Subgroups — #12665

Order 256.

163 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	1	yes	maximal, characteristic
2	#2108	C₂	1	2	yes	maximal
3	C₂×#346	C₂	1	1	yes	maximal, characteristic
4	#1772	C₂	1	1	yes	maximal, characteristic
5	#1379	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#108	C₄	1	1	yes	characteristic
10	#427	C₄	1	1	yes	characteristic
11	#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#364	C₄	1	4	yes	complemented
14	#346	—	2	2	no	—
15	#346	C₄	1	4	yes	complemented
16	C₄×C₁₆	—	2	2	no	—
17	#339	C₄	1	2	yes	—
18	C₂×M₁₆	C₂×C₄	1	2	yes	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	#117	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	M₃₂	—	2	2	no	—
24	M₃₂	C₂×C₄	1	4	yes	—
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₆	D₄	1	2	yes	—
31	C₂×C₁₆	Q₈	1	2	yes	—
32	#108	C₈	1	4	yes	—
33	#107	C₈	1	2	yes	—
34	#106	C₈	1	2	yes	—
35	#97	—	2	2	no	—
36	C₄×C₈	—	2	2	no	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
41	M₁₆	—	2	2	no	—
42	M₁₆	C₂×C₈	1	4	yes	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	C₄²	1	2	yes	—
45	C₂×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈	#41	1	1	yes	characteristic
53	C₂×C₈	#41	1	1	yes	characteristic
54	C₄⋊C₄	C₂×C₈	1	4	yes	—
55	C₂²⋊C₄	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	—	2	1	no	—
58	C₄²	C₄²	1	2	yes	—
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	4	no	—
61	C₁₆	C₄⋊C₄	1	4	yes	—
62	C₂³	#131	1	1	yes	characteristic
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄	#131	1	1	yes	characteristic
72	C₈	—	2	4	no	—
73	C₈	C₄×C₈	1	2	yes	—
74	C₈	C₄×C₈	1	2	yes	—
75	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈	#117	1	1	yes	characteristic
77	C₈	C₄×D₄	1	1	yes	characteristic
78	C₈	C₄×Q₈	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	#405	1	1	yes	characteristic
81	C₂²	C₈×D₄	1	1	yes	characteristic
82	C₂²	C₈×Q₈	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	—	4	4	no	—
85	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₄	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄	C₈×D₄	1	1	yes	characteristic
88	C₄	C₈×Q₈	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2	1	no	—
91	C₂	C₈×C₄⋊C₄	1	1	yes	characteristic
92	C₂	#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₈×D₄
prime radical	C₂³	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ 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. #364	C₄	semidirect
15. #346	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
#2108	2
C₂×#346	2
#1772	2
#1379	2