Subgroups — Q₈×#322 #184374

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#8261	C₂	1	6	yes	maximal
4	#8260	C₂	1	3	yes	maximal
5	C₄×#322	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈	C₄	1	4	yes	—
7	C₂×C₈×Q₈	C₄	1	2	yes	—
8	C₄²×Q₈	C₂²	1	1	yes	characteristic
9	C₈×C₄⋊C₄	C₄	1	12	yes	—
10	C₈×C₄⋊C₄	C₄	1	6	yes	—
11	C₄×C₄⋊C₈	C₂²	1	6	yes	—
12	C₄×C₄⋊C₈	C₂²	1	3	yes	—
13	C₄²×C₈	C₂²	1	3	yes	—
14	C₄²×C₈	C₂²	1	6	yes	—
15	C₂×#322	C₂²	1	3	yes	—
16	C₂×#322	C₂²	1	6	yes	—
17	C₂×#322	C₂²	1	1	yes	characteristic
18	#1273	C₂²	1	6	yes	—
19	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
21	C₈×Q₈	—	2	16	no	—
22	C₈×Q₈	C₈	1	16	yes	complemented
23	C₂×C₄⋊C₈	C₂×C₄	1	12	yes	—
24	C₂×C₄⋊C₈	C₂×C₄	1	6	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
27	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
29	C₂×C₄×C₈	C₂³	1	6	yes	—
30	C₂×C₄×C₈	C₂³	1	3	yes	—
31	C₂×C₄×C₈	C₂³	1	2	yes	—
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₄⋊C₄	C₂×C₄	1	3	yes	—
34	C₄×C₄⋊C₄	C₂×C₄	1	6	yes	—
35	C₄³	C₂³	1	3	yes	—
36	#336	C₂×C₄	1	12	yes	—
37	#336	C₂×C₄	1	6	yes	—
38	#322	—	2	6	no	—
39	#322	—	2	12	no	—
40	#322	Q₈	1	4	yes	direct product factor
41	C₂²×Q₈	C₄²	1	1	yes	characteristic
42	C₂²×C₈	C₂²×C₄	1	12	yes	—
43	C₂²×C₈	C₂²×C₄	1	4	yes	—
44	C₂²×C₈	C₂²×C₄	1	6	yes	—
45	C₂²×C₈	C₂²×C₄	1	2	yes	—
46	C₄×Q₈	C₂×C₈	1	8	yes	—
47	C₄×Q₈	M₁₆	1	16	yes	—
48	C₂×C₄⋊C₄	C₄²	1	3	yes	—
49	C₂×C₄⋊C₄	C₄²	1	6	yes	—
50	C₂×C₄²	C₂²×C₄	1	3	yes	—
51	C₂×C₄²	C₂²×C₄	1	6	yes	—
52	C₂×C₄²	C₂²×C₄	1	3	yes	—
53	C₂×C₄²	C₂²×C₄	1	6	yes	—
54	C₂×C₄²	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₈	—	2	24	no	—
56	C₄⋊C₈	C₂×C₈	1	24	yes	—
57	C₄×C₈	—	2	12	no	—
58	C₄×C₈	—	2	24	no	—
59	C₄×C₈	—	2	6	no	—
60	C₄×C₈	—	2	6	no	—
61	C₄×C₈	—	2	6	no	—
62	C₄×C₈	—	4	12	no	—
63	C₄×C₈	C₂×C₈	1	24	yes	—
64	C₄×C₈	C₂×Q₈	1	4	yes	—
65	C₄×C₈	C₂×Q₈	1	2	yes	—
66	C₄×C₈	#41	1	4	yes	—
67	C₄×C₈	#41	1	2	yes	—
68	#95	C₄²	1	6	yes	—
69	C₂×Q₈	C₄×C₈	1	4	yes	—
70	C₂×Q₈	#97	1	8	yes	—
71	C₂²×C₄	C₂×C₄²	1	3	yes	—
72	C₂²×C₄	C₂×C₄²	1	6	yes	—
73	C₂²×C₄	C₂×C₄²	1	3	yes	—
74	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂³×C₄	1	2	yes	—
76	C₂×C₈	—	2	6	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	12	no	—
81	C₂×C₈	—	2	12	no	—
82	C₂×C₈	—	2	12	no	—
83	C₂×C₈	C₄×Q₈	1	8	yes	—
84	C₂×C₈	C₄×Q₈	1	4	yes	—
85	C₂×C₈	C₂²×C₈	1	12	yes	—
86	C₂×C₈	C₂²×C₈	1	4	yes	—
87	C₂×C₈	#131	1	4	yes	—
88	C₄⋊C₄	C₄×C₈	1	12	yes	—
89	C₄⋊C₄	#97	1	24	yes	—
90	C₄²	—	2	6	no	—
91	C₄²	—	2	12	no	—
92	C₄²	C₂²×C₈	1	12	yes	—
93	C₄²	C₂×M₁₆	1	24	yes	—
94	C₄²	C₂²×Q₈	1	1	yes	characteristic
95	C₄²	C₂×#41	1	1	yes	characteristic
96	C₄²	C₂×#41	1	2	yes	—
97	C₂³	C₂²×C₄²	1	1	yes	characteristic
98	Q₈	#322	1	16	yes	direct product factor
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	C₂×C₄×C₈	1	6	yes	—
103	C₂×C₄	C₂×C₄×C₈	1	6	yes	—
104	C₂×C₄	C₂×#97	1	12	yes	—
105	C₂×C₄	C₂×#97	1	12	yes	—
106	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×Q₈	1	2	yes	—
108	C₂×C₄	C₄×#41	1	1	yes	characteristic
109	C₂×C₄	C₄×#41	1	2	yes	—
110	C₂×C₄	C₂³×C₈	1	2	yes	—
111	C₂×C₄	C₂²×M₁₆	1	4	yes	—
112	C₂×C₄	C₂×#131	1	2	yes	—
113	C₂×C₄	#568	1	4	yes	—
114	C₈	—	2	8	no	—
115	C₈	—	2	12	no	—
116	C₈	—	4	12	no	—
117	C₈	C₈×Q₈	1	8	yes	complemented
118	C₂²	C₄²×Q₈	1	1	yes	characteristic
119	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
120	C₂²	C₂²×#97	1	2	yes	—
121	C₂²	C₄×#131	1	1	yes	characteristic
122	C₂²	#2875	1	2	yes	—
123	C₄	—	2	6	no	—
124	C₄	—	2	12	no	—
125	C₄	C₂×#322	1	12	yes	—
126	C₄	C₂×C₈×Q₈	1	2	yes	—
127	C₄	Q₈×M₁₆	1	4	yes	—
128	C₄	C₈×#41	1	2	yes	—
129	C₄	#2971	1	4	yes	—
130	C₂	C₂²×#322	1	1	yes	characteristic
131	C₂	#17316	1	1	yes	characteristic
132	C₂	#17320	1	2	yes	—
133	C₂	C₄×C₈×Q₈	1	1	yes	characteristic
134	C₂	Q₈×#97	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle, prime radical	C₂³	C₂²×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
22. C₈×Q₈	C₈	semidirect
40. #322	Q₈	direct (N × H)
98. Q₈	#322	direct (N × H)
117. C₈	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Q₈	2
#8261	2
#8260	2
C₄×#322	2

Subgroups — Q8×#322 #184374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#322 #184374