Subgroups — #12032

Order 256.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	C₂	1	1	yes	maximal, characteristic
2	#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	#1371	C₂	1	4	yes	maximal, complemented
5	#423	—	2	1	no	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₁₆	C₂²	1	2	yes	—
11	#335	—	2	1	no	—
12	#335	—	2	1	no	—
13	#335	C₂²	1	2	yes	—
14	#335	C₂²	1	2	yes	—
15	C₂×M₁₆	—	4	1	no	—
16	C₂²×C₈	—	4	1	no	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₁₆	C₂×C₄	1	4	yes	—
21	C₄⋊C₈	—	2	2	no	—
22	C₄⋊C₈	—	2	2	no	—
23	C₄⋊C₈	—	4	1	no	—
24	C₄⋊C₈	—	4	1	no	—
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₄×C₈	Q₈	1	1	yes	characteristic
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄	D₈	1	1	yes	characteristic
31	C₂²×C₄	Q₁₆	1	1	yes	characteristic
32	M₁₆	—	8	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	—	4	1	no	—
35	C₂×C₈	—	4	2	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	—	8	2	no	—
38	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₄²	C₄⋊C₄	1	1	yes	characteristic
43	C₄²	D₈	1	1	yes	characteristic
44	C₄²	Q₁₆	1	1	yes	characteristic
45	C₄²	C₂×D₄	1	1	yes	characteristic
46	C₁₆	—	2	4	no	—
47	C₂³	#107	1	1	yes	characteristic
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	#107	1	1	yes	characteristic
52	C₂×C₄	#107	1	1	yes	characteristic
53	C₂×C₄	#107	1	1	yes	characteristic
54	C₂×C₄	D₁₆	1	2	yes	—
55	C₂×C₄	Q₃₂	1	2	yes	—
56	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	2	1	no	—
61	C₈	—	2	1	no	—
62	C₈	—	8	2	no	—
63	C₈	—	8	2	no	—
64	C₈	#131	1	2	yes	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	#366	1	2	yes	—
68	C₂²	C₂×#107	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	#366	1	2	yes	—
73	C₄	#424	1	1	yes	characteristic
74	C₄	#430	1	1	yes	characteristic
75	C₄	C₂×D₁₆	1	1	yes	characteristic
76	C₄	C₂×Q₃₂	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	#1571	1	1	yes	characteristic
79	C₂	C₂×#366	1	1	yes	characteristic
80	C₂	#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1371	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
#1567	2
C₂×#335	2
#1371	2