Subgroups — C₂³×C₈×Q₈ #10574210

Order 512.

9314 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₈	C₂	1	3	yes	maximal
4	C₂³×C₄×C₈	C₂	1	3	yes	maximal
5	C₂⁴×Q₈	C₄	1	1	yes	characteristic
6	C₂⁴×C₈	C₂²	1	3	yes	—
7	C₂⁴×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Q₈	C₄	1	28	yes	—
9	C₂²×C₄×Q₈	C₂²	1	28	yes	—
10	C₂³×C₄⋊C₄	C₄	1	3	yes	—
11	C₂³×C₄²	C₂²	1	3	yes	—
12	C₂×C₈×Q₈	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊C₈	C₂²	1	84	yes	—
14	C₂²×C₄×C₈	C₂²	1	84	yes	—
15	C₂³×Q₈	C₈	1	32	yes	direct product factor
16	C₂³×Q₈	C₂×C₄	1	28	yes	—
17	C₂⁴×C₄	C₂×C₄	1	3	yes	—
18	C₂⁴×C₄	C₂×C₄	1	3	yes	—
19	C₂⁴×C₄	C₂³	1	1	yes	characteristic
20	C₂³×C₈	—	2	24	no	—
21	C₂³×C₈	Q₈	1	16	yes	direct product factor
22	C₂³×C₈	C₂³	1	42	yes	—
23	C₂³×C₈	C₂³	1	14	yes	—
24	C₂×C₄×Q₈	C₂×C₄	1	336	yes	—
25	C₂×C₄×Q₈	C₂³	1	112	yes	—
26	C₂²×C₄⋊C₄	C₂×C₄	1	84	yes	—
27	C₂²×C₄²	C₂×C₄	1	42	yes	—
28	C₂²×C₄²	C₂³	1	42	yes	—
29	C₈×Q₈	C₂³	1	512	yes	direct product factor
30	C₂×C₄⋊C₈	C₂³	1	336	yes	—
31	C₂×C₄×C₈	C₂³	1	336	yes	—
32	C₂⁵	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Q₈	C₂×C₈	1	448	yes	direct product factor
34	C₂²×Q₈	C₂²×C₄	1	112	yes	—
35	C₂³×C₄	—	2	24	no	—
36	C₂³×C₄	C₂×C₈	1	48	yes	—
37	C₂³×C₄	C₂²×C₄	1	42	yes	—
38	C₂³×C₄	C₂²×C₄	1	7	yes	—
39	C₂³×C₄	C₂²×C₄	1	42	yes	—
40	C₂³×C₄	C₂×Q₈	1	8	yes	—
41	C₂³×C₄	#41	1	8	yes	—
42	C₂³×C₄	C₂⁴	1	7	yes	—
43	C₂²×C₈	—	2	168	no	—
44	C₂²×C₈	C₂×Q₈	1	112	yes	direct product factor
45	C₂²×C₈	C₂⁴	1	84	yes	—
46	C₂²×C₈	C₂⁴	1	28	yes	—
47	C₄×Q₈	C₂²×C₄	1	448	yes	—
48	C₄×Q₈	C₂⁴	1	64	yes	—
49	C₂×C₄⋊C₄	C₂²×C₄	1	336	yes	—
50	C₂×C₄²	C₂²×C₄	1	252	yes	—
51	C₂×C₄²	C₂⁴	1	84	yes	—
52	C₄⋊C₈	C₂⁴	1	192	yes	—
53	C₄×C₈	C₂⁴	1	192	yes	—
54	C₂⁴	C₄×Q₈	1	8	yes	—
55	C₂⁴	C₂²×C₈	1	8	yes	—
56	C₂⁴	#131	1	8	yes	—
57	C₂⁴	C₂³×C₄	1	7	yes	—
58	C₂×Q₈	C₂²×C₈	1	896	yes	direct product factor
59	C₂×Q₈	C₂³×C₄	1	64	yes	—
60	C₂²×C₄	—	2	168	no	—
61	C₂²×C₄	C₄×Q₈	1	56	yes	—
62	C₂²×C₄	C₂²×C₈	1	336	yes	—
63	C₂²×C₄	C₂³×C₄	1	84	yes	—
64	C₂²×C₄	C₂³×C₄	1	21	yes	—
65	C₂²×C₄	C₂³×C₄	1	84	yes	—
66	C₂²×C₄	C₂²×Q₈	1	28	yes	—
67	C₂²×C₄	C₂×#41	1	28	yes	—
68	C₂²×C₄	C₂⁵	1	7	yes	—
69	C₂×C₈	—	2	168	no	—
70	C₂×C₈	C₂²×Q₈	1	112	yes	direct product factor
71	C₂×C₈	C₂⁵	1	24	yes	—
72	C₂×C₈	C₂⁵	1	8	yes	—
73	C₄⋊C₄	C₂³×C₄	1	192	yes	—
74	C₄²	C₂³×C₄	1	168	yes	—
75	C₄²	C₂⁵	1	24	yes	—
76	C₂³	C₈×Q₈	1	64	yes	direct product factor
77	C₂³	C₂×C₄×Q₈	1	28	yes	—
78	C₂³	C₂³×C₈	1	28	yes	—
79	C₂³	C₂×#131	1	28	yes	—
80	C₂³	C₂⁴×C₄	1	7	yes	—
81	Q₈	C₂³×C₈	1	256	yes	direct product factor
82	C₂×C₄	—	2	168	no	—
83	C₂×C₄	C₂×C₄×Q₈	1	84	yes	—
84	C₂×C₄	C₂³×C₈	1	336	yes	—
85	C₂×C₄	C₂⁴×C₄	1	24	yes	—
86	C₂×C₄	C₂⁴×C₄	1	7	yes	—
87	C₂×C₄	C₂⁴×C₄	1	24	yes	—
88	C₂×C₄	C₂³×Q₈	1	14	yes	—
89	C₂×C₄	C₂²×#41	1	14	yes	—
90	C₂×C₄	C₂⁶	1	1	yes	characteristic
91	C₈	—	2	24	no	—
92	C₈	C₂³×Q₈	1	16	yes	direct product factor
93	C₂²	C₂×C₈×Q₈	1	112	yes	direct product factor
94	C₂²	C₂²×C₄×Q₈	1	14	yes	—
95	C₂²	C₂⁴×C₈	1	14	yes	—
96	C₂²	C₂²×#131	1	14	yes	—
97	C₂²	C₂⁵×C₄	1	1	yes	characteristic
98	C₄	—	2	24	no	—
99	C₄	C₂²×C₄×Q₈	1	14	yes	—
100	C₄	C₂⁴×C₈	1	48	yes	—
101	C₄	C₂⁴×Q₈	1	1	yes	characteristic
102	C₄	C₂³×#41	1	1	yes	characteristic
103	C₂	C₂²×C₈×Q₈	1	28	yes	direct product factor
104	C₂	C₂³×C₄×Q₈	1	1	yes	characteristic
105	C₂	C₂⁵×C₈	1	1	yes	characteristic
106	C₂	C₂³×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₈	C₂²
commutator	C₂	C₂⁵×C₈
Frattini, cosocle	C₂×C₄	C₂⁶
socle, prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×Q₈	C₂	direct (N × H)
12. C₂×C₈×Q₈	C₂²	direct (N × H)
15. C₂³×Q₈	C₈	direct (N × H)
21. C₂³×C₈	Q₈	direct (N × H)
29. C₈×Q₈	C₂³	direct (N × H)
33. C₂²×Q₈	C₂×C₈	direct (N × H)
44. C₂²×C₈	C₂×Q₈	direct (N × H)
58. C₂×Q₈	C₂²×C₈	direct (N × H)
70. C₂×C₈	C₂²×Q₈	direct (N × H)
76. C₂³	C₈×Q₈	direct (N × H)
81. Q₈	C₂³×C₈	direct (N × H)
92. C₈	C₂³×Q₈	direct (N × H)
93. C₂²	C₂×C₈×Q₈	direct (N × H)
103. C₂	C₂²×C₈×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×Q₈	2
C₂²×C₈×Q₈	2
C₂³×C₄⋊C₈	2
C₂³×C₄×C₈	2

Subgroups — C23×C8×Q8 #10574210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₈×Q₈ #10574210