Subgroups — #72312489

Order 2048.

At least 9070 conjugacy classes of subgroups; only the 294 normal subgroups are shown.

The subgroup lattice has 238 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312214	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266835	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266378	C₂	1	1	yes	maximal, characteristic, complemented
4	#72264770	C₂	1	1	yes	maximal, characteristic, complemented
5	#72259136	C₂	1	1	yes	maximal, characteristic, complemented
6	#72258368	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255971	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251530	C₂	1	1	yes	maximal, characteristic, complemented
9	#72246450	C₂	1	1	yes	maximal, characteristic, complemented
10	#72238342	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237467	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236678	C₂	1	1	yes	maximal, characteristic, complemented
13	#72235130	C₂	1	1	yes	maximal, characteristic, complemented
14	#72234953	C₂	1	1	yes	maximal, characteristic, complemented
15	#72234100	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic, complemented
17	#10586404	C₂²	1	1	yes	characteristic, complemented
18	C₂×#23895	C₂²	1	1	yes	characteristic
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic
21	#6437621	C₂²	1	1	yes	characteristic
22	C₂×#23897	C₂²	1	1	yes	characteristic
23	#6437445	C₂²	1	1	yes	characteristic
24	#6437445	C₂²	1	1	yes	characteristic
25	#6431795	C₂²	1	1	yes	characteristic
26	#763333	C₂²	1	1	yes	characteristic
27	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
28	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
29	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
31	#493370	C₂²	1	1	yes	characteristic, complemented
32	#493357	C₂²	1	1	yes	characteristic, complemented
33	#382436	C₂²	1	1	yes	characteristic, complemented
34	#382434	C₂²	1	1	yes	characteristic, complemented
35	#359558	C₂²	1	1	yes	characteristic, complemented
36	#359549	C₂²	1	1	yes	characteristic, complemented
37	#359509	C₂²	1	1	yes	characteristic, complemented
38	#359442	C₂²	1	1	yes	characteristic, complemented
39	#188109	C₂²	1	1	yes	characteristic, complemented
40	#188050	C₂²	1	1	yes	characteristic, complemented
41	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
42	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
43	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
44	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
45	#146475	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
50	#108222	C₂²	1	1	yes	characteristic, complemented
51	#62842	D₄	1	1	yes	characteristic, complemented
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂²×#535	C₂³	1	1	yes	characteristic
57	C₂²×#534	C₂³	1	1	yes	characteristic
58	C₂×#3026	D₄	1	2	yes	complemented
59	C₂×#3019	D₄	1	2	yes	complemented
60	#60359	D₄	1	2	yes	complemented
61	C₂×#2883	D₄	1	1	yes	characteristic, complemented
62	C₂×#2883	D₄	1	1	yes	characteristic, complemented
63	#34980	D₄	1	2	yes	complemented
64	#34171	C₂³	1	1	yes	characteristic
65	#23907	D₄	1	2	yes	complemented
66	#23906	D₄	1	2	yes	complemented
67	#23901	D₄	1	2	yes	complemented
68	#23900	D₄	1	2	yes	complemented
69	#23895	D₄	1	4	yes	complemented
70	C₂×#1913	D₄	1	2	yes	complemented
71	C₂×#1911	D₄	1	2	yes	complemented
72	C₂×#1898	D₄	1	2	yes	complemented
73	#20979	D₄	1	2	yes	complemented
74	C₂×#1783	C₂³	1	1	yes	characteristic
75	C₂×#1783	C₂³	1	1	yes	characteristic
76	C₂×#1783	C₂³	1	1	yes	characteristic
77	C₂×#1781	C₂³	1	1	yes	characteristic
78	C₂×#1781	C₂³	1	1	yes	characteristic
79	C₂×#1781	C₂³	1	1	yes	characteristic
80	C₂×#1781	C₂³	1	1	yes	characteristic
81	#20354	D₄	1	1	yes	characteristic, complemented
82	#15262	D₄	1	1	yes	characteristic, complemented
83	#15261	D₄	1	1	yes	characteristic, complemented
84	#15259	D₄	1	1	yes	characteristic, complemented
85	#15258	D₄	1	1	yes	characteristic, complemented
86	C₂×#1438	C₂³	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	C₂³×#41	C₂×D₄	1	1	yes	characteristic
92	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
94	#3484	C₂×D₄	1	2	yes	—
95	#3484	C₂×D₄	1	2	yes	—
96	#3484	C₂×D₄	1	2	yes	—
97	#3484	C₂×D₄	1	2	yes	—
98	C₂×#535	C₂×D₄	1	2	yes	—
99	C₂×#534	C₂×D₄	1	2	yes	—
100	C₂²×#127	C₂×D₄	1	1	yes	characteristic
101	C₂²×#127	C₂×D₄	1	1	yes	characteristic
102	C₂²×#124	C₂×D₄	1	1	yes	characteristic
103	C₂²×#124	C₂×D₄	1	1	yes	characteristic
104	C₂×#525	C₂×D₄	1	2	yes	—
105	C₂²×#123	C₂×D₄	1	1	yes	characteristic
106	C₂²×#123	C₂×D₄	1	1	yes	characteristic
107	C₂²×#120	C₂×D₄	1	2	yes	—
108	C₂²×#120	C₂×D₄	1	1	yes	characteristic
109	C₂²×#120	C₂×D₄	1	1	yes	characteristic
110	C₂×#518	C₂×D₄	1	1	yes	characteristic
111	C₂×#518	C₂×D₄	1	1	yes	characteristic
112	C₂×#518	C₂×D₄	1	1	yes	characteristic
113	C₂×#518	C₂×D₄	1	1	yes	characteristic
114	C₂×#518	C₂×D₄	1	1	yes	characteristic
115	C₂×#518	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	C₂²×#100	C₂×D₄	1	2	yes	—
119	C₂²×#99	C₂×D₄	1	2	yes	—
120	C₂×#407	C₂×D₄	1	2	yes	—
121	C₂×#379	C₂×D₄	1	2	yes	—
122	#1783	C₂×D₄	1	2	yes	—
123	#1781	C₂×D₄	1	2	yes	—
124	#1781	C₂×D₄	1	2	yes	—
125	#1438	C₂×D₄	1	1	yes	characteristic
126	#1438	C₂×D₄	1	1	yes	characteristic
127	C₂⁶	#120	1	1	yes	characteristic
128	C₂⁶	C₂²×D₄	1	1	yes	characteristic
129	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×H(5,𝔽₂)	#120	1	2	yes	—
131	C₂×H(5,𝔽₂)	#120	1	2	yes	—
132	C₂²×#41	#120	1	2	yes	—
133	C₂³×Q₈	#120	1	1	yes	characteristic
134	C₂³×D₄	#120	1	1	yes	characteristic
135	C₂³×D₄	#120	1	2	yes	—
136	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
137	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
138	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
140	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
141	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂⁴×C₄	#120	1	1	yes	characteristic
149	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
150	#518	#120	1	2	yes	—
151	#518	#120	1	2	yes	—
152	C₂×#117	#120	1	2	yes	—
153	C₂²×C₂²⋊C₄	#120	1	2	yes	—
154	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
156	C₂²×C₄²	#120	1	1	yes	characteristic
157	C₂²×C₄²	#120	1	1	yes	characteristic
158	C₂⁵	B(4,𝔽₂)	1	2	yes	—
159	C₂⁵	C₂×#120	1	1	yes	characteristic
160	C₂⁵	C₂×#120	1	1	yes	characteristic
161	C₂⁵	#534	1	1	yes	characteristic
162	C₂⁵	#534	1	1	yes	characteristic
163	C₂⁵	#534	1	1	yes	characteristic
164	C₂⁵	#534	1	1	yes	characteristic
165	C₂⁵	D₄²	1	1	yes	characteristic
166	C₂⁵	D₄²	1	1	yes	characteristic
167	C₂⁵	D₄²	1	1	yes	characteristic
168	C₂⁵	D₄²	1	1	yes	characteristic
169	C₂⁵	D₄²	1	1	yes	characteristic
170	C₂⁵	D₄²	1	1	yes	characteristic
171	C₂⁵	H(3,𝔽₄)	1	1	yes	characteristic
172	C₂²×Q₈	B(4,𝔽₂)	1	2	yes	complemented
173	C₂²×D₄	B(4,𝔽₂)	1	2	yes	complemented
174	C₂²×D₄	C₂×#120	1	1	yes	characteristic
175	C₂²×D₄	C₂×#120	1	1	yes	characteristic
176	C₂²×D₄	#534	1	1	yes	characteristic
177	C₂²×D₄	#534	1	1	yes	characteristic
178	C₂²×D₄	#534	1	1	yes	characteristic
179	C₂²×D₄	#534	1	1	yes	characteristic
180	C₂²×D₄	D₄²	1	1	yes	characteristic
181	C₂²×D₄	D₄²	1	1	yes	characteristic
182	C₂²×D₄	D₄²	1	1	yes	characteristic
183	C₂²×D₄	D₄²	1	1	yes	characteristic
184	C₂²×D₄	D₄²	1	1	yes	characteristic
185	C₂²×D₄	D₄²	1	1	yes	characteristic
186	C₂²×D₄	D₄²	1	1	yes	characteristic
187	C₂²×D₄	D₄²	1	1	yes	characteristic
188	C₂²×D₄	D₄²	1	1	yes	characteristic
189	C₂²×D₄	H(3,𝔽₄)	1	1	yes	characteristic
190	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
191	C₂³×C₄	C₂×#120	1	1	yes	characteristic
192	C₂³×C₄	C₂×#120	1	1	yes	characteristic
193	C₂³×C₄	#534	1	1	yes	characteristic
194	C₂³×C₄	D₄²	1	1	yes	characteristic
195	C₂³×C₄	D₄²	1	1	yes	characteristic
196	C₂³×C₄	D₄²	1	1	yes	characteristic
197	#117	B(4,𝔽₂)	1	4	yes	complemented
198	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
199	C₂⁴	#2403	1	1	yes	characteristic
200	C₂⁴	#2403	1	1	yes	characteristic
201	C₂⁴	#2403	1	1	yes	characteristic
202	C₂⁴	#2403	1	1	yes	characteristic
203	C₂⁴	#2403	1	1	yes	characteristic
204	C₂⁴	#2403	1	1	yes	characteristic
205	C₂⁴	#2679	1	1	yes	characteristic
206	C₂⁴	#2679	1	1	yes	characteristic
207	C₂⁴	#2679	1	1	yes	characteristic
208	C₂⁴	#2679	1	1	yes	characteristic
209	C₂⁴	#2679	1	1	yes	characteristic
210	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
211	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
212	C₂²×C₄	#2403	1	1	yes	characteristic
213	C₂²×C₄	#2403	1	1	yes	characteristic
214	C₂²×C₄	#2403	1	1	yes	characteristic
215	C₂²×C₄	#2679	1	1	yes	characteristic
216	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
217	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
218	C₂³	#15947	1	1	yes	characteristic
219	C₂³	#23895	1	1	yes	characteristic
220	C₂³	#23896	1	1	yes	characteristic
221	C₂³	#23900	1	2	yes	—
222	C₂³	#23900	1	2	yes	—
223	C₂³	#23900	1	1	yes	characteristic
224	C₂³	#23900	1	1	yes	characteristic
225	C₂³	#23900	1	1	yes	characteristic
226	C₂³	#23900	1	1	yes	characteristic
227	C₂×C₄	#23900	1	2	yes	—
228	C₂×C₄	#23900	1	2	yes	—
229	C₂²	#245225	1	1	yes	characteristic
230	C₂²	#245229	1	1	yes	characteristic
231	C₂²	#245229	1	1	yes	characteristic
232	C₂²	#245233	1	1	yes	characteristic
233	C₂²	#449469	1	1	yes	characteristic
234	C₂²	#449475	1	1	yes	characteristic
235	C₂²	#449475	1	1	yes	characteristic
236	C₂	#72219165	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72219165
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ 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. #72312214	C₂	semidirect
2. #72266835	C₂	semidirect
3. #72266378	C₂	semidirect
4. #72264770	C₂	semidirect
5. #72259136	C₂	semidirect
6. #72258368	C₂	semidirect
7. #72255971	C₂	semidirect
8. #72251530	C₂	semidirect
9. #72246450	C₂	semidirect
10. #72238342	C₂	semidirect
11. #72237467	C₂	semidirect
12. #72236678	C₂	semidirect
13. #72235130	C₂	semidirect
14. #72234953	C₂	semidirect
15. #72234100	C₂	semidirect
16. #10586422	C₂²	semidirect
17. #10586404	C₂²	semidirect
27. C₂×#12707	C₂²	semidirect
28. C₂×#12705	C₂²	semidirect
29. C₂×#12705	C₂²	semidirect
30. C₂×#12705	C₂²	semidirect
31. #493370	C₂²	semidirect
32. #493357	C₂²	semidirect
33. #382436	C₂²	semidirect
34. #382434	C₂²	semidirect
35. #359558	C₂²	semidirect
36. #359549	C₂²	semidirect
37. #359509	C₂²	semidirect
38. #359442	C₂²	semidirect
39. #188109	C₂²	semidirect
40. #188050	C₂²	semidirect
41. C₂×#8534	C₂²	semidirect
42. C₂×#8534	C₂²	semidirect
43. C₂×#8453	C₂²	semidirect
44. C₂×#8453	C₂²	semidirect
45. #146475	C₂²	semidirect
46. C₂×#7304	C₂²	semidirect
47. C₂×#7316	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7311	C₂²	semidirect
50. #108222	C₂²	semidirect
51. #62842	D₄	semidirect
58. C₂×#3026	D₄	semidirect
59. C₂×#3019	D₄	semidirect
60. #60359	D₄	semidirect
61. C₂×#2883	D₄	semidirect
62. C₂×#2883	D₄	semidirect
63. #34980	D₄	semidirect
65. #23907	D₄	semidirect
66. #23906	D₄	semidirect
67. #23901	D₄	semidirect
68. #23900	D₄	semidirect
69. #23895	D₄	semidirect
70. C₂×#1913	D₄	semidirect
71. C₂×#1911	D₄	semidirect
72. C₂×#1898	D₄	semidirect
73. #20979	D₄	semidirect
81. #20354	D₄	semidirect
82. #15262	D₄	semidirect
83. #15261	D₄	semidirect
84. #15259	D₄	semidirect
85. #15258	D₄	semidirect
172. C₂²×Q₈	B(4,𝔽₂)	semidirect
173. C₂²×D₄	B(4,𝔽₂)	semidirect
197. #117	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72312214	2
#72266835	2
#72266378	2
#72264770	2
#72259136	2
#72258368	2
#72255971	2
#72251530	2
#72246450	2
#72238342	2
#72237467	2
#72236678	2
#72235130	2
#72234953	2
#72234100	2