Subgroups — #72341347

Order 2048.

At least 9009 conjugacy classes of subgroups; only the 342 normal subgroups are shown.

The subgroup lattice has 284 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72341312	C₂	1	1	yes	maximal, characteristic, complemented
2	#72311315	C₂	1	1	yes	maximal, characteristic, complemented
3	#72311115	C₂	1	1	yes	maximal, characteristic, complemented
4	#72303066	C₂	1	1	yes	maximal, characteristic, complemented
5	#72303055	C₂	1	1	yes	maximal, characteristic, complemented
6	#72285407	C₂	1	1	yes	maximal, characteristic, complemented
7	#72283640	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238016	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238015	C₂	1	1	yes	maximal, characteristic, complemented
10	#72238014	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238013	C₂	1	1	yes	maximal, characteristic, complemented
12	#72238012	C₂	1	1	yes	maximal, characteristic, complemented
13	#72238010	C₂	1	1	yes	maximal, characteristic, complemented
14	#72213523	C₂	1	1	yes	maximal, characteristic, complemented
15	#72213504	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586097	C₂²	1	1	yes	characteristic
17	#10574841	C₂²	1	1	yes	characteristic
18	#7743195	C₂²	1	1	yes	characteristic
19	#6628897	C₂²	1	1	yes	characteristic
20	#6628896	C₂²	1	1	yes	characteristic
21	#6470235	C₂²	1	1	yes	characteristic
22	#6470235	C₂²	1	1	yes	characteristic, complemented
23	#6470234	C₂²	1	1	yes	characteristic
24	#6470234	C₂²	1	1	yes	characteristic, complemented
25	#6437630	C₂²	1	1	yes	characteristic
26	#6437452	C₂²	1	1	yes	characteristic
27	#6385980	C₂²	1	1	yes	characteristic
28	#496241	C₂²	1	1	yes	characteristic, complemented
29	#496240	C₂²	1	1	yes	characteristic, complemented
30	#494682	C₂²	1	1	yes	characteristic, complemented
31	#494681	C₂²	1	1	yes	characteristic, complemented
32	#493430	C₂²	1	1	yes	characteristic
33	#493430	C₂²	1	1	yes	characteristic
34	#493397	C₂²	1	1	yes	characteristic
35	#493397	C₂²	1	1	yes	characteristic
36	#493397	C₂²	1	1	yes	characteristic, complemented
37	#493397	C₂²	1	1	yes	characteristic, complemented
38	#392138	C₂²	1	1	yes	characteristic
39	#392135	C₂²	1	1	yes	characteristic
40	#392135	C₂²	1	1	yes	characteristic
41	#359605	C₂²	1	1	yes	characteristic
42	#359529	C₂²	1	1	yes	characteristic
43	#213855	C₂²	1	1	yes	characteristic, complemented
44	#213854	C₂²	1	1	yes	characteristic, complemented
45	#146577	C₂²	1	1	yes	characteristic, complemented
46	#146577	C₂²	1	1	yes	characteristic, complemented
47	#128079	C₂²	1	1	yes	characteristic, complemented
48	#128079	C₂²	1	1	yes	characteristic, complemented
49	#128062	C₂²	1	1	yes	characteristic, complemented
50	#128062	C₂²	1	1	yes	characteristic, complemented
51	C₂×H(7,𝔽₂)	D₄	1	1	yes	characteristic, complemented
52	C₂×H(7,𝔽₂)	C₂³	1	1	yes	characteristic
53	#62852	D₄	1	1	yes	characteristic, complemented
54	#62852	C₂³	1	1	yes	characteristic
55	#62852	C₂³	1	1	yes	characteristic
56	#62812	D₄	1	1	yes	characteristic, complemented
57	C₂×#3498	C₂³	1	1	yes	characteristic
58	C₂²×#560	D₄	1	1	yes	characteristic, complemented
59	C₄×#585	D₄	1	1	yes	characteristic, complemented
60	C₂²×#458	D₄	1	2	yes	complemented
61	C₂²×B(4,𝔽₂)	D₄	1	2	yes	complemented
62	C₂×#3021	D₄	1	1	yes	characteristic, complemented
63	C₂×#3018	D₄	1	1	yes	characteristic, complemented
64	C₂×#2888	C₂³	1	1	yes	characteristic
65	#36955	D₄	1	1	yes	characteristic, complemented
66	#36702	D₄	1	1	yes	characteristic, complemented
67	#35446	C₂³	1	1	yes	characteristic
68	#35346	C₂³	1	1	yes	characteristic
69	#34364	D₄	1	1	yes	characteristic, complemented
70	C₂×#2127	D₄	1	2	yes	complemented
71	C₂²×#353	D₄	1	2	yes	complemented
72	#24058	D₄	1	2	yes	—
73	#24058	D₄	1	2	yes	complemented
74	#23901	D₄	1	2	yes	complemented
75	#23900	D₄	1	2	yes	—
76	#21794	D₄	1	2	yes	complemented
77	#21750	D₄	1	2	yes	complemented
78	#20956	D₄	1	1	yes	characteristic, complemented
79	#20952	D₄	1	1	yes	characteristic, complemented
80	#20609	D₄	1	2	yes	complemented
81	#20609	D₄	1	2	yes	complemented
82	#20592	C₂³	1	1	yes	characteristic
83	#20586	C₂³	1	1	yes	characteristic
84	#20585	C₂³	1	1	yes	characteristic
85	#20585	C₂³	1	1	yes	characteristic
86	#20585	C₂³	1	1	yes	characteristic
87	#20585	C₂³	1	1	yes	characteristic
88	#20585	C₂³	1	1	yes	characteristic
89	#20585	C₂³	1	1	yes	characteristic
90	#10989	D₄	1	2	yes	complemented
91	#10975	D₄	1	2	yes	complemented
92	C₂×#585	C₂×D₄	1	1	yes	characteristic
93	C₂×#585	C₂×D₄	1	1	yes	characteristic
94	C₂×#585	C₂×D₄	1	1	yes	characteristic
95	C₂×#585	C₂×D₄	1	1	yes	characteristic
96	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
97	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
98	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
99	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
100	#3484	C₂×D₄	1	2	yes	—
101	#3484	C₂×D₄	1	2	yes	—
102	C₂×#535	C₂×D₄	1	1	yes	characteristic
103	C₂×#535	C₂×D₄	1	1	yes	characteristic
104	C₂×#535	C₂×D₄	1	1	yes	characteristic
105	C₂×#535	C₂×D₄	1	1	yes	characteristic
106	C₂×#535	C₂×D₄	1	1	yes	characteristic
107	C₂×#535	#41	1	1	yes	characteristic
108	C₂×#532	C₂×D₄	1	1	yes	characteristic
109	C₂×#532	C₂×D₄	1	1	yes	characteristic
110	C₂×#532	#41	1	1	yes	characteristic
111	C₂²×#127	C₂×D₄	1	1	yes	characteristic
112	C₂²×#127	C₂×D₄	1	1	yes	characteristic
113	C₂²×#127	C₂×D₄	1	1	yes	characteristic
114	C₂²×#123	C₂×D₄	1	1	yes	characteristic
115	C₂²×#123	C₂×D₄	1	1	yes	characteristic
116	C₂²×#123	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	#3428	C₂×D₄	1	1	yes	characteristic
121	#3428	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic
123	C₂×#413	C₂×D₄	1	1	yes	characteristic
124	C₂×#413	C₂×D₄	1	1	yes	characteristic
125	C₂²×#100	C₂×D₄	1	2	yes	—
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	#2309	C₂×D₄	1	2	yes	—
131	#2308	C₂×D₄	1	1	yes	characteristic
132	#2308	C₂×D₄	1	1	yes	characteristic
133	#2308	C₂×D₄	1	1	yes	characteristic
134	#2308	C₂×D₄	1	1	yes	characteristic
135	#2308	C₂×D₄	1	1	yes	characteristic
136	#2308	C₂×D₄	1	1	yes	characteristic
137	#1782	C₂×D₄	1	2	yes	—
138	#1782	C₂×D₄	1	2	yes	—
139	#1781	C₂×D₄	1	2	yes	—
140	#1781	C₂×D₄	1	2	yes	—
141	C₂⁶	#120	1	1	yes	characteristic
142	C₂×H(5,𝔽₂)	#120	1	2	yes	—
143	C₂×H(5,𝔽₂)	#120	1	2	yes	—
144	C₂×H(5,𝔽₂)	#120	1	2	yes	—
145	C₂²×#41	#120	1	1	yes	characteristic
146	C₂²×#41	#120	1	1	yes	characteristic
147	C₂²×#41	#121	1	1	yes	characteristic
148	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
149	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
150	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
151	C₂²×#41	C₂×#41	1	1	yes	characteristic
152	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂³×D₄	#120	1	1	yes	characteristic
155	C₂³×D₄	#120	1	1	yes	characteristic
156	C₂³×D₄	#121	1	1	yes	characteristic
157	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
158	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
159	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
160	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
161	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
162	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
163	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
164	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂⁴×C₄	#120	1	1	yes	characteristic
167	C₂×C₄⋊Q₈	#121	1	1	yes	characteristic
168	C₂×#127	#120	1	1	yes	characteristic
169	C₂×#127	#120	1	1	yes	characteristic
170	C₂×#127	#120	1	1	yes	characteristic
171	C₂×#127	#121	1	1	yes	characteristic
172	C₂×#127	#121	1	1	yes	characteristic
173	C₂×#127	#121	1	1	yes	characteristic
174	C₂×#127	#127	1	1	yes	characteristic
175	C₂×#127	#127	1	1	yes	characteristic
176	C₂×#127	#127	1	1	yes	characteristic
177	C₂×#124	#120	1	1	yes	characteristic
178	C₂×#124	#121	1	1	yes	characteristic
179	C₂×#124	#121	1	1	yes	characteristic
180	C₂×#124	#121	1	1	yes	characteristic
181	C₂×#124	#121	1	1	yes	characteristic
182	C₂×#124	#127	1	1	yes	characteristic
183	#518	#120	1	2	yes	—
184	C₂×#117	#120	1	1	yes	characteristic
185	C₂×#117	#120	1	1	yes	characteristic
186	C₂×#117	#121	1	1	yes	characteristic
187	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
188	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
190	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
193	C₂²×C₄²	#120	1	1	yes	characteristic
194	C₂²×C₄²	#120	1	1	yes	characteristic
195	C₂²×C₄²	#121	1	1	yes	characteristic
196	C₂×#99	#120	1	4	yes	—
197	C₂×#99	#120	1	4	yes	—
198	C₂⁵	C₂×#120	1	1	yes	characteristic
199	C₂⁵	C₂×#120	1	1	yes	characteristic
200	C₂⁵	D₄²	1	1	yes	characteristic
201	C₂⁵	D₄²	1	1	yes	characteristic
202	C₂⁵	D₄²	1	1	yes	characteristic
203	C₂⁵	D₄²	1	1	yes	characteristic
204	H(5,𝔽₂)	B(4,𝔽₂)	1	4	yes	—
205	C₂²×Q₈	#392	1	1	yes	characteristic
206	C₂²×Q₈	C₂×#121	1	1	yes	characteristic
207	C₂²×Q₈	#534	1	1	yes	characteristic
208	C₂²×D₄	#392	1	1	yes	characteristic
209	C₂²×D₄	#392	1	1	yes	characteristic
210	C₂²×D₄	#392	1	1	yes	characteristic
211	C₂²×D₄	C₂×#120	1	1	yes	characteristic
212	C₂²×D₄	C₂×#120	1	1	yes	characteristic
213	C₂²×D₄	C₂×#120	1	1	yes	characteristic
214	C₂²×D₄	C₂×#120	1	1	yes	characteristic
215	C₂²×D₄	C₂×#120	1	1	yes	characteristic
216	C₂²×D₄	C₂×#121	1	1	yes	characteristic
217	C₂²×D₄	C₂×#127	1	1	yes	characteristic
218	C₂²×D₄	#534	1	1	yes	characteristic
219	C₂²×D₄	#534	1	1	yes	characteristic
220	C₂²×D₄	#534	1	1	yes	characteristic
221	C₂²×D₄	D₄²	1	1	yes	characteristic
222	C₂²×D₄	D₄²	1	1	yes	characteristic
223	C₂²×D₄	D₄²	1	1	yes	characteristic
224	C₂²×D₄	D₄²	1	1	yes	characteristic
225	C₂²×D₄	D₄²	1	1	yes	characteristic
226	C₂²×D₄	D₄²	1	1	yes	characteristic
227	C₂²×D₄	#546	1	1	yes	characteristic
228	C₂²×D₄	#546	1	1	yes	characteristic
229	C₂³×C₄	C₂×#120	1	1	yes	characteristic
230	C₂³×C₄	C₂×#120	1	1	yes	characteristic
231	C₂³×C₄	C₂×#121	1	1	yes	characteristic
232	C₂³×C₄	#535	1	1	yes	characteristic
233	C₂³×C₄	#535	1	1	yes	characteristic
234	C₂³×C₄	D₄²	1	1	yes	characteristic
235	C₂³×C₄	D₄²	1	1	yes	characteristic
236	C₂³×C₄	#546	1	1	yes	characteristic
237	C₂³×C₄	#546	1	1	yes	characteristic
238	#127	B(4,𝔽₂)	1	2	yes	—
239	#127	B(4,𝔽₂)	1	2	yes	—
240	#127	B(4,𝔽₂)	1	4	yes	—
241	C₂×C₄²	#392	1	1	yes	characteristic
242	C₂×C₄²	#392	1	1	yes	characteristic
243	C₂×C₄²	#392	1	1	yes	characteristic
244	C₂×C₄²	#392	1	1	yes	characteristic
245	C₂⁴	#2403	1	1	yes	characteristic
246	C₂⁴	#2403	1	1	yes	characteristic
247	C₂⁴	#2403	1	1	yes	characteristic
248	C₂⁴	#2410	1	1	yes	characteristic
249	C₂⁴	#2410	1	1	yes	characteristic
250	C₂⁴	#2410	1	1	yes	characteristic
251	C₂⁴	#2433	1	1	yes	characteristic
252	C₂⁴	#2433	1	1	yes	characteristic
253	C₂⁴	#2846	1	1	yes	characteristic
254	C₂×D₄	#2021	1	2	yes	—
255	C₂×D₄	#2021	1	2	yes	—
256	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
257	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
258	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
259	C₂×D₄	#3026	1	2	yes	—
260	C₂²×C₄	C₂×#392	1	1	yes	characteristic
261	C₂²×C₄	#2403	1	1	yes	characteristic
262	C₂²×C₄	#2408	1	1	yes	characteristic
263	C₂²×C₄	#2408	1	1	yes	characteristic
264	C₂²×C₄	#2410	1	1	yes	characteristic
265	C₂²×C₄	#2613	1	1	yes	characteristic
266	C₄²	#2021	1	1	yes	characteristic
267	C₄²	#2021	1	1	yes	characteristic
268	C₄²	#2021	1	2	yes	—
269	C₂³	#14715	1	1	yes	characteristic
270	C₂³	#23895	1	2	yes	—
271	C₂³	#23900	1	2	yes	—
272	C₂³	#23900	1	2	yes	—
273	C₂³	#24404	1	2	yes	—
274	C₂³	#32898	1	2	yes	—
275	C₂×C₄	C₂×#2021	1	1	yes	characteristic
276	C₂×C₄	C₂×#2021	1	1	yes	characteristic
277	C₂×C₄	#23900	1	1	yes	characteristic
278	C₂×C₄	#23903	1	1	yes	characteristic
279	C₂²	#394058	1	1	yes	characteristic
280	C₂²	#394064	1	1	yes	characteristic
281	C₂²	#394066	1	1	yes	characteristic
282	C₂	#72210600	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72210600
commutator, Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72341312	C₂	semidirect
2. #72311315	C₂	semidirect
3. #72311115	C₂	semidirect
4. #72303066	C₂	semidirect
5. #72303055	C₂	semidirect
6. #72285407	C₂	semidirect
7. #72283640	C₂	semidirect
8. #72238016	C₂	semidirect
9. #72238015	C₂	semidirect
10. #72238014	C₂	semidirect
11. #72238013	C₂	semidirect
12. #72238012	C₂	semidirect
13. #72238010	C₂	semidirect
14. #72213523	C₂	semidirect
15. #72213504	C₂	semidirect
22. #6470235	C₂²	semidirect
24. #6470234	C₂²	semidirect
28. #496241	C₂²	semidirect
29. #496240	C₂²	semidirect
30. #494682	C₂²	semidirect
31. #494681	C₂²	semidirect
36. #493397	C₂²	semidirect
37. #493397	C₂²	semidirect
43. #213855	C₂²	semidirect
44. #213854	C₂²	semidirect
45. #146577	C₂²	semidirect
46. #146577	C₂²	semidirect
47. #128079	C₂²	semidirect
48. #128079	C₂²	semidirect
49. #128062	C₂²	semidirect
50. #128062	C₂²	semidirect
51. C₂×H(7,𝔽₂)	D₄	semidirect
53. #62852	D₄	semidirect
56. #62812	D₄	semidirect
58. C₂²×#560	D₄	semidirect
59. C₄×#585	D₄	semidirect
60. C₂²×#458	D₄	semidirect
61. C₂²×B(4,𝔽₂)	D₄	semidirect
62. C₂×#3021	D₄	semidirect
63. C₂×#3018	D₄	semidirect
65. #36955	D₄	semidirect
66. #36702	D₄	semidirect
69. #34364	D₄	semidirect
70. C₂×#2127	D₄	semidirect
71. C₂²×#353	D₄	semidirect
73. #24058	D₄	semidirect
74. #23901	D₄	semidirect
76. #21794	D₄	semidirect
77. #21750	D₄	semidirect
78. #20956	D₄	semidirect
79. #20952	D₄	semidirect
80. #20609	D₄	semidirect
81. #20609	D₄	semidirect
90. #10989	D₄	semidirect
91. #10975	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72341312	2
#72311315	2
#72311115	2
#72303066	2
#72303055	2
#72285407	2
#72283640	2
#72238016	2
#72238015	2
#72238014	2
#72238013	2
#72238012	2
#72238010	2
#72213523	2
#72213504	2