Subgroups — #12840

Order 256.

177 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	C₂×#368	C₂	1	2	yes	maximal, complemented
3	#1813	C₂	1	2	yes	maximal
4	#1385	C₂	1	2	yes	maximal, complemented
5	C₂×#131	C₂²	1	1	yes	characteristic
6	#504	—	2	4	no	—
7	C₂×M₃₂	C₄	1	2	yes	—
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₄	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	4	yes	—
12	#413	—	2	2	no	—
13	#368	—	2	4	no	—
14	#338	—	2	2	no	—
15	C₂×#41	D₄	1	1	yes	characteristic
16	#131	—	2	2	no	—
17	#131	—	2	2	no	—
18	C₂×M₁₆	—	2	4	no	—
19	C₂×M₁₆	D₄	1	1	yes	characteristic
20	C₂×M₁₆	Q₈	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	M₃₂	—	2	2	no	—
24	M₃₂	—	2	4	no	—
25	C₂×C₁₆	—	2	4	no	—
26	C₂×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆	D₄	1	4	yes	—
30	#108	—	2	4	no	—
31	#108	—	2	4	no	—
32	#101	—	4	2	no	—
33	#100	—	4	2	no	—
34	#41	—	2	2	no	—
35	#41	—	2	2	no	—
36	C₂×Q₈	D₈	1	1	yes	characteristic
37	C₂×D₄	D₈	1	1	yes	characteristic
38	C₂×D₄	Q₁₆	1	2	yes	—
39	C₂²×C₄	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	M₁₆	—	2	2	no	—
42	M₁₆	—	2	4	no	—
43	M₁₆	—	4	4	no	—
44	M₁₆	—	4	4	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	4	4	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	C₁₆	—	2	2	no	—
54	C₁₆	C₂²⋊C₄	1	2	yes	—
55	C₁₆	C₂²⋊C₄	1	2	yes	—
56	C₂³	#107	1	2	yes	—
57	C₂³	C₂×Q₁₆	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	#107	1	2	yes	—
64	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄	#121	1	2	yes	—
66	C₂×C₄	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄	#135	1	2	yes	—
68	C₈	—	2	2	no	—
69	C₈	—	8	4	no	—
70	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈	#117	1	1	yes	characteristic
72	C₈	C₂²⋊Q₈	1	2	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	—	4	1	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×#107	1	1	yes	characteristic
77	C₂²	#467	1	2	yes	—
78	C₄	—	2	2	no	—
79	C₄	#380	1	1	yes	characteristic
80	C₄	#427	1	1	yes	characteristic
81	C₄	#466	1	2	yes	—
82	C₂	—	2	1	no	—
83	C₂	—	2	2	no	—
84	C₂	—	4	2	no	—
85	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. C₂×#368	C₂	semidirect
4. #1385	C₂	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
C₂×#368	2
#1813	2
#1385	2