Subgroups — #72258989

Order 2048.

At least 7180 conjugacy classes of subgroups; only the 356 normal subgroups are shown.

The subgroup lattice has 271 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72258988	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258986	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258983	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237601	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237562	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233611	C₂	1	1	yes	maximal, characteristic, complemented
7	#72233441	C₂	1	1	yes	maximal, characteristic, complemented
8	#72225782	C₂	1	1	yes	maximal, characteristic, complemented
9	#72225653	C₂	1	1	yes	maximal, characteristic, complemented
10	#72219304	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219260	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219162	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218840	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218838	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#144366	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#3490	C₂²	1	1	yes	characteristic
17	C₂×#34198	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	#6437621	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	C₂×#23900	C₂²	1	1	yes	characteristic
22	#6371504	C₂²	1	1	yes	characteristic
23	#763512	C₂²	1	1	yes	characteristic
24	#617161	C₂²	1	1	yes	characteristic
25	#616960	C₂²	1	1	yes	characteristic
26	C₂×#15947	C₂²	1	1	yes	characteristic
27	#517594	C₂²	1	1	yes	characteristic
28	#493879	C₂²	1	1	yes	characteristic
29	#493879	C₂²	1	1	yes	characteristic
30	#493357	C₂²	1	1	yes	characteristic
31	#493357	C₂²	1	1	yes	characteristic
32	C₂×#12687	C₂²	1	1	yes	characteristic
33	C₂×#12687	C₂²	1	1	yes	characteristic
34	C₂×#12687	C₂²	1	1	yes	characteristic
35	C₂×#12687	C₂²	1	1	yes	characteristic
36	#392233	C₂²	1	1	yes	characteristic
37	#392231	C₂²	1	1	yes	characteristic
38	#392231	C₂²	1	1	yes	characteristic
39	#359811	C₂²	1	1	yes	characteristic
40	#359501	C₂²	1	1	yes	characteristic
41	#359501	C₂²	1	1	yes	characteristic
42	#359442	C₂²	1	1	yes	characteristic
43	#186835	C₂²	1	1	yes	characteristic
44	C₂×#8453	C₂²	1	1	yes	characteristic
45	#128009	C₂²	1	1	yes	characteristic
46	#128009	C₂²	1	1	yes	characteristic
47	C₂×#7307	C₂²	1	1	yes	characteristic
48	C₂×#7307	C₂²	1	1	yes	characteristic
49	#108238	C₂²	1	1	yes	characteristic
50	#107130	C₂²	1	1	yes	characteristic
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂²×#543	D₄	1	1	yes	characteristic, complemented
53	C₂²×#534	C₂³	1	1	yes	characteristic
54	C₂³×#120	D₄	1	1	yes	characteristic, complemented
55	C₂³×#120	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂²×#518	D₄	1	1	yes	characteristic, complemented
58	C₂²×#518	C₂³	1	1	yes	characteristic
59	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
60	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
61	C₂×#2883	D₄	1	2	yes	complemented
62	#34171	C₂³	1	1	yes	characteristic
63	C₂×#2315	D₄	1	1	yes	characteristic, complemented
64	C₂×#2308	C₂³	1	1	yes	characteristic
65	C₂²×#379	D₄	1	1	yes	characteristic, complemented
66	C₂×#2273	D₄	1	1	yes	characteristic, complemented
67	C₂×#1828	D₄	1	4	yes	complemented
68	C₂×#1783	C₂³	1	1	yes	characteristic
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	#20354	D₄	1	2	yes	complemented
75	#15948	D₄	1	1	yes	characteristic, complemented
76	#15948	D₄	1	1	yes	characteristic, complemented
77	#15947	D₄	1	1	yes	characteristic, complemented
78	#15947	D₄	1	1	yes	characteristic, complemented
79	#15947	D₄	1	1	yes	characteristic, complemented
80	#15947	D₄	1	1	yes	characteristic, complemented
81	#15947	D₄	1	1	yes	characteristic, complemented
82	#15947	D₄	1	1	yes	characteristic, complemented
83	#15463	D₄	1	2	yes	complemented
84	#15266	D₄	1	2	yes	complemented
85	#15266	D₄	1	2	yes	complemented
86	#15266	D₄	1	2	yes	complemented
87	#14335	C₂³	1	1	yes	characteristic
88	#13962	D₄	1	2	yes	complemented
89	C₂×#1438	C₂³	1	1	yes	characteristic
90	#13799	D₄	1	2	yes	complemented
91	#10768	D₄	1	4	yes	complemented
92	C₂⁷	C₂×D₄	1	1	yes	characteristic
93	C₂⁷	C₂×D₄	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
96	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
97	C₂²×#123	C₂×D₄	1	1	yes	characteristic
98	C₂²×#123	C₂×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂×D₄	1	1	yes	characteristic
100	C₂²×#120	C₂×D₄	1	1	yes	characteristic
101	C₂²×#120	C₂×D₄	1	1	yes	characteristic
102	C₂²×#120	C₂×D₄	1	1	yes	characteristic
103	C₂²×#120	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	C₂×D₄	1	1	yes	characteristic
105	C₂²×#120	C₂×D₄	1	1	yes	characteristic
106	C₂²×#120	C₂×D₄	1	1	yes	characteristic
107	C₂²×#120	C₂×D₄	1	1	yes	characteristic
108	C₂²×#120	C₂×D₄	1	1	yes	characteristic
109	C₂×#518	C₂×D₄	1	1	yes	characteristic
110	C₂×#518	C₂×D₄	1	2	yes	—
111	C₂×#518	C₂×D₄	1	2	yes	—
112	C₂×#518	C₂×D₄	1	2	yes	—
113	C₂×#518	C₂×D₄	1	2	yes	—
114	C₂×#518	#41	1	2	yes	—
115	C₂²×#117	C₂×D₄	1	1	yes	characteristic
116	C₂²×#117	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
119	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
125	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₂²×#99	C₂×D₄	1	2	yes	—
127	C₂²×#99	C₂×D₄	1	2	yes	—
128	C₂²×#99	C₂×D₄	1	2	yes	—
129	C₂²×#99	C₂×D₄	1	2	yes	—
130	C₂×#386	C₂×D₄	1	2	yes	—
131	C₂×#386	C₂×D₄	1	2	yes	—
132	C₂×#381	C₂×D₄	1	2	yes	—
133	C₂²×#95	C₂×D₄	1	2	yes	—
134	#1438	C₂×D₄	1	1	yes	characteristic
135	#1438	C₂×D₄	1	1	yes	characteristic
136	#1438	C₂×D₄	1	2	yes	—
137	#1438	C₂×D₄	1	1	yes	characteristic
138	#1438	C₂×D₄	1	2	yes	—
139	#1438	C₂×D₄	1	1	yes	characteristic
140	C₂⁶	#120	1	1	yes	characteristic
141	C₂⁶	#120	1	1	yes	characteristic
142	C₂⁶	#120	1	1	yes	characteristic
143	C₂⁶	#120	1	1	yes	characteristic
144	C₂⁶	#120	1	1	yes	characteristic
145	C₂⁶	C₂²×D₄	1	1	yes	characteristic
146	C₂⁶	C₂²×D₄	1	1	yes	characteristic
147	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
148	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
149	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
150	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
151	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
152	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
154	C₂³×D₄	C₂×#41	1	1	yes	characteristic
155	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂⁴×C₄	#120	1	1	yes	characteristic
160	C₂⁴×C₄	#120	1	2	yes	—
161	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂×#117	#121	1	2	yes	—
163	C₂×#117	#121	1	2	yes	—
164	C₂×#117	#127	1	2	yes	—
165	C₂×#117	#127	1	2	yes	—
166	C₂²×C₂²⋊C₄	#120	1	2	yes	—
167	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	#120	1	2	yes	—
170	C₂²×C₂²⋊C₄	#120	1	2	yes	—
171	C₂²×C₂²⋊C₄	#120	1	2	yes	—
172	C₂²×C₂²⋊C₄	#120	1	2	yes	—
173	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
174	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
175	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
180	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
181	C₂²×C₂²⋊C₄	#121	1	2	yes	—
182	C₂²×C₂²⋊C₄	#121	1	2	yes	—
183	C₂²×C₂²⋊C₄	#121	1	2	yes	—
184	C₂²×C₂²⋊C₄	#121	1	2	yes	—
185	C₂×#95	#120	1	2	yes	—
186	C₂×#95	#120	1	2	yes	—
187	C₂×#95	#120	1	2	yes	—
188	C₂×#95	#120	1	2	yes	—
189	C₂⁵	C₂×#120	1	1	yes	characteristic
190	C₂⁵	C₂×#120	1	1	yes	characteristic
191	C₂⁵	C₂×#120	1	1	yes	characteristic
192	C₂⁵	C₂×#120	1	1	yes	characteristic
193	C₂⁵	C₂×#120	1	1	yes	characteristic
194	C₂⁵	C₂×#120	1	1	yes	characteristic
195	C₂⁵	C₂×#120	1	1	yes	characteristic
196	C₂⁵	C₂×#121	1	1	yes	characteristic
197	C₂⁵	C₂×#121	1	1	yes	characteristic
198	C₂⁵	#534	1	1	yes	characteristic
199	C₂⁵	#534	1	1	yes	characteristic
200	C₂⁵	#534	1	1	yes	characteristic
201	C₂⁵	#534	1	1	yes	characteristic
202	C₂²×D₄	D₄²	1	1	yes	characteristic
203	C₂²×D₄	D₄²	1	1	yes	characteristic
204	C₂²×D₄	D₄²	1	1	yes	characteristic
205	C₂²×D₄	D₄²	1	1	yes	characteristic
206	C₂²×D₄	D₄²	1	1	yes	characteristic
207	C₂²×D₄	D₄²	1	1	yes	characteristic
208	C₂²×D₄	D₄²	1	1	yes	characteristic
209	C₂²×D₄	D₄²	1	1	yes	characteristic
210	C₂²×D₄	D₄²	1	1	yes	characteristic
211	C₂²×D₄	D₄²	1	1	yes	characteristic
212	C₂²×D₄	D₄²	1	1	yes	characteristic
213	C₂²×D₄	D₄²	1	1	yes	characteristic
214	C₂²×D₄	#546	1	1	yes	characteristic
215	C₂²×D₄	#546	1	1	yes	characteristic
216	C₂²×D₄	#546	1	1	yes	characteristic
217	C₂²×D₄	#546	1	1	yes	characteristic
218	C₂³×C₄	C₂×#120	1	1	yes	characteristic
219	C₂³×C₄	C₂×#120	1	1	yes	characteristic
220	C₂³×C₄	C₂×#121	1	1	yes	characteristic
221	C₂³×C₄	C₂×#127	1	1	yes	characteristic
222	C₂³×C₄	#535	1	1	yes	characteristic
223	C₂³×C₄	#535	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	#392	1	2	yes	—
225	C₂×C₂²⋊C₄	#392	1	2	yes	—
226	C₂×C₂²⋊C₄	#392	1	2	yes	—
227	C₂×C₂²⋊C₄	#392	1	2	yes	—
228	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	8	yes	—
229	C₂⁴	C₂×#392	1	1	yes	characteristic
230	C₂⁴	#2403	1	1	yes	characteristic
231	C₂⁴	#2403	1	1	yes	characteristic
232	C₂⁴	#2403	1	1	yes	characteristic
233	C₂⁴	#2403	1	1	yes	characteristic
234	C₂⁴	#2410	1	1	yes	characteristic
235	C₂⁴	#2410	1	1	yes	characteristic
236	C₂⁴	#2410	1	1	yes	characteristic
237	C₂⁴	#2410	1	1	yes	characteristic
238	C₂⁴	#2613	1	1	yes	characteristic
239	C₂⁴	#2846	1	1	yes	characteristic
240	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
241	C₂⁴	#3025	1	2	yes	—
242	C₂⁴	#3025	1	4	yes	—
243	C₂⁴	#3027	1	2	yes	—
244	C₂²×C₄	#2408	1	1	yes	characteristic
245	C₂²×C₄	#2408	1	1	yes	characteristic
246	C₂²×C₄	#2433	1	1	yes	characteristic
247	C₂²×C₄	#2433	1	1	yes	characteristic
248	C₂²⋊C₄	#2021	1	4	yes	—
249	C₂²⋊C₄	#2021	1	4	yes	—
250	C₂³	#14715	1	1	yes	characteristic
251	C₂³	C₂×#2021	1	2	yes	—
252	C₂³	#23443	1	1	yes	characteristic
253	C₂³	#23443	1	2	yes	—
254	C₂³	#23445	1	1	yes	characteristic
255	C₂³	#23900	1	2	yes	—
256	C₂³	#23900	1	2	yes	—
257	C₂³	#23902	1	2	yes	—
258	C₂³	#23902	1	2	yes	—
259	C₂³	#29636	1	2	yes	—
260	C₂³	#32899	1	2	yes	—
261	C₂²	#233538	1	2	yes	—
262	C₂²	#233543	1	2	yes	—
263	C₂²	#393930	1	1	yes	characteristic
264	C₂²	#394058	1	1	yes	characteristic
265	C₂²	#394060	1	1	yes	characteristic
266	C₂²	#511853	1	2	yes	—
267	C₂²	#511856	1	2	yes	—
268	C₂	#72237972	1	2	yes	—
269	C₂	#72257488	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#393930
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×#518 ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72258988	C₂	semidirect
2. #72258986	C₂	semidirect
3. #72258983	C₂	semidirect
4. #72237601	C₂	semidirect
5. #72237562	C₂	semidirect
6. #72233611	C₂	semidirect
7. #72233441	C₂	semidirect
8. #72225782	C₂	semidirect
9. #72225653	C₂	semidirect
10. #72219304	C₂	semidirect
11. #72219260	C₂	semidirect
12. #72219162	C₂	semidirect
13. #72218840	C₂	semidirect
14. #72218838	C₂	semidirect
15. C₂×#144366	C₂	semidirect
52. C₂²×#543	D₄	semidirect
54. C₂³×#120	D₄	semidirect
57. C₂²×#518	D₄	semidirect
59. C₂⁴×C₂²⋊C₄	D₄	semidirect
60. C₂⁴×C₂²⋊C₄	D₄	semidirect
61. C₂×#2883	D₄	semidirect
63. C₂×#2315	D₄	semidirect
65. C₂²×#379	D₄	semidirect
66. C₂×#2273	D₄	semidirect
67. C₂×#1828	D₄	semidirect
74. #20354	D₄	semidirect
75. #15948	D₄	semidirect
76. #15948	D₄	semidirect
77. #15947	D₄	semidirect
78. #15947	D₄	semidirect
79. #15947	D₄	semidirect
80. #15947	D₄	semidirect
81. #15947	D₄	semidirect
82. #15947	D₄	semidirect
83. #15463	D₄	semidirect
84. #15266	D₄	semidirect
85. #15266	D₄	semidirect
86. #15266	D₄	semidirect
88. #13962	D₄	semidirect
90. #13799	D₄	semidirect
91. #10768	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72258988	2
#72258986	2
#72258983	2
#72237601	2
#72237562	2
#72233611	2
#72233441	2
#72225782	2
#72225653	2
#72219304	2
#72219260	2
#72219162	2
#72218840	2
#72218838	2
C₂×#144366	2