Subgroups — #7018

Order 256.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1831	C₂	1	2	yes	maximal
2	#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈	—	2	2	no	—
4	C₂×C₄×C₈	C₄	1	2	yes	—
5	C₄³	C₂²	1	1	yes	characteristic
6	#336	—	2	2	no	—
7	#336	—	2	2	no	—
8	#336	C₄	1	4	yes	—
9	C₂²×C₈	—	2	2	no	—
10	C₂²×C₈	—	2	4	no	—
11	C₂×C₄²	C₂×C₄	1	2	yes	—
12	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²	D₄	1	2	yes	—
14	C₂×C₄²	D₄	1	1	yes	characteristic
15	C₂×C₄²	Q₈	1	1	yes	characteristic
16	C₄⋊C₈	—	4	2	no	—
17	C₄⋊C₈	—	4	2	no	—
18	C₄×C₈	—	2	2	no	—
19	C₄×C₈	C₈	1	4	yes	complemented
20	C₂²×C₄	C₄²	1	1	yes	characteristic
21	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
23	C₂²×C₄	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈	—	2	4	no	—
26	C₂×C₈	—	4	4	no	—
27	C₂×C₈	—	4	4	no	—
28	C₂×C₈	—	4	4	no	—
29	C₂×C₈	—	4	4	no	—
30	C₄²	—	2	2	no	—
31	C₄²	—	4	2	no	—
32	C₄²	—	4	2	no	—
33	C₄²	C₂×C₈	1	2	yes	—
34	C₄²	M₁₆	1	2	yes	—
35	C₄²	D₈	1	1	yes	characteristic
36	C₄²	SD₁₆	1	2	yes	—
37	C₄²	Q₁₆	1	1	yes	characteristic
38	C₂³	#95	1	1	yes	characteristic
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄	#97	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊C₈	1	2	yes	—
49	C₂×C₄	#102	1	2	yes	—
50	C₂×C₄	#103	1	2	yes	—
51	C₂×C₄	C₄⋊C₈	1	2	yes	—
52	C₂×C₄	#106	1	1	yes	characteristic
53	C₂×C₄	#107	1	1	yes	characteristic
54	C₂×C₄	#108	1	2	yes	—
55	C₈	—	4	4	no	—
56	C₈	—	8	4	no	—
57	C₂²	#336	1	1	yes	characteristic
58	C₂²	H(3,ℤ/4)	1	2	yes	—
59	C₂²	#340	1	1	yes	characteristic
60	C₂²	#341	1	1	yes	characteristic
61	C₂²	#344	1	2	yes	—
62	C₄	—	2	1	no	—
63	C₄	—	2	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	2	2	no	—
66	C₄	—	2	2	no	—
67	C₄	—	4	2	no	—
68	C₄	#334	1	2	yes	—
69	C₄	#335	1	2	yes	—
70	C₂	#1274	1	1	yes	characteristic
71	C₂	#1277	1	1	yes	characteristic
72	C₂	#1281	1	1	yes	characteristic
73	C₂	#1286	1	2	yes	—
74	C₂	#1287	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#95
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₄³	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. 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
#1744	2