Subgroups — C₂³×C₃×D₃×Q₈ #11919174

Order 1152.

12953 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₃×Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₃×C₄×D₃	C₂	1	3	yes	maximal
4	C₂³×C₃×Dic₆	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₃×Q₈	—	3	1	no	maximal
6	C₂³×D₃×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂⁴×C₃×D₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×Q₈	C₂²	1	28	yes	complemented
9	C₂³×C₃²×C₄	C₂²	1	3	yes	—
10	C₂³×C₃×Dic₃	C₂²	1	3	yes	—
11	C₂×C₃×D₃×Q₈	C₂²	1	448	yes	direct product factor
12	C₂²×C₃×C₄×D₃	C₂²	1	84	yes	—
13	C₂²×C₃×Dic₆	C₂²	1	84	yes	complemented
14	C₂³×C₃×Q₈	—	2	1	no	—
15	C₂³×C₃×Q₈	—	3	56	no	—
16	C₂³×C₃×Q₈	—	3	3	no	—
17	C₂³×C₃×Q₈	D₃	1	1	yes	characteristic, direct product factor
18	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁴×C₃×C₄	—	3	3	no	—
20	C₂²×D₃×Q₈	C₂×C₃	1	56	yes	direct product factor
21	C₂³×C₄×D₃	C₂×C₃	1	3	yes	—
22	C₂³×Dic₆	C₂×C₃	1	3	yes	complemented
23	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
24	C₂³×C₃×D₃	Q₈	1	16	yes	direct product factor
25	C₂³×C₃×D₃	C₂³	1	14	yes	—
26	C₂×C₃²×Q₈	C₂³	1	112	yes	complemented
27	C₂²×C₃²×C₄	C₂³	1	42	yes	—
28	C₂²×C₃×Dic₃	C₂³	1	42	yes	—
29	C₃×D₃×Q₈	C₂³	1	512	yes	direct product factor
30	C₂×C₃×C₄×D₃	C₂³	1	336	yes	—
31	C₂×C₃×Dic₆	C₂³	1	336	yes	complemented
32	C₂⁴×Q₈	—	3	1	no	Sylow
33	C₂⁵×C₃	—	3	1	no	—
34	C₂⁴×D₃	C₂²×C₃	1	1	yes	characteristic
35	C₂²×C₃×Q₈	—	2	28	no	—
36	C₂²×C₃×Q₈	—	3	448	no	—
37	C₂²×C₃×Q₈	—	3	84	no	—
38	C₂²×C₃×Q₈	C₂×D₃	1	28	yes	direct product factor
39	C₂²×C₃×Q₈	C₂²×C₃	1	28	yes	complemented
40	C₂³×C₃×C₄	—	2	3	no	—
41	C₂³×C₃×C₄	—	3	3	no	—
42	C₂³×C₃×C₄	—	3	84	no	—
43	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
44	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
45	C₂³×Dic₃	C₂²×C₃	1	3	yes	—
46	C₂×D₃×Q₈	C₂²×C₃	1	448	yes	direct product factor
47	C₂²×C₄×D₃	C₂²×C₃	1	84	yes	—
48	C₂²×Dic₆	C₂²×C₃	1	84	yes	complemented
49	C₂³×C₃²	C₂×Q₈	1	8	yes	complemented
50	C₂³×C₃²	C₂⁴	1	7	yes	—
51	C₂²×C₃×D₃	C₂×Q₈	1	112	yes	direct product factor
52	C₂²×C₃×D₃	C₂⁴	1	28	yes	—
53	C₃²×Q₈	C₂⁴	1	64	yes	complemented
54	C₂×C₃²×C₄	C₂⁴	1	84	yes	—
55	C₂×C₃×Dic₃	C₂⁴	1	84	yes	—
56	C₃×C₄×D₃	C₂⁴	1	192	yes	—
57	C₃×Dic₆	C₂⁴	1	192	yes	complemented
58	C₂³×Q₈	—	3	56	no	—
59	C₂³×Q₈	—	3	3	no	—
60	C₂³×Q₈	C₃×D₃	1	1	yes	characteristic, direct product factor
61	C₂⁴×C₄	—	3	3	no	—
62	C₂⁴×C₃	—	2	1	no	—
63	C₂⁴×C₃	—	3	16	no	—
64	C₂⁴×C₃	—	3	14	no	—
65	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
66	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
67	C₂³×D₃	C₃×Q₈	1	16	yes	direct product factor
68	C₂³×D₃	C₂³×C₃	1	14	yes	—
69	C₂×C₃×Q₈	—	2	112	no	—
70	C₂×C₃×Q₈	—	3	512	no	—
71	C₂×C₃×Q₈	—	3	336	no	—
72	C₂×C₃×Q₈	C₂²×D₃	1	112	yes	direct product factor
73	C₂×C₃×Q₈	C₂³×C₃	1	112	yes	complemented
74	C₂²×C₃×C₄	—	2	42	no	—
75	C₂²×C₃×C₄	—	3	42	no	—
76	C₂²×C₃×C₄	—	3	336	no	—
77	C₂²×C₃×C₄	C₂²×D₃	1	42	yes	—
78	C₂²×C₃×C₄	C₂³×C₃	1	42	yes	—
79	C₂²×Dic₃	C₂³×C₃	1	42	yes	—
80	D₃×Q₈	C₂³×C₃	1	512	yes	direct product factor
81	C₂×C₄×D₃	C₂³×C₃	1	336	yes	—
82	C₂×Dic₆	C₂³×C₃	1	336	yes	complemented
83	C₂²×C₃²	C₂²×Q₈	1	28	yes	complemented
84	C₂²×C₃²	C₂⁵	1	7	yes	—
85	C₂×C₃×D₃	C₂²×Q₈	1	112	yes	direct product factor
86	C₂×C₃×D₃	C₂⁵	1	8	yes	—
87	C₃²×C₄	C₂⁵	1	24	yes	—
88	C₃×Dic₃	C₂⁵	1	24	yes	—
89	C₂⁵	—	3	1	no	—
90	C₂²×Q₈	—	3	448	no	—
91	C₂²×Q₈	—	3	84	no	—
92	C₂²×Q₈	C₂×C₃×D₃	1	28	yes	direct product factor
93	C₂³×C₄	—	3	3	no	—
94	C₂³×C₄	—	3	84	no	—
95	C₂³×C₄	C₂×C₃×D₃	1	3	yes	—
96	C₂³×C₃	—	2	7	no	—
97	C₂³×C₃	—	2	8	no	—
98	C₂³×C₃	—	3	112	no	—
99	C₂³×C₃	—	3	28	no	—
100	C₂³×C₃	D₃×Q₈	1	8	yes	direct product factor
101	C₂³×C₃	C₂×C₃×Q₈	1	8	yes	complemented
102	C₂³×C₃	C₂³×D₃	1	7	yes	—
103	C₂³×C₃	C₂⁴×C₃	1	7	yes	—
104	C₂²×D₃	C₂×C₃×Q₈	1	112	yes	direct product factor
105	C₂²×D₃	C₂⁴×C₃	1	28	yes	—
106	C₃×Q₈	—	2	64	no	—
107	C₃×Q₈	—	3	192	no	—
108	C₃×Q₈	C₂³×D₃	1	64	yes	direct product factor
109	C₃×Q₈	C₂⁴×C₃	1	64	yes	complemented
110	C₂×C₃×C₄	—	2	84	no	—
111	C₂×C₃×C₄	—	3	84	no	—
112	C₂×C₃×C₄	—	3	192	no	—
113	C₂×C₃×C₄	C₂³×D₃	1	84	yes	—
114	C₂×C₃×C₄	C₂⁴×C₃	1	84	yes	—
115	C₂×Dic₃	C₂⁴×C₃	1	84	yes	—
116	C₄×D₃	C₂⁴×C₃	1	192	yes	—
117	Dic₆	C₂⁴×C₃	1	192	yes	complemented
118	C₂×C₃²	C₂³×Q₈	1	14	yes	complemented
119	C₂×C₃²	C₂⁶	1	1	yes	characteristic
120	C₃×D₃	C₂³×Q₈	1	16	yes	direct product factor
121	C₂⁴	—	3	14	no	—
122	C₂⁴	—	3	16	no	—
123	C₂⁴	C₂²×C₃×D₃	1	1	yes	characteristic
124	C₂×Q₈	—	3	512	no	—
125	C₂×Q₈	—	3	336	no	—
126	C₂×Q₈	C₂²×C₃×D₃	1	112	yes	direct product factor
127	C₂²×C₄	—	3	42	no	—
128	C₂²×C₄	—	3	336	no	—
129	C₂²×C₄	C₂²×C₃×D₃	1	42	yes	—
130	C₂²×C₃	—	2	28	no	—
131	C₂²×C₃	—	2	7	no	—
132	C₂²×C₃	—	3	112	no	—
133	C₂²×C₃	—	3	8	no	—
134	C₂²×C₃	C₂×D₃×Q₈	1	28	yes	direct product factor
135	C₂²×C₃	C₂²×C₃×Q₈	1	28	yes	complemented
136	C₂²×C₃	C₂⁴×D₃	1	7	yes	—
137	C₂²×C₃	C₂⁵×C₃	1	7	yes	—
138	C₂×D₃	C₂²×C₃×Q₈	1	112	yes	direct product factor
139	C₂×D₃	C₂⁵×C₃	1	8	yes	—
140	C₃×C₄	—	2	24	no	—
141	C₃×C₄	—	3	24	no	—
142	C₃×C₄	C₂⁴×D₃	1	24	yes	—
143	C₃×C₄	C₂⁵×C₃	1	24	yes	—
144	Dic₃	C₂⁵×C₃	1	24	yes	—
145	C₃²	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	3	112	no	—
147	C₂³	—	3	28	no	—
148	C₂³	C₃×D₃×Q₈	1	8	yes	direct product factor
149	C₂³	C₂³×C₃×D₃	1	7	yes	—
150	Q₈	—	3	192	no	—
151	Q₈	C₂³×C₃×D₃	1	64	yes	direct product factor
152	C₂×C₄	—	3	84	no	—
153	C₂×C₄	—	3	192	no	—
154	C₂×C₄	C₂³×C₃×D₃	1	84	yes	—
155	C₂×C₃	—	2	14	no	—
156	C₂×C₃	—	2	1	no	—
157	C₂×C₃	—	3	16	no	—
158	C₂×C₃	C₂²×D₃×Q₈	1	14	yes	direct product factor
159	C₂×C₃	C₂³×C₃×Q₈	1	14	yes	complemented
160	C₂×C₃	C₂⁵×D₃	1	1	yes	characteristic
161	C₂×C₃	C₂⁶×C₃	1	1	yes	characteristic
162	D₃	C₂³×C₃×Q₈	1	16	yes	direct product factor
163	C₂²	—	3	112	no	—
164	C₂²	—	3	8	no	—
165	C₂²	C₂×C₃×D₃×Q₈	1	28	yes	direct product factor
166	C₂²	C₂⁴×C₃×D₃	1	7	yes	—
167	C₄	—	3	24	no	—
168	C₄	C₂⁴×C₃×D₃	1	24	yes	—
169	C₃	—	2	1	no	—
170	C₃	C₂³×D₃×Q₈	1	1	yes	characteristic, direct product factor
171	C₃	C₂⁴×C₃×Q₈	1	1	yes	characteristic, complemented
172	C₂	—	3	16	no	—
173	C₂	C₂²×C₃×D₃×Q₈	1	14	yes	direct product factor
174	C₂	C₂⁵×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁶×C₃
Frattini	C₂	C₂⁵×C₃×D₃
Fitting	C₂³×C₃²×Q₈	C₂
socle	C₂⁴×C₃²	C₂³
prime radical	C₂⁴×C₃×D₃	C₂²
hypercenter	C₂³×C₃×Q₈	D₃
hypocenter	C₃	C₂⁴×C₃×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²×Q₈	C₂	semidirect
2. C₂²×C₃×D₃×Q₈	C₂	direct (N × H)
4. C₂³×C₃×Dic₆	C₂	semidirect
6. C₂³×D₃×Q₈	C₃	direct (N × H)
8. C₂²×C₃²×Q₈	C₂²	semidirect
11. C₂×C₃×D₃×Q₈	C₂²	direct (N × H)
13. C₂²×C₃×Dic₆	C₂²	semidirect
17. C₂³×C₃×Q₈	D₃	direct (N × H)
18. C₂³×C₃×Q₈	C₂×C₃	semidirect
20. C₂²×D₃×Q₈	C₂×C₃	direct (N × H)
22. C₂³×Dic₆	C₂×C₃	semidirect
24. C₂³×C₃×D₃	Q₈	direct (N × H)
26. C₂×C₃²×Q₈	C₂³	semidirect
29. C₃×D₃×Q₈	C₂³	direct (N × H)
31. C₂×C₃×Dic₆	C₂³	semidirect
38. C₂²×C₃×Q₈	C₂×D₃	direct (N × H)
39. C₂²×C₃×Q₈	C₂²×C₃	semidirect
46. C₂×D₃×Q₈	C₂²×C₃	direct (N × H)
48. C₂²×Dic₆	C₂²×C₃	semidirect
49. C₂³×C₃²	C₂×Q₈	semidirect
51. C₂²×C₃×D₃	C₂×Q₈	direct (N × H)
53. C₃²×Q₈	C₂⁴	semidirect
57. C₃×Dic₆	C₂⁴	semidirect
60. C₂³×Q₈	C₃×D₃	direct (N × H)
67. C₂³×D₃	C₃×Q₈	direct (N × H)
72. C₂×C₃×Q₈	C₂²×D₃	direct (N × H)
73. C₂×C₃×Q₈	C₂³×C₃	semidirect
80. D₃×Q₈	C₂³×C₃	direct (N × H)
82. C₂×Dic₆	C₂³×C₃	semidirect
83. C₂²×C₃²	C₂²×Q₈	semidirect
85. C₂×C₃×D₃	C₂²×Q₈	direct (N × H)
92. C₂²×Q₈	C₂×C₃×D₃	direct (N × H)
100. C₂³×C₃	D₃×Q₈	direct (N × H)
101. C₂³×C₃	C₂×C₃×Q₈	semidirect
104. C₂²×D₃	C₂×C₃×Q₈	direct (N × H)
108. C₃×Q₈	C₂³×D₃	direct (N × H)
109. C₃×Q₈	C₂⁴×C₃	semidirect
117. Dic₆	C₂⁴×C₃	semidirect
118. C₂×C₃²	C₂³×Q₈	semidirect
120. C₃×D₃	C₂³×Q₈	direct (N × H)
126. C₂×Q₈	C₂²×C₃×D₃	direct (N × H)
134. C₂²×C₃	C₂×D₃×Q₈	direct (N × H)
135. C₂²×C₃	C₂²×C₃×Q₈	semidirect
138. C₂×D₃	C₂²×C₃×Q₈	direct (N × H)
145. C₃²	C₂⁴×Q₈	semidirect
148. C₂³	C₃×D₃×Q₈	direct (N × H)
151. Q₈	C₂³×C₃×D₃	direct (N × H)
158. C₂×C₃	C₂²×D₃×Q₈	direct (N × H)
159. C₂×C₃	C₂³×C₃×Q₈	semidirect
162. D₃	C₂³×C₃×Q₈	direct (N × H)
165. C₂²	C₂×C₃×D₃×Q₈	direct (N × H)
170. C₃	C₂³×D₃×Q₈	direct (N × H)
171. C₃	C₂⁴×C₃×Q₈	semidirect
173. C₂	C₂²×C₃×D₃×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×C₃²×Q₈	2
C₂²×C₃×D₃×Q₈	2
C₂³×C₃×C₄×D₃	2
C₂³×C₃×Dic₆	2
C₂⁴×C₃×Q₈	3
C₂³×D₃×Q₈	3

Subgroups — C23×C3×D3×Q8 #11919174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×D₃×Q₈ #11919174