Subgroups — #72309519

Order 2048.

At least 7504 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

The subgroup lattice has 265 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72309515	C₂	1	8	yes	maximal, complemented
2	C₂×#115348	C₂	1	1	yes	maximal, characteristic, complemented
3	#72224136	C₂	1	1	yes	maximal, characteristic
4	#72219300	C₂	1	1	yes	maximal, characteristic, complemented
5	#72219243	C₂	1	1	yes	maximal, characteristic, complemented
6	#72219237	C₂	1	1	yes	maximal, characteristic, complemented
7	#72218069	C₂	1	1	yes	maximal, characteristic, complemented
8	#72218057	C₂	1	1	yes	maximal, characteristic, complemented
9	#10586404	C₄	1	1	yes	characteristic, complemented
10	#7630939	C₂²	1	1	yes	characteristic
11	#7630939	C₂²	1	1	yes	characteristic
12	C₂×#34980	C₂²	1	1	yes	characteristic
13	#6499892	C₄	1	4	yes	complemented
14	#6499892	C₂²	1	4	yes	—
15	#6374210	C₄	1	1	yes	characteristic, complemented
16	#6374210	C₄	1	1	yes	characteristic, complemented
17	#6374128	C₄	1	1	yes	characteristic, complemented
18	C₂²×#1781	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#1438	C₂²	1	1	yes	characteristic, complemented
20	#392113	C₂²	1	1	yes	characteristic, complemented
21	#392109	C₂²	1	1	yes	characteristic, complemented
22	#147569	C₂²	1	4	yes	complemented
23	#147567	C₂²	1	4	yes	complemented
24	#147567	C₂²	1	4	yes	complemented
25	#115348	C₂²	1	4	yes	complemented
26	#115348	C₂²	1	4	yes	complemented
27	#113037	C₂²	1	4	yes	complemented
28	C₂⁵×D₄	C₂³	1	1	yes	characteristic
29	#62862	C₂×C₄	1	4	yes	complemented
30	C₂×#3484	C₂×C₄	1	1	yes	characteristic
31	C₂×#3484	C₂×C₄	1	1	yes	characteristic
32	C₂×#3484	D₄	1	1	yes	characteristic
33	C₂×#3484	D₄	1	1	yes	characteristic
34	C₂×#3484	D₄	1	1	yes	characteristic
35	C₂×#3484	D₄	1	1	yes	characteristic
36	C₂²×#534	C₂×C₄	1	1	yes	characteristic
37	C₂²×#534	D₄	1	1	yes	characteristic
38	C₂³×#120	D₄	1	1	yes	characteristic
39	#36614	D₄	1	2	yes	—
40	C₂×#2846	D₄	1	2	yes	—
41	#35222	C₂×C₄	1	2	yes	—
42	#35222	D₄	1	2	yes	—
43	#35222	C₂³	1	2	yes	—
44	#34980	C₂×C₄	1	2	yes	—
45	#34980	C₂×C₄	1	2	yes	—
46	#34980	D₄	1	2	yes	—
47	#34980	D₄	1	2	yes	—
48	#34980	D₄	1	2	yes	—
49	#34980	C₂³	1	2	yes	—
50	#34980	C₂³	1	2	yes	—
51	#34171	C₂×C₄	1	1	yes	characteristic
52	#34171	C₂×C₄	1	1	yes	characteristic
53	#34171	D₄	1	1	yes	characteristic
54	#34171	D₄	1	1	yes	characteristic
55	#34171	D₄	1	1	yes	characteristic
56	#34171	D₄	1	1	yes	characteristic
57	C₂×#2308	C₂×C₄	1	1	yes	characteristic
58	C₂×#2308	D₄	1	1	yes	characteristic
59	C₂²×#379	D₄	1	1	yes	characteristic
60	#21750	C₂×C₄	1	4	yes	complemented
61	#21749	C₂×C₄	1	4	yes	complemented
62	#21749	C₂×C₄	1	4	yes	complemented
63	C₂×#1783	C₂³	1	2	yes	complemented
64	C₂×#1781	C₂³	1	2	yes	complemented
65	C₂×#1781	C₂³	1	2	yes	complemented
66	C₂×#1438	C₂³	1	2	yes	complemented
67	C₂⁷	C₂×D₄	1	1	yes	characteristic
68	C₂⁷	C₂×D₄	1	1	yes	characteristic
69	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
71	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
73	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
74	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
76	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
77	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
79	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
84	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×#535	C₂²⋊C₄	1	2	yes	—
86	C₂×#535	C₂²×C₄	1	2	yes	—
87	C₂×#535	C₂×D₄	1	2	yes	—
88	C₂×#534	C₂²⋊C₄	1	2	yes	—
89	C₂×#534	C₂²⋊C₄	1	2	yes	—
90	C₂×#534	C₂²⋊C₄	1	2	yes	—
91	C₂×#534	C₂²×C₄	1	2	yes	—
92	C₂×#534	C₂²×C₄	1	2	yes	—
93	C₂×#534	C₂×D₄	1	2	yes	—
94	C₂×#534	C₂×D₄	1	2	yes	—
95	C₂×#525	C₂²⋊C₄	1	2	yes	—
96	C₂×#525	C₂²⋊C₄	1	2	yes	—
97	C₂×#525	C₂×D₄	1	2	yes	—
98	C₂²×#120	C₂²⋊C₄	1	2	yes	—
99	C₂²×#120	C₂²⋊C₄	1	2	yes	—
100	C₂²×#120	C₂²⋊C₄	1	2	yes	—
101	C₂²×#120	C₂×D₄	1	2	yes	—
102	C₂²×#120	C₂×D₄	1	2	yes	—
103	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
111	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
112	#2309	C₂²⋊C₄	1	2	yes	—
113	#2309	C₂²×C₄	1	2	yes	—
114	#2309	C₂×D₄	1	2	yes	—
115	#2308	C₂²⋊C₄	1	2	yes	—
116	#2308	C₂²⋊C₄	1	2	yes	—
117	#2308	C₂²⋊C₄	1	2	yes	—
118	#2308	C₂²×C₄	1	2	yes	—
119	#2308	C₂²×C₄	1	2	yes	—
120	#2308	C₂×D₄	1	2	yes	—
121	#2308	C₂×D₄	1	2	yes	—
122	#2280	C₂²⋊C₄	1	2	yes	—
123	#2280	C₂²⋊C₄	1	2	yes	—
124	#2280	C₂×D₄	1	2	yes	—
125	C₂×#379	C₂²⋊C₄	1	2	yes	—
126	C₂×#379	C₂²⋊C₄	1	2	yes	—
127	C₂×#379	C₂²⋊C₄	1	2	yes	—
128	C₂×#379	C₂×D₄	1	2	yes	—
129	C₂×#379	C₂×D₄	1	2	yes	—
130	C₂⁶	#99	1	1	yes	characteristic, complemented
131	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂⁶	#120	1	1	yes	characteristic
136	C₂⁶	#120	1	1	yes	characteristic
137	C₂⁶	#120	1	1	yes	characteristic
138	C₂⁶	#120	1	1	yes	characteristic
139	C₂⁶	#120	1	1	yes	characteristic
140	C₂⁶	#120	1	1	yes	characteristic
141	C₂⁶	#120	1	1	yes	characteristic
142	C₂⁶	C₂²×D₄	1	1	yes	characteristic
143	C₂⁶	C₂²×D₄	1	1	yes	characteristic
144	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂³×D₄	#99	1	2	yes	complemented
146	C₂³×D₄	#99	1	4	yes	complemented
147	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
154	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂³×D₄	#120	1	1	yes	characteristic
156	C₂³×D₄	#120	1	1	yes	characteristic
157	C₂³×D₄	#120	1	1	yes	characteristic
158	C₂³×D₄	#120	1	1	yes	characteristic
159	C₂³×D₄	#120	1	1	yes	characteristic
160	C₂³×D₄	#120	1	1	yes	characteristic
161	C₂³×D₄	C₂³×C₄	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	C₂²×D₄	1	1	yes	characteristic
165	C₂⁴×C₄	#99	1	1	yes	characteristic, complemented
166	C₂⁴×C₄	#99	1	2	yes	complemented
167	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂⁴×C₄	#120	1	1	yes	characteristic
170	C₂⁴×C₄	#120	1	1	yes	characteristic
171	C₂⁴×C₄	#120	1	1	yes	characteristic
172	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₂²⋊Q₈	#99	1	4	yes	complemented
174	C₂×#121	#99	1	4	yes	complemented
175	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
176	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₂²⋊C₄	#99	1	2	yes	complemented
179	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
180	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂⁵	#379	1	1	yes	characteristic
187	C₂⁵	#379	1	1	yes	characteristic
188	C₂⁵	#379	1	1	yes	characteristic
189	C₂⁵	#379	1	1	yes	characteristic
190	C₂⁵	#379	1	1	yes	characteristic
191	C₂⁵	#379	1	1	yes	characteristic
192	C₂⁵	C₂×#99	1	1	yes	characteristic
193	C₂⁵	C₂×#99	1	1	yes	characteristic
194	C₂⁵	C₂×#99	1	1	yes	characteristic
195	C₂⁵	C₂×#99	1	1	yes	characteristic
196	C₂⁵	C₂×#99	1	4	yes	—
197	C₂⁵	C₂×#99	1	2	yes	—
198	C₂⁵	C₂×#99	1	2	yes	complemented
199	C₂⁵	B(4,𝔽₂)	1	2	yes	—
200	C₂⁵	B(4,𝔽₂)	1	2	yes	—
201	C₂⁵	#458	1	2	yes	—
202	C₂⁵	#458	1	2	yes	—
203	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
204	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₂⁵	C₂×#120	1	1	yes	characteristic
206	C₂⁵	C₂×#120	1	1	yes	characteristic
207	C₂⁵	C₂×#120	1	1	yes	characteristic
208	C₂²×Q₈	C₂×#99	1	2	yes	complemented
209	C₂²×D₄	C₂×#99	1	2	yes	complemented
210	C₂³×C₄	#379	1	1	yes	characteristic
211	C₂³×C₄	#379	1	1	yes	characteristic
212	C₂³×C₄	C₂×#99	1	4	yes	—
213	C₂³×C₄	C₂×#99	1	2	yes	—
214	C₂³×C₄	C₂×#99	1	4	yes	—
215	C₂³×C₄	C₂×#99	1	2	yes	complemented
216	C₂³×C₄	C₂×#99	1	4	yes	complemented
217	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
218	C₂³×C₄	C₂×#120	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×#99	1	4	yes	complemented
220	C₂⁴	#1781	1	1	yes	characteristic
221	C₂⁴	#1781	1	2	yes	—
222	C₂⁴	#1781	1	2	yes	—
223	C₂⁴	#1781	1	2	yes	—
224	C₂⁴	#1781	1	1	yes	characteristic
225	C₂⁴	#1781	1	1	yes	characteristic
226	C₂⁴	#1781	1	1	yes	characteristic
227	C₂⁴	#1790	1	2	yes	—
228	C₂⁴	#1790	1	2	yes	—
229	C₂⁴	#1790	1	2	yes	—
230	C₂⁴	#1790	1	2	yes	—
231	C₂⁴	C₂×#379	1	1	yes	characteristic
232	C₂⁴	C₂²×#99	1	1	yes	characteristic
233	C₂⁴	C₂²×#99	1	2	yes	—
234	C₂⁴	C₂²×#99	1	1	yes	characteristic
235	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
236	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
237	C₂⁴	C₂×#458	1	1	yes	characteristic
238	C₂⁴	C₂×#458	1	1	yes	characteristic
239	C₂⁴	#3026	1	2	yes	—
240	C₂²×C₄	#1781	1	2	yes	—
241	C₂²×C₄	C₂²×#99	1	2	yes	—
242	C₂²×C₄	#3026	1	2	yes	—
243	C₂³	#8534	1	2	yes	—
244	C₂³	#8534	1	2	yes	—
245	C₂³	#8541	1	2	yes	—
246	C₂³	#8541	1	2	yes	—
247	C₂³	C₂×#1781	1	1	yes	characteristic
248	C₂³	C₂×#1781	1	1	yes	characteristic
249	C₂³	C₂×#1781	1	1	yes	characteristic
250	C₂³	C₂×#1790	1	1	yes	characteristic
251	C₂³	C₂×#1790	1	1	yes	characteristic
252	C₂³	#20609	1	1	yes	characteristic
253	C₂³	#20609	1	1	yes	characteristic
254	C₂²	#133839	1	4	yes	—
255	C₂²	C₂×#8534	1	1	yes	characteristic
256	C₂²	#187928	1	1	yes	characteristic
257	C₂²	#187928	1	1	yes	characteristic
258	C₂²	#187928	1	1	yes	characteristic
259	C₂²	#187928	1	1	yes	characteristic
260	C₂²	#187944	1	1	yes	characteristic
261	C₂²	C₂×#8541	1	1	yes	characteristic
262	C₂	C₂×#133839	1	2	yes	—
263	C₂	#72235054	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#8541
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72224136	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72309515	C₂	semidirect
2. C₂×#115348	C₂	semidirect
4. #72219300	C₂	semidirect
5. #72219243	C₂	semidirect
6. #72219237	C₂	semidirect
7. #72218069	C₂	semidirect
8. #72218057	C₂	semidirect
9. #10586404	C₄	semidirect
13. #6499892	C₄	semidirect
15. #6374210	C₄	semidirect
16. #6374210	C₄	semidirect
17. #6374128	C₄	semidirect
18. C₂²×#1781	C₂²	semidirect
19. C₂²×#1438	C₂²	semidirect
20. #392113	C₂²	semidirect
21. #392109	C₂²	semidirect
22. #147569	C₂²	semidirect
23. #147567	C₂²	semidirect
24. #147567	C₂²	semidirect
25. #115348	C₂²	semidirect
26. #115348	C₂²	semidirect
27. #113037	C₂²	semidirect
29. #62862	C₂×C₄	semidirect
60. #21750	C₂×C₄	semidirect
61. #21749	C₂×C₄	semidirect
62. #21749	C₂×C₄	semidirect
63. C₂×#1783	C₂³	semidirect
64. C₂×#1781	C₂³	semidirect
65. C₂×#1781	C₂³	semidirect
66. C₂×#1438	C₂³	semidirect
130. C₂⁶	#99	semidirect
145. C₂³×D₄	#99	semidirect
146. C₂³×D₄	#99	semidirect
165. C₂⁴×C₄	#99	semidirect
166. C₂⁴×C₄	#99	semidirect
173. C₂×C₂²⋊Q₈	#99	semidirect
174. C₂×#121	#99	semidirect
178. C₂²×C₂²⋊C₄	#99	semidirect
179. C₂²×C₂²⋊C₄	#99	semidirect
198. C₂⁵	C₂×#99	semidirect
208. C₂²×Q₈	C₂×#99	semidirect
209. C₂²×D₄	C₂×#99	semidirect
215. C₂³×C₄	C₂×#99	semidirect
216. C₂³×C₄	C₂×#99	semidirect
219. C₂×C₄⋊C₄	C₂×#99	semidirect

Maximal subgroups

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

subgroup	index
#72309515	2
C₂×#115348	2
#72224136	2
#72219300	2
#72219243	2
#72219237	2
#72218069	2
#72218057	2