Subgroups — C₂²×Q₈×C₃⋊C₈ #11659539

Order 768.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₃⋊C₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	3	yes	maximal
4	C₂²×C₄×C₃⋊C₈	C₂	1	3	yes	maximal
5	C₂²×C₈×Q₈	—	3	1	no	maximal, Sylow
6	C₂³×C₃×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Q₈	C₄	1	12	yes	—
8	C₂×C₃×C₄×Q₈	C₂²	1	12	yes	—
9	C₂²×C₃×C₄⋊C₄	C₄	1	3	yes	—
10	C₂²×C₃×C₄²	C₂²	1	3	yes	—
11	C₂³×C₃⋊C₈	C₂²	1	3	yes	—
12	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×C₃⋊C₈	C₂²	1	64	yes	direct product factor
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	36	yes	—
15	C₂×C₄×C₃⋊C₈	C₂²	1	36	yes	—
16	C₂²×C₄×Q₈	D₃	1	1	yes	characteristic
17	C₂×C₈×Q₈	—	3	24	no	—
18	C₂²×C₄⋊C₈	—	3	3	no	—
19	C₂²×C₄×C₈	—	3	3	no	—
20	C₂²×C₃×Q₈	C₈	1	16	yes	complemented
21	C₂²×C₃×Q₈	C₂×C₄	1	12	yes	—
22	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
23	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
24	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₃×C₄×Q₈	C₂×C₄	1	48	yes	—
26	C₃×C₄×Q₈	C₂³	1	16	yes	—
27	C₂×C₃×C₄⋊C₄	C₂×C₄	1	36	yes	—
28	C₂×C₃×C₄²	C₂×C₄	1	18	yes	—
29	C₂×C₃×C₄²	C₂³	1	18	yes	—
30	C₂²×C₃⋊C₈	—	2	12	no	—
31	C₂²×C₃⋊C₈	Q₈	1	8	yes	direct product factor
32	C₂²×C₃⋊C₈	C₂³	1	18	yes	—
33	C₂²×C₃⋊C₈	C₂³	1	6	yes	—
34	C₄⋊(C₃⋊C₈)	C₂³	1	48	yes	—
35	C₄×C₃⋊C₈	C₂³	1	48	yes	—
36	C₂³×Q₈	Dic₃	1	1	yes	characteristic
37	C₂³×C₈	—	3	1	no	—
38	C₂³×C₈	—	3	3	no	—
39	C₂×C₄×Q₈	Dic₃	1	12	yes	—
40	C₂×C₄×Q₈	C₂×D₃	1	12	yes	—
41	C₂²×C₄⋊C₄	Dic₃	1	3	yes	—
42	C₂²×C₄²	C₂×D₃	1	3	yes	—
43	C₈×Q₈	—	3	64	no	—
44	C₂×C₄⋊C₈	—	3	36	no	—
45	C₂×C₄×C₈	—	3	36	no	—
46	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×Q₈	C₂×C₈	1	96	yes	complemented
48	C₂×C₃×Q₈	C₂²×C₄	1	16	yes	—
49	C₂²×C₃×C₄	—	2	12	no	—
50	C₂²×C₃×C₄	C₂×C₈	1	24	yes	—
51	C₂²×C₃×C₄	C₂²×C₄	1	18	yes	—
52	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
53	C₂²×C₃×C₄	C₂²×C₄	1	18	yes	—
54	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
55	C₂²×C₃×C₄	#41	1	4	yes	—
56	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
57	C₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
58	C₃×C₄²	C₂²×C₄	1	36	yes	—
59	C₃×C₄²	C₂⁴	1	12	yes	—
60	C₂×C₃⋊C₈	—	2	36	no	—
61	C₂×C₃⋊C₈	C₂×Q₈	1	24	yes	direct product factor
62	C₂×C₃⋊C₈	C₂⁴	1	12	yes	—
63	C₂×C₃⋊C₈	C₂⁴	1	4	yes	—
64	C₂²×Q₈	C₃⋊C₈	1	16	yes	direct product factor
65	C₂²×Q₈	C₂×Dic₃	1	12	yes	—
66	C₂³×C₄	C₂×Dic₃	1	3	yes	—
67	C₂³×C₄	C₂×Dic₃	1	3	yes	—
68	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
69	C₂²×C₈	—	3	8	no	—
70	C₂²×C₈	—	3	6	no	—
71	C₂²×C₈	—	3	18	no	—
72	C₂²×C₈	—	6	12	no	—
73	C₄×Q₈	C₂×Dic₃	1	48	yes	—
74	C₄×Q₈	C₂²×D₃	1	16	yes	—
75	C₂×C₄⋊C₄	C₂×Dic₃	1	36	yes	—
76	C₂×C₄²	C₂×Dic₃	1	18	yes	—
77	C₂×C₄²	C₂²×D₃	1	18	yes	—
78	C₄⋊C₈	—	3	48	no	—
79	C₄×C₈	—	3	48	no	—
80	C₂³×C₃	C₄×Q₈	1	4	yes	—
81	C₂³×C₃	C₂²×C₈	1	4	yes	—
82	C₂³×C₃	#131	1	4	yes	—
83	C₂³×C₃	C₂³×C₄	1	3	yes	—
84	C₃×Q₈	C₂²×C₈	1	64	yes	complemented
85	C₂×C₃×C₄	—	2	36	no	—
86	C₂×C₃×C₄	C₄×Q₈	1	12	yes	—
87	C₂×C₃×C₄	C₂²×C₈	1	72	yes	—
88	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
89	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
90	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
91	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
92	C₂×C₃×C₄	C₂×#41	1	6	yes	—
93	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
94	C₃⋊C₈	—	2	12	no	—
95	C₃⋊C₈	C₂²×Q₈	1	8	yes	direct product factor
96	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
97	C₂×Q₈	C₂×C₃⋊C₈	1	96	yes	direct product factor
98	C₂×Q₈	C₂²×Dic₃	1	16	yes	—
99	C₂²×C₄	—	2	12	no	—
100	C₂²×C₄	C₂×C₃⋊C₈	1	24	yes	—
101	C₂²×C₄	D₃×Q₈	1	4	yes	—
102	C₂²×C₄	#239	1	4	yes	—
103	C₂²×C₄	C₂²×Dic₃	1	18	yes	—
104	C₂²×C₄	C₂²×Dic₃	1	18	yes	—
105	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
106	C₂²×C₄	C₂³×D₃	1	3	yes	—
107	C₂×C₈	—	3	24	no	—
108	C₂×C₈	—	3	4	no	—
109	C₂×C₈	—	3	12	no	—
110	C₂×C₈	—	6	36	no	—
111	C₄⋊C₄	C₂²×Dic₃	1	48	yes	—
112	C₄²	C₂²×Dic₃	1	36	yes	—
113	C₄²	C₂³×D₃	1	12	yes	—
114	C₂²×C₃	C₈×Q₈	1	16	yes	complemented
115	C₂²×C₃	C₂×C₄×Q₈	1	6	yes	—
116	C₂²×C₃	C₂³×C₈	1	6	yes	—
117	C₂²×C₃	C₂×#131	1	6	yes	—
118	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
119	C₃×C₄	—	2	12	no	—
120	C₃×C₄	C₂×C₄×Q₈	1	6	yes	—
121	C₃×C₄	C₂³×C₈	1	24	yes	—
122	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
123	C₃×C₄	C₂²×#41	1	1	yes	characteristic
124	C₂³	C₂²×C₃⋊C₈	1	4	yes	—
125	C₂³	Q₈×Dic₃	1	4	yes	—
126	C₂³	#948	1	4	yes	—
127	C₂³	C₂³×Dic₃	1	3	yes	—
128	Q₈	C₂²×C₃⋊C₈	1	64	yes	direct product factor
129	C₂×C₄	—	2	36	no	—
130	C₂×C₄	C₂²×C₃⋊C₈	1	72	yes	—
131	C₂×C₄	Q₈×Dic₃	1	12	yes	—
132	C₂×C₄	C₂×D₃×Q₈	1	6	yes	—
133	C₂×C₄	C₂×#239	1	6	yes	—
134	C₂×C₄	C₂³×Dic₃	1	12	yes	—
135	C₂×C₄	C₂³×Dic₃	1	12	yes	—
136	C₂×C₄	C₂³×Dic₃	1	3	yes	—
137	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
138	C₈	—	3	8	no	—
139	C₈	—	6	12	no	—
140	C₂×C₃	C₂×C₈×Q₈	1	12	yes	complemented
141	C₂×C₃	C₂²×C₄×Q₈	1	1	yes	characteristic
142	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
143	C₂×C₃	C₂²×#131	1	1	yes	characteristic
144	C₂²	Q₈×C₃⋊C₈	1	16	yes	direct product factor
145	C₂²	C₂³×C₃⋊C₈	1	6	yes	—
146	C₂²	C₂×Q₈×Dic₃	1	6	yes	—
147	C₂²	C₂×#948	1	6	yes	—
148	C₂²	C₂⁴×Dic₃	1	1	yes	characteristic
149	C₄	—	2	12	no	—
150	C₄	C₂³×C₃⋊C₈	1	24	yes	—
151	C₄	C₂×Q₈×Dic₃	1	6	yes	—
152	C₄	C₂²×D₃×Q₈	1	1	yes	characteristic
153	C₄	C₂²×#239	1	1	yes	characteristic
154	C₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
155	C₂	C₂×Q₈×C₃⋊C₈	1	12	yes	direct product factor
156	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic
157	C₂	C₂²×Q₈×Dic₃	1	1	yes	characteristic
158	C₂	C₂²×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₃
commutator	C₂×C₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄×Q₈	C₂
socle, prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₃
hypocenter	C₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×C₃⋊C₈	C₂	direct (N × H)
13. Q₈×C₃⋊C₈	C₂²	direct (N × H)
20. C₂²×C₃×Q₈	C₈	semidirect
31. C₂²×C₃⋊C₈	Q₈	direct (N × H)
47. C₂×C₃×Q₈	C₂×C₈	semidirect
61. C₂×C₃⋊C₈	C₂×Q₈	direct (N × H)
64. C₂²×Q₈	C₃⋊C₈	direct (N × H)
84. C₃×Q₈	C₂²×C₈	semidirect
95. C₃⋊C₈	C₂²×Q₈	direct (N × H)
97. C₂×Q₈	C₂×C₃⋊C₈	direct (N × H)
114. C₂²×C₃	C₈×Q₈	semidirect
128. Q₈	C₂²×C₃⋊C₈	direct (N × H)
140. C₂×C₃	C₂×C₈×Q₈	semidirect
144. C₂²	Q₈×C₃⋊C₈	direct (N × H)
154. C₃	C₂²×C₈×Q₈	semidirect
155. C₂	C₂×Q₈×C₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×Q₈	2
C₂×Q₈×C₃⋊C₈	2
C₂²×C₄⋊(C₃⋊C₈)	2
C₂²×C₄×C₃⋊C₈	2
C₂²×C₈×Q₈	3

Subgroups — C22×Q8×C3⋊C8 #11659539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×C₃⋊C₈ #11659539