Subgroups — #12670

Order 256.

159 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368	C₂	1	1	yes	maximal, characteristic
2	#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1772	C₂	1	2	yes	maximal
5	#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#108	C₂²	1	2	yes	—
9	#427	C₂²	1	2	yes	—
10	#405	—	2	2	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#368	C₄	1	4	yes	complemented
13	#367	C₄	1	2	yes	complemented
14	#366	C₄	1	2	yes	complemented
15	C₄×C₁₆	C₄	1	4	yes	complemented
16	#339	—	2	2	no	—
17	C₂×M₁₆	—	2	2	no	—
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	#117	—	2	2	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆	C₂×C₄	1	2	yes	—
29	#108	—	2	2	no	—
30	#108	C₂×C₄	1	4	yes	—
31	#107	C₂×C₄	1	4	yes	—
32	#106	—	2	2	no	—
33	#97	—	4	2	no	—
34	C₄×C₈	—	4	2	no	—
35	C₄×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈	Q₈	1	2	yes	—
37	C₂²×C₄	—	2	1	no	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	M₁₆	—	2	4	no	—
40	M₁₆	—	4	2	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₈	D₈	1	1	yes	characteristic
44	C₂×C₈	D₈	1	1	yes	characteristic
45	C₂×C₈	Q₁₆	1	2	yes	—
46	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈	#41	1	1	yes	characteristic
51	C₂×C₈	#41	1	1	yes	characteristic
52	C₄⋊C₄	—	2	4	no	—
53	C₂²⋊C₄	—	4	2	no	—
54	C₄²	—	2	1	no	—
55	C₄²	—	4	2	no	—
56	C₄²	C₄⋊C₄	1	2	yes	—
57	C₁₆	—	2	2	no	—
58	C₁₆	C₄²	1	2	yes	—
59	C₁₆	C₄²	1	2	yes	—
60	C₂³	#135	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	4	4	no	—
64	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄	#135	1	1	yes	characteristic
70	C₈	—	4	4	no	—
71	C₈	#107	1	2	yes	—
72	C₈	#107	1	2	yes	—
73	C₈	C₂×C₄²	1	1	yes	characteristic
74	C₈	#117	1	1	yes	characteristic
75	C₈	C₄×Q₈	1	1	yes	characteristic
76	C₈	C₄×Q₈	1	1	yes	characteristic
77	C₂²	—	2	2	no	—
78	C₂²	#427	1	1	yes	characteristic
79	C₂²	C₄×D₈	1	1	yes	characteristic
80	C₂²	C₄×Q₁₆	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	8	4	no	—
83	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄	C₂×#107	1	1	yes	characteristic
85	C₄	C₄×D₈	1	1	yes	characteristic
86	C₄	C₄×Q₁₆	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	2	1	no	—
89	C₂	C₄×#107	1	1	yes	characteristic
90	C₂	#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₄×D₈
prime radical	C₂³	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×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
12. #368	C₄	semidirect
13. #367	C₄	semidirect
14. #366	C₄	semidirect
15. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#368	2
#2158	2
C₂×C₄×C₁₆	2
#1772	2
#1382	2