Subgroups — C₃×C₄×C₈×Dic₃ #11783006

Order 1152.

897 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×Dic₃	C₂	1	4	yes	maximal
3	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄²×C₈	—	3	1	no	maximal
6	C₄×C₈×Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×Dic₃	C₄	1	4	yes	—
10	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₄	1	4	yes	complemented
13	C₃×C₈×Dic₃	C₄	1	16	yes	direct product factor
14	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
15	C₂×C₃×C₄×C₈	—	2	1	no	—
16	C₂×C₃×C₄×C₈	—	3	4	no	—
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄³	—	3	1	no	—
21	C₂×C₈×Dic₃	C₂×C₃	1	4	yes	—
22	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
27	C₂×C₃²×C₈	C₂×C₄	1	8	yes	—
28	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
29	C₃²×C₄²	C₂×C₄	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₃×C₄×Dic₃	C₈	1	16	yes	direct product factor
32	C₃×C₄×Dic₃	C₂×C₄	1	8	yes	—
33	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	8	yes	—
35	C₄²×C₈	—	3	1	no	Sylow
36	C₂²×C₃×C₈	—	2	2	no	—
37	C₂²×C₃×C₈	—	3	2	no	—
38	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
39	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	—	3	4	no	—
42	C₂×C₃×C₄²	—	3	2	no	—
43	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₄×Dic₃	C₃×C₄	1	4	yes	—
46	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
47	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
48	C₃×C₄×C₈	—	2	4	no	—
49	C₃×C₄×C₈	—	3	16	no	—
50	C₃×C₄×C₈	—	3	4	no	—
51	C₃×C₄×C₈	Dic₃	1	4	yes	direct product factor
52	C₃×C₄×C₈	C₃×C₄	1	4	yes	complemented
53	C₈×Dic₃	C₃×C₄	1	16	yes	direct product factor
54	C₄×C₃⋊C₈	C₃×C₄	1	4	yes	complemented
55	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₄²	1	4	yes	—
57	C₂×C₃²×C₄	C₂×C₈	1	8	yes	—
58	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
60	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×Dic₃	C₄²	1	8	yes	—
62	C₂×C₃×Dic₃	C₂×C₈	1	16	yes	—
63	C₃²×C₈	C₄²	1	8	yes	complemented
64	C₃×C₃⋊C₈	C₄²	1	8	yes	complemented
65	C₂×C₄×C₈	—	3	4	no	—
66	C₂×C₄×C₈	—	3	1	no	—
67	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
68	C₄³	—	3	1	no	—
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	3	4	no	—
72	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
73	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
74	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂²×Dic₃	C₂×C₃×C₄	1	4	yes	—
77	C₂×C₃×C₈	—	2	8	no	—
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	—	3	4	no	—
80	C₂×C₃×C₈	—	3	8	no	—
81	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
82	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
83	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
84	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
85	C₃×C₄²	—	2	2	no	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	—	3	16	no	—
88	C₃×C₄²	—	3	8	no	—
89	C₃×C₄²	C₄×D₃	1	2	yes	—
90	C₃×C₄²	C₂×Dic₃	1	2	yes	—
91	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
92	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
93	C₄×Dic₃	C₃×C₈	1	16	yes	direct product factor
94	C₄×Dic₃	C₂×C₃×C₄	1	8	yes	—
95	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
96	C₂×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
97	C₂²×C₃²	C₂×C₄²	1	2	yes	—
98	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
99	C₂²×C₃²	C₂²×C₈	1	4	yes	—
100	C₃²×C₄	C₄×C₈	1	8	yes	complemented
101	C₃²×C₄	C₂×C₄²	1	2	yes	—
102	C₃²×C₄	C₂×C₄²	1	2	yes	—
103	C₃×Dic₃	C₄×C₈	1	16	yes	direct product factor
104	C₂²×C₈	—	3	2	no	—
105	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
106	C₂×C₄²	—	3	4	no	—
107	C₂×C₄²	—	3	2	no	—
108	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
109	C₄×C₈	—	3	16	no	—
110	C₄×C₈	—	3	4	no	—
111	C₄×C₈	C₃×Dic₃	1	4	yes	direct product factor
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
114	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	8	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	3	8	no	—
121	C₂×C₃×C₄	—	3	16	no	—
122	C₂×C₃×C₄	D₃×C₈	1	8	yes	—
123	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
124	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
125	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
126	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
128	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
131	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₂×Dic₃	C₃×C₄²	1	8	yes	—
133	C₂×Dic₃	C₂×C₃×C₈	1	16	yes	—
134	C₃×C₈	—	2	8	no	—
135	C₃×C₈	—	3	8	no	—
136	C₃×C₈	C₄×Dic₃	1	8	yes	direct product factor
137	C₃×C₈	C₃×C₄²	1	8	yes	complemented
138	C₃⋊C₈	C₃×C₄²	1	8	yes	complemented
139	C₂×C₃²	C₄³	1	1	yes	characteristic
140	C₂×C₃²	C₂×C₄×C₈	1	4	yes	—
141	C₂×C₃²	C₂×C₄×C₈	1	2	yes	—
142	C₂²×C₄	—	3	4	no	—
143	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
144	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
145	C₂×C₈	—	3	8	no	—
146	C₂×C₈	—	3	4	no	—
147	C₂×C₈	C₃×C₄×D₃	1	8	yes	—
148	C₂×C₈	C₂×C₃×Dic₃	1	4	yes	—
149	C₄²	—	3	16	no	—
150	C₄²	—	3	8	no	—
151	C₄²	C₃×C₄×D₃	1	2	yes	—
152	C₄²	C₂×C₃×Dic₃	1	2	yes	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	C₄²×D₃	1	2	yes	—
157	C₂²×C₃	C₂×D₃×C₈	1	4	yes	—
158	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
159	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
160	C₂²×C₃	C₂×C₃×C₄²	1	2	yes	—
161	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	8	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	3	16	no	—
166	C₃×C₄	C₈×Dic₃	1	8	yes	direct product factor
167	C₃×C₄	C₃×C₄×C₈	1	8	yes	complemented
168	C₃×C₄	C₄²×D₃	1	2	yes	—
169	C₃×C₄	C₂×C₄×Dic₃	1	2	yes	—
170	C₃×C₄	C₂×C₃×C₄²	1	2	yes	—
171	C₃×C₄	C₂×C₃×C₄²	1	2	yes	—
172	Dic₃	C₃×C₄×C₈	1	16	yes	direct product factor
173	C₃²	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
174	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
175	C₂×C₄	—	3	16	no	—
176	C₂×C₄	—	3	8	no	—
177	C₂×C₄	C₃×D₃×C₈	1	8	yes	—
178	C₂×C₄	C₃×C₄×Dic₃	1	4	yes	—
179	C₂×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
180	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
182	C₈	—	3	8	no	—
183	C₈	C₃×C₄×Dic₃	1	8	yes	direct product factor
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	2	4	no	—
186	C₂×C₃	—	2	2	no	—
187	C₂×C₃	C₄×D₃×C₈	1	4	yes	—
188	C₂×C₃	C₄²×Dic₃	1	1	yes	characteristic
189	C₂×C₃	C₂×C₈×Dic₃	1	2	yes	—
190	C₂×C₃	C₃×C₄³	1	1	yes	characteristic
191	C₂×C₃	C₂×C₃×C₄×C₈	1	4	yes	—
192	C₂×C₃	C₂×C₃×C₄×C₈	1	2	yes	—
193	C₂²	C₃×C₄²×D₃	1	2	yes	—
194	C₂²	C₂×C₃×D₃×C₈	1	4	yes	—
195	C₂²	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
196	C₄	—	3	16	no	—
197	C₄	C₃×C₈×Dic₃	1	8	yes	direct product factor
198	C₄	C₃×C₄²×D₃	1	2	yes	—
199	C₄	C₂×C₃×C₄×Dic₃	1	2	yes	—
200	C₃	—	2	1	no	—
201	C₃	C₄×C₈×Dic₃	1	1	yes	characteristic, direct product factor
202	C₃	C₃×C₄²×C₈	1	1	yes	characteristic, complemented
203	C₂	C₃×C₄×D₃×C₈	1	4	yes	—
204	C₂	C₃×C₄²×Dic₃	1	1	yes	characteristic
205	C₂	C₂×C₃×C₈×Dic₃	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄×C₈	D₃
commutator, hypocenter	C₃	C₃×C₄²×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₄×C₈×Dic₃	C₃	direct (N × H)
12. C₃²×C₄×C₈	C₄	semidirect
13. C₃×C₈×Dic₃	C₄	direct (N × H)
14. C₃×C₄×C₃⋊C₈	C₄	semidirect
31. C₃×C₄×Dic₃	C₈	direct (N × H)
51. C₃×C₄×C₈	Dic₃	direct (N × H)
52. C₃×C₄×C₈	C₃×C₄	semidirect
53. C₈×Dic₃	C₃×C₄	direct (N × H)
54. C₄×C₃⋊C₈	C₃×C₄	semidirect
63. C₃²×C₈	C₄²	semidirect
64. C₃×C₃⋊C₈	C₄²	semidirect
93. C₄×Dic₃	C₃×C₈	direct (N × H)
100. C₃²×C₄	C₄×C₈	semidirect
103. C₃×Dic₃	C₄×C₈	direct (N × H)
111. C₄×C₈	C₃×Dic₃	direct (N × H)
136. C₃×C₈	C₄×Dic₃	direct (N × H)
137. C₃×C₈	C₃×C₄²	semidirect
138. C₃⋊C₈	C₃×C₄²	semidirect
166. C₃×C₄	C₈×Dic₃	direct (N × H)
167. C₃×C₄	C₃×C₄×C₈	semidirect
172. Dic₃	C₃×C₄×C₈	direct (N × H)
173. C₃²	C₄²×C₈	semidirect
183. C₈	C₃×C₄×Dic₃	direct (N × H)
197. C₄	C₃×C₈×Dic₃	direct (N × H)
201. C₃	C₄×C₈×Dic₃	direct (N × H)
202. C₃	C₃×C₄²×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×C₈×Dic₃	2
C₃×C₄²×Dic₃	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×C₄²×C₈	3
C₄×C₈×Dic₃	3

Subgroups — C3×C4×C8×Dic3 #11783006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₈×Dic₃ #11783006