Subgroups — #11997

Order 256.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	#1367	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂	—	2	1	no	—
6	C₂²×C₁₆	—	2	1	no	—
7	#431	C₄	1	1	yes	characteristic
8	#431	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆	—	2	1	no	—
11	C₄⋊C₁₆	—	2	1	no	—
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₄⋊C₁₆	C₂²	1	2	yes	—
14	#322	C₄	1	2	yes	—
15	C₈²	C₂²	1	2	yes	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	Q₈	1	1	yes	characteristic
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	M₃₂	—	4	2	no	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	—	4	2	no	—
23	C₄⋊C₈	—	2	1	no	—
24	C₄⋊C₈	C₈	1	2	yes	—
25	C₂²⋊C₈	—	2	1	no	—
26	C₂²⋊C₈	C₈	1	2	yes	—
27	C₄×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄	M₁₆	1	1	yes	characteristic
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₂×C₈	1	4	yes	—
41	C₂×C₈	SD₁₆	1	2	yes	—
42	C₂×C₈	SD₁₆	1	2	yes	—
43	C₂×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₄²	C₄⋊C₄	1	1	yes	characteristic
46	C₄²	C₂×C₈	1	1	yes	characteristic
47	C₄²	M₁₆	1	1	yes	characteristic
48	C₄²	C₂²×C₄	1	1	yes	characteristic
49	C₁₆	—	4	2	no	—
50	C₁₆	—	4	2	no	—
51	C₂³	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄	#106	1	2	yes	—
58	C₂×C₄	#108	1	2	yes	—
59	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	2	1	no	—
64	C₈	—	2	4	no	—
65	C₈	—	4	2	no	—
66	C₈	#106	1	2	yes	—
67	C₈	C₂×SD₁₆	1	1	yes	characteristic
68	C₈	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	#334	1	2	yes	—
72	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#334	1	2	yes	—
77	C₄	C₂×#106	1	1	yes	characteristic
78	C₄	C₂×#108	1	1	yes	characteristic
79	C₄	#504	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂×#334	1	1	yes	characteristic
82	C₂	#2151	1	1	yes	characteristic
83	C₂	#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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. #1367	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2150	2
C₂×C₄⋊C₁₆	2
#1450	2
#1367	2