Subgroups — #21737

Order 256.

331 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3162	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	2	yes	maximal
3	#1995	C₂	1	2	yes	maximal
4	#1945	C₂	1	4	yes	maximal
5	#1849	C₂	1	1	yes	maximal, characteristic
6	#1773	C₂	1	4	yes	maximal
7	#1726	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	2	yes	—
9	#501	C₄	1	8	yes	complemented
10	#446	—	2	4	no	—
11	C₈×Q₈	—	2	4	no	—
12	C₂×#108	C₂²	1	4	yes	—
13	#427	C₂²	1	4	yes	—
14	#424	C₂²	1	4	yes	—
15	#423	C₂²	1	2	yes	—
16	#405	C₂²	1	4	yes	—
17	C₂×#97	C₂²	1	1	yes	characteristic
18	C₂×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#376	C₂²	1	1	yes	characteristic
22	#339	C₂²	1	8	yes	—
23	C₂×M₁₆	C₂³	1	4	yes	—
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₈	C₂³	1	2	yes	—
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈	C₂×C₄	1	4	yes	—
28	#125	C₂×C₄	1	4	yes	—
29	C₂²⋊Q₈	—	2	4	no	—
30	C₄×Q₈	—	2	4	no	—
31	#117	C₂³	1	4	yes	—
32	C₂×C₄²	C₂³	1	2	yes	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	#108	—	2	8	no	—
35	#107	C₂×C₄	1	8	yes	—
36	#106	C₂×C₄	1	8	yes	—
37	C₄⋊C₈	—	2	8	no	—
38	C₄⋊C₈	—	2	4	no	—
39	#97	—	2	4	no	—
40	#97	C₂×C₄	1	4	yes	—
41	#97	Q₈	1	8	yes	complemented
42	C₄×C₈	—	2	4	no	—
43	C₄×C₈	—	2	4	no	—
44	C₄×C₈	D₄	1	4	yes	—
45	#95	—	2	2	no	—
46	C₂×Q₈	—	2	4	no	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	#41	1	2	yes	—
49	C₂²×C₄	C₂⁴	1	1	yes	characteristic
50	M₁₆	—	2	8	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	C₂²×C₄	1	4	yes	—
54	C₂×C₈	C₂×D₄	1	2	yes	—
55	C₂×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₈	C₂×Q₈	1	4	yes	—
57	C₂×C₈	C₂×Q₈	1	4	yes	—
58	C₂×C₈	#41	1	4	yes	—
59	C₄⋊C₄	—	2	8	no	—
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	C₂²×C₄	1	8	yes	—
62	C₂²⋊C₄	—	2	4	no	—
63	C₄²	—	2	4	no	—
64	C₄²	—	2	2	no	—
65	C₄²	C₂²×C₄	1	2	yes	—
66	C₄²	C₂×D₄	1	2	yes	—
67	C₄²	C₂×Q₈	1	4	yes	—
68	C₂³	C₂×#41	1	1	yes	characteristic
69	Q₈	—	4	8	no	—
70	C₂×C₄	—	2	8	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	C₄×D₄	1	4	yes	—
75	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄	C₂²×Q₈	1	2	yes	—
78	C₂×C₄	C₂×#41	1	2	yes	—
79	C₈	—	2	2	no	—
80	C₈	—	4	4	no	—
81	C₈	C₄×Q₈	1	8	yes	—
82	C₈	C₄⋊Q₈	1	4	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
85	C₂²	#533	1	2	yes	—
86	C₄	—	2	4	no	—
87	C₄	—	4	4	no	—
88	C₄	—	4	8	no	—
89	C₄	C₂×C₄×Q₈	1	2	yes	—
90	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₄	#532	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
94	C₂	#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. #501	C₄	semidirect
41. #97	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3162	2
#2960	2
#1995	2
#1945	2
#1849	2
#1773	2
#1726	2