Subgroups — #8785

Order 256.

191 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1831	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1277	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	2	yes	—
9	C₄³	C₂²	1	1	yes	characteristic
10	#336	—	2	4	no	—
11	C₂²×C₈	—	2	4	no	—
12	C₂²×C₈	C₂×C₄	1	2	yes	—
13	C₂²×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²	D₄	1	2	yes	—
17	C₂×C₄²	Q₈	1	2	yes	—
18	C₂×C₄²	C₂³	1	1	yes	characteristic
19	C₄⋊C₈	—	2	8	no	—
20	C₄×C₈	—	2	4	no	—
21	C₄×C₈	—	2	2	no	—
22	C₄×C₈	—	2	4	no	—
23	C₄×C₈	C₈	1	4	yes	complemented
24	C₂²×C₄	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	4	4	no	—
34	C₂×C₈	—	4	8	no	—
35	C₂×C₈	C₂×C₈	1	4	yes	—
36	C₂×C₈	M₁₆	1	4	yes	—
37	C₄²	—	2	4	no	—
38	C₄²	—	2	2	no	—
39	C₄²	—	2	4	no	—
40	C₄²	C₂×C₈	1	2	yes	—
41	C₄²	M₁₆	1	2	yes	—
42	C₄²	#41	1	4	yes	—
43	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	C₄⋊C₈	1	4	yes	—
50	C₂×C₄	#108	1	4	yes	—
51	C₂×C₄	#117	1	2	yes	—
52	C₂×C₄	#124	1	2	yes	—
53	C₂×C₄	#125	1	2	yes	—
54	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄	C₂×M₁₆	1	2	yes	—
56	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄	#135	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	2	4	no	—
60	C₈	—	8	4	no	—
61	C₂²	#382	1	1	yes	characteristic
62	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²	#423	1	1	yes	characteristic
64	C₂²	#427	1	2	yes	—
65	C₂²	C₂×#108	1	2	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	2	4	no	—
68	C₄	—	2	2	no	—
69	C₄	#431	1	2	yes	—
70	C₄	#432	1	2	yes	—
71	C₄	#488	1	4	yes	—
72	C₂	#1564	1	2	yes	—
73	C₂	#1568	1	2	yes	—
74	C₂	#1831	1	1	yes	characteristic
75	C₂	#1835	1	1	yes	characteristic
76	C₂	#1837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
23. C₄×C₈	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1831	2
C₄²×C₈	2
#1277	2