#34980	C₂⁴
LCS	G ▷ C₂×#34980 ▷ 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. #72323244	C₂	semidirect
2. #72323242	C₂	semidirect
3. #72323241	C₂	semidirect
4. #72323238	C₂	semidirect
5. #72323229	C₂	semidirect
6. #72323228	C₂²	semidirect
7. #72323226	C₂²	semidirect
8. #72323220	C₂	semidirect
9. #72323219	C₂²	semidirect
10. #72323217	C₂²	semidirect
11. #72317088	C₂	semidirect
12. #72313692	C₂	semidirect
13. #72313608	C₂²	semidirect
14. #72313556	C₂²	semidirect
15. #72310246	C₂	semidirect
16. #72309737	C₂²	semidirect
17. #72309645	C₂²	semidirect
18. #72309490	C₂²	semidirect
19. #72309421	C₂²	semidirect
20. #72266165	C₂²	semidirect
21. #72266163	C₂²	semidirect
26. #72256435	C₂	semidirect
27. #72237886	C₂²	semidirect
28. #72237608	D₄	semidirect
30. #72234789	C₂²	semidirect
31. #72234751	C₂	semidirect
34. #72234670	D₄	semidirect
39. #72234638	D₄	semidirect
40. #72234637	D₄	semidirect
43. #72234588	D₄	semidirect
45. C₂×#8932885	D₄	semidirect
48. #72225868	D₄	semidirect
50. #72225736	D₄	semidirect
51. #72225735	D₄	semidirect
52. #72225713	D₄	semidirect
54. #72224136	D₄	semidirect
59. #72219301	D₄	semidirect
61. #72218518	D₄	semidirect
62. #72218518	D₄	semidirect
63. #72218291	D₄	semidirect
64. C₂×#134033	D₄	semidirect
65. #72211186	D₄	semidirect
66. #72210578	D₄	semidirect
68. #10586404	C₂×D₄	semidirect
87. #6374251	C₂×D₄	semidirect
89. #6374128	C₂×D₄	semidirect
90. #6374128	C₂×D₄	semidirect
96. C₂×#11499	C₂×D₄	semidirect
99. C₂×#8547	C₂×D₄	semidirect
100. C₂×#8541	C₂×D₄	semidirect
128. C₂×#2840	#120	semidirect
131. C₂×#2403	#120	semidirect
146. C₂×#1781	#120	semidirect
147. #14679	#120	semidirect
148. #13827	#120	semidirect
149. C₂²⋊C₄²	#120	semidirect

Maximal subgroups

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

subgroup	index
#72323244	2
#72323242	2
#72323241	2
#72323238	2
#72323229	2
#72323220	2
#72317088	2
#72313692	2
#72310246	2
#72256435	2
#72234751	2