Subgroups — #72225316

Order 1024.

294 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1398	C₂	1	1	yes	maximal, characteristic
2	#129101	C₂	1	1	yes	maximal, characteristic
3	#129088	C₂	1	1	yes	maximal, characteristic
4	C₂×#1398	—	2	1	no	—
5	C₂×#1398	C₄	1	2	yes	—
6	C₄×M₆₄	—	2	1	no	—
7	C₄×#364	—	2	1	no	—
8	C₄×#346	C₂²	1	1	yes	characteristic
9	C₂×#1373	C₄	1	1	yes	characteristic
10	C₂×#1373	C₄	1	1	yes	characteristic
11	#11999	C₄	1	2	yes	—
12	#8266	—	2	1	no	—
13	#7160	—	2	1	no	—
14	#7157	—	2	1	no	—
15	C₂×M₆₄	—	2	2	no	—
16	C₂×#364	—	2	1	no	—
17	C₂×#364	—	2	1	no	—
18	C₄×M₃₂	—	4	1	no	—
19	C₂×#346	C₂×C₄	1	1	yes	characteristic
20	C₂×#346	C₂×C₄	1	1	yes	characteristic
21	C₂×#346	D₄	1	1	yes	characteristic
22	C₂×#346	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₆	D₄	1	1	yes	characteristic
25	C₄²×C₈	D₄	1	1	yes	characteristic
26	#1564	—	2	2	no	—
27	#1450	—	4	1	no	—
28	#1398	—	2	4	no	—
29	#1398	—	2	2	no	—
30	#1398	C₈	1	4	yes	complemented
31	#1397	—	2	2	no	—
32	C₄×C₃₂	—	2	2	no	—
33	#1376	—	4	1	no	—
34	#1373	—	2	1	no	—
35	#1373	—	2	1	no	—
36	#1373	C₈	1	2	yes	complemented
37	#1373	C₈	1	2	yes	complemented
38	#1368	—	2	2	no	—
39	#1368	C₈	1	4	yes	—
40	#1276	—	4	1	no	—
41	C₂×M₃₂	—	4	1	no	—
42	C₂×M₃₂	—	4	1	no	—
43	C₂²×C₁₆	C₄²	1	1	yes	characteristic
44	C₂²×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	#431	—	4	2	no	—
48	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₈	1	1	yes	characteristic
52	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₄×C₈	Q₁₆	1	1	yes	characteristic
54	C₄³	SD₁₆	1	1	yes	characteristic
55	M₆₄	—	2	4	no	—
56	M₆₄	—	2	4	no	—
57	C₂×C₃₂	—	2	4	no	—
58	#364	—	2	2	no	—
59	#364	—	2	2	no	—
60	#364	—	4	1	no	—
61	#364	—	4	1	no	—
62	#346	—	2	2	no	—
63	#346	—	2	2	no	—
64	#346	—	8	1	no	—
65	#346	C₂×C₈	1	2	yes	—
66	#346	C₂×C₈	1	2	yes	—
67	#346	M₁₆	1	2	yes	—
68	#346	M₁₆	1	2	yes	—
69	C₄×C₁₆	—	2	1	no	—
70	C₄×C₁₆	—	2	1	no	—
71	C₄×C₁₆	—	8	1	no	—
72	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
73	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
74	C₄×C₁₆	M₁₆	1	1	yes	characteristic
75	C₄×C₁₆	M₁₆	1	1	yes	characteristic
76	#335	—	4	2	no	—
77	#334	—	2	4	no	—
78	#322	—	8	1	no	—
79	C₈²	—	8	1	no	—
80	C₂²×C₈	#95	1	1	yes	characteristic
81	C₂²×C₈	#102	1	1	yes	characteristic
82	C₂²×C₈	#103	1	1	yes	characteristic
83	C₂²×C₈	#107	1	1	yes	characteristic
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	#102	1	1	yes	characteristic
86	C₂×C₄²	#103	1	1	yes	characteristic
87	C₂×C₄²	#106	1	1	yes	characteristic
88	M₃₂	—	4	2	no	—
89	M₃₂	—	4	2	no	—
90	M₃₂	—	8	1	no	—
91	M₃₂	—	8	1	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	8	1	no	—
99	C₂×C₁₆	—	8	1	no	—
100	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
101	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
102	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
103	C₂×C₁₆	#97	1	1	yes	characteristic
104	C₂×C₁₆	#97	1	1	yes	characteristic
105	C₂×C₁₆	#97	1	1	yes	characteristic
106	C₂×C₁₆	C₂²⋊C₈	1	2	yes	—
107	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₁₆	C₄⋊C₈	1	2	yes	—
110	C₄⋊C₈	—	4	4	no	—
111	C₂²⋊C₈	—	8	2	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	8	2	no	—
118	C₄×C₈	C₂²⋊C₈	1	2	yes	—
119	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₈	#104	1	2	yes	—
122	C₄×C₈	#104	1	1	yes	characteristic
123	C₄×C₈	#104	1	1	yes	characteristic
124	C₄×C₈	C₄⋊C₈	1	2	yes	—
125	C₄×C₈	#108	1	2	yes	—
126	C₃₂	—	2	8	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	#340	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	8	4	no	—
136	C₂×C₈	D₄⋊C₈	1	1	yes	characteristic
137	C₂×C₈	D₄⋊C₈	1	1	yes	characteristic
138	C₂×C₈	Q₈⋊C₈	1	2	yes	—
139	C₂×C₈	#335	1	2	yes	—
140	C₂×C₈	#336	1	1	yes	characteristic
141	C₂×C₈	#336	1	1	yes	characteristic
142	C₂×C₈	#336	1	1	yes	characteristic
143	C₂×C₈	#339	1	1	yes	characteristic
144	C₂×C₈	#339	1	1	yes	characteristic
145	C₂×C₈	#341	1	1	yes	characteristic
146	C₂×C₈	#360	1	1	yes	characteristic
147	C₂×C₈	#360	1	1	yes	characteristic
148	C₂×C₈	#361	1	1	yes	characteristic
149	C₂×C₈	#362	1	1	yes	characteristic
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	8	2	no	—
154	C₄²	D₄⋊C₈	1	2	yes	—
155	C₄²	Q₈⋊C₈	1	1	yes	characteristic
156	C₄²	Q₈⋊C₈	1	1	yes	characteristic
157	C₄²	#334	1	2	yes	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	2	2	no	—
160	C₁₆	—	2	1	no	—
161	C₁₆	—	2	1	no	—
162	C₁₆	—	8	2	no	—
163	C₁₆	—	8	2	no	—
164	C₁₆	C₈²	1	1	yes	characteristic
165	C₁₆	#322	1	1	yes	characteristic
166	C₁₆	#322	1	1	yes	characteristic
167	C₁₆	#322	1	1	yes	characteristic
168	C₂³	#1389	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	#1277	1	1	yes	characteristic
177	C₂×C₄	#1279	1	1	yes	characteristic
178	C₂×C₄	#1279	1	1	yes	characteristic
179	C₂×C₄	#1386	1	1	yes	characteristic
180	C₂×C₄	#1387	1	1	yes	characteristic
181	C₂×C₄	#1388	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	2	no	—
184	C₈	—	16	4	no	—
185	C₈	#1273	1	1	yes	characteristic
186	C₈	#1276	1	1	yes	characteristic
187	C₈	#1278	1	1	yes	characteristic
188	C₈	#1278	1	1	yes	characteristic
189	C₈	#1333	1	1	yes	characteristic
190	C₈	#1333	1	1	yes	characteristic
191	C₈	#1334	1	1	yes	characteristic
192	C₈	#1334	1	1	yes	characteristic
193	C₂²	—	4	1	no	—
194	C₂²	#7159	1	1	yes	characteristic
195	C₂²	#7175	1	1	yes	characteristic
196	C₂²	#7177	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	4	2	no	—
201	C₄	#7015	1	1	yes	characteristic
202	C₄	#7158	1	1	yes	characteristic
203	C₄	#7176	1	1	yes	characteristic
204	C₄	#7178	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#93270	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#1333
commutator	C₁₆	C₈²
Frattini, cosocle	C₄×#346	C₂²
socle	C₂	#93270
prime radical	C₂³	#1389
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
30. #1398	C₈	semidirect
36. #1373	C₈	semidirect
37. #1373	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#1398	2
#129101	2
#129088	2