Subgroups — #12559

Order 256.

277 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	2	yes	maximal
3	#1381	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆	—	2	2	no	—
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂×M₃₂	—	2	4	no	—
7	C₂×M₃₂	C₄	1	4	yes	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	C₂×#108	—	2	2	no	—
10	C₂×#108	—	2	4	no	—
11	C₂×#108	C₄	1	8	yes	—
12	C₂×#108	C₂²	1	4	yes	—
13	C₂³×C₄	D₄	1	1	yes	characteristic
14	C₂×M₁₆	—	2	8	no	—
15	C₂×M₁₆	—	2	4	no	—
16	C₂×M₁₆	—	4	2	no	—
17	C₂×M₁₆	—	4	4	no	—
18	C₂²×C₈	—	4	2	no	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₈	D₄	1	4	yes	—
23	C₂²×C₈	Q₈	1	2	yes	—
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	M₃₂	—	2	8	no	—
26	C₂×C₁₆	—	2	4	no	—
27	C₂×C₁₆	C₂×C₄	1	4	yes	—
28	#108	—	2	8	no	—
29	#108	—	2	4	no	—
30	#108	C₂×C₄	1	8	yes	—
31	C₂⁴	Q₁₆	1	1	yes	characteristic
32	C₂²×C₄	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄	D₈	1	2	yes	—
35	C₂²×C₄	SD₁₆	1	4	yes	—
36	C₂²×C₄	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
38	M₁₆	—	2	8	no	—
39	M₁₆	—	4	8	no	—
40	M₁₆	—	4	4	no	—
41	C₂×C₈	—	4	8	no	—
42	C₂×C₈	—	4	4	no	—
43	C₂×C₈	C₄²	1	4	yes	—
44	C₂×C₈	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₈	C₄⋊C₄	1	8	yes	—
48	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₈	C₂×D₄	1	2	yes	—
51	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₁₆	—	2	4	no	—
53	C₂³	—	4	2	no	—
54	C₂³	#103	1	4	yes	—
55	C₂³	#107	1	2	yes	—
56	C₂³	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄	#95	1	4	yes	—
58	C₂×C₄	#102	1	4	yes	—
59	C₂×C₄	#102	1	4	yes	—
60	C₂×C₄	#103	1	4	yes	—
61	C₂×C₄	#106	1	4	yes	—
62	C₂×C₄	#107	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄	C₂×SD₁₆	1	2	yes	—
67	C₈	—	4	8	no	—
68	C₈	#95	1	4	yes	—
69	C₈	C₂×C₄²	1	1	yes	characteristic
70	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈	C₂×C₄⋊C₄	1	2	yes	—
72	C₂²	—	2	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	4	4	no	—
75	C₂²	#340	1	4	yes	—
76	C₂²	C₂×#103	1	2	yes	—
77	C₂²	C₂×#107	1	1	yes	characteristic
78	C₄	#340	1	4	yes	—
79	C₄	C₂×#95	1	1	yes	characteristic
80	C₄	C₂×#102	1	2	yes	—
81	C₄	C₂×#106	1	1	yes	characteristic
82	C₂	—	2	4	no	—
83	C₂	—	2	2	no	—
84	C₂	C₂×#340	1	1	yes	characteristic
85	C₂	#2161	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂⁴	Q₁₆
LCS	G ▷ 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
3. #1381	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂²×#108	2
#1381	2