Subgroups — C₃²×#4830 #13162264

Order 1728.

764 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×Dic₃⋊C₈	C₂	1	2	yes	maximal
4	C₃²×C₈×Dic₃	C₂	1	2	yes	maximal
5	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃²×C₈×Q₈	—	3	1	no	maximal
7	C₃×#4830	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×Dic₆	C₄	1	1	yes	characteristic
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₃³×C₄²	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊Dic₃	C₄	1	1	yes	characteristic
12	C₃²×Dic₃⋊C₄	C₄	1	2	yes	—
13	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×Q₈	—	3	1	no	—
17	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
18	C₃²×C₄⋊C₈	—	2	4	no	—
19	C₃²×C₄⋊C₈	—	3	2	no	—
20	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
22	C₃²×C₄×C₈	—	3	2	no	—
23	C₃²×C₄×C₈	—	3	1	no	—
24	C₃×Dic₃⋊C₈	C₂×C₃	1	8	yes	—
25	C₃×C₈×Dic₃	C₂×C₃	1	8	yes	—
26	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
32	C₃²×Dic₆	C₈	1	4	yes	complemented
33	C₃³×C₈	—	2	2	no	—
34	C₃²×C₃⋊C₈	—	2	1	no	—
35	C₃²×C₃⋊C₈	Q₈	1	2	yes	complemented
36	C₃×C₈×Q₈	—	3	4	no	—
37	#4830	C₃²	1	1	yes	characteristic, direct product factor
38	C₂×C₃²×Q₈	—	3	1	no	—
39	C₂×C₃×Dic₆	C₃×C₄	1	4	yes	—
40	C₂×C₃²×C₈	—	2	8	no	—
41	C₂×C₃²×C₈	—	3	1	no	—
42	C₂×C₃²×C₈	—	3	1	no	—
43	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
44	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
45	C₃²×C₄⋊C₄	—	3	2	no	—
46	C₃²×C₄⋊C₄	—	3	1	no	—
47	C₃²×C₄²	—	2	4	no	—
48	C₃²×C₄²	—	3	2	no	—
49	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₃²×C₄²	C₂²×C₃	1	4	yes	—
51	C₃×C₄⋊Dic₃	C₃×C₄	1	4	yes	—
52	C₃×Dic₃⋊C₄	C₃×C₄	1	8	yes	—
53	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
54	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
55	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
56	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
57	C₃³×C₄	—	2	1	no	—
58	C₃³×C₄	C₂×C₈	1	2	yes	—
59	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₃³×C₄	#41	1	1	yes	characteristic
61	C₃²×Dic₃	—	2	2	no	—
62	C₃²×Dic₃	C₂×C₈	1	4	yes	—
63	C₃×C₄×Q₈	—	3	4	no	—
64	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
65	C₃×C₄⋊C₈	—	2	4	no	—
66	C₃×C₄⋊C₈	—	3	8	no	—
67	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	—
68	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
69	C₃×C₄×C₈	—	3	8	no	—
70	C₃×C₄×C₈	—	3	4	no	—
71	Dic₃⋊C₈	C₂×C₃²	1	2	yes	—
72	C₈×Dic₃	C₂×C₃²	1	2	yes	—
73	C₄×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic
74	C₃²×Q₈	—	3	4	no	—
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	3	2	no	—
79	C₂×C₃²×C₄	—	3	2	no	—
80	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
86	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
87	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
88	C₃×Dic₆	C₃×C₈	1	16	yes	complemented
89	C₃²×C₈	—	2	2	no	—
90	C₃²×C₈	—	2	8	no	—
91	C₃²×C₈	—	3	2	no	—
92	C₃²×C₈	—	4	8	no	—
93	C₃²×C₈	—	6	1	no	—
94	C₃×C₃⋊C₈	—	2	4	no	—
95	C₃×C₃⋊C₈	C₃×Q₈	1	8	yes	complemented
96	C₈×Q₈	—	3	1	no	Sylow
97	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₃³	#131	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	3	4	no	—
101	C₂×Dic₆	C₃²×C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	—	2	8	no	—
103	C₂×C₃×C₈	—	3	4	no	—
104	C₂×C₃×C₈	—	3	4	no	—
105	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
106	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
107	C₃×C₄⋊C₄	—	3	8	no	—
108	C₃×C₄⋊C₄	—	3	4	no	—
109	C₃×C₄²	—	2	4	no	—
110	C₃×C₄²	—	3	8	no	—
111	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
112	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
113	C₄⋊Dic₃	C₃²×C₄	1	1	yes	characteristic
114	Dic₃⋊C₄	C₃²×C₄	1	2	yes	—
115	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
116	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
118	C₂²×C₃²	—	2	4	no	—
119	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	—	2	8	no	—
123	C₃²×C₄	—	2	4	no	—
124	C₃²×C₄	—	2	4	no	—
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	—	3	4	no	—
127	C₃²×C₄	—	4	4	no	—
128	C₃²×C₄	—	6	2	no	—
129	C₃²×C₄	D₃×C₈	1	2	yes	—
130	C₃²×C₄	C₂×C₃×C₈	1	8	yes	—
131	C₃²×C₄	#237	1	1	yes	characteristic
132	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
133	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
134	C₃²×C₄	C₃×#41	1	4	yes	—
135	C₃×Dic₃	—	2	8	no	—
136	C₃×Dic₃	C₂×C₃×C₈	1	16	yes	—
137	C₄×Q₈	—	3	1	no	—
138	C₄⋊C₈	—	3	2	no	—
139	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic
140	C₄×C₈	—	3	2	no	—
141	C₄×C₈	—	3	1	no	—
142	C₃³	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
143	C₃×Q₈	—	3	16	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	3	8	no	—
148	C₂×C₃×C₄	—	3	8	no	—
149	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
150	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
154	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
155	C₂×Dic₃	C₂×C₃²×C₄	1	2	yes	—
156	C₂×Dic₃	C₂×C₃²×C₄	1	2	yes	—
157	Dic₆	C₃²×C₈	1	4	yes	complemented
158	C₃×C₈	—	2	8	no	—
159	C₃×C₈	—	2	2	no	—
160	C₃×C₈	—	3	8	no	—
161	C₃×C₈	—	4	8	no	—
162	C₃×C₈	—	6	4	no	—
163	C₃⋊C₈	—	2	1	no	—
164	C₃⋊C₈	C₃²×Q₈	1	2	yes	complemented
165	C₂×C₃²	—	2	4	no	—
166	C₂×C₃²	—	2	4	no	—
167	C₂×C₃²	—	2	4	no	—
168	C₂×C₃²	#887	1	1	yes	characteristic
169	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
170	C₂×C₃²	#907	1	1	yes	characteristic
171	C₂×C₃²	C₃×C₄×Q₈	1	4	yes	—
172	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
173	C₂×C₃²	C₃×#131	1	4	yes	—
174	C₂×Q₈	—	3	1	no	—
175	C₂×C₈	—	3	1	no	—
176	C₂×C₈	—	3	1	no	—
177	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
178	C₄⋊C₄	—	3	2	no	—
179	C₄⋊C₄	—	3	1	no	—
180	C₄²	—	3	2	no	—
181	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
184	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	8	no	—
190	C₃×C₄	—	3	16	no	—
191	C₃×C₄	—	4	4	no	—
192	C₃×C₄	—	6	8	no	—
193	C₃×C₄	C₃×D₃×C₈	1	8	yes	—
194	C₃×C₄	C₂×C₃²×C₈	1	2	yes	—
195	C₃×C₄	C₃×#237	1	4	yes	—
196	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
197	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
198	C₃×C₄	C₃²×#41	1	1	yes	characteristic
199	Dic₃	—	2	2	no	—
200	Dic₃	C₂×C₃²×C₈	1	4	yes	—
201	C₃²	—	2	4	no	—
202	C₃²	#4830	1	1	yes	characteristic, direct product factor
203	C₃²	C₃×C₈×Q₈	1	4	yes	complemented
204	Q₈	—	3	4	no	—
205	C₂×C₄	—	3	2	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	3	2	no	—
212	C₈	—	6	1	no	—
213	C₂×C₃	—	2	4	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	C₃×#887	1	4	yes	—
217	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
218	C₂×C₃	C₃×#907	1	4	yes	—
219	C₂×C₃	C₃²×C₄×Q₈	1	1	yes	characteristic
220	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
221	C₂×C₃	C₃²×#131	1	1	yes	characteristic
222	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	3	4	no	—
225	C₄	—	6	2	no	—
226	C₄	C₃²×D₃×C₈	1	2	yes	—
227	C₄	C₃²×#237	1	1	yes	characteristic
228	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
229	C₃	—	2	4	no	—
230	C₃	C₃×#4830	1	4	yes	direct product factor
231	C₃	C₃²×C₈×Q₈	1	1	yes	characteristic, complemented
232	C₂	C₃²×#887	1	1	yes	characteristic
233	C₂	C₂×C₃²×D₃×C₈	1	1	yes	characteristic
234	C₂	C₃²×#907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃³	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
7. C₃×#4830	C₃	direct (N × H)
32. C₃²×Dic₆	C₈	semidirect
35. C₃²×C₃⋊C₈	Q₈	semidirect
37. #4830	C₃²	direct (N × H)
88. C₃×Dic₆	C₃×C₈	semidirect
95. C₃×C₃⋊C₈	C₃×Q₈	semidirect
142. C₃³	C₈×Q₈	semidirect
157. Dic₆	C₃²×C₈	semidirect
164. C₃⋊C₈	C₃²×Q₈	semidirect
202. C₃²	#4830	direct (N × H)
203. C₃²	C₃×C₈×Q₈	semidirect
230. C₃	C₃×#4830	direct (N × H)
231. C₃	C₃²×C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×Dic₆	2
C₃³×C₄⋊C₈	2
C₃²×Dic₃⋊C₈	2
C₃²×C₈×Dic₃	2
C₃²×C₄×C₃⋊C₈	2
C₃²×C₈×Q₈	3
C₃×#4830	3

Subgroups — C32×#4830 #13162264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4830 #13162264