Subgroups — C₂×Dic₃×C₃⋊(C₃⋊C₈) #13160276

Order 1728.

1276 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₃⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	maximal
6	C₂×Dic₃×C₃⋊C₈	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×Dic₃	C₄	1	2	yes	—
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
11	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₃²×C₄×Dic₃	C₄	1	2	yes	—
14	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
15	C₂×C₃²×C₄²	—	3	1	no	—
16	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
17	C₂²×C₃⋊(C₃⋊C₈)	—	3	4	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	6	4	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
20	C₂×C₃×C₄×Dic₃	D₃	1	4	yes	—
21	C₂²×C₃×C₃⋊C₈	—	3	4	no	—
22	C₄×C₃⋊(C₃⋊C₈)	—	3	4	no	—
23	Dic₃×C₃⋊C₈	—	3	16	no	—
24	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃²×Dic₃	C₈	1	8	yes	complemented
30	C₂×C₃²×Dic₃	C₂×C₄	1	4	yes	—
31	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	4	yes	complemented
32	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	complemented
33	C₂×C₈×Dic₃	—	9	1	no	—
34	C₂×C₄×C₃⋊C₈	—	9	4	no	—
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	—	3	2	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	Dic₃	1	8	yes	—
40	C₃²×C₄²	—	3	2	no	—
41	C₃²×C₄²	—	3	2	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	16	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
46	C₂×C₃⋊(C₃⋊C₈)	—	6	16	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
48	C₂×C₃⋊(C₃⋊C₈)	Dic₃	1	4	yes	direct product factor
49	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
50	C₃×C₄×Dic₃	Dic₃	1	8	yes	—
51	C₃×C₄×Dic₃	C₂×D₃	1	8	yes	—
52	C₂×C₃×C₃⋊C₈	—	3	16	no	—
53	C₂×C₃×C₃⋊C₈	—	3	8	no	—
54	C₂²×C₃³	C₄²	1	2	yes	—
55	C₂²×C₃³	C₂×C₈	1	4	yes	—
56	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
57	C₃³×C₄	C₄²	1	2	yes	—
58	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃²×Dic₃	C₂×C₈	1	8	yes	complemented
61	C₂²×C₃×C₈	—	9	1	no	—
62	C₂×C₃×C₄²	—	3	4	no	—
63	C₂×C₄×Dic₃	C₃⋊D₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	—	3	4	no	—
65	C₂²×C₃⋊C₈	—	9	4	no	—
66	C₂²×C₃⋊C₈	—	9	1	no	—
67	C₂²×C₃⋊C₈	—	18	4	no	—
68	C₈×Dic₃	—	9	4	no	—
69	C₄×C₃⋊C₈	—	9	16	no	—
70	C₂³×C₃²	—	2	4	no	—
71	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
72	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
73	C₂×C₃²×C₄	—	2	4	no	—
74	C₂×C₃²×C₄	—	2	8	no	—
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	3	8	no	—
78	C₂×C₃²×C₄	—	3	4	no	—
79	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
80	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
81	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
83	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
84	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₃⋊C₈	1	32	yes	complemented
88	C₂×C₃×Dic₃	C₂×Dic₃	1	16	yes	—
89	C₃⋊(C₃⋊C₈)	—	3	4	no	—
90	C₃⋊(C₃⋊C₈)	—	3	16	no	—
91	C₃⋊(C₃⋊C₈)	—	6	16	no	—
92	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	4	yes	direct product factor
93	C₃×C₃⋊C₈	—	3	16	no	—
94	C₂×C₄×C₈	—	27	1	no	Sylow
95	C₂×C₃³	C₄×C₈	1	4	yes	complemented
96	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₃³	C₂²×C₈	1	2	yes	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	3	8	no	—
100	C₂²×C₃×C₄	D₃²	1	4	yes	—
101	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
102	C₂²×Dic₃	C₃⋊Dic₃	1	2	yes	—
103	C₂×C₃×C₈	—	9	4	no	—
104	C₂×C₃×C₈	—	9	2	no	—
105	C₃×C₄²	—	3	8	no	—
106	C₃×C₄²	—	3	8	no	—
107	C₄×Dic₃	C₃⋊Dic₃	1	2	yes	—
108	C₄×Dic₃	C₂×C₃⋊D₃	1	2	yes	—
109	C₂×C₃⋊C₈	—	3	16	no	—
110	C₂×C₃⋊C₈	—	3	8	no	—
111	C₂×C₃⋊C₈	—	9	16	no	—
112	C₂×C₃⋊C₈	—	9	4	no	—
113	C₂×C₃⋊C₈	—	9	8	no	—
114	C₂×C₃⋊C₈	—	9	2	no	—
115	C₂×C₃⋊C₈	—	18	16	no	—
116	C₂×C₃⋊C₈	—	18	8	no	—
117	C₂²×C₃²	—	2	16	no	—
118	C₂²×C₃²	—	2	8	no	—
119	C₂²×C₃²	—	2	4	no	—
120	C₂²×C₃²	D₃×C₈	1	4	yes	—
121	C₂²×C₃²	C₂×C₃⋊C₈	1	16	yes	—
122	C₂²×C₃²	C₄×Dic₃	1	8	yes	—
123	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
124	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
126	C₃²×C₄	—	2	8	no	—
127	C₃²×C₄	—	2	4	no	—
128	C₃²×C₄	—	2	4	no	—
129	C₃²×C₄	—	3	8	no	—
130	C₃²×C₄	C₄×Dic₃	1	8	yes	—
131	C₃²×C₄	C₄×Dic₃	1	2	yes	—
132	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
133	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
134	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
136	C₃×Dic₃	C₂×C₃⋊C₈	1	32	yes	complemented
137	C₂²×C₈	—	9	1	no	—
138	C₂²×C₈	—	27	1	no	—
139	C₂×C₄²	—	3	1	no	—
140	C₄×C₈	—	27	4	no	—
141	C₃³	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	D₃×Dic₃	1	4	yes	—
144	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	3	32	no	—
150	C₂×C₃×C₄	—	3	16	no	—
151	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
152	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
153	C₂×C₃×C₄	#624	1	8	yes	—
154	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
155	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
158	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
159	C₂×Dic₃	C₃⋊(C₃⋊C₈)	1	8	yes	direct product factor
160	C₂×Dic₃	C₂×C₃⋊Dic₃	1	4	yes	—
161	C₃×C₈	—	9	4	no	—
162	C₃⋊C₈	—	3	16	no	—
163	C₃⋊C₈	—	9	16	no	—
164	C₃⋊C₈	—	9	4	no	—
165	C₃⋊C₈	—	18	16	no	—
166	C₂×C₃²	—	2	16	no	—
167	C₂×C₃²	—	2	8	no	—
168	C₂×C₃²	—	2	4	no	—
169	C₂×C₃²	C₄×C₃⋊C₈	1	16	yes	complemented
170	C₂×C₃²	C₈×Dic₃	1	4	yes	complemented
171	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
172	C₂×C₃²	C₂²×C₃⋊C₈	1	8	yes	—
173	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
174	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
175	C₂²×C₄	—	3	2	no	—
176	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
177	C₂×C₈	—	9	4	no	—
178	C₂×C₈	—	9	2	no	—
179	C₂×C₈	—	27	4	no	—
180	C₂×C₈	—	27	2	no	—
181	C₄²	—	3	2	no	—
182	C₄²	—	3	2	no	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	16	no	—
185	C₂²×C₃	—	2	8	no	—
186	C₂²×C₃	D₃×C₃⋊C₈	1	16	yes	—
187	C₂²×C₃	Dic₃²	1	8	yes	—
188	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
189	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
190	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
191	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	8	no	—
195	C₃×C₄	—	3	32	no	—
196	C₃×C₄	Dic₃²	1	8	yes	—
197	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
198	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
199	C₃×C₄	C₂×#624	1	4	yes	—
200	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
201	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
202	Dic₃	C₂×C₃⋊(C₃⋊C₈)	1	8	yes	direct product factor
203	C₃²	—	2	4	no	—
204	C₃²	C₂×C₄×C₃⋊C₈	1	4	yes	complemented
205	C₃²	C₂×C₈×Dic₃	1	1	yes	characteristic, complemented
206	C₂³	D₃×C₃⋊Dic₃	1	1	yes	characteristic
207	C₂×C₄	—	3	4	no	—
208	C₂×C₄	—	3	8	no	—
209	C₂×C₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
210	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
211	C₂×C₄	#6289	1	2	yes	—
212	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
213	C₈	—	9	4	no	—
214	C₈	—	27	4	no	—
215	C₂×C₃	—	2	16	no	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	2	8	no	—
218	C₂×C₃	Dic₃×C₃⋊C₈	1	16	yes	complemented
219	C₂×C₃	C₄×C₃⋊(C₃⋊C₈)	1	4	yes	complemented
220	C₂×C₃	C₂×D₃×C₃⋊C₈	1	8	yes	—
221	C₂×C₃	C₂×Dic₃²	1	4	yes	—
222	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	2	yes	—
223	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
224	C₂²	D₃×C₃⋊(C₃⋊C₈)	1	4	yes	—
225	C₂²	Dic₃×C₃⋊Dic₃	1	2	yes	—
226	C₂²	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
227	C₄	—	3	8	no	—
228	C₄	Dic₃×C₃⋊Dic₃	1	2	yes	—
229	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
230	C₄	C₂×#6289	1	1	yes	characteristic
231	C₃	—	2	4	no	—
232	C₃	C₂×Dic₃×C₃⋊C₈	1	4	yes	complemented
233	C₃	C₂×C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented
234	C₂	Dic₃×C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
235	C₂	C₂×D₃×C₃⋊(C₃⋊C₈)	1	2	yes	—
236	C₂	C₂×Dic₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×C₃⋊D₃
commutator, hypocenter	C₃³	C₂×C₄×C₈
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle, prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. Dic₃×C₃⋊(C₃⋊C₈)	C₂	direct (N × H)
9. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
11. C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	semidirect
29. C₂×C₃²×Dic₃	C₈	semidirect
31. C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	semidirect
32. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
48. C₂×C₃⋊(C₃⋊C₈)	Dic₃	direct (N × H)
60. C₃²×Dic₃	C₂×C₈	semidirect
87. C₂×C₃×Dic₃	C₃⋊C₈	semidirect
92. C₃⋊(C₃⋊C₈)	C₂×Dic₃	direct (N × H)
95. C₂×C₃³	C₄×C₈	semidirect
136. C₃×Dic₃	C₂×C₃⋊C₈	semidirect
141. C₃³	C₂×C₄×C₈	semidirect
159. C₂×Dic₃	C₃⋊(C₃⋊C₈)	direct (N × H)
169. C₂×C₃²	C₄×C₃⋊C₈	semidirect
170. C₂×C₃²	C₈×Dic₃	semidirect
202. Dic₃	C₂×C₃⋊(C₃⋊C₈)	direct (N × H)
204. C₃²	C₂×C₄×C₃⋊C₈	semidirect
205. C₃²	C₂×C₈×Dic₃	semidirect
218. C₂×C₃	Dic₃×C₃⋊C₈	semidirect
219. C₂×C₃	C₄×C₃⋊(C₃⋊C₈)	semidirect
232. C₃	C₂×Dic₃×C₃⋊C₈	semidirect
233. C₃	C₂×C₄×C₃⋊(C₃⋊C₈)	semidirect
234. C₂	Dic₃×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
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₂×C₃²×C₄×Dic₃	2
Dic₃×C₃⋊(C₃⋊C₈)	2
C₂×C₄×C₃⋊(C₃⋊C₈)	3
C₂×Dic₃×C₃⋊C₈	3

Subgroups — C2×Dic3×C3⋊(C3⋊C8) #13160276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₃×C₃⋊(C₃⋊C₈) #13160276