Subgroups — #13264

Order 256.

193 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#346	C₂	1	1	yes	maximal, characteristic
4	#1374	C₂	1	4	yes	maximal
5	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	C₈×Q₈	—	2	2	no	—
9	C₈×Q₈	—	2	2	no	—
10	C₂×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	2	no	—
15	#346	—	2	2	no	—
16	C₂²×Q₈	C₈	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	C₄×Q₈	—	2	2	no	—
21	C₄×Q₈	—	2	2	no	—
22	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₈	1	2	yes	—
24	C₂×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	4	no	—
30	C₂×C₁₆	Q₈	1	4	yes	complemented
31	C₄⋊C₈	—	2	4	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₄×C₈	D₄	1	2	yes	—
37	C₂×Q₈	—	2	2	no	—
38	C₂×Q₈	—	2	4	no	—
39	C₂²×C₄	C₂×C₈	1	2	yes	—
40	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×C₈	1	2	yes	—
43	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	4	no	—
47	C₂×C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×Q₈	1	2	yes	—
51	C₂×C₈	#41	1	2	yes	—
52	C₄⋊C₄	—	2	4	no	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄²	—	2	2	no	—
55	C₄²	—	2	2	no	—
56	C₄²	C₂²⋊C₄	1	2	yes	—
57	C₄²	M₁₆	1	2	yes	—
58	C₁₆	—	2	4	no	—
59	C₁₆	—	4	2	no	—
60	C₂³	C₂²×C₈	1	1	yes	characteristic
61	Q₈	—	2	8	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄	M₃₂	1	4	yes	—
71	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄	C₄×Q₈	1	2	yes	—
73	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄	#131	1	2	yes	—
75	C₈	—	2	2	no	—
76	C₈	—	4	2	no	—
77	C₈	C₂²⋊Q₈	1	2	yes	—
78	C₈	#124	1	1	yes	characteristic
79	C₈	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂²	C₈×Q₈	1	2	yes	—
82	C₂²	C₂×M₃₂	1	2	yes	—
83	C₂²	#504	1	2	yes	—
84	C₄	—	2	2	no	—
85	C₄	—	2	4	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	2	no	—
88	C₄	#391	1	1	yes	characteristic
89	C₄	#408	1	1	yes	characteristic
90	C₄	#446	1	2	yes	—
91	C₂	#1984	1	1	yes	characteristic
92	C₂	#2112	1	1	yes	characteristic
93	C₂	#2113	1	1	yes	characteristic
94	C₂	#2183	1	4	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂³
commutator, socle, prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×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
30. C₂×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈	2
C₂×C₄⋊C₁₆	2
C₂×#346	2
#1374	2