Subgroups — #7178

Order 256.

137 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346	C₂	1	1	yes	maximal, characteristic
2	C₄×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆	—	2	1	no	—
5	C₂×#107	C₄	1	2	yes	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄×C₄⋊C₄	—	2	1	no	—
9	#346	—	2	1	no	—
10	#346	C₄	1	2	yes	complemented
11	#340	—	2	1	no	—
12	#334	—	2	1	no	—
13	#334	C₄	1	2	yes	complemented
14	C₂²×C₈	—	4	1	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄²	—	4	1	no	—
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₁₆	—	2	4	no	—
21	C₂×C₁₆	—	2	2	no	—
22	#107	—	2	2	no	—
23	#107	C₈	1	4	yes	complemented
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	4	1	no	—
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	Q₈	1	1	yes	characteristic
30	#95	—	4	1	no	—
31	C₂²×C₄	—	4	2	no	—
32	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	D₈	1	1	yes	characteristic
34	C₂²×C₄	SD₁₆	1	1	yes	characteristic
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	—	8	2	no	—
38	C₂×C₈	C₄²	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₈	M₁₆	1	2	yes	—
44	C₄⋊C₄	—	2	4	no	—
45	C₄⋊C₄	—	4	2	no	—
46	C₄²	—	8	2	no	—
47	C₄²	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²	C₄⋊C₄	1	1	yes	characteristic
49	C₄²	SD₁₆	1	1	yes	characteristic
50	C₄²	Q₁₆	1	1	yes	characteristic
51	C₁₆	—	2	4	no	—
52	C₂³	#102	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	4	no	—
57	C₂×C₄	—	8	2	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	#95	1	1	yes	characteristic
60	C₂×C₄	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄	#102	1	1	yes	characteristic
62	C₂×C₄	#103	1	1	yes	characteristic
63	C₂×C₄	#103	1	1	yes	characteristic
64	C₂×C₄	#104	1	2	yes	—
65	C₂×C₄	#106	1	1	yes	characteristic
66	C₂×C₄	#107	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₈	—	8	2	no	—
70	C₈	C₄×C₈	1	1	yes	characteristic
71	C₈	#97	1	1	yes	characteristic
72	C₈	C₄⋊C₈	1	2	yes	—
73	C₂²	D₄⋊C₈	1	2	yes	—
74	C₂²	#340	1	1	yes	characteristic
75	C₂²	#360	1	2	yes	—
76	C₂²	#361	1	1	yes	characteristic
77	C₂²	#362	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	8	4	no	—
83	C₄	Q₈⋊C₈	1	2	yes	—
84	C₄	#336	1	1	yes	characteristic
85	C₄	#339	1	1	yes	characteristic
86	C₂	#1279	1	1	yes	characteristic
87	C₂	#1333	1	2	yes	—
88	C₂	#1334	1	2	yes	—
89	C₂	#1386	1	1	yes	characteristic
90	C₂	#1387	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#102
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. #346	C₄	semidirect
13. #334	C₄	semidirect
23. #107	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#346	2
C₄×#107	2
C₂×#334	2