Subgroups — #94116

Order 512.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13639	C₂	1	2	yes	maximal
2	C₂×#1312	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
4	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×C₈²	C₄	1	1	yes	characteristic
6	C₄⋊C₃₂	—	2	2	no	—
7	C₂²⋊C₃₂	C₄	1	4	yes	—
8	C₄×C₃₂	—	2	2	no	—
9	#1312	—	2	2	no	—
10	C₂²×C₁₆	C₈	1	2	yes	—
11	C₂²×C₁₆	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₈	C₈	1	1	yes	characteristic
13	C₂×C₄×C₈	C₈	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃₂	—	2	4	no	—
16	C₄×C₁₆	—	2	1	no	—
17	C₄×C₁₆	—	2	1	no	—
18	C₄×C₁₆	D₄	1	2	yes	—
19	C₄×C₁₆	D₄	1	1	yes	characteristic
20	C₄×C₁₆	Q₈	1	1	yes	characteristic
21	C₈²	—	2	1	no	—
22	C₈²	—	2	1	no	—
23	C₂²×C₈	C₁₆	1	2	yes	—
24	C₂²×C₈	C₁₆	1	2	yes	—
25	C₂²×C₈	C₄²	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	4	2	no	—
30	C₂×C₁₆	—	4	2	no	—
31	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
32	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
33	C₄×C₈	—	2	1	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	—	2	1	no	—
37	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₈	M₁₆	1	1	yes	characteristic
40	C₄×C₈	M₁₆	1	1	yes	characteristic
41	C₃₂	—	4	4	no	—
42	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
43	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
44	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	C₃₂	1	4	yes	complemented
52	C₂×C₈	#95	1	1	yes	characteristic
53	C₂×C₈	#97	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
56	C₄²	C₂²⋊C₈	1	1	yes	characteristic
57	C₄²	C₄⋊C₈	1	1	yes	characteristic
58	C₄²	M₃₂	1	1	yes	characteristic
59	C₄²	M₃₂	1	1	yes	characteristic
60	C₁₆	—	2	2	no	—
61	C₁₆	—	2	2	no	—
62	C₁₆	—	4	2	no	—
63	C₂³	C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	#336	1	1	yes	characteristic
68	C₂×C₄	#346	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
71	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
72	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
73	C₂×C₄	C₂×C₃₂	1	2	yes	—
74	C₂×C₄	M₆₄	1	2	yes	—
75	C₈	—	2	1	no	—
76	C₈	—	2	1	no	—
77	C₈	—	2	4	no	—
78	C₈	—	4	2	no	—
79	C₈	#344	1	2	yes	—
80	C₈	#349	1	2	yes	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	#1374	1	1	yes	characteristic
84	C₂²	C₄×C₃₂	1	1	yes	characteristic
85	C₂²	#1397	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	#1281	1	1	yes	characteristic
89	C₄	#1377	1	1	yes	characteristic
90	C₄	C₂²⋊C₃₂	1	2	yes	—
91	C₄	#1400	1	2	yes	—
92	C₄	C₄⋊C₃₂	1	2	yes	—
93	C₂	—	2	1	no	—
94	C₂	—	2	1	no	—
95	C₂	#7195	1	1	yes	characteristic
96	C₂	#7465	1	1	yes	characteristic
97	C₂	#7468	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂²	C₄×C₃₂
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	C₄×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
51. C₂×C₈	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
#13639	2
C₂×#1312	2