Subgroups — #72310462

Order 4096.

At least 15059 conjugacy classes of subgroups; only the 379 normal subgroups are shown.

The subgroup lattice has 235 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72259029	C₂	1	1	yes	maximal, characteristic
2	#72259008	C₄	1	1	yes	characteristic, complemented
3	#72259002	C₄	1	2	yes	complemented
4	#72237038	C₂	1	2	yes	maximal, complemented
5	#72235847	C₄	1	1	yes	characteristic, complemented
6	#72234688	C₂	1	1	yes	maximal, characteristic
7	#72234620	C₄	1	2	yes	complemented
8	#72225776	C₄	1	2	yes	complemented
9	#72225629	C₂	1	1	yes	maximal, characteristic
10	C₂×#111462	C₄	1	2	yes	complemented
11	#72219298	C₂	1	2	yes	maximal, complemented
12	#72219227	C₂²	1	1	yes	characteristic
13	C₂×#106076	C₄	1	2	yes	complemented
14	C₂×#7670203	C₂²	1	1	yes	characteristic
15	#72218518	C₂²	1	2	yes	—
16	#72218375	C₂²	1	1	yes	characteristic
17	#72217789	C₂²	1	2	yes	—
18	C₂²×#3484	D₄	1	1	yes	characteristic
19	C₂²×#3484	D₄	1	1	yes	characteristic
20	C₂⁴×#120	D₄	1	1	yes	characteristic
21	C₂⁴×#120	C₂³	1	1	yes	characteristic
22	C₂²×#2846	D₄	1	1	yes	characteristic
23	C₂×#35390	Q₈	1	1	yes	characteristic
24	C₂×#35348	C₂×C₄	1	1	yes	characteristic
25	C₂×#35348	Q₈	1	1	yes	characteristic
26	C₂×#35346	C₂×C₄	1	1	yes	characteristic
27	C₂×#34981	C₂×C₄	1	1	yes	characteristic
28	C₂×#34981	C₂×C₄	1	1	yes	characteristic
29	C₂×#34980	C₂×C₄	1	1	yes	characteristic
30	C₂×#34980	D₄	1	1	yes	characteristic
31	C₂×#34198	C₂×C₄	1	1	yes	characteristic
32	C₂×#34198	D₄	1	1	yes	characteristic
33	C₂×#21746	C₂×C₄	1	2	yes	complemented
34	C₂×#21746	C₂×C₄	1	2	yes	complemented
35	#392109	D₄	1	4	yes	—
36	C₂×#10725	C₂×C₄	1	2	yes	complemented
37	C₂×#8541	C₂×C₄	1	4	yes	—
38	C₂×#8541	D₄	1	4	yes	—
39	C₂×#7555	C₂×C₄	1	2	yes	complemented
40	C₂⁸	C₂×D₄	1	1	yes	characteristic
41	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
42	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
43	C₂⁵×D₄	C₂×D₄	1	2	yes	—
44	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
45	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×#534	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×#534	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×#534	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×#534	C₄⋊C₄	1	1	yes	characteristic
50	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
51	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
52	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
55	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
56	C₂³×#120	C₄⋊C₄	1	1	yes	characteristic
57	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
58	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
59	C₂³×#120	C₂×D₄	1	1	yes	characteristic
60	C₂³×#120	C₂×D₄	1	2	yes	—
61	C₂³×#120	C₂×Q₈	1	1	yes	characteristic
62	C₂³×#120	#41	1	2	yes	—
63	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×#518	C₄⋊C₄	1	1	yes	characteristic
65	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂⁴×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂⁴×C₂²⋊C₄	#41	1	2	yes	—
68	C₂×#2315	C₄²	1	1	yes	characteristic
69	C₂×#2315	C₄⋊C₄	1	1	yes	characteristic
70	C₂×#2308	C₄²	1	1	yes	characteristic
71	C₂×#2308	C₄²	1	1	yes	characteristic
72	C₂×#2308	C₄²	1	1	yes	characteristic
73	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×#2308	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×#379	C₄⋊C₄	1	1	yes	characteristic
78	C₂×#2273	C₄⋊C₄	1	1	yes	characteristic
79	C₂×#1781	C₂²⋊C₄	1	8	yes	complemented
80	C₂⁷	#99	1	1	yes	characteristic
81	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂⁷	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂⁷	#120	1	1	yes	characteristic
85	C₂⁷	#121	1	2	yes	—
86	C₂⁷	#121	1	2	yes	—
87	C₂²×H(5,𝔽₂)	#99	1	2	yes	—
88	C₂²×H(5,𝔽₂)	#99	1	2	yes	—
89	C₂⁴×D₄	#95	1	1	yes	characteristic
90	C₂⁴×D₄	#95	1	1	yes	characteristic
91	C₂⁴×D₄	#95	1	1	yes	characteristic
92	C₂⁴×D₄	#99	1	1	yes	characteristic
93	C₂⁴×D₄	#99	1	1	yes	characteristic
94	C₂⁴×D₄	#99	1	2	yes	—
95	C₂⁴×D₄	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂⁴×D₄	C₄×D₄	1	2	yes	—
99	C₂⁴×D₄	#120	1	1	yes	characteristic
100	C₂⁴×D₄	#120	1	2	yes	—
101	C₂⁴×D₄	C₂²⋊Q₈	1	2	yes	—
102	C₂⁴×D₄	#123	1	1	yes	characteristic
103	C₂⁵×C₄	#99	1	1	yes	characteristic
104	C₂⁵×C₄	C₄×D₄	1	2	yes	—
105	C₂⁵×C₄	#123	1	1	yes	characteristic
106	C₂²×#120	#99	1	2	yes	—
107	C₂²×#120	#99	1	2	yes	—
108	C₂²×#120	#99	1	4	yes	—
109	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
110	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
111	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
112	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
113	C₂³×C₂²⋊C₄	#95	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	#99	1	2	yes	—
115	C₂³×C₂²⋊C₄	C₂×C₄²	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	#117	1	2	yes	—
118	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
119	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
120	C₂³×C₂²⋊C₄	C₂²⋊Q₈	1	2	yes	—
121	C₂³×C₂²⋊C₄	#123	1	2	yes	—
122	#2308	#99	1	4	yes	—
123	C₂⁶	#342	1	1	yes	characteristic
124	C₂⁶	#342	1	1	yes	characteristic
125	C₂⁶	#342	1	1	yes	characteristic
126	C₂⁶	C₂×#95	1	1	yes	characteristic
127	C₂⁶	#379	1	1	yes	characteristic
128	C₂⁶	#380	1	2	yes	—
129	C₂⁶	#385	1	1	yes	characteristic
130	C₂⁶	#386	1	1	yes	characteristic
131	C₂⁶	#386	1	1	yes	characteristic
132	C₂⁶	#386	1	2	yes	—
133	C₂⁶	C₂×#99	1	1	yes	characteristic
134	C₂⁶	C₂×#99	1	2	yes	—
135	C₂⁶	C₂×#99	1	2	yes	—
136	C₂⁶	C₂×#99	1	1	yes	characteristic
137	C₂⁶	C₂×#99	1	2	yes	—
138	C₂³×D₄	#342	1	1	yes	characteristic
139	C₂³×D₄	#342	1	1	yes	characteristic
140	C₂³×D₄	#342	1	1	yes	characteristic
141	C₂³×D₄	#342	1	1	yes	characteristic
142	C₂³×D₄	#342	1	2	yes	—
143	C₂³×D₄	#342	1	2	yes	—
144	C₂³×D₄	C₂×#99	1	2	yes	—
145	C₂³×D₄	C₂×#99	1	2	yes	—
146	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
147	C₂³×D₄	#458	1	4	yes	—
148	C₂⁴×C₄	#342	1	1	yes	characteristic
149	C₂⁴×C₄	#342	1	1	yes	characteristic
150	C₂⁴×C₄	#342	1	2	yes	—
151	C₂⁴×C₄	#342	1	1	yes	characteristic
152	C₂⁴×C₄	C₄×C₂²⋊C₄	1	2	yes	—
153	C₂⁴×C₄	#381	1	1	yes	characteristic
154	C₂⁴×C₄	#385	1	1	yes	characteristic
155	C₂⁴×C₄	#385	1	2	yes	—
156	C₂×#120	#353	1	4	yes	—
157	C₂×#120	#354	1	4	yes	—
158	C₂²×C₂²⋊C₄	#342	1	2	yes	—
159	C₂²×C₂²⋊C₄	#342	1	2	yes	—
160	C₂²×C₂²⋊C₄	#342	1	2	yes	—
161	C₂²×C₂²⋊C₄	#342	1	2	yes	—
162	C₂⁵	#1304	1	1	yes	characteristic
163	C₂⁵	#1304	1	1	yes	characteristic
164	C₂⁵	#1304	1	2	yes	—
165	C₂⁵	C₂²⋊#95	1	1	yes	characteristic
166	C₂⁵	C₂×#342	1	1	yes	characteristic
167	C₂⁵	C₂×#342	1	1	yes	characteristic
168	C₂⁵	C₂×#342	1	1	yes	characteristic
169	C₂⁵	C₂×#342	1	1	yes	characteristic
170	C₂⁵	C₂×#342	1	1	yes	characteristic
171	C₂⁵	C₂×#342	1	1	yes	characteristic
172	C₂⁵	#1781	1	2	yes	—
173	C₂⁵	#1790	1	4	yes	—
174	C₂⁵	#1889	1	2	yes	—
175	C₂⁵	#1890	1	2	yes	—
176	C₂⁵	#1898	1	2	yes	—
177	C₂⁵	#1898	1	2	yes	—
178	C₂⁵	#1898	1	2	yes	—
179	C₂⁵	#1898	1	2	yes	—
180	C₂⁵	#1913	1	2	yes	—
181	C₂⁵	#1913	1	2	yes	—
182	C₂⁵	C₂×#353	1	2	yes	—
183	C₂⁵	C₂×#354	1	2	yes	—
184	C₂⁵	#2127	1	2	yes	—
185	C₂⁵	#2127	1	2	yes	—
186	C₂²×D₄	#1393	1	4	yes	—
187	C₂³×C₄	#1304	1	1	yes	characteristic
188	C₂³×C₄	#1304	1	1	yes	characteristic
189	C₂³×C₄	#1304	1	2	yes	—
190	C₂³×C₄	#1819	1	2	yes	—
191	C₂³×C₄	#1867	1	4	yes	—
192	C₂×C₂²⋊C₄	#1393	1	4	yes	—
193	C₂⁴	#7307	1	2	yes	complemented
194	C₂⁴	C₂×#1304	1	1	yes	characteristic
195	C₂⁴	C₂×#1304	1	1	yes	characteristic
196	C₂⁴	#8101	1	1	yes	characteristic
197	C₂⁴	#8101	1	1	yes	characteristic
198	C₂⁴	#8534	1	4	yes	—
199	C₂⁴	#9495	1	2	yes	—
200	C₂⁴	#9495	1	2	yes	—
201	C₂⁴	#9499	1	2	yes	—
202	C₂⁴	#9499	1	2	yes	—
203	C₂⁴	#10725	1	1	yes	characteristic
204	C₂⁴	#10735	1	1	yes	characteristic
205	C₂⁴	#10735	1	1	yes	characteristic
206	C₂⁴	#10745	1	2	yes	—
207	C₂⁴	#10754	1	2	yes	—
208	C₂⁴	C₂×#1393	1	2	yes	—
209	C₂⁴	#12582	1	2	yes	—
210	C₂⁴	#13088	1	2	yes	—
211	C₂⁴	#13088	1	2	yes	—
212	C₂⁴	#13090	1	2	yes	—
213	C₂⁴	#13090	1	2	yes	—
214	C₂²×C₄	#7307	1	2	yes	complemented
215	C₂³	C₂²⋊#1304	1	1	yes	characteristic
216	C₂³	#106542	1	2	yes	—
217	C₂³	#106766	1	2	yes	—
218	C₂³	#107114	1	1	yes	characteristic
219	C₂³	#108222	1	1	yes	characteristic
220	C₂³	C₂×#7307	1	1	yes	characteristic
221	C₂³	#128009	1	1	yes	characteristic
222	C₂³	#128058	1	2	yes	—
223	C₂³	#139743	1	4	yes	—
224	C₂³	#146569	1	2	yes	—
225	C₂³	#146581	1	2	yes	—
226	C₂²	#72218854	1	1	yes	characteristic
227	C₂²	#72219296	1	1	yes	characteristic
228	C₂²	#72246287	1	2	yes	—
229	C₂²	#72255793	1	2	yes	—
230	C₂²	#72258221	1	4	yes	—
231	C₂²	#72258276	1	1	yes	characteristic
232	C₂	#72258227	1	2	yes	—
233	C₂	#72258414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72218854
commutator	C₂³×C₂²⋊C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×#120	C₂³
prime radical	C₂×#7670203	C₂²
LCS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂⁴×#120 ▷ C₂⁵ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72259008	C₄	semidirect
3. #72259002	C₄	semidirect
4. #72237038	C₂	semidirect
5. #72235847	C₄	semidirect
7. #72234620	C₄	semidirect
8. #72225776	C₄	semidirect
10. C₂×#111462	C₄	semidirect
11. #72219298	C₂	semidirect
13. C₂×#106076	C₄	semidirect
33. C₂×#21746	C₂×C₄	semidirect
34. C₂×#21746	C₂×C₄	semidirect
36. C₂×#10725	C₂×C₄	semidirect
39. C₂×#7555	C₂×C₄	semidirect
79. C₂×#1781	C₂²⋊C₄	semidirect
193. C₂⁴	#7307	semidirect
214. C₂²×C₄	#7307	semidirect

Maximal subgroups

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

subgroup	index
#72259029	2
#72237038	2
#72234688	2
#72225629	2
#72219298	2