Subgroups — #72323156

Order 2048.

272 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72286505	C₂	1	1	yes	maximal, characteristic
2	C₂×#94509	C₂	1	1	yes	maximal, characteristic
3	#72231924	C₂	1	1	yes	maximal, characteristic
4	C₂×#7511	—	2	1	no	—
5	#151996	C₄	1	1	yes	characteristic
6	C₂×#7464	C₄	1	1	yes	characteristic
7	C₂×#7330	C₂²	1	1	yes	characteristic
8	#129143	—	2	1	no	—
9	#122919	—	2	1	no	—
10	#94509	—	2	1	no	—
11	#94509	C₄	1	2	yes	complemented
12	#94055	—	2	1	no	—
13	#93877	—	2	1	no	—
14	#93877	C₄	1	2	yes	complemented
15	C₂×M₁₂₈	—	2	2	no	—
16	C₂×#1425	C₈	1	1	yes	characteristic
17	#13633	C₈	1	1	yes	characteristic
18	C₂×#1397	C₂×C₄	1	1	yes	characteristic
19	C₂×#1397	D₄	1	1	yes	characteristic
20	#12670	—	2	1	no	—
21	#12664	—	4	1	no	—
22	#12012	—	4	1	no	—
23	C₂×#1373	—	2	1	no	—
24	C₂×#1311	D₄	1	1	yes	characteristic
25	#7511	—	2	2	no	—
26	#7511	—	2	2	no	—
27	#7473	—	2	1	no	—
28	#7464	—	2	1	no	—
29	#7464	C₈	1	2	yes	—
30	#7330	C₂×C₄	1	1	yes	characteristic
31	#7330	C₂×C₄	1	1	yes	characteristic
32	#7330	D₄	1	1	yes	characteristic
33	#7330	Q₈	1	1	yes	characteristic
34	#7179	—	4	1	no	—
35	#7099	—	2	2	no	—
36	#7099	—	4	1	no	—
37	C₂²×C₃₂	—	2	1	no	—
38	C₂²×C₃₂	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₃₂	M₁₆	1	1	yes	characteristic
40	#2162	—	8	1	no	—
41	C₂×#368	—	2	1	no	—
42	#2158	—	2	1	no	—
43	C₂×#364	—	4	1	no	—
44	#2108	—	8	1	no	—
45	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆	SD₁₆	1	1	yes	characteristic
47	#1772	—	4	1	no	—
48	C₂×C₈²	D₈	1	1	yes	characteristic
49	M₁₂₈	—	2	4	no	—
50	M₁₂₈	—	4	2	no	—
51	C₂×C₆₄	—	4	2	no	—
52	#1425	—	2	1	no	—
53	#1425	C₁₆	1	2	yes	complemented
54	#1424	C₁₆	1	2	yes	complemented
55	#1423	—	2	1	no	—
56	#1397	—	2	1	no	—
57	#1397	C₄²	1	1	yes	characteristic
58	#1397	C₂²⋊C₄	1	1	yes	characteristic
59	#1397	C₄⋊C₄	1	1	yes	characteristic
60	#1397	C₄⋊C₄	1	1	yes	characteristic
61	#1397	C₂×C₈	1	1	yes	characteristic
62	#1397	M₁₆	1	1	yes	characteristic
63	#1382	—	4	1	no	—
64	#1379	—	8	1	no	—
65	#1373	—	2	2	no	—
66	#1373	—	4	1	no	—
67	#1370	—	8	1	no	—
68	#1367	—	4	2	no	—
69	#1311	C₂²⋊C₄	1	1	yes	characteristic
70	#1311	C₄⋊C₄	1	1	yes	characteristic
71	#1311	SD₁₆	1	1	yes	characteristic
72	#1311	Q₁₆	1	1	yes	characteristic
73	C₂×M₃₂	—	8	1	no	—
74	C₂²×C₁₆	—	2	1	no	—
75	C₂²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
76	C₂²×C₁₆	#104	1	1	yes	characteristic
77	C₂×#108	—	4	1	no	—
78	#427	—	4	1	no	—
79	#405	—	8	1	no	—
80	C₂×C₄×C₈	—	2	1	no	—
81	C₂×C₄×C₈	#102	1	1	yes	characteristic
82	C₂×C₃₂	—	2	2	no	—
83	C₂×C₃₂	—	2	2	no	—
84	C₂×C₃₂	—	2	1	no	—
85	C₂×C₃₂	—	2	1	no	—
86	C₂×C₃₂	—	2	1	no	—
87	C₂×C₃₂	—	2	1	no	—
88	C₂×C₃₂	C₄×C₈	1	1	yes	characteristic
89	C₂×C₃₂	#97	1	1	yes	characteristic
90	C₂×C₃₂	C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₃₂	C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₃₂	C₂×C₁₆	1	1	yes	characteristic
93	C₂×C₃₂	C₂×C₁₆	1	1	yes	characteristic
94	C₂×C₃₂	M₃₂	1	1	yes	characteristic
95	C₂×C₃₂	M₃₂	1	1	yes	characteristic
96	#368	—	2	2	no	—
97	#368	—	4	1	no	—
98	#367	—	4	1	no	—
99	#366	—	2	2	no	—
100	#364	—	4	2	no	—
101	#364	—	8	1	no	—
102	C₄⋊C₁₆	—	8	2	no	—
103	C₂²⋊C₁₆	—	16	1	no	—
104	#346	—	16	1	no	—
105	C₄×C₁₆	—	2	1	no	—
106	C₄×C₁₆	—	16	1	no	—
107	C₄×C₁₆	—	16	1	no	—
108	C₄×C₁₆	#95	1	1	yes	characteristic
109	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
110	C₄×C₁₆	#103	1	1	yes	characteristic
111	C₄×C₁₆	#103	1	1	yes	characteristic
112	C₄×C₁₆	#104	1	1	yes	characteristic
113	C₄×C₁₆	#106	1	1	yes	characteristic
114	#339	—	8	1	no	—
115	C₈²	—	2	1	no	—
116	C₈²	#102	1	1	yes	characteristic
117	C₈²	#107	1	1	yes	characteristic
118	C₆₄	—	4	4	no	—
119	C₂×M₁₆	—	8	1	no	—
120	C₂²×C₈	—	4	1	no	—
121	C₂²×C₈	D₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₈	#361	1	1	yes	characteristic
123	#117	—	8	1	no	—
124	C₂×C₄²	#360	1	1	yes	characteristic
125	M₃₂	—	8	2	no	—
126	M₃₂	—	16	1	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	4	1	no	—
132	C₂×C₁₆	—	4	1	no	—
133	C₂×C₁₆	—	16	1	no	—
134	C₂×C₁₆	—	16	2	no	—
135	C₂×C₁₆	Q₈⋊C₈	1	1	yes	characteristic
136	C₂×C₁₆	Q₈⋊C₈	1	1	yes	characteristic
137	C₂×C₁₆	#336	1	1	yes	characteristic
138	C₂×C₁₆	#339	1	1	yes	characteristic
139	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
140	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
141	C₂×C₁₆	#350	1	1	yes	characteristic
142	C₂×C₁₆	#350	1	1	yes	characteristic
143	#108	—	4	2	no	—
144	#108	—	8	1	no	—
145	#107	—	4	2	no	—
146	#106	—	8	1	no	—
147	#97	—	16	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	4	1	no	—
151	C₄×C₈	—	16	1	no	—
152	C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
153	C₄×C₈	#340	1	1	yes	characteristic
154	C₄×C₈	#360	1	1	yes	characteristic
155	C₄×C₈	#362	1	1	yes	characteristic
156	C₃₂	—	2	2	no	—
157	C₃₂	—	2	2	no	—
158	C₃₂	—	2	1	no	—
159	C₃₂	—	2	1	no	—
160	C₃₂	C₄×C₁₆	1	1	yes	characteristic
161	C₃₂	#346	1	1	yes	characteristic
162	C₃₂	C₄⋊C₁₆	1	1	yes	characteristic
163	C₃₂	C₄⋊C₁₆	1	1	yes	characteristic
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	#1333	1	1	yes	characteristic
166	M₁₆	—	8	2	no	—
167	M₁₆	—	16	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	16	1	no	—
173	C₂×C₈	—	16	1	no	—
174	C₂×C₈	#1279	1	1	yes	characteristic
175	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
176	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
177	C₂×C₈	#1333	1	1	yes	characteristic
178	C₂×C₈	#1334	1	1	yes	characteristic
179	C₂×C₈	#1336	1	1	yes	characteristic
180	C₂×C₈	#1336	1	1	yes	characteristic
181	C₂×C₈	#1387	1	1	yes	characteristic
182	C₄⋊C₄	—	8	2	no	—
183	C₂²⋊C₄	—	16	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	16	1	no	—
186	C₄²	#1334	1	1	yes	characteristic
187	C₄²	#1386	1	1	yes	characteristic
188	C₁₆	—	2	1	no	—
189	C₁₆	—	2	1	no	—
190	C₁₆	—	4	1	no	—
191	C₁₆	—	4	1	no	—
192	C₁₆	—	16	2	no	—
193	C₁₆	—	32	2	no	—
194	C₁₆	Q₈⋊C₁₆	1	1	yes	characteristic
195	C₁₆	Q₈⋊C₁₆	1	1	yes	characteristic
196	C₁₆	#1374	1	1	yes	characteristic
197	C₁₆	#1379	1	1	yes	characteristic
198	C₂³	#7389	1	1	yes	characteristic
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	—	16	2	no	—
203	C₂×C₄	#7070	1	1	yes	characteristic
204	C₂×C₄	#7070	1	1	yes	characteristic
205	C₂×C₄	#7178	1	1	yes	characteristic
206	C₂×C₄	#7389	1	1	yes	characteristic
207	C₂×C₄	#7390	1	1	yes	characteristic
208	C₂×C₄	#7391	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	8	1	no	—
212	C₈	—	8	1	no	—
213	C₈	—	16	2	no	—
214	C₈	#7071	1	1	yes	characteristic
215	C₈	#7071	1	1	yes	characteristic
216	C₈	#7167	1	1	yes	characteristic
217	C₈	#7182	1	1	yes	characteristic
218	C₂²	—	4	1	no	—
219	C₂²	#93757	1	1	yes	characteristic
220	C₂²	#93758	1	1	yes	characteristic
221	C₂²	#94057	1	1	yes	characteristic
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	32	2	no	—
225	C₄	#92942	1	1	yes	characteristic
226	C₄	#93757	1	1	yes	characteristic
227	C₄	#93758	1	1	yes	characteristic
228	C₄	#94059	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#72286530	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#93757
commutator	C₃₂	C₄×C₁₆
Frattini, cosocle	C₂×#7330	C₂²
socle	C₂	#72286530
prime radical	C₂³	#7389
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #94509	C₄	semidirect
14. #93877	C₄	semidirect
53. #1425	C₁₆	semidirect
54. #1424	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72286505	2
C₂×#94509	2
#72231924	2