Subgroups — #12837

Order 256.

197 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#368	C₂	1	2	yes	maximal, complemented
3	#1810	C₂	1	2	yes	maximal
4	#1381	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆	—	2	4	no	—
7	C₂²×C₁₆	C₄	1	4	yes	—
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#108	C₂²	1	4	yes	—
11	#407	—	2	2	no	—
12	#368	—	2	4	no	—
13	#341	—	2	2	no	—
14	C₂³×C₄	D₄	1	1	yes	characteristic
15	C₂×M₁₆	—	2	4	no	—
16	C₂²×C₈	—	2	2	no	—
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	Q₈	1	2	yes	—
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₁₆	—	2	2	no	—
23	C₂×C₁₆	—	2	8	no	—
24	C₂×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₆	D₄	1	4	yes	—
27	#108	—	2	4	no	—
28	#108	—	2	4	no	—
29	C₂²⋊C₈	—	4	4	no	—
30	C₂⁴	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	—	2	2	no	—
33	C₂²×C₄	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄	D₈	1	2	yes	—
35	C₂²×C₄	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
37	M₁₆	—	4	4	no	—
38	M₁₆	—	4	4	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	—	4	4	no	—
44	C₂×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	2	yes	—
47	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈	#41	1	2	yes	—
49	C₁₆	—	2	2	no	—
50	C₁₆	C₂²⋊C₄	1	4	yes	—
51	C₂³	—	2	4	no	—
52	C₂³	—	2	2	no	—
53	C₂³	#107	1	2	yes	—
54	C₂³	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	#107	1	2	yes	—
60	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄	#121	1	2	yes	—
62	C₂×C₄	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄	#135	1	2	yes	—
64	C₈	—	2	2	no	—
65	C₈	—	8	4	no	—
66	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₈	#117	1	1	yes	characteristic
68	C₈	C₂²⋊Q₈	1	2	yes	—
69	C₂²	—	2	8	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	2	no	—
73	C₂²	#368	1	4	yes	complemented
74	C₂²	C₂×#107	1	1	yes	characteristic
75	C₂²	#467	1	2	yes	—
76	C₄	—	2	2	no	—
77	C₄	#380	1	1	yes	characteristic
78	C₄	#427	1	1	yes	characteristic
79	C₄	#466	1	2	yes	—
80	C₂	—	2	4	no	—
81	C₂	—	2	2	no	—
82	C₂	#1808	1	1	yes	characteristic
83	C₂	C₂×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×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
2. C₂×#368	C₂	semidirect
4. #1381	C₂	semidirect
73. C₂²	#368	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆	2
C₂×#368	2
#1810	2
#1381	2