Subgroups — #33500

Order 256.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₆	C₂	1	4	yes	maximal, complemented
4	#2164	C₂	1	4	yes	maximal
5	#2162	C₂	1	2	yes	maximal
6	#533	C₄	1	1	yes	characteristic
7	#446	C₄	1	4	yes	—
8	C₈×Q₈	C₂²	1	4	yes	—
9	#424	C₄	1	2	yes	—
10	#423	C₄	1	1	yes	characteristic
11	#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₄⋊C₁₆	C₂²	1	8	yes	—
16	C₂²⋊C₁₆	C₂²	1	4	yes	—
17	#346	C₂²	1	4	yes	—
18	C₄×C₁₆	C₂²	1	4	yes	—
19	C₄×C₁₆	C₂²	1	4	yes	—
20	C₂×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈	C₈	1	2	yes	—
24	#125	C₈	1	2	yes	—
25	C₂²⋊Q₈	C₈	1	4	yes	—
26	C₄×Q₈	C₂×C₄	1	4	yes	—
27	#117	C₂×C₄	1	2	yes	—
28	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆	C₂³	1	4	yes	—
30	C₂×C₁₆	C₂³	1	4	yes	—
31	C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₄⋊C₈	C₂×C₄	1	8	yes	—
33	#97	C₂×C₄	1	4	yes	—
34	C₄×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₈	C₂³	1	2	yes	—
36	C₄×C₈	C₂³	1	4	yes	—
37	C₂×Q₈	C₂×C₈	1	4	yes	—
38	C₂²×C₄	C₂×C₈	1	2	yes	—
39	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
40	M₁₆	—	4	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈	C₂²×C₄	1	2	yes	—
45	C₂×C₈	C₂²×C₄	1	4	yes	—
46	C₂×C₈	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄	C₂×C₈	1	4	yes	—
48	C₄⋊C₄	C₂×C₈	1	8	yes	—
49	C₄⋊C₄	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄	C₂×C₈	1	4	yes	—
51	C₄²	C₂×C₈	1	1	yes	characteristic
52	C₄²	C₂×C₈	1	1	yes	characteristic
53	C₄²	C₂²×C₄	1	2	yes	—
54	C₄²	C₂²×C₄	1	4	yes	—
55	C₁₆	—	2	4	no	—
56	C₁₆	—	2	4	no	—
57	C₂³	C₂²×C₈	1	1	yes	characteristic
58	Q₈	—	2	8	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄	C₂²×C₈	1	2	yes	—
64	C₂×C₄	C₂²×C₈	1	8	yes	—
65	C₂×C₄	C₂²×C₈	1	2	yes	—
66	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	2	no	—
69	C₈	—	2	4	no	—
70	C₈	2₋¹⁺⁴	1	2	yes	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	C₂³×C₈	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	8	no	—
76	C₄	—	2	1	no	—
77	C₄	#504	1	4	yes	—
78	C₄	#519	1	1	yes	characteristic
79	C₄	#568	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	#2982	1	1	yes	characteristic
82	C₂	C₂×#504	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2183	C₂	semidirect
3. Q₈×C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2960	2
#2183	2
Q₈×C₁₆	2
#2164	2
#2162	2