Subgroups — #7228

Order 256.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2113	C₂	1	1	yes	maximal, characteristic
2	#1812	C₂	1	2	yes	maximal
3	C₂×#131	C₂²	1	1	yes	characteristic
4	C₂×M₃₂	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₁₆	C₄	1	1	yes	characteristic, complemented
6	#427	—	2	2	no	—
7	#410	—	2	2	no	—
8	C₂²⋊C₁₆	—	2	2	no	—
9	H(3,ℤ/4)	C₄	1	4	yes	complemented
10	C₂×#41	D₄	1	1	yes	characteristic
11	#131	—	4	1	no	—
12	#131	—	4	1	no	—
13	C₂×M₁₆	C₂×C₄	1	2	yes	—
14	C₂×M₁₆	Q₈	1	1	yes	characteristic
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	D₄	1	2	yes	—
17	#117	—	2	4	no	—
18	M₃₂	—	4	1	no	—
19	C₂×C₁₆	—	2	1	no	—
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	—	4	1	no	—
22	#107	—	4	2	no	—
23	#106	—	4	2	no	—
24	#99	—	4	2	no	—
25	#99	—	4	2	no	—
26	#41	—	4	1	no	—
27	#41	—	4	1	no	—
28	C₂×Q₈	Q₁₆	1	1	yes	characteristic
29	C₂×D₄	D₈	1	1	yes	characteristic
30	C₂×D₄	SD₁₆	1	2	yes	—
31	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
33	M₁₆	—	4	2	no	—
34	M₁₆	—	4	1	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	C₄²	1	1	yes	characteristic
40	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₄⋊C₄	1	2	yes	—
42	C₄⋊C₄	—	4	4	no	—
43	C₂²⋊C₄	—	4	4	no	—
44	C₄²	—	4	4	no	—
45	C₁₆	—	4	1	no	—
46	C₁₆	—	4	1	no	—
47	C₂³	#102	1	2	yes	—
48	C₂³	#106	1	1	yes	characteristic
49	Q₈	—	4	1	no	—
50	D₄	—	4	1	no	—
51	D₄	—	4	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	8	4	no	—
57	C₂×C₄	#95	1	1	yes	characteristic
58	C₂×C₄	#103	1	2	yes	—
59	C₂×C₄	#107	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	2	2	no	—
62	C₈	#99	1	1	yes	characteristic
63	C₈	#99	1	1	yes	characteristic
64	C₂²	—	2	2	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	#340	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	16	4	no	—
72	C₄	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄	#342	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	2	no	—
76	C₂	—	4	1	no	—
77	C₂	#1289	1	1	yes	characteristic

	subgroup	G/H
center	C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₂×#131	C₂²
socle	C₂	#1289
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₂×M₃₂	C₄	semidirect
5. C₂²×C₁₆	C₄	semidirect
9. H(3,ℤ/4)	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2113	2
#1812	2