Subgroups — #13457

Order 256.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×#365	C₂	1	2	yes	maximal
3	#1945	C₂	1	1	yes	maximal, characteristic
4	#1386	C₂	1	2	yes	maximal
5	#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	—	2	1	no	—
7	C₂×M₃₂	C₂²	1	2	yes	—
8	#498	—	2	4	no	—
9	C₂×#108	C₂²	1	1	yes	characteristic
10	#427	C₂²	1	1	yes	characteristic
11	C₂×#106	—	2	1	no	—
12	C₂×#106	C₂²	1	2	yes	—
13	#424	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#365	—	2	4	no	—
16	#339	—	2	1	no	—
17	C₂²×Q₈	—	4	1	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈	—	2	4	no	—
22	C₄⋊Q₈	—	4	2	no	—
23	#117	—	2	1	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	—	4	1	no	—
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	M₃₂	—	2	2	no	—
28	M₃₂	Q₈	1	4	yes	complemented
29	C₂×C₁₆	—	2	2	no	—
30	#108	—	2	2	no	—
31	#107	Q₈	1	2	yes	complemented
32	#106	—	2	4	no	—
33	#106	—	2	2	no	—
34	#106	—	2	1	no	—
35	#106	D₄	1	4	yes	—
36	C₄⋊C₈	—	4	2	no	—
37	C₄×C₈	—	2	2	no	—
38	C₂×Q₈	—	4	4	no	—
39	C₂×Q₈	—	8	4	no	—
40	C₂²×C₄	—	2	1	no	—
41	C₂²×C₄	—	4	2	no	—
42	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
43	M₁₆	—	4	1	no	—
44	M₁₆	—	4	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	SD₁₆	1	2	yes	—
49	C₂×C₈	C₂×D₄	1	2	yes	—
50	C₂×C₈	C₂×Q₈	1	2	yes	—
51	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈	#41	1	1	yes	characteristic
53	C₄⋊C₄	—	2	4	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	4	4	no	—
56	C₄⋊C₄	—	4	2	no	—
57	C₂²⋊C₄	—	4	1	no	—
58	C₄²	—	2	1	no	—
59	C₄²	—	4	1	no	—
60	C₄²	SD₁₆	1	2	yes	—
61	C₁₆	—	2	4	no	—
62	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Q₈	—	8	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	#120	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄	#137	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	8	1	no	—
77	C₈	C₂²⋊Q₈	1	2	yes	—
78	C₈	C₄⋊Q₈	1	2	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	#450	1	1	yes	characteristic
81	C₂²	#476	1	1	yes	characteristic
82	C₂²	#501	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	8	4	no	—
85	C₄	—	8	2	no	—
86	C₄	#395	1	1	yes	characteristic
87	C₄	#448	1	1	yes	characteristic
88	C₄	#475	1	1	yes	characteristic
89	C₄	#498	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	#2058	1	1	yes	characteristic
93	C₂	#2230	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×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
28. M₃₂	Q₈	semidirect
31. #107	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#498	2
C₂×#365	2
#1945	2
#1386	2
#1376	2