Subgroups — #12839

Order 256.

181 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	1	yes	maximal, characteristic
2	#2158	C₂	1	2	yes	maximal, complemented
3	#1812	C₂	1	2	yes	maximal
4	#1384	C₂	1	2	yes	maximal, complemented
5	C₂×#131	C₂²	1	1	yes	characteristic
6	#504	—	2	2	no	—
7	#504	—	2	2	no	—
8	C₂×M₃₂	C₄	1	2	yes	complemented
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#427	C₂²	1	4	yes	—
13	#410	—	2	2	no	—
14	#367	—	2	2	no	—
15	#366	—	2	2	no	—
16	H(3,ℤ/4)	—	2	2	no	—
17	C₂×#41	D₄	1	1	yes	characteristic
18	#131	—	2	2	no	—
19	#131	—	2	2	no	—
20	C₂×M₁₆	D₄	1	1	yes	characteristic
21	C₂×M₁₆	Q₈	1	2	yes	—
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	#117	—	2	4	no	—
25	M₃₂	—	2	2	no	—
26	M₃₂	—	2	4	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆	D₄	1	2	yes	—
33	C₂×C₁₆	D₄	1	2	yes	—
34	#107	—	2	4	no	—
35	#106	—	2	4	no	—
36	#99	—	4	4	no	—
37	#41	—	2	2	no	—
38	#41	—	2	2	no	—
39	C₂×Q₈	Q₁₆	1	1	yes	characteristic
40	C₂×D₄	D₈	1	2	yes	—
41	C₂×D₄	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
44	M₁₆	—	2	2	no	—
45	M₁₆	—	2	4	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	2	yes	—
51	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈	#41	1	2	yes	—
53	C₄⋊C₄	—	4	4	no	—
54	C₂²⋊C₄	—	4	4	no	—
55	C₄²	—	4	4	no	—
56	C₁₆	—	2	2	no	—
57	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
58	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₆	C₂²⋊C₄	1	2	yes	—
60	C₂³	#107	1	2	yes	—
61	C₂³	C₂×D₈	1	1	yes	characteristic
62	Q₈	—	2	2	no	—
63	D₄	—	2	4	no	—
64	D₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	8	4	no	—
68	C₂×C₄	#107	1	2	yes	—
69	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄	#121	1	2	yes	—
71	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄	#135	1	2	yes	—
73	C₈	—	2	2	no	—
74	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₈	#117	1	1	yes	characteristic
76	C₈	C₂²⋊Q₈	1	2	yes	—
77	C₂²	—	2	2	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	2	no	—
80	C₂²	C₂×#107	1	1	yes	characteristic
81	C₂²	#466	1	2	yes	—
82	C₄	—	2	2	no	—
83	C₄	—	16	4	no	—
84	C₄	#380	1	1	yes	characteristic
85	C₄	#427	1	1	yes	characteristic
86	C₄	#467	1	2	yes	—
87	C₂	—	2	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	4	2	no	—
90	C₂	#1808	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#427
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1808
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2158	C₂	semidirect
4. #1384	C₂	semidirect
8. C₂×M₃₂	C₄	semidirect
10. C₂²×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×#504	2
#2158	2
#1812	2
#1384	2