Subgroups — #72337587

Order 4096.

At least 12110 conjugacy classes of subgroups; only the 401 normal subgroups are shown.

The subgroup lattice has 272 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72337581	C₂	1	2	yes	maximal, complemented
2	#72337580	C₂	1	2	yes	maximal, complemented
3	#72337577	C₂	1	2	yes	maximal, complemented
4	#72337575	C₂	1	2	yes	maximal, complemented
5	#72327288	C₂	1	1	yes	maximal, characteristic
6	#72327248	C₂²	1	1	yes	characteristic
7	#72327244	C₄	1	1	yes	characteristic, complemented
8	#72327231	C₂²	1	1	yes	characteristic
9	#72326786	C₄	1	1	yes	characteristic, complemented
10	#72318215	C₂²	1	1	yes	characteristic
11	#72309335	C₄	1	1	yes	characteristic, complemented
12	#72302047	C₂	1	1	yes	maximal, characteristic, complemented
13	#72283093	C₄	1	1	yes	characteristic, complemented
14	#72265913	C₂²	1	2	yes	complemented
15	#72265913	C₂²	1	2	yes	complemented
16	#72237771	C₂	1	1	yes	maximal, characteristic, complemented
17	#72237770	C₂	1	2	yes	maximal, complemented
18	#72237734	C₂²	1	4	yes	complemented
19	#72237733	C₂²	1	2	yes	complemented
20	#72237726	C₂²	1	2	yes	complemented
21	#72237725	C₂²	1	4	yes	complemented
22	#72226352	C₂	1	2	yes	maximal, complemented
23	#72226351	C₂²	1	4	yes	complemented
24	#72226351	C₂²	1	4	yes	complemented
25	#72225873	C₂²	1	2	yes	complemented
26	#72225776	C₂²	1	2	yes	complemented
27	#72225733	C₂²	1	1	yes	characteristic
28	#72225660	C₂²	1	1	yes	characteristic
29	#72225655	C₂²	1	1	yes	characteristic
30	#72218920	C₄	1	1	yes	characteristic, complemented
31	#72217730	C₄	1	1	yes	characteristic, complemented
32	C₂×#449475	C₂²	1	1	yes	characteristic
33	#72216646	C₄	1	1	yes	characteristic, complemented
34	#72210024	C₄	1	1	yes	characteristic, complemented
35	#10586066	D₄	1	1	yes	characteristic
36	#8093108	C₂×C₄	1	1	yes	characteristic, complemented
37	#8093107	C₂×C₄	1	1	yes	characteristic, complemented
38	#7670230	C₂×C₄	1	1	yes	characteristic, complemented
39	#7670203	C₂×C₄	1	1	yes	characteristic, complemented
40	#7642273	C₂×C₄	1	1	yes	characteristic
41	#7642244	C₂×C₄	1	1	yes	characteristic
42	#7642244	C₂×C₄	1	1	yes	characteristic
43	C₂²×#2846	C₂×C₄	1	1	yes	characteristic
44	C₂²×#2846	D₄	1	2	yes	—
45	C₂×#35346	C₂³	1	1	yes	characteristic
46	#6629262	C₂×C₄	1	1	yes	characteristic
47	#6629262	D₄	1	1	yes	characteristic
48	#6628896	C₂×C₄	1	1	yes	characteristic
49	#6628896	D₄	1	1	yes	characteristic
50	#6451874	D₄	1	2	yes	—
51	#6432953	D₄	1	1	yes	characteristic
52	#6432953	D₄	1	1	yes	characteristic
53	#6432953	C₂³	1	1	yes	characteristic
54	#6432953	C₂³	1	1	yes	characteristic
55	#6374210	D₄	1	1	yes	characteristic
56	#6374210	D₄	1	1	yes	characteristic
57	#6370693	D₄	1	1	yes	characteristic
58	#449521	C₂×C₄	1	1	yes	characteristic, complemented
59	#449521	C₂×C₄	1	1	yes	characteristic, complemented
60	#449475	C₂×C₄	1	1	yes	characteristic
61	#449475	C₂×C₄	1	1	yes	characteristic
62	#449475	C₂×C₄	1	1	yes	characteristic, complemented
63	#449475	C₂×C₄	1	1	yes	characteristic, complemented
64	#449475	D₄	1	2	yes	—
65	#449475	D₄	1	2	yes	—
66	#449475	C₂³	1	1	yes	characteristic
67	#449475	C₂³	1	1	yes	characteristic
68	#418213	C₂×C₄	1	1	yes	characteristic, complemented
69	#418132	C₂×C₄	1	1	yes	characteristic, complemented
70	#417896	C₂×C₄	1	1	yes	characteristic
71	#417896	C₂×C₄	1	1	yes	characteristic, complemented
72	#417896	D₄	1	2	yes	—
73	#417896	C₂³	1	1	yes	characteristic
74	#417824	C₂×C₄	1	1	yes	characteristic
75	#417824	C₂×C₄	1	1	yes	characteristic, complemented
76	#417824	D₄	1	2	yes	—
77	#417824	C₂³	1	1	yes	characteristic
78	#391735	C₂×C₄	1	1	yes	characteristic, complemented
79	#391709	C₂×C₄	1	1	yes	characteristic, complemented
80	#382422	C₂×C₄	1	1	yes	characteristic
81	#375666	C₂×C₄	1	1	yes	characteristic, complemented
82	#374004	C₂×C₄	1	1	yes	characteristic, complemented
83	#359705	D₄	1	2	yes	—
84	C₂×#10965	C₂×C₄	1	1	yes	characteristic
85	C₂×#10965	D₄	1	2	yes	—
86	#213829	C₂³	1	4	yes	—
87	#134045	C₂³	1	4	yes	—
88	C₂⁸	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×H(7,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	#62842	C₂×D₄	1	1	yes	characteristic
92	#62842	C₂×D₄	1	1	yes	characteristic
93	C₂×#3490	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×#3484	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×#3484	C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
97	C₂³×#120	C₂×D₄	1	2	yes	—
98	#62652	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×#2846	C₂²⋊C₄	1	2	yes	—
100	C₂×#2846	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#2846	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#2846	C₂²⋊C₄	1	2	yes	—
103	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
104	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
105	C₂×#2846	C₂×D₄	1	2	yes	—
106	C₂×#2846	C₂×D₄	1	2	yes	—
107	#35390	C₂²⋊C₄	1	1	yes	characteristic
108	#35390	C₂²×C₄	1	1	yes	characteristic
109	#35367	C₂²⋊C₄	1	1	yes	characteristic
110	#35367	C₂²×C₄	1	1	yes	characteristic
111	#35350	C₂²⋊C₄	1	1	yes	characteristic
112	#35350	C₂²⋊C₄	1	1	yes	characteristic
113	#35350	C₂²×C₄	1	1	yes	characteristic
114	#35350	C₂²×C₄	1	1	yes	characteristic
115	#35346	C₂²⋊C₄	1	1	yes	characteristic
116	#35346	C₂²⋊C₄	1	1	yes	characteristic
117	#35346	C₂²×C₄	1	1	yes	characteristic
118	#35346	C₂²×C₄	1	1	yes	characteristic
119	#35346	C₂²×C₄	1	1	yes	characteristic
120	#35346	C₂×D₄	1	1	yes	characteristic
121	#35346	C₂×D₄	1	2	yes	—
122	#35346	C₂×D₄	1	2	yes	—
123	#35346	C₂×D₄	1	1	yes	characteristic
124	#35346	C₂⁴	1	1	yes	characteristic
125	#34176	C₂²⋊C₄	1	1	yes	characteristic
126	#34171	C₂²⋊C₄	1	1	yes	characteristic
127	#34171	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×#2308	C₂×D₄	1	1	yes	characteristic
129	C₂×#2308	C₂×D₄	1	2	yes	—
130	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×#379	C₂²⋊C₄	1	2	yes	—
132	#23468	C₂²⋊C₄	1	2	yes	—
133	#23468	C₂×D₄	1	2	yes	—
134	#23444	C₂²⋊C₄	1	2	yes	—
135	#23444	C₂×D₄	1	2	yes	—
136	#20689	C₂²⋊C₄	1	1	yes	characteristic
137	#20689	C₂²⋊C₄	1	1	yes	characteristic
138	#20689	C₂×D₄	1	1	yes	characteristic
139	#20689	C₂×D₄	1	1	yes	characteristic
140	C₂×#1819	C₂²⋊C₄	1	2	yes	—
141	#20585	C₂²⋊C₄	1	1	yes	characteristic
142	#20585	C₂²⋊C₄	1	1	yes	characteristic
143	#20585	C₂²⋊C₄	1	1	yes	characteristic
144	#20585	C₂²⋊C₄	1	1	yes	characteristic
145	#20585	C₂×D₄	1	1	yes	characteristic
146	#20585	C₂×D₄	1	1	yes	characteristic
147	#20585	C₂×D₄	1	1	yes	characteristic
148	#20585	C₂×D₄	1	1	yes	characteristic
149	#10965	C₂²⋊C₄	1	2	yes	—
150	#10965	C₂²⋊C₄	1	2	yes	—
151	#10965	C₂²⋊C₄	1	2	yes	—
152	#10965	C₂²⋊C₄	1	2	yes	—
153	#10965	C₂²×C₄	1	1	yes	characteristic
154	#10965	C₂²×C₄	1	1	yes	characteristic
155	#10965	C₂²×C₄	1	1	yes	characteristic
156	#10965	C₂²×C₄	1	1	yes	characteristic
157	#10965	C₂×D₄	1	2	yes	—
158	#10965	C₂×D₄	1	2	yes	—
159	#10938	C₂²⋊C₄	1	2	yes	—
160	#10938	C₂²⋊C₄	1	2	yes	—
161	#10938	C₂×D₄	1	2	yes	—
162	#10938	C₂×D₄	1	2	yes	—
163	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
170	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
171	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂⁴×D₄	#120	1	2	yes	—
173	C₂×#543	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×#535	#99	1	2	yes	—
175	C₂×#534	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×#525	#99	1	2	yes	—
177	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×#120	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×#120	#120	1	2	yes	—
182	C₂²×#120	#120	1	2	yes	—
183	C₂²×#120	C₂³×C₄	1	1	yes	characteristic
184	C₂²×#120	C₂²×D₄	1	2	yes	—
185	C₂×#518	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×#518	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂³×C₂²⋊C₄	#120	1	2	yes	—
189	#2309	#99	1	2	yes	—
190	#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	#2308	C₂×C₂²⋊C₄	1	2	yes	—
195	#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	#2308	#120	1	2	yes	—
197	#2308	#120	1	2	yes	—
198	#2308	#120	1	2	yes	—
199	#2308	C₂²×D₄	1	2	yes	—
200	#2308	C₂²×D₄	1	1	yes	characteristic
201	C₂×#379	#99	1	2	yes	—
202	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×#379	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂×#379	C₂×C₂²⋊C₄	1	2	yes	—
206	#2273	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	#2273	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂⁶	#379	1	1	yes	characteristic
211	C₂⁶	#379	1	1	yes	characteristic
212	C₂⁶	C₂×#99	1	1	yes	characteristic
213	C₂⁶	B(4,𝔽₂)	1	2	yes	—
214	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×#41	C₂×#99	1	2	yes	—
216	C₂³×D₄	#379	1	1	yes	characteristic
217	C₂³×D₄	#379	1	2	yes	—
218	C₂³×D₄	C₂×#99	1	1	yes	characteristic
219	C₂³×D₄	C₂×#99	1	1	yes	characteristic
220	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
221	C₂³×D₄	#458	1	2	yes	—
222	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×D₄	C₂×#120	1	2	yes	—
224	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
225	C₂⁴×C₄	#458	1	2	yes	—
226	C₂×#127	AΣL(1,𝔽₁₆)	1	4	yes	complemented
227	C₂×#124	AΣL(1,𝔽₁₆)	1	4	yes	complemented
228	C₂×#117	C₂×#99	1	2	yes	—
229	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
230	C₂²×C₂²⋊C₄	#379	1	2	yes	—
231	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
233	#379	C₂×#99	1	2	yes	—
234	#379	C₂×#99	1	2	yes	—
235	C₂⁵	#1781	1	1	yes	characteristic
236	C₂⁵	#1781	1	1	yes	characteristic
237	C₂⁵	#1781	1	1	yes	characteristic
238	C₂⁵	#1790	1	2	yes	—
239	C₂⁵	#1790	1	2	yes	—
240	C₂⁵	#1790	1	2	yes	—
241	C₂⁵	C₂×#379	1	1	yes	characteristic
242	C₂⁵	C₂²×#99	1	1	yes	characteristic
243	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
244	C₂²×Q₈	C₂×AΣL(1,𝔽₁₆)	1	2	yes	complemented
245	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
246	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	2	yes	complemented
247	C₂³×C₄	#1781	1	1	yes	characteristic
248	C₂³×C₄	#1790	1	2	yes	—
249	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
250	C₂³×C₄	C₂×#458	1	2	yes	—
251	C₂×C₄²	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
252	C₂⁴	#8534	1	1	yes	characteristic
253	C₂⁴	#8534	1	1	yes	characteristic
254	C₂⁴	#8534	1	2	yes	—
255	C₂⁴	#12687	1	2	yes	—
256	C₂⁴	#12687	1	2	yes	—
257	C₂⁴	#12705	1	2	yes	—
258	C₂⁴	#12753	1	4	yes	—
259	C₂⁴	C₂×#1781	1	1	yes	characteristic
260	C₂⁴	C₂×#1790	1	2	yes	—
261	C₂²×C₄	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
262	C₂³	#133278	1	4	yes	—
263	C₂³	C₂×#8534	1	1	yes	characteristic
264	C₂³	C₂×#12687	1	1	yes	characteristic
265	C₂³	C₂×#12705	1	1	yes	characteristic
266	C₂³	C₂×#12753	1	2	yes	—
267	C₂³	#493594	1	2	yes	—
268	C₂²	C₂×#133278	1	1	yes	characteristic
269	C₂²	#72258234	1	2	yes	—
270	C₂	#72259010	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72259010
commutator	C₂²×#120	C₂³×C₄
Frattini, cosocle	#35346	C₂⁴
prime radical	#72327288	C₂
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#35346 ◁ G
LECS	G ▷ #35346 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72337581	C₂	semidirect
2. #72337580	C₂	semidirect
3. #72337577	C₂	semidirect
4. #72337575	C₂	semidirect
7. #72327244	C₄	semidirect
9. #72326786	C₄	semidirect
11. #72309335	C₄	semidirect
12. #72302047	C₂	semidirect
13. #72283093	C₄	semidirect
14. #72265913	C₂²	semidirect
15. #72265913	C₂²	semidirect
16. #72237771	C₂	semidirect
17. #72237770	C₂	semidirect
18. #72237734	C₂²	semidirect
19. #72237733	C₂²	semidirect
20. #72237726	C₂²	semidirect
21. #72237725	C₂²	semidirect
22. #72226352	C₂	semidirect
23. #72226351	C₂²	semidirect
24. #72226351	C₂²	semidirect
25. #72225873	C₂²	semidirect
26. #72225776	C₂²	semidirect
30. #72218920	C₄	semidirect
31. #72217730	C₄	semidirect
33. #72216646	C₄	semidirect
34. #72210024	C₄	semidirect
36. #8093108	C₂×C₄	semidirect
37. #8093107	C₂×C₄	semidirect
38. #7670230	C₂×C₄	semidirect
39. #7670203	C₂×C₄	semidirect
58. #449521	C₂×C₄	semidirect
59. #449521	C₂×C₄	semidirect
62. #449475	C₂×C₄	semidirect
63. #449475	C₂×C₄	semidirect
68. #418213	C₂×C₄	semidirect
69. #418132	C₂×C₄	semidirect
71. #417896	C₂×C₄	semidirect
75. #417824	C₂×C₄	semidirect
78. #391735	C₂×C₄	semidirect
79. #391709	C₂×C₄	semidirect
81. #375666	C₂×C₄	semidirect
82. #374004	C₂×C₄	semidirect
226. C₂×#127	AΣL(1,𝔽₁₆)	semidirect
227. C₂×#124	AΣL(1,𝔽₁₆)	semidirect
244. C₂²×Q₈	C₂×AΣL(1,𝔽₁₆)	semidirect
246. C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	semidirect
251. C₂×C₄²	C₂×AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#72337581	2
#72337580	2
#72337577	2
#72337575	2
#72327288	2
#72302047	2
#72237771	2
#72237770	2
#72226352	2