Subgroups — #94013

Order 512.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13642	C₂	1	2	yes	maximal
2	C₄×#347	C₂	1	1	yes	maximal, characteristic
3	C₂×#347	—	2	2	no	—
4	C₄×M₃₂	C₄	1	1	yes	characteristic
5	C₄×M₃₂	C₂²	1	1	yes	characteristic
6	C₄×#97	C₄	1	1	yes	characteristic
7	#1422	C₄	1	4	yes	complemented
8	#1400	—	2	2	no	—
9	#1398	—	2	2	no	—
10	C₂×M₃₂	—	2	1	no	—
11	C₂×M₃₂	D₄	1	2	yes	—
12	C₂×#97	—	2	1	no	—
13	C₂×#97	—	2	1	no	—
14	C₂×C₄×C₈	C₈	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₄³	C₈	1	1	yes	characteristic
17	M₆₄	—	2	4	no	—
18	#347	—	2	8	no	—
19	#346	—	2	1	no	—
20	#346	D₄	1	1	yes	characteristic
21	#346	Q₈	1	1	yes	characteristic
22	C₄×C₁₆	C₈	1	2	yes	—
23	C₄×C₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₈	M₁₆	1	1	yes	characteristic
27	C₂×C₄²	—	2	1	no	—
28	C₂×C₄²	—	4	1	no	—
29	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
30	M₃₂	—	2	4	no	—
31	M₃₂	—	2	4	no	—
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
34	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
35	#97	—	2	4	no	—
36	#97	—	4	2	no	—
37	C₄×C₈	—	4	1	no	—
38	C₄×C₈	C₄²	1	1	yes	characteristic
39	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₄×C₈	C₂×C₈	1	1	yes	characteristic
41	C₄×C₈	M₁₆	1	1	yes	characteristic
42	C₃₂	—	4	4	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	4	1	no	—
45	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	#95	1	1	yes	characteristic
52	C₂×C₈	#97	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
55	C₄²	—	2	1	no	—
56	C₄²	—	2	2	no	—
57	C₄²	—	4	1	no	—
58	C₄²	—	8	2	no	—
59	C₄²	C₄×C₈	1	1	yes	characteristic
60	C₄²	C₄⋊C₈	1	1	yes	characteristic
61	C₁₆	—	2	4	no	—
62	C₁₆	—	2	2	no	—
63	C₁₆	#104	1	4	yes	—
64	C₂³	#349	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	8	2	no	—
72	C₂×C₄	#336	1	1	yes	characteristic
73	C₂×C₄	#349	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	2	1	no	—
76	C₈	—	4	2	no	—
77	C₈	#339	1	2	yes	—
78	C₈	#347	1	2	yes	—
79	C₂²	—	2	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	#1377	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	2	no	—
86	C₄	#1276	1	1	yes	characteristic
87	C₄	#1375	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	4	1	no	—
90	C₂	#7149	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1276
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄×M₃₂	C₂²
socle	C₂	#7149
prime radical	C₂³	#349
LCS	G ▷ C₄² ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
7. #1422	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13642	2
C₄×#347	2