Subgroups — #72315335

Order 2048.

At least 7896 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

The subgroup lattice has 269 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310283	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266378	C₂	1	1	yes	maximal, characteristic, complemented
3	#72263490	C₂	1	1	yes	maximal, characteristic, complemented
4	#72260537	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237045	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233800	C₂	1	1	yes	maximal, characteristic, complemented
7	#72225778	C₂	1	1	yes	maximal, characteristic, complemented
8	#72225657	C₂	1	1	yes	maximal, characteristic, complemented
9	#72219293	C₂	1	1	yes	maximal, characteristic, complemented
10	#72219268	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219240	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219040	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218980	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218840	C₂	1	1	yes	maximal, characteristic, complemented
15	#72218838	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#3532	C₂²	1	1	yes	characteristic
17	#7642244	C₂²	1	1	yes	characteristic, complemented
18	C₂×#35390	C₂²	1	1	yes	characteristic
19	#6437621	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23900	C₂²	1	1	yes	characteristic
22	#6437445	C₂²	1	1	yes	characteristic
23	#763333	C₂²	1	1	yes	characteristic
24	#616961	C₂²	1	1	yes	characteristic
25	#616960	C₂²	1	1	yes	characteristic
26	C₂×#15947	C₂²	1	1	yes	characteristic
27	#552677	C₂²	1	1	yes	characteristic
28	#493370	C₂²	1	1	yes	characteristic
29	#493370	C₂²	1	1	yes	characteristic
30	#493357	C₂²	1	1	yes	characteristic
31	#493357	C₂²	1	1	yes	characteristic
32	#493357	C₂²	1	1	yes	characteristic, complemented
33	#493357	C₂²	1	1	yes	characteristic, complemented
34	C₂×#12687	C₂²	1	1	yes	characteristic
35	C₂×#12687	C₂²	1	1	yes	characteristic
36	#382422	C₂²	1	1	yes	characteristic, complemented
37	#359509	C₂²	1	1	yes	characteristic
38	#359501	C₂²	1	1	yes	characteristic
39	#359442	C₂²	1	1	yes	characteristic
40	#359442	C₂²	1	1	yes	characteristic, complemented
41	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
42	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
43	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
44	#186835	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
47	#128009	C₂²	1	1	yes	characteristic, complemented
48	#128009	C₂²	1	1	yes	characteristic, complemented
49	#108238	C₂²	1	1	yes	characteristic, complemented
50	#108222	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂²×#551	D₄	1	1	yes	characteristic, complemented
53	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
54	C₂²×H(3,𝔽₄)	D₄	1	1	yes	characteristic, complemented
55	C₂²×#560	D₄	1	1	yes	characteristic, complemented
56	C₂²×#535	C₂³	1	1	yes	characteristic
57	C₂²×#534	D₄	1	1	yes	characteristic, complemented
58	C₂²×#534	C₂³	1	1	yes	characteristic
59	C₂²×#534	C₂³	1	1	yes	characteristic
60	C₂³×#120	D₄	1	1	yes	characteristic, complemented
61	C₂³×#120	D₄	1	1	yes	characteristic, complemented
62	C₂³×#120	D₄	1	2	yes	complemented
63	C₂³×#120	C₂³	1	1	yes	characteristic
64	C₂³×#120	C₂³	1	1	yes	characteristic
65	C₂×#3026	D₄	1	2	yes	complemented
66	C₂×#2404	D₄	1	2	yes	complemented
67	#34171	C₂³	1	1	yes	characteristic
68	C₂×#2273	D₄	1	1	yes	characteristic, complemented
69	C₂×#1913	D₄	1	4	yes	complemented
70	#20960	D₄	1	2	yes	complemented
71	C₂×#1783	C₂³	1	1	yes	characteristic
72	C₂×#1783	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	#15948	D₄	1	1	yes	characteristic, complemented
78	#15948	D₄	1	1	yes	characteristic, complemented
79	#15947	D₄	1	1	yes	characteristic, complemented
80	#15947	D₄	1	1	yes	characteristic, complemented
81	#15262	D₄	1	1	yes	characteristic, complemented
82	#15261	D₄	1	1	yes	characteristic, complemented
83	#15259	D₄	1	1	yes	characteristic, complemented
84	#15258	D₄	1	1	yes	characteristic, complemented
85	#14506	D₄	1	2	yes	complemented
86	#14358	D₄	1	2	yes	complemented
87	#14335	C₂³	1	1	yes	characteristic
88	#13868	D₄	1	2	yes	complemented
89	C₂×#1438	D₄	1	2	yes	complemented
90	C₂×#1438	C₂³	1	1	yes	characteristic
91	#12690	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	C₂³×#41	C₂×D₄	1	1	yes	characteristic
96	C₂⁴×D₄	C₂×D₄	1	2	yes	—
97	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
100	C₂×#534	C₂×D₄	1	2	yes	—
101	C₂×#534	C₂×D₄	1	2	yes	—
102	C₂×#534	C₂×D₄	1	2	yes	—
103	C₂×#534	#41	1	2	yes	—
104	C₂²×#127	C₂×D₄	1	1	yes	characteristic
105	C₂²×#127	C₂×D₄	1	1	yes	characteristic
106	C₂²×#124	C₂×D₄	1	1	yes	characteristic
107	C₂²×#124	C₂×D₄	1	1	yes	characteristic
108	C₂²×#123	C₂×D₄	1	1	yes	characteristic
109	C₂²×#123	C₂×D₄	1	1	yes	characteristic
110	C₂²×#123	C₂×D₄	1	1	yes	characteristic
111	C₂²×#123	C₂×D₄	1	1	yes	characteristic
112	C₂²×#120	C₂×D₄	1	1	yes	characteristic
113	C₂²×#120	C₂×D₄	1	1	yes	characteristic
114	C₂²×#120	C₂×D₄	1	1	yes	characteristic
115	C₂²×#120	C₂×D₄	1	1	yes	characteristic
116	C₂²×#120	C₂×D₄	1	1	yes	characteristic
117	C₂²×#120	C₂×D₄	1	1	yes	characteristic
118	C₂²×#120	C₂×D₄	1	1	yes	characteristic
119	C₂²×#120	C₂×D₄	1	1	yes	characteristic
120	C₂×#518	C₂×D₄	1	1	yes	characteristic
121	C₂×#518	C₂×D₄	1	2	yes	—
122	C₂²×#117	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	2	yes	—
126	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
129	C₂²×#99	C₂×D₄	1	2	yes	—
130	C₂²×#99	C₂×D₄	1	2	yes	—
131	C₂×#381	C₂×D₄	1	2	yes	—
132	#1783	C₂×D₄	1	2	yes	—
133	#1781	C₂×D₄	1	2	yes	—
134	#1438	C₂×D₄	1	2	yes	—
135	#1438	C₂×D₄	1	1	yes	characteristic
136	#1438	C₂×D₄	1	1	yes	characteristic
137	#1438	C₂×D₄	1	2	yes	—
138	#1438	C₂×D₄	1	1	yes	characteristic
139	#1438	C₂×D₄	1	1	yes	characteristic
140	C₂⁶	#120	1	2	yes	—
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×Q₈	#120	1	1	yes	characteristic
148	C₂³×D₄	#120	1	1	yes	characteristic
149	C₂³×D₄	#120	1	2	yes	—
150	C₂³×D₄	#121	1	2	yes	—
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	C₂²×D₄	1	1	yes	characteristic
155	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
156	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
157	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
158	C₂³×D₄	C₂×#41	1	1	yes	characteristic
159	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂⁴×C₄	#120	1	2	yes	—
164	C₂⁴×C₄	#120	1	1	yes	characteristic
165	C₂⁴×C₄	#120	1	1	yes	characteristic
166	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×#123	#120	1	2	yes	—
168	C₂×#123	#121	1	2	yes	—
169	C₂×#123	#121	1	2	yes	—
170	C₂×#123	#127	1	2	yes	—
171	C₂×#120	#120	1	2	yes	—
172	C₂×#120	#121	1	2	yes	—
173	C₂×#120	#121	1	2	yes	—
174	C₂×#120	#127	1	2	yes	—
175	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	#120	1	2	yes	—
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	#120	1	1	yes	characteristic
182	C₂²×C₂²⋊C₄	#120	1	2	yes	—
183	C₂²×C₂²⋊C₄	#120	1	2	yes	—
184	C₂²×C₂²⋊C₄	#121	1	2	yes	—
185	C₂²×C₄²	#120	1	1	yes	characteristic
186	C₂²×C₄²	#120	1	1	yes	characteristic
187	C₂×#95	#120	1	2	yes	—
188	C₂×#95	#120	1	2	yes	—
189	C₂⁵	#392	1	2	yes	—
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	#534	1	1	yes	characteristic
198	C₂⁵	#534	1	1	yes	characteristic
199	C₂⁵	D₄²	1	1	yes	characteristic
200	C₂⁵	D₄²	1	1	yes	characteristic
201	C₂⁵	D₄²	1	1	yes	characteristic
202	C₂⁵	#546	1	1	yes	characteristic
203	C₂²×D₄	C₂×#120	1	1	yes	characteristic
204	C₂²×D₄	C₂×#121	1	1	yes	characteristic
205	C₂²×D₄	C₂×#121	1	1	yes	characteristic
206	C₂²×D₄	C₂×#127	1	1	yes	characteristic
207	C₂²×D₄	#534	1	1	yes	characteristic
208	C₂²×D₄	#534	1	1	yes	characteristic
209	C₂²×D₄	#535	1	1	yes	characteristic
210	C₂²×D₄	#535	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	D₄²	1	1	yes	characteristic
215	C₂²×D₄	D₄²	1	1	yes	characteristic
216	C₂²×D₄	D₄²	1	1	yes	characteristic
217	C₂²×D₄	#546	1	1	yes	characteristic
218	C₂²×D₄	#546	1	1	yes	characteristic
219	C₂³×C₄	#392	1	2	yes	—
220	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
221	C₂³×C₄	C₂×#120	1	1	yes	characteristic
222	C₂³×C₄	C₂×#120	1	1	yes	characteristic
223	C₂³×C₄	D₄²	1	1	yes	characteristic
224	C₂³×C₄	D₄²	1	1	yes	characteristic
225	C₂³×C₄	D₄²	1	1	yes	characteristic
226	C₂³×C₄	#546	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	#392	1	2	yes	—
228	C₂×C₂²⋊C₄	#392	1	2	yes	—
229	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	8	yes	—
230	C₂⁴	C₂×#392	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	#2408	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	#2433	1	1	yes	characteristic
239	C₂⁴	#2613	1	1	yes	characteristic
240	C₂⁴	#2846	1	1	yes	characteristic
241	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
242	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
243	C₂⁴	#3026	1	2	yes	—
244	C₂²×C₄	#2403	1	1	yes	characteristic
245	C₂²×C₄	#2408	1	1	yes	characteristic
246	C₂²×C₄	#2410	1	1	yes	characteristic
247	C₂²×C₄	#2433	1	1	yes	characteristic
248	C₂²⋊C₄	#2021	1	4	yes	complemented
249	C₂²⋊C₄	#2021	1	4	yes	complemented
250	C₂³	#14715	1	1	yes	characteristic
251	C₂³	C₂×#2021	1	2	yes	—
252	C₂³	#23895	1	2	yes	—
253	C₂³	#23900	1	1	yes	characteristic
254	C₂³	#23900	1	2	yes	—
255	C₂³	#23900	1	2	yes	—
256	C₂³	#23903	1	1	yes	characteristic
257	C₂³	#24404	1	2	yes	—
258	C₂³	#32898	1	2	yes	—
259	C₂²	#245233	1	2	yes	—
260	C₂²	#245255	1	2	yes	—
261	C₂²	#394058	1	1	yes	characteristic
262	C₂²	#394064	1	1	yes	characteristic
263	C₂²	#394066	1	1	yes	characteristic
264	C₂²	#511790	1	2	yes	—
265	C₂²	#511793	1	2	yes	—
266	C₂	#72233086	1	2	yes	—
267	C₂	#72236724	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#394066
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ 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. #72310283	C₂	semidirect
2. #72266378	C₂	semidirect
3. #72263490	C₂	semidirect
4. #72260537	C₂	semidirect
5. #72237045	C₂	semidirect
6. #72233800	C₂	semidirect
7. #72225778	C₂	semidirect
8. #72225657	C₂	semidirect
9. #72219293	C₂	semidirect
10. #72219268	C₂	semidirect
11. #72219240	C₂	semidirect
12. #72219040	C₂	semidirect
13. #72218980	C₂	semidirect
14. #72218840	C₂	semidirect
15. #72218838	C₂	semidirect
17. #7642244	C₂²	semidirect
20. #6437621	C₂²	semidirect
32. #493357	C₂²	semidirect
33. #493357	C₂²	semidirect
36. #382422	C₂²	semidirect
40. #359442	C₂²	semidirect
41. C₂×#8547	C₂²	semidirect
42. C₂×#8541	C₂²	semidirect
43. C₂×#8534	C₂²	semidirect
44. #186835	C₂²	semidirect
45. C₂×#7304	C₂²	semidirect
46. C₂×#7304	C₂²	semidirect
47. #128009	C₂²	semidirect
48. #128009	C₂²	semidirect
49. #108238	C₂²	semidirect
50. #108222	C₂²	semidirect
52. C₂²×#551	D₄	semidirect
53. C₂²×D₄²	D₄	semidirect
54. C₂²×H(3,𝔽₄)	D₄	semidirect
55. C₂²×#560	D₄	semidirect
57. C₂²×#534	D₄	semidirect
60. C₂³×#120	D₄	semidirect
61. C₂³×#120	D₄	semidirect
62. C₂³×#120	D₄	semidirect
65. C₂×#3026	D₄	semidirect
66. C₂×#2404	D₄	semidirect
68. C₂×#2273	D₄	semidirect
69. C₂×#1913	D₄	semidirect
70. #20960	D₄	semidirect
77. #15948	D₄	semidirect
78. #15948	D₄	semidirect
79. #15947	D₄	semidirect
80. #15947	D₄	semidirect
81. #15262	D₄	semidirect
82. #15261	D₄	semidirect
83. #15259	D₄	semidirect
84. #15258	D₄	semidirect
85. #14506	D₄	semidirect
86. #14358	D₄	semidirect
88. #13868	D₄	semidirect
89. C₂×#1438	D₄	semidirect
91. #12690	D₄	semidirect
248. C₂²⋊C₄	#2021	semidirect
249. C₂²⋊C₄	#2021	semidirect

Maximal subgroups

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

subgroup	index
#72310283	2
#72266378	2
#72263490	2
#72260537	2
#72237045	2
#72233800	2
#72225778	2
#72225657	2
#72219293	2
#72219268	2
#72219240	2
#72219040	2
#72218980	2
#72218840	2
#72218838	2