Subgroups — H(3,ℤ/8) #93252

Order 512.

191 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12593	C₂	1	1	yes	maximal, characteristic
2	#8137	C₂	1	2	yes	maximal
3	C₄×M₃₂	C₄	1	2	yes	—
4	#1726	C₂²	1	1	yes	characteristic
5	#1450	C₄	1	4	yes	—
6	#1375	—	2	2	no	—
7	#1275	—	2	2	no	—
8	#1274	—	2	2	no	—
9	C₂×M₃₂	—	2	2	no	—
10	#431	—	2	4	no	—
11	C₂×#97	D₄	1	2	yes	—
12	C₂×#97	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	#376	D₄	1	1	yes	characteristic
16	#346	—	2	2	no	—
17	C₄×C₁₆	C₈	1	4	yes	complemented
18	#322	—	2	4	no	—
19	C₈²	C₈	1	8	yes	complemented
20	C₂²×C₈	C₄²	1	1	yes	characteristic
21	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₈	C₄⋊C₄	1	2	yes	—
23	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
25	M₃₂	—	4	4	no	—
26	C₂×C₁₆	—	2	4	no	—
27	C₄⋊C₈	—	4	4	no	—
28	C₂²⋊C₈	—	4	4	no	—
29	#97	—	4	2	no	—
30	#97	—	4	1	no	—
31	C₄×C₈	—	2	8	no	—
32	C₄×C₈	C₂×C₈	1	4	yes	—
33	C₄×C₈	C₂×C₈	1	2	yes	—
34	C₄×C₈	M₁₆	1	4	yes	—
35	C₄×C₈	M₁₆	1	2	yes	—
36	#95	—	4	1	no	—
37	C₂²×C₄	—	2	2	no	—
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	#95	1	1	yes	characteristic
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	4	8	no	—
44	C₂×C₈	C₄×C₈	1	1	yes	characteristic
45	C₂×C₈	C₄×C₈	1	2	yes	—
46	C₂×C₈	#97	1	1	yes	characteristic
47	C₂×C₈	#97	1	2	yes	—
48	C₂×C₈	C₂²⋊C₈	1	2	yes	—
49	C₂×C₈	C₄⋊C₈	1	4	yes	—
50	C₄²	—	2	2	no	—
51	C₄²	—	2	1	no	—
52	C₄²	C₂²⋊C₈	1	4	yes	—
53	C₄²	C₄⋊C₈	1	2	yes	—
54	C₁₆	—	4	4	no	—
55	C₂³	H(3,ℤ/4)	1	1	yes	characteristic
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	#336	1	1	yes	characteristic
61	C₂×C₄	#336	1	2	yes	—
62	C₂×C₄	H(3,ℤ/4)	1	2	yes	—
63	C₂×C₄	#338	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	2	4	no	—
66	C₈	—	8	8	no	—
67	C₈	C₈²	1	1	yes	characteristic
68	C₈	#322	1	1	yes	characteristic
69	C₈	#322	1	2	yes	—
70	C₂²	—	4	1	no	—
71	C₂²	#1274	1	2	yes	—
72	C₂²	#1275	1	1	yes	characteristic
73	C₄	—	4	4	no	—
74	C₄	—	4	2	no	—
75	C₄	#1273	1	1	yes	characteristic
76	C₄	#1274	1	1	yes	characteristic
77	C₄	#1275	1	2	yes	—
78	C₂	—	2	2	no	—
79	C₂	—	2	1	no	—
80	C₂	#7014	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₈	C₈²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7014
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
17. C₄×C₁₆	C₈	semidirect
19. C₈²	C₈	semidirect

Maximal subgroups

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

subgroup	index
#12593	2
#8137	2