Subgroups — #12911

Order 256.

133 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	#2155	C₂	1	2	yes	maximal
4	#2108	C₂	1	1	yes	maximal, characteristic
5	#1833	C₂	1	2	yes	maximal
6	C₂×M₃₂	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	C₂²	1	1	yes	characteristic
11	#368	—	2	1	no	—
12	#368	—	2	1	no	—
13	#367	—	2	1	no	—
14	#366	—	2	1	no	—
15	#365	—	2	4	no	—
16	#346	C₄	1	2	yes	complemented
17	C₄×C₁₆	C₄	1	2	yes	complemented
18	#344	—	2	2	no	—
19	H(3,ℤ/4)	—	2	2	no	—
20	C₂×M₁₆	—	2	2	no	—
21	C₂×M₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	#117	—	2	2	no	—
24	#117	D₄	1	1	yes	characteristic
25	M₃₂	D₄	1	2	yes	—
26	M₃₂	Q₈	1	2	yes	—
27	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₆	D₄	1	1	yes	characteristic
32	C₂×C₁₆	Q₈	1	1	yes	characteristic
33	C₂×C₁₆	Q₈	1	1	yes	characteristic
34	#108	—	2	2	no	—
35	#108	—	2	2	no	—
36	#107	—	2	2	no	—
37	#106	—	2	2	no	—
38	#97	Q₈	1	2	yes	—
39	C₄×C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	M₁₆	—	2	1	no	—
42	M₁₆	—	2	1	no	—
43	M₁₆	—	4	2	no	—
44	M₁₆	—	4	2	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₂²⋊C₄	—	2	1	no	—
56	C₂²⋊C₄	—	4	2	no	—
57	C₄²	—	4	2	no	—
58	C₄²	C₄⋊C₄	1	2	yes	—
59	C₁₆	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆	C₄⋊C₄	1	1	yes	characteristic
63	C₁₆	C₄⋊C₄	1	4	yes	—
64	C₂³	#135	1	1	yes	characteristic
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	8	2	no	—
67	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄	#127	1	1	yes	characteristic
69	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄	#135	1	1	yes	characteristic
71	C₂×C₄	#135	1	1	yes	characteristic
72	C₂×C₄	#135	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	8	2	no	—
75	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈	C₄⋊Q₈	1	1	yes	characteristic
78	C₈	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²	—	4	1	no	—
80	C₂²	#427	1	1	yes	characteristic
81	C₂²	#495	1	1	yes	characteristic
82	C₂²	#499	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	—	16	2	no	—
85	C₄	#384	1	1	yes	characteristic
86	C₄	#427	1	1	yes	characteristic
87	C₄	#495	1	1	yes	characteristic
88	C₄	#499	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	#1846	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#427
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1846
prime radical	C₂³	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #346	C₄	semidirect
17. 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
#2155	2
#2108	2
#1833	2