Subgroups — #33505

Order 256.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
3	#2169	C₂	1	6	yes	maximal, complemented
4	#2162	C₂	1	3	yes	maximal, complemented
5	#2150	C₂	1	3	yes	maximal, complemented
6	C₄×#41	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	6	yes	—
8	C₈×Q₈	C₄	1	1	yes	characteristic
9	C₈×Q₈	C₂²	1	1	yes	characteristic
10	C₈×D₄	C₄	1	3	yes	—
11	C₈×D₄	C₂²	1	3	yes	—
12	#431	C₄	1	3	yes	—
13	C₂×C₄×C₈	C₂²	1	3	yes	—
14	C₄⋊C₁₆	C₂²	1	3	yes	—
15	C₄⋊C₁₆	C₂²	1	6	yes	—
16	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²⋊C₁₆	C₂²	1	6	yes	—
18	C₄×C₁₆	C₂²	1	6	yes	—
19	C₂×#41	C₈	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₈	C₂³	1	3	yes	—
22	C₄×Q₈	C₈	1	1	yes	characteristic
23	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₄×D₄	C₈	1	3	yes	—
25	C₄×D₄	C₂×C₄	1	3	yes	—
26	#117	C₈	1	3	yes	—
27	C₂×C₄²	C₂×C₄	1	3	yes	—
28	M₃₂	—	4	6	no	—
29	C₂×C₁₆	C₂³	1	2	yes	—
30	C₂×C₁₆	C₂³	1	6	yes	—
31	C₄⋊C₈	C₂×C₄	1	3	yes	—
32	C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈	C₂×C₄	1	6	yes	—
34	C₄×C₈	C₂×C₄	1	3	yes	—
35	C₄×C₈	C₂×C₄	1	3	yes	—
36	C₄×C₈	C₂³	1	3	yes	—
37	C₄×C₈	C₂³	1	1	yes	characteristic
38	#41	—	2	4	no	—
39	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄	C₂×C₈	1	3	yes	—
41	C₂²×C₄	C₂×C₈	1	3	yes	—
42	C₂²×C₄	C₂×C₈	1	3	yes	—
43	C₂²×C₄	C₂²×C₄	1	3	yes	—
44	C₂×C₈	—	2	3	no	—
45	C₂×C₈	—	2	3	no	—
46	C₂×C₈	—	4	3	no	—
47	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₄	1	3	yes	—
49	C₂×C₈	C₂²×C₄	1	3	yes	—
50	C₂×C₈	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄	C₂×C₈	1	3	yes	—
52	C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₂²⋊C₄	C₂×C₈	1	6	yes	—
54	C₄²	C₂×C₈	1	3	yes	—
55	C₄²	C₂×C₈	1	3	yes	—
56	C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₄²	C₂²×C₄	1	3	yes	—
58	C₁₆	—	2	6	no	—
59	C₁₆	—	2	2	no	—
60	C₂³	C₂²×C₈	1	3	yes	—
61	Q₈	M₃₂	1	4	yes	complemented
62	D₄	—	2	6	no	—
63	C₂×C₄	—	2	3	no	—
64	C₂×C₄	—	2	3	no	—
65	C₂×C₄	—	2	6	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄	C₂²×C₈	1	3	yes	—
69	C₂×C₄	C₂²×C₈	1	3	yes	—
70	C₂×C₄	C₂²×C₈	1	3	yes	—
71	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
73	C₈	—	2	3	no	—
74	C₈	—	2	1	no	—
75	C₈	—	2	3	no	—
76	C₈	H(5,𝔽₂)	1	1	yes	characteristic
77	C₈	2₋¹⁺⁴	1	1	yes	characteristic
78	C₂²	—	2	3	no	—
79	C₂²	—	2	3	no	—
80	C₂²	—	2	3	no	—
81	C₂²	C₂³×C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	3	no	—
84	C₄	—	2	3	no	—
85	C₄	C₂×M₃₂	1	6	yes	—
86	C₄	#504	1	2	yes	—
87	C₄	#520	1	1	yes	characteristic
88	C₄	#568	1	1	yes	characteristic
89	C₂	—	4	3	no	—
90	C₂	#2988	1	1	yes	characteristic
91	C₂	C₂²×M₃₂	1	1	yes	characteristic
92	C₂	C₂×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂×#41	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. Q₈×C₁₆	C₂	semidirect
3. #2169	C₂	semidirect
4. #2162	C₂	semidirect
5. #2150	C₂	semidirect
61. Q₈	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41	2
Q₈×C₁₆	2
#2169	2
#2162	2
#2150	2