Subgroups — #93250

Order 512.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12591	C₂	1	2	yes	maximal
2	#8131	C₂	1	1	yes	maximal, characteristic
3	C₂×#346	C₄	1	4	yes	—
4	C₄²×C₈	C₄	1	2	yes	—
5	C₄²×C₈	C₂²	1	1	yes	characteristic
6	#1374	—	2	2	no	—
7	#1374	—	2	2	no	—
8	#1273	—	2	2	no	—
9	C₂²×C₁₆	—	2	4	no	—
10	C₂×C₄×C₈	—	2	2	no	—
11	C₂×C₄×C₈	—	2	2	no	—
12	C₂×C₄×C₈	C₈	1	4	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	D₄	1	2	yes	—
15	C₂×C₄×C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
17	C₄³	C₂×C₄	1	1	yes	characteristic
18	#346	—	2	4	no	—
19	#346	C₈	1	8	yes	complemented
20	C₂²×C₈	—	2	4	no	—
21	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₈	C₄⋊C₄	1	2	yes	—
23	C₂×C₄²	C₄²	1	1	yes	characteristic
24	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄²	C₂×C₈	1	2	yes	—
27	C₂×C₄²	M₁₆	1	2	yes	—
28	C₂×C₁₆	—	2	8	no	—
29	C₂×C₁₆	—	4	4	no	—
30	C₂×C₁₆	—	4	4	no	—
31	C₄×C₈	—	2	8	no	—
32	C₄×C₈	—	4	2	no	—
33	C₄×C₈	—	4	1	no	—
34	C₄×C₈	—	4	1	no	—
35	C₄×C₈	—	4	4	no	—
36	C₄×C₈	—	4	4	no	—
37	C₄×C₈	C₂×C₈	1	4	yes	—
38	C₄×C₈	M₁₆	1	4	yes	—
39	C₂²×C₄	#95	1	1	yes	characteristic
40	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
41	C₂²×C₄	#97	1	1	yes	characteristic
42	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
43	C₂²×C₄	C₄⋊C₈	1	2	yes	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	8	no	—
49	C₂×C₈	—	4	4	no	—
50	C₂×C₈	—	4	4	no	—
51	C₂×C₈	C₂²⋊C₈	1	4	yes	—
52	C₂×C₈	C₄⋊C₈	1	4	yes	—
53	C₄²	—	2	2	no	—
54	C₄²	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₄²	—	2	4	no	—
57	C₄²	—	2	4	no	—
58	C₄²	C₄×C₈	1	2	yes	—
59	C₄²	#97	1	2	yes	—
60	C₁₆	—	4	8	no	—
61	C₂³	#336	1	1	yes	characteristic
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₈²	1	1	yes	characteristic
69	C₂×C₄	#322	1	2	yes	—
70	C₂×C₄	#322	1	1	yes	characteristic
71	C₂×C₄	#336	1	2	yes	—
72	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
73	C₂×C₄	H(3,ℤ/4)	1	2	yes	—
74	C₂×C₄	#338	1	1	yes	characteristic
75	C₂×C₄	#349	1	4	yes	—
76	C₂×C₄	#364	1	4	yes	—
77	C₈	—	2	4	no	—
78	C₈	—	2	4	no	—
79	C₈	—	4	8	no	—
80	C₂²	#1273	1	1	yes	characteristic
81	C₂²	#1274	1	1	yes	characteristic
82	C₂²	#1275	1	1	yes	characteristic
83	C₂²	#1376	1	2	yes	—
84	C₂²	#1377	1	2	yes	—
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	2	4	no	—
89	C₄	—	2	4	no	—
90	C₄	#1274	1	2	yes	—
91	C₄	#1275	1	2	yes	—
92	C₂	#7014	1	1	yes	characteristic
93	C₂	#7138	1	1	yes	characteristic
94	C₂	#7139	1	1	yes	characteristic
95	C₂	#7143	1	2	yes	—
96	C₂	#7146	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. #346	C₈	semidirect

Maximal subgroups

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

subgroup	index
#12591	2
#8131	2