Subgroups — #13170

Order 256.

139 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×#346	C₂	1	1	yes	maximal, characteristic
4	#1945	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	#424	—	2	2	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#368	—	2	2	no	—
13	#367	C₄	1	2	yes	complemented
14	#366	C₄	1	2	yes	complemented
15	#346	—	2	2	no	—
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	1	yes	characteristic
25	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆	D₄	1	2	yes	—
27	C₂×C₁₆	Q₈	1	2	yes	—
28	C₂×C₁₆	Q₈	1	2	yes	—
29	#108	—	2	2	no	—
30	#108	—	2	2	no	—
31	#107	C₂×C₄	1	4	yes	—
32	#106	—	2	2	no	—
33	C₄⋊C₈	—	4	4	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	—	2	1	no	—
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
38	M₁₆	—	4	2	no	—
39	M₁₆	—	4	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	D₈	1	2	yes	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈	#41	1	1	yes	characteristic
49	C₂×C₈	#41	1	1	yes	characteristic
50	C₄⋊C₄	—	2	4	no	—
51	C₂²⋊C₄	—	4	2	no	—
52	C₄²	—	2	1	no	—
53	C₄²	—	4	2	no	—
54	C₄²	Q₁₆	1	2	yes	—
55	C₁₆	—	2	2	no	—
56	C₁₆	C₄⋊C₄	1	2	yes	—
57	C₁₆	C₄⋊C₄	1	2	yes	—
58	C₂³	#135	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	4	4	no	—
62	C₂×C₄	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄	#121	1	1	yes	characteristic
64	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄	#135	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	8	2	no	—
70	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₈	C₄×Q₈	1	1	yes	characteristic
72	C₈	C₂²⋊Q₈	1	1	yes	characteristic
73	C₈	#125	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	C₄×D₈	1	1	yes	characteristic
76	C₂²	#467	1	1	yes	characteristic
77	C₂²	#499	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	8	4	no	—
80	C₄	#389	1	1	yes	characteristic
81	C₄	C₄×Q₁₆	1	1	yes	characteristic
82	C₄	#466	1	1	yes	characteristic
83	C₄	#500	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	1	no	—
86	C₂	#1942	1	1	yes	characteristic
87	C₂	#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₄×D₈
prime radical	C₂³	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
13. #367	C₄	semidirect
14. #366	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×#346	2
#1945	2
#1382	2