Subgroups — #33401

Order 256.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407	C₂	1	2	yes	maximal
2	C₄×#131	C₂	1	1	yes	maximal, characteristic
3	#2153	C₂	1	6	yes	maximal, complemented
4	C₂×#364	C₂	1	6	yes	maximal, complemented
5	C₂×#131	C₄	1	1	yes	characteristic
6	C₂×#131	C₂²	1	1	yes	characteristic
7	C₄×#41	C₄	1	1	yes	characteristic
8	#504	—	2	8	no	—
9	C₂×M₃₂	C₂²	1	12	yes	—
10	#405	C₄	1	3	yes	—
11	#405	C₂²	1	3	yes	—
12	C₄×M₁₆	C₄	1	3	yes	—
13	C₂×C₄×C₈	C₂²	1	3	yes	—
14	#364	C₂²	1	4	yes	—
15	#364	C₂²	1	12	yes	—
16	C₂×#41	C₂×C₄	1	1	yes	characteristic
17	#131	C₈	1	8	yes	—
18	#131	D₄	1	4	yes	—
19	#131	Q₈	1	4	yes	—
20	C₂×M₁₆	C₂×C₄	1	3	yes	—
21	C₂×M₁₆	C₂×C₄	1	3	yes	—
22	C₂²×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₈	C₂³	1	3	yes	—
24	C₄×Q₈	—	2	1	no	—
25	C₄×D₄	—	2	3	no	—
26	#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²	C₂×C₄	1	3	yes	—
28	M₃₂	—	2	12	no	—
29	M₃₂	C₂³	1	8	yes	—
30	C₂×C₁₆	—	2	12	no	—
31	#97	C₂×C₄	1	6	yes	—
32	C₄×C₈	C₂×C₄	1	3	yes	—
33	C₄×C₈	C₂×C₄	1	3	yes	—
34	C₄×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₈	C₂³	1	3	yes	—
36	#41	C₄⋊C₄	1	4	yes	—
37	#41	C₂×C₈	1	4	yes	—
38	C₂×Q₈	M₁₆	1	1	yes	characteristic
39	C₂×D₄	M₁₆	1	3	yes	—
40	C₂²×C₄	—	2	3	no	—
41	C₂²×C₄	C₂²×C₄	1	3	yes	—
42	M₁₆	C₄⋊C₄	1	12	yes	—
43	M₁₆	C₂×C₈	1	12	yes	—
44	C₂×C₈	C₂×C₈	1	12	yes	—
45	C₂×C₈	C₂²×C₄	1	3	yes	—
46	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈	C₂²×C₄	1	3	yes	—
48	C₂×C₈	C₂×D₄	1	6	yes	—
49	C₂×C₈	C₂×Q₈	1	6	yes	—
50	C₂×C₈	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄	—	2	3	no	—
52	C₂²⋊C₄	—	2	3	no	—
53	C₄²	—	2	3	no	—
54	C₄²	C₂²×C₄	1	1	yes	characteristic
55	C₄²	C₂²×C₄	1	3	yes	—
56	C₁₆	—	2	8	no	—
57	C₂³	C₂×M₁₆	1	3	yes	—
58	Q₈	C₄⋊C₈	1	4	yes	—
59	D₄	C₄⋊C₈	1	12	yes	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	3	no	—
62	C₂×C₄	—	4	6	no	—
63	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄	C₂²×C₈	1	6	yes	—
65	C₂×C₄	C₂×M₁₆	1	3	yes	—
66	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
67	C₈	C₂×C₄⋊C₄	1	6	yes	—
68	C₈	C₂²×C₈	1	2	yes	—
69	C₈	C₂²×C₈	1	6	yes	—
70	C₈	C₂²×D₄	1	1	yes	characteristic
71	C₈	C₂²×Q₈	1	1	yes	characteristic
72	C₂²	—	4	3	no	—
73	C₂²	C₂×C₄⋊C₈	1	6	yes	—
74	C₂²	C₂²×M₁₆	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	4	3	no	—
77	C₄	C₂×C₄⋊C₈	1	6	yes	—
78	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₄	C₂³×C₈	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	2	6	no	—
82	C₂	C₂²×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂²×C₄⋊C₈
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2153	C₂	semidirect
4. C₂×#364	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3407	2
C₄×#131	2
#2153	2
C₂×#364	2