Subgroups — C₂×Q₈×C₃⋊(C₃⋊C₈) #11861253

Order 1152.

1320 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊(C₃⋊C₈)	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	3	yes	maximal
4	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	3	yes	maximal
5	C₂×Q₈×C₃⋊C₈	—	3	4	no	maximal
6	C₂²×C₃²×Q₈	C₄	1	1	yes	characteristic
7	C₃²×C₄×Q₈	C₄	1	4	yes	—
8	C₃²×C₄×Q₈	C₂²	1	4	yes	—
9	C₂×C₃²×C₄⋊C₄	C₄	1	3	yes	—
10	C₂×C₃²×C₄²	C₂²	1	3	yes	—
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	—
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	12	yes	—
14	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	12	yes	—
15	C₂×C₃×C₄×Q₈	D₃	1	4	yes	—
16	Q₈×C₃⋊C₈	—	3	32	no	—
17	C₂×C₄⋊(C₃⋊C₈)	—	3	12	no	—
18	C₂×C₄×C₃⋊C₈	—	3	12	no	—
19	C₂×C₃²×Q₈	C₈	1	8	yes	complemented
20	C₂×C₃²×Q₈	C₂×C₄	1	4	yes	—
21	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
22	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₃²×C₄⋊C₄	C₂×C₄	1	12	yes	—
25	C₃²×C₄²	C₂×C₄	1	6	yes	—
26	C₃²×C₄²	C₂³	1	6	yes	—
27	C₂×C₃⋊(C₃⋊C₈)	—	2	6	no	—
28	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	4	yes	direct product factor
29	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	6	yes	—
30	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
31	C₂×C₈×Q₈	—	9	1	no	Sylow
32	C₂²×C₃×Q₈	Dic₃	1	4	yes	—
33	C₃×C₄×Q₈	Dic₃	1	16	yes	—
34	C₃×C₄×Q₈	C₂×D₃	1	16	yes	—
35	C₂×C₃×C₄⋊C₄	Dic₃	1	12	yes	—
36	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
37	C₂²×C₃⋊C₈	—	3	4	no	—
38	C₂²×C₃⋊C₈	—	3	12	no	—
39	C₄⋊(C₃⋊C₈)	—	3	48	no	—
40	C₄×C₃⋊C₈	—	3	48	no	—
41	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
42	C₃²×Q₈	C₂×C₈	1	16	yes	complemented
43	C₂×C₃²×C₄	—	2	6	no	—
44	C₂×C₃²×C₄	C₂×C₈	1	12	yes	—
45	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
46	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
48	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
49	C₂×C₃²×C₄	#41	1	2	yes	—
50	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
51	C₃⋊(C₃⋊C₈)	—	2	6	no	—
52	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	4	yes	direct product factor
53	C₂×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
54	C₈×Q₈	—	9	8	no	—
55	C₂×C₄⋊C₈	—	9	3	no	—
56	C₂×C₄×C₈	—	9	3	no	—
57	C₂×C₃×Q₈	C₃⋊C₈	1	32	yes	complemented
58	C₂×C₃×Q₈	C₂×Dic₃	1	16	yes	—
59	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
60	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
61	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
62	C₃×C₄⋊C₄	C₂×Dic₃	1	48	yes	—
63	C₃×C₄²	C₂×Dic₃	1	24	yes	—
64	C₃×C₄²	C₂²×D₃	1	24	yes	—
65	C₂×C₃⋊C₈	—	3	16	no	—
66	C₂×C₃⋊C₈	—	3	8	no	—
67	C₂×C₃⋊C₈	—	3	24	no	—
68	C₂×C₃⋊C₈	—	6	24	no	—
69	C₂²×C₃²	C₄×Q₈	1	2	yes	—
70	C₂²×C₃²	C₂²×C₈	1	2	yes	—
71	C₂²×C₃²	#131	1	2	yes	—
72	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
73	C₃²×C₄	—	2	6	no	—
74	C₃²×C₄	C₄×Q₈	1	2	yes	—
75	C₃²×C₄	C₂²×C₈	1	12	yes	—
76	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
77	C₃²×C₄	C₂×#41	1	1	yes	characteristic
78	C₂²×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
79	C₂²×C₈	—	9	1	no	—
80	C₂²×C₈	—	9	3	no	—
81	C₄×Q₈	C₃⋊Dic₃	1	4	yes	—
82	C₄×Q₈	C₂×C₃⋊D₃	1	4	yes	—
83	C₂×C₄⋊C₄	C₃⋊Dic₃	1	3	yes	—
84	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
85	C₄⋊C₈	—	9	12	no	—
86	C₄×C₈	—	9	12	no	—
87	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
88	C₃×Q₈	C₂×C₃⋊C₈	1	64	yes	complemented
89	C₂×C₃×C₄	—	2	24	no	—
90	C₂×C₃×C₄	C₂×C₃⋊C₈	1	48	yes	—
91	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
92	C₂×C₃×C₄	#239	1	8	yes	—
93	C₂×C₃×C₄	C₂²×Dic₃	1	24	yes	—
94	C₂×C₃×C₄	C₂²×Dic₃	1	24	yes	—
95	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
96	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
97	C₃⋊C₈	—	3	16	no	—
98	C₃⋊C₈	—	6	24	no	—
99	C₂×C₃²	C₈×Q₈	1	4	yes	complemented
100	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
102	C₂×C₃²	C₂×#131	1	1	yes	characteristic
103	C₂×Q₈	C₃⋊(C₃⋊C₈)	1	8	yes	direct product factor
104	C₂×Q₈	C₂×C₃⋊Dic₃	1	4	yes	—
105	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
106	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
107	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
108	C₂×C₈	—	9	2	no	—
109	C₂×C₈	—	9	4	no	—
110	C₂×C₈	—	9	6	no	—
111	C₂×C₈	—	18	6	no	—
112	C₄⋊C₄	C₂×C₃⋊Dic₃	1	12	yes	—
113	C₄²	C₂×C₃⋊Dic₃	1	6	yes	—
114	C₄²	C₂²×C₃⋊D₃	1	6	yes	—
115	C₂²×C₃	C₂²×C₃⋊C₈	1	8	yes	—
116	C₂²×C₃	Q₈×Dic₃	1	8	yes	—
117	C₂²×C₃	#948	1	8	yes	—
118	C₂²×C₃	C₂³×Dic₃	1	4	yes	—
119	C₃×C₄	—	2	24	no	—
120	C₃×C₄	C₂²×C₃⋊C₈	1	48	yes	—
121	C₃×C₄	Q₈×Dic₃	1	8	yes	—
122	C₃×C₄	C₂×D₃×Q₈	1	4	yes	—
123	C₃×C₄	C₂×#239	1	4	yes	—
124	C₃²	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
125	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
126	Q₈	C₂×C₃⋊(C₃⋊C₈)	1	16	yes	direct product factor
127	C₂×C₄	—	2	6	no	—
128	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	12	yes	—
129	C₂×C₄	Q₈×C₃⋊D₃	1	2	yes	—
130	C₂×C₄	#3832	1	2	yes	—
131	C₂×C₄	C₂²×C₃⋊Dic₃	1	6	yes	—
132	C₂×C₄	C₂²×C₃⋊Dic₃	1	6	yes	—
133	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
134	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
135	C₈	—	9	4	no	—
136	C₈	—	18	6	no	—
137	C₂×C₃	Q₈×C₃⋊C₈	1	16	yes	complemented
138	C₂×C₃	C₂³×C₃⋊C₈	1	4	yes	—
139	C₂×C₃	C₂×Q₈×Dic₃	1	4	yes	—
140	C₂×C₃	C₂×#948	1	4	yes	—
141	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	2	yes	—
142	C₂²	Q₈×C₃⋊Dic₃	1	2	yes	—
143	C₂²	#64157	1	2	yes	—
144	C₂²	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
145	C₄	—	2	6	no	—
146	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	12	yes	—
147	C₄	Q₈×C₃⋊Dic₃	1	2	yes	—
148	C₄	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
149	C₄	C₂×#3832	1	1	yes	characteristic
150	C₃	C₂×Q₈×C₃⋊C₈	1	4	yes	complemented
151	C₂	Q₈×C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
152	C₂	C₂³×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
153	C₂	C₂×Q₈×C₃⋊Dic₃	1	1	yes	characteristic
154	C₂	C₂×#64157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×C₄×Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₈×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊(C₃⋊C₈)	C₂	direct (N × H)
19. C₂×C₃²×Q₈	C₈	semidirect
28. C₂×C₃⋊(C₃⋊C₈)	Q₈	direct (N × H)
42. C₃²×Q₈	C₂×C₈	semidirect
52. C₃⋊(C₃⋊C₈)	C₂×Q₈	direct (N × H)
57. C₂×C₃×Q₈	C₃⋊C₈	semidirect
88. C₃×Q₈	C₂×C₃⋊C₈	semidirect
99. C₂×C₃²	C₈×Q₈	semidirect
103. C₂×Q₈	C₃⋊(C₃⋊C₈)	direct (N × H)
124. C₃²	C₂×C₈×Q₈	semidirect
126. Q₈	C₂×C₃⋊(C₃⋊C₈)	direct (N × H)
137. C₂×C₃	Q₈×C₃⋊C₈	semidirect
150. C₃	C₂×Q₈×C₃⋊C₈	semidirect
151. C₂	Q₈×C₃⋊(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
C₂×C₃²×C₄×Q₈	2
Q₈×C₃⋊(C₃⋊C₈)	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
C₂×Q₈×C₃⋊C₈	3

Subgroups — C2×Q8×C3⋊(C3⋊C8) #11861253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₃⋊(C₃⋊C₈) #11861253