Subgroups — #12556

Order 256.

303 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#427	C₂	1	2	yes	maximal
3	#1380	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈	C₂²	1	1	yes	characteristic
5	C₂×#117	—	2	2	no	—
6	C₂²×C₁₆	—	2	4	no	—
7	C₂²×C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	#427	—	2	4	no	—
10	#427	C₄	1	8	yes	complemented
11	C₂×#107	C₂²	1	4	yes	—
12	C₂×#106	—	2	2	no	—
13	C₂³×C₄	D₄	1	1	yes	characteristic
14	C₂²×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	D₄	1	4	yes	—
18	C₂²×C₈	Q₈	1	2	yes	—
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	#117	—	2	8	no	—
21	#117	—	4	4	no	—
22	C₂×C₄⋊C₄	—	2	4	no	—
23	C₂×C₂²⋊C₄	—	4	2	no	—
24	C₂×C₄²	—	4	2	no	—
25	C₂×C₁₆	—	2	8	no	—
26	C₂×C₁₆	—	2	4	no	—
27	C₂×C₁₆	C₂×C₄	1	4	yes	—
28	#107	—	2	4	no	—
29	#107	C₂×C₄	1	8	yes	—
30	#106	—	2	8	no	—
31	C₂⁴	D₈	1	1	yes	characteristic
32	C₂²×C₄	—	4	4	no	—
33	C₂²×C₄	C₂²⋊C₄	1	4	yes	—
34	C₂²×C₄	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄	D₈	1	1	yes	characteristic
36	C₂²×C₄	SD₁₆	1	4	yes	—
37	C₂²×C₄	Q₁₆	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	C₄²	1	4	yes	—
40	C₂×C₈	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₈	C₄⋊C₄	1	8	yes	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂²×C₄	1	2	yes	—
46	C₂×C₈	C₂×D₄	1	2	yes	—
47	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄	—	2	8	no	—
49	C₄⋊C₄	—	4	4	no	—
50	C₂²⋊C₄	—	4	8	no	—
51	C₄²	—	4	8	no	—
52	C₁₆	—	2	4	no	—
53	C₂³	—	2	4	no	—
54	C₂³	#102	1	4	yes	—
55	C₂³	#107	1	2	yes	—
56	C₂³	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄	—	4	8	no	—
58	C₂×C₄	—	8	8	no	—
59	C₂×C₄	#95	1	4	yes	—
60	C₂×C₄	#102	1	4	yes	—
61	C₂×C₄	#103	1	8	yes	—
62	C₂×C₄	#106	1	4	yes	—
63	C₂×C₄	#107	1	2	yes	—
64	C₂×C₄	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
68	C₈	#95	1	4	yes	—
69	C₈	C₂×C₄²	1	1	yes	characteristic
70	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈	C₂×C₄⋊C₄	1	2	yes	—
72	C₂²	—	2	8	no	—
73	C₂²	—	2	4	no	—
74	C₂²	#340	1	4	yes	—
75	C₂²	C₂×#102	1	2	yes	—
76	C₂²	C₂×#107	1	1	yes	characteristic
77	C₂²	#508	1	4	yes	—
78	C₄	—	8	8	no	—
79	C₄	#340	1	4	yes	—
80	C₄	C₂×#95	1	1	yes	characteristic
81	C₄	C₂×#103	1	2	yes	—
82	C₄	C₂×#106	1	1	yes	characteristic
83	C₂	—	2	4	no	—
84	C₂	C₂×#340	1	1	yes	characteristic
85	C₂	#2138	1	4	yes	—
86	C₂	#2158	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂³	C₂×D₈
prime radical	C₂⁴	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1380	C₂	semidirect
7. C₂²×C₁₆	C₄	semidirect
10. #427	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆	2
C₂×#427	2
#1380	2