Subgroups — C₂²×C₃×Q₈×C₃⋊D₃ #13181737

Order 1728.

5724 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×Q₈×C₃⋊D₃	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃×C₄×C₃⋊D₃	C₂	1	3	yes	maximal
4	C₂²×C₃×#634	C₂	1	3	yes	maximal, complemented
5	C₂²×Q₈×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×D₃×Q₈	—	3	4	no	maximal
7	C₂³×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃³×Q₈	C₂²	1	12	yes	complemented
9	C₂²×C₃³×C₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₃⋊Dic₃	C₂²	1	3	yes	—
11	C₃×Q₈×C₃⋊D₃	C₂²	1	64	yes	direct product factor
12	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	36	yes	—
13	C₂×C₃×#634	C₂²	1	36	yes	complemented
14	C₂²×C₃²×Q₈	—	2	4	no	—
15	C₂²×C₃²×Q₈	D₃	1	4	yes	complemented
16	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×Q₈×C₃⋊D₃	C₂×C₃	1	24	yes	direct product factor
18	C₂²×C₄×C₃⋊D₃	C₂×C₃	1	3	yes	—
19	C₂²×#634	C₂×C₃	1	3	yes	complemented
20	C₂×C₃×D₃×Q₈	—	3	96	no	—
21	C₂²×C₃×C₄×D₃	—	3	12	no	—
22	C₂²×C₃×Dic₆	—	3	12	no	—
23	C₂³×C₃³	C₂³	1	1	yes	characteristic
24	C₂²×C₃×C₃⋊D₃	Q₈	1	8	yes	direct product factor
25	C₂²×C₃×C₃⋊D₃	C₂³	1	6	yes	—
26	C₃³×Q₈	C₂³	1	16	yes	complemented
27	C₂×C₃³×C₄	C₂³	1	18	yes	—
28	C₂×C₃×C₃⋊Dic₃	C₂³	1	18	yes	—
29	C₃×C₄×C₃⋊D₃	C₂³	1	48	yes	—
30	C₃×#634	C₂³	1	48	yes	complemented
31	C₂³×C₃×Q₈	—	9	1	no	—
32	C₂²×D₃×Q₈	—	3	4	no	—
33	C₂³×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
34	C₂³×C₃×D₃	—	3	4	no	—
35	C₂×C₃²×Q₈	—	2	48	no	—
36	C₂×C₃²×Q₈	C₂×D₃	1	48	yes	complemented
37	C₂×C₃²×Q₈	C₂²×C₃	1	12	yes	complemented
38	C₂²×C₃²×C₄	—	2	12	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
40	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
41	C₂²×C₃⋊Dic₃	C₂²×C₃	1	3	yes	—
42	Q₈×C₃⋊D₃	C₂²×C₃	1	64	yes	direct product factor
43	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	36	yes	—
44	C₂×#634	C₂²×C₃	1	36	yes	complemented
45	C₂²×C₃×Dic₃	—	3	12	no	—
46	C₃×D₃×Q₈	—	3	256	no	—
47	C₂×C₃×C₄×D₃	—	3	144	no	—
48	C₂×C₃×Dic₆	—	3	144	no	—
49	C₂²×C₃³	C₂×Q₈	1	4	yes	complemented
50	C₂²×C₃³	C₂⁴	1	3	yes	—
51	C₂×C₃×C₃⋊D₃	C₂×Q₈	1	24	yes	direct product factor
52	C₂×C₃×C₃⋊D₃	C₂⁴	1	4	yes	—
53	C₃³×C₄	C₂⁴	1	12	yes	—
54	C₃×C₃⋊Dic₃	C₂⁴	1	12	yes	—
55	C₂²×C₃×Q₈	—	2	4	no	—
56	C₂²×C₃×Q₈	—	9	24	no	—
57	C₂²×C₃×Q₈	—	9	3	no	—
58	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	complemented
59	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic, direct product factor
60	C₂³×C₃×C₄	—	9	3	no	—
61	C₂×D₃×Q₈	—	3	96	no	—
62	C₂²×C₄×D₃	—	3	12	no	—
63	C₂²×Dic₆	—	3	12	no	—
64	C₂³×C₃²	—	2	4	no	—
65	C₂³×C₃²	C₂²×D₃	1	4	yes	—
66	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
67	C₂²×C₃⋊D₃	C₃×Q₈	1	8	yes	direct product factor
68	C₂²×C₃⋊D₃	C₂³×C₃	1	6	yes	—
69	C₂²×C₃×D₃	—	3	32	no	—
70	C₂²×C₃×D₃	—	3	24	no	—
71	C₃²×Q₈	—	2	64	no	—
72	C₃²×Q₈	C₂²×D₃	1	64	yes	complemented
73	C₃²×Q₈	C₂³×C₃	1	16	yes	complemented
74	C₂×C₃²×C₄	—	2	72	no	—
75	C₂×C₃²×C₄	C₂²×D₃	1	72	yes	—
76	C₂×C₃²×C₄	C₂³×C₃	1	18	yes	—
77	C₂×C₃⋊Dic₃	C₂³×C₃	1	18	yes	—
78	C₄×C₃⋊D₃	C₂³×C₃	1	48	yes	—
79	#634	C₂³×C₃	1	48	yes	complemented
80	C₂×C₃×Dic₃	—	3	72	no	—
81	C₃×C₄×D₃	—	3	192	no	—
82	C₃×Dic₆	—	3	192	no	—
83	C₂³×Q₈	—	9	1	no	Sylow
84	C₂×C₃³	C₂²×Q₈	1	6	yes	complemented
85	C₂×C₃³	C₂⁵	1	1	yes	characteristic
86	C₃×C₃⋊D₃	C₂²×Q₈	1	8	yes	direct product factor
87	C₂⁴×C₃	—	9	1	no	—
88	C₂³×D₃	—	3	4	no	—
89	C₂×C₃×Q₈	—	2	48	no	—
90	C₂×C₃×Q₈	—	9	64	no	—
91	C₂×C₃×Q₈	—	9	36	no	—
92	C₂×C₃×Q₈	C₂×C₃×D₃	1	48	yes	complemented
93	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	12	yes	direct product factor
94	C₂²×C₃×C₄	—	2	12	no	—
95	C₂²×C₃×C₄	—	9	3	no	—
96	C₂²×C₃×C₄	—	9	36	no	—
97	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	—
98	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
99	C₂²×Dic₃	—	3	12	no	—
100	D₃×Q₈	—	3	256	no	—
101	C₂×C₄×D₃	—	3	144	no	—
102	C₂×Dic₆	—	3	144	no	—
103	C₂²×C₃²	—	2	12	no	—
104	C₂²×C₃²	—	2	16	no	—
105	C₂²×C₃²	D₃×Q₈	1	16	yes	complemented
106	C₂²×C₃²	C₂×C₃×Q₈	1	4	yes	complemented
107	C₂²×C₃²	C₂³×D₃	1	12	yes	—
108	C₂²×C₃²	C₂⁴×C₃	1	3	yes	—
109	C₂×C₃⋊D₃	C₂×C₃×Q₈	1	24	yes	direct product factor
110	C₂×C₃⋊D₃	C₂⁴×C₃	1	4	yes	—
111	C₂×C₃×D₃	—	3	96	no	—
112	C₂×C₃×D₃	—	3	16	no	—
113	C₃²×C₄	—	2	48	no	—
114	C₃²×C₄	C₂³×D₃	1	48	yes	—
115	C₃²×C₄	C₂⁴×C₃	1	12	yes	—
116	C₃⋊Dic₃	C₂⁴×C₃	1	12	yes	—
117	C₃×Dic₃	—	3	48	no	—
118	C₂²×Q₈	—	9	24	no	—
119	C₂²×Q₈	—	9	3	no	—
120	C₂²×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, direct product factor
121	C₂³×C₄	—	9	3	no	—
122	C₃³	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	9	6	no	—
125	C₂³×C₃	—	9	8	no	—
126	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
127	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
128	C₂²×D₃	—	3	24	no	—
129	C₂²×D₃	—	3	32	no	—
130	C₃×Q₈	—	2	64	no	—
131	C₃×Q₈	—	9	48	no	—
132	C₃×Q₈	C₂²×C₃×D₃	1	64	yes	complemented
133	C₃×Q₈	C₂²×C₃⋊D₃	1	16	yes	direct product factor
134	C₂×C₃×C₄	—	2	72	no	—
135	C₂×C₃×C₄	—	9	18	no	—
136	C₂×C₃×C₄	—	9	48	no	—
137	C₂×C₃×C₄	C₂²×C₃×D₃	1	72	yes	—
138	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	18	yes	—
139	C₂×Dic₃	—	3	72	no	—
140	C₄×D₃	—	3	192	no	—
141	Dic₆	—	3	192	no	—
142	C₂×C₃²	—	2	24	no	—
143	C₂×C₃²	—	2	4	no	—
144	C₂×C₃²	C₂×D₃×Q₈	1	24	yes	complemented
145	C₂×C₃²	C₂²×C₃×Q₈	1	6	yes	complemented
146	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
147	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
148	C₃⋊D₃	C₂²×C₃×Q₈	1	8	yes	direct product factor
149	C₃×D₃	—	3	32	no	—
150	C₂⁴	—	9	1	no	—
151	C₂×Q₈	—	9	64	no	—
152	C₂×Q₈	—	9	36	no	—
153	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	12	yes	direct product factor
154	C₂²×C₄	—	9	3	no	—
155	C₂²×C₄	—	9	36	no	—
156	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
157	C₂²×C₃	—	2	12	no	—
158	C₂²×C₃	—	2	16	no	—
159	C₂²×C₃	—	9	24	no	—
160	C₂²×C₃	—	9	4	no	—
161	C₂²×C₃	C₃×D₃×Q₈	1	16	yes	complemented
162	C₂²×C₃	Q₈×C₃⋊D₃	1	4	yes	direct product factor
163	C₂²×C₃	C₂³×C₃×D₃	1	12	yes	—
164	C₂²×C₃	C₂³×C₃⋊D₃	1	3	yes	—
165	C₂×D₃	—	3	96	no	—
166	C₂×D₃	—	3	16	no	—
167	C₃×C₄	—	2	48	no	—
168	C₃×C₄	—	9	12	no	—
169	C₃×C₄	C₂³×C₃×D₃	1	48	yes	—
170	C₃×C₄	C₂³×C₃⋊D₃	1	12	yes	—
171	Dic₃	—	3	48	no	—
172	C₃²	—	2	4	no	—
173	C₃²	C₂²×D₃×Q₈	1	4	yes	complemented
174	C₃²	C₂³×C₃×Q₈	1	1	yes	characteristic, complemented
175	C₂³	—	9	6	no	—
176	C₂³	—	9	8	no	—
177	C₂³	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
178	Q₈	—	9	48	no	—
179	Q₈	C₂²×C₃×C₃⋊D₃	1	16	yes	direct product factor
180	C₂×C₄	—	9	18	no	—
181	C₂×C₄	—	9	48	no	—
182	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	18	yes	—
183	C₂×C₃	—	2	24	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	—	9	8	no	—
186	C₂×C₃	C₂×C₃×D₃×Q₈	1	24	yes	complemented
187	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	6	yes	direct product factor
188	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
189	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
190	D₃	—	3	32	no	—
191	C₂²	—	9	24	no	—
192	C₂²	—	9	4	no	—
193	C₂²	C₃×Q₈×C₃⋊D₃	1	4	yes	direct product factor
194	C₂²	C₂³×C₃×C₃⋊D₃	1	3	yes	—
195	C₄	—	9	12	no	—
196	C₄	C₂³×C₃×C₃⋊D₃	1	12	yes	—
197	C₃	—	2	4	no	—
198	C₃	C₂²×C₃×D₃×Q₈	1	4	yes	complemented
199	C₃	C₂²×Q₈×C₃⋊D₃	1	1	yes	characteristic, direct product factor
200	C₂	—	9	8	no	—
201	C₂	C₂×C₃×Q₈×C₃⋊D₃	1	6	yes	direct product factor
202	C₂	C₂⁴×C₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁵×C₃
Frattini	C₂	C₂⁴×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×Q₈	C₂
socle	C₂³×C₃³	C₂³
prime radical	C₂³×C₃×C₃⋊D₃	C₂²
hypercenter	C₂²×C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂³×C₃×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
2. C₂×C₃×Q₈×C₃⋊D₃	C₂	direct (N × H)
4. C₂²×C₃×#634	C₂	semidirect
5. C₂²×Q₈×C₃⋊D₃	C₃	direct (N × H)
8. C₂×C₃³×Q₈	C₂²	semidirect
11. C₃×Q₈×C₃⋊D₃	C₂²	direct (N × H)
13. C₂×C₃×#634	C₂²	semidirect
15. C₂²×C₃²×Q₈	D₃	semidirect
16. C₂²×C₃²×Q₈	C₂×C₃	semidirect
17. C₂×Q₈×C₃⋊D₃	C₂×C₃	direct (N × H)
19. C₂²×#634	C₂×C₃	semidirect
24. C₂²×C₃×C₃⋊D₃	Q₈	direct (N × H)
26. C₃³×Q₈	C₂³	semidirect
30. C₃×#634	C₂³	semidirect
36. C₂×C₃²×Q₈	C₂×D₃	semidirect
37. C₂×C₃²×Q₈	C₂²×C₃	semidirect
42. Q₈×C₃⋊D₃	C₂²×C₃	direct (N × H)
44. C₂×#634	C₂²×C₃	semidirect
49. C₂²×C₃³	C₂×Q₈	semidirect
51. C₂×C₃×C₃⋊D₃	C₂×Q₈	direct (N × H)
58. C₂²×C₃×Q₈	C₃×D₃	semidirect
59. C₂²×C₃×Q₈	C₃⋊D₃	direct (N × H)
67. C₂²×C₃⋊D₃	C₃×Q₈	direct (N × H)
72. C₃²×Q₈	C₂²×D₃	semidirect
73. C₃²×Q₈	C₂³×C₃	semidirect
79. #634	C₂³×C₃	semidirect
84. C₂×C₃³	C₂²×Q₈	semidirect
86. C₃×C₃⋊D₃	C₂²×Q₈	direct (N × H)
92. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
93. C₂×C₃×Q₈	C₂×C₃⋊D₃	direct (N × H)
105. C₂²×C₃²	D₃×Q₈	semidirect
106. C₂²×C₃²	C₂×C₃×Q₈	semidirect
109. C₂×C₃⋊D₃	C₂×C₃×Q₈	direct (N × H)
120. C₂²×Q₈	C₃×C₃⋊D₃	direct (N × H)
122. C₃³	C₂³×Q₈	semidirect
132. C₃×Q₈	C₂²×C₃×D₃	semidirect
133. C₃×Q₈	C₂²×C₃⋊D₃	direct (N × H)
144. C₂×C₃²	C₂×D₃×Q₈	semidirect
145. C₂×C₃²	C₂²×C₃×Q₈	semidirect
148. C₃⋊D₃	C₂²×C₃×Q₈	direct (N × H)
153. C₂×Q₈	C₂×C₃×C₃⋊D₃	direct (N × H)
161. C₂²×C₃	C₃×D₃×Q₈	semidirect
162. C₂²×C₃	Q₈×C₃⋊D₃	direct (N × H)
173. C₃²	C₂²×D₃×Q₈	semidirect
174. C₃²	C₂³×C₃×Q₈	semidirect
179. Q₈	C₂²×C₃×C₃⋊D₃	direct (N × H)
186. C₂×C₃	C₂×C₃×D₃×Q₈	semidirect
187. C₂×C₃	C₂×Q₈×C₃⋊D₃	direct (N × H)
193. C₂²	C₃×Q₈×C₃⋊D₃	direct (N × H)
198. C₃	C₂²×C₃×D₃×Q₈	semidirect
199. C₃	C₂²×Q₈×C₃⋊D₃	direct (N × H)
201. C₂	C₂×C₃×Q₈×C₃⋊D₃	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×C₃³×Q₈	2
C₂×C₃×Q₈×C₃⋊D₃	2
C₂²×C₃×C₄×C₃⋊D₃	2
C₂²×C₃×#634	2
C₂²×Q₈×C₃⋊D₃	3
C₂²×C₃×D₃×Q₈	3

Subgroups — C22×C3×Q8×C3⋊D3 #13181737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Q₈×C₃⋊D₃ #13181737