Subgroups — #17316

Order 256.

445 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41	C₂	1	2	yes	maximal
2	C₄×#131	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	3	yes	maximal, complemented
4	C₈×M₁₆	C₂	1	6	yes	maximal, complemented
5	C₂×#322	C₂	1	3	yes	maximal, complemented
6	C₂×#131	C₄	1	2	yes	—
7	C₄×#41	C₂²	1	1	yes	characteristic
8	C₈×Q₈	C₄	1	4	yes	—
9	C₈×D₄	C₄	1	12	yes	—
10	#431	C₂²	1	6	yes	—
11	#405	C₄	1	6	yes	—
12	C₄×M₁₆	C₂²	1	3	yes	—
13	C₂×C₄×C₈	C₂²	1	3	yes	—
14	C₂×C₄×C₈	C₂²	1	6	yes	—
15	#322	C₂²	1	1	yes	characteristic
16	#322	C₂²	1	6	yes	—
17	#322	C₂²	1	3	yes	—
18	C₈²	C₂²	1	6	yes	—
19	C₂×#41	C₂×C₄	1	1	yes	characteristic
20	#131	C₈	1	16	yes	complemented
21	C₂×M₁₆	C₂×C₄	1	6	yes	—
22	C₂²×C₈	C₂×C₄	1	12	yes	—
23	C₂²×C₈	C₂×C₄	1	6	yes	—
24	C₄×Q₈	C₂×C₄	1	2	yes	—
25	C₄×D₄	C₂×C₄	1	6	yes	—
26	#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²	C₂³	1	3	yes	—
28	C₄⋊C₈	C₂×C₄	1	12	yes	—
29	C₂²⋊C₈	C₂×C₄	1	12	yes	—
30	#97	C₂×C₄	1	6	yes	—
31	C₄×C₈	C₂×C₄	1	6	yes	—
32	C₄×C₈	C₂×C₄	1	12	yes	—
33	C₄×C₈	C₂³	1	3	yes	—
34	C₄×C₈	C₂³	1	6	yes	—
35	C₄×C₈	C₂³	1	2	yes	—
36	C₄×C₈	C₂³	1	1	yes	characteristic
37	#41	C₂×C₈	1	8	yes	—
38	C₂×Q₈	C₄²	1	1	yes	characteristic
39	C₂×D₄	C₄²	1	3	yes	—
40	C₂²×C₄	C₂²×C₄	1	6	yes	—
41	C₂²×C₄	C₂²×C₄	1	3	yes	—
42	M₁₆	C₂×C₈	1	24	yes	complemented
43	C₂×C₈	—	4	12	no	—
44	C₂×C₈	C₂×C₈	1	24	yes	complemented
45	C₂×C₈	C₂²×C₄	1	4	yes	—
46	C₂×C₈	C₂²×C₄	1	12	yes	—
47	C₂×C₈	C₂²×C₄	1	2	yes	—
48	C₂×C₈	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄	C₄²	1	6	yes	—
50	C₂²⋊C₄	C₄²	1	6	yes	—
51	C₄²	C₂²×C₄	1	3	yes	—
52	C₄²	C₂²×C₄	1	6	yes	—
53	C₄²	C₂⁴	1	1	yes	characteristic
54	C₂³	C₂×C₄²	1	3	yes	—
55	Q₈	C₄×C₈	1	4	yes	—
56	D₄	C₄×C₈	1	12	yes	—
57	C₂×C₄	—	2	12	no	—
58	C₂×C₄	C₂×C₄²	1	6	yes	—
59	C₂×C₄	C₂×C₄²	1	3	yes	—
60	C₂×C₄	C₂²×C₈	1	12	yes	—
61	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄	C₂³×C₄	1	2	yes	—
63	C₈	—	2	4	no	—
64	C₈	—	2	12	no	—
65	C₈	C₂²×C₈	1	4	yes	—
66	C₈	C₂²×C₈	1	12	yes	—
67	C₂²	—	2	6	no	—
68	C₂²	C₂×C₄×C₈	1	6	yes	—
69	C₂²	C₂²×C₄²	1	1	yes	characteristic
70	C₄	—	2	6	no	—
71	C₄	C₂×C₄×C₈	1	6	yes	—
72	C₄	C₂³×C₈	1	2	yes	—
73	C₄	#568	1	4	yes	—
74	C₂	—	2	6	no	—
75	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
76	C₂	#2875	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁴
commutator	C₂	C₂²×C₄×C₈
socle	C₂²	C₂²×C₄²
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1450	C₂	semidirect
4. C₈×M₁₆	C₂	semidirect
5. C₂×#322	C₂	semidirect
20. #131	C₈	semidirect
42. M₁₆	C₂×C₈	semidirect
44. C₂×C₈	C₂×C₈	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
C₄×#131	2
#1450	2
C₈×M₁₆	2
C₂×#322	2