Subgroups — C₃²×#5090 #13162528

Order 1728.

920 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#216	C₂	1	2	yes	maximal
3	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
4	C₃²×#835	C₂	1	2	yes	maximal
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₃²×#462	—	3	1	no	maximal
7	C₃×#5090	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
10	C₃³×C₄⋊C₄	—	2	1	no	—
11	C₃³×C₄⋊C₄	—	2	1	no	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×#216	—	2	4	no	—
14	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₃²×Dic₃⋊C₄	—	2	1	no	—
16	C₃²×C₄×Dic₃	—	2	1	no	—
17	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃²×Q₁₆	—	3	2	no	—
19	C₃²×C₄⋊Q₈	—	2	4	no	—
20	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
21	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	—
22	C₃²×C₄×Q₈	—	3	1	no	—
23	C₂×C₃×#216	C₂×C₃	1	8	yes	—
24	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
25	C₃²×C₄⋊C₈	—	3	1	no	—
26	C₃²×#103	—	3	2	no	—
27	C₃×#835	C₂×C₃	1	8	yes	—
28	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	—
29	C₃³×Q₈	—	2	2	no	—
30	C₃³×Q₈	—	2	2	no	—
31	C₂×C₃³×C₄	—	2	2	no	—
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
34	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃²×Dic₃	—	2	1	no	—
36	C₂×C₃²×Dic₃	—	2	1	no	—
37	C₃²×Dic₆	—	2	1	no	—
38	C₃²×Dic₆	D₄	1	2	yes	—
39	C₃²×C₃⋊C₈	—	2	2	no	—
40	C₃×#462	—	3	4	no	—
41	#5090	C₃²	1	1	yes	characteristic, direct product factor
42	C₂×C₃²×Q₈	—	2	8	no	—
43	C₂×C₃²×Q₈	—	3	1	no	—
44	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
45	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
46	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
47	C₃²×Q₁₆	—	6	4	no	—
48	C₂×C₃²×C₈	—	3	2	no	—
49	C₃²×C₄⋊C₄	—	2	8	no	—
50	C₃²×C₄⋊C₄	—	2	8	no	—
51	C₃²×C₄⋊C₄	—	2	1	no	—
52	C₃²×C₄⋊C₄	—	2	1	no	—
53	C₃²×C₄⋊C₄	—	2	4	no	—
54	C₃²×C₄⋊C₄	—	2	4	no	—
55	C₃²×C₄⋊C₄	—	3	1	no	—
56	C₃²×C₄⋊C₄	—	6	1	no	—
57	C₃²×C₄²	—	2	4	no	—
58	C₃²×C₄²	—	6	1	no	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂²×C₃	1	4	yes	—
61	C₃×#216	—	2	16	no	—
62	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
63	C₃×Dic₃⋊C₄	—	2	4	no	—
64	C₃×C₄×Dic₃	—	2	4	no	—
65	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	—	4	2	no	—
69	C₃³×C₄	Q₁₆	1	2	yes	complemented
70	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₃³×C₄	#41	1	1	yes	characteristic
72	C₃²×Dic₃	—	2	2	no	—
73	C₃²×Dic₃	—	4	1	no	—
74	C₂×C₃×Q₁₆	—	3	8	no	—
75	C₃×C₄⋊Q₈	—	2	4	no	—
76	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	—
77	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
78	C₃×C₄×Q₈	—	3	4	no	—
79	C₂×#216	C₂×C₃²	1	2	yes	—
80	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
81	C₃×C₄⋊C₈	—	3	4	no	—
82	C₃×#103	—	3	8	no	—
83	#835	C₂×C₃²	1	2	yes	—
84	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic
85	C₃²×Q₈	—	2	2	no	—
86	C₃²×Q₈	—	2	2	no	—
87	C₃²×Q₈	—	2	8	no	—
88	C₃²×Q₈	—	2	8	no	—
89	C₃²×Q₈	—	3	2	no	—
90	C₃²×Q₈	—	4	8	no	—
91	C₃²×Q₈	—	4	8	no	—
92	C₃²×Q₈	—	6	1	no	—
93	C₂×C₃²×C₄	—	2	16	no	—
94	C₂×C₃²×C₄	—	2	2	no	—
95	C₂×C₃²×C₄	—	2	8	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	—	6	1	no	—
100	C₂×C₃²×C₄	—	6	1	no	—
101	C₂×C₃²×C₄	#67	1	1	yes	characteristic
102	C₂×C₃²×C₄	#67	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
104	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
107	C₂×C₃×Dic₃	—	2	4	no	—
108	C₂×C₃×Dic₃	—	2	4	no	—
109	C₃×Dic₆	—	2	4	no	—
110	C₃×Dic₆	C₃×D₄	1	8	yes	—
111	C₃²×C₈	—	6	2	no	—
112	C₃×C₃⋊C₈	—	2	8	no	—
113	#462	—	3	1	no	Sylow
114	C₂×C₃³	#121	1	1	yes	characteristic
115	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₃³	#137	1	1	yes	characteristic
117	C₂×C₃×Q₈	—	2	8	no	—
118	C₂×C₃×Q₈	—	3	4	no	—
119	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
120	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
121	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
122	C₃×Q₁₆	—	6	16	no	—
123	C₂×C₃×C₈	—	3	8	no	—
124	C₃×C₄⋊C₄	—	2	8	no	—
125	C₃×C₄⋊C₄	—	2	8	no	—
126	C₃×C₄⋊C₄	—	2	4	no	—
127	C₃×C₄⋊C₄	—	2	4	no	—
128	C₃×C₄⋊C₄	—	2	1	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	—	3	4	no	—
131	C₃×C₄⋊C₄	—	6	4	no	—
132	C₃×C₄²	—	2	4	no	—
133	C₃×C₄²	—	6	4	no	—
134	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
135	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
136	#216	—	2	4	no	—
137	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
138	Dic₃⋊C₄	—	2	1	no	—
139	C₄×Dic₃	—	2	1	no	—
140	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
143	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
144	C₃²×C₄	—	2	8	no	—
145	C₃²×C₄	—	2	1	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	2	8	no	—
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	4	16	no	—
151	C₃²×C₄	—	4	2	no	—
152	C₃²×C₄	—	4	8	no	—
153	C₃²×C₄	—	6	2	no	—
154	C₃²×C₄	—	12	1	no	—
155	C₃²×C₄	#216	1	2	yes	complemented
156	C₃²×C₄	C₃×Q₁₆	1	8	yes	complemented
157	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
158	C₃²×C₄	#237	1	1	yes	characteristic
159	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
160	C₃²×C₄	C₃×#41	1	4	yes	—
161	C₃×Dic₃	—	2	8	no	—
162	C₃×Dic₃	—	4	4	no	—
163	C₂×Q₁₆	—	3	2	no	—
164	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic
165	C₄×Q₈	—	3	1	no	—
166	C₄⋊C₈	—	3	1	no	—
167	#103	—	3	2	no	—
168	C₃³	#462	1	1	yes	characteristic, complemented, Sylow
169	C₃×Q₈	—	2	8	no	—
170	C₃×Q₈	—	2	8	no	—
171	C₃×Q₈	—	2	2	no	—
172	C₃×Q₈	—	2	2	no	—
173	C₃×Q₈	—	3	8	no	—
174	C₃×Q₈	—	4	8	no	—
175	C₃×Q₈	—	4	8	no	—
176	C₃×Q₈	—	6	4	no	—
177	C₂×C₃×C₄	—	2	16	no	—
178	C₂×C₃×C₄	—	2	8	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	6	4	no	—
184	C₂×C₃×C₄	—	6	4	no	—
185	C₂×C₃×C₄	C₃×#67	1	4	yes	—
186	C₂×C₃×C₄	C₃×#67	1	4	yes	—
187	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
190	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
191	C₂×Dic₃	—	2	1	no	—
192	C₂×Dic₃	—	2	1	no	—
193	Dic₆	—	2	1	no	—
194	Dic₆	C₃²×D₄	1	2	yes	—
195	C₃×C₈	—	6	8	no	—
196	C₃⋊C₈	—	2	2	no	—
197	C₂×C₃²	—	2	4	no	—
198	C₂×C₃²	—	2	4	no	—
199	C₂×C₃²	—	2	4	no	—
200	C₂×C₃²	#938	1	1	yes	characteristic
201	C₂×C₃²	#942	1	1	yes	characteristic
202	C₂×C₃²	C₂×#216	1	1	yes	characteristic
203	C₂×C₃²	C₃×#121	1	4	yes	—
204	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
205	C₂×C₃²	C₃×#137	1	4	yes	—
206	C₂×Q₈	—	3	1	no	—
207	C₂×Q₈	C₂×C₃²×D₃	1	2	yes	—
208	Q₁₆	—	6	4	no	—
209	C₂×C₈	—	3	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	6	1	no	—
214	C₄²	—	6	1	no	—
215	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
218	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
219	C₃×C₄	—	2	8	no	—
220	C₃×C₄	—	2	4	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	8	no	—
225	C₃×C₄	—	4	16	no	—
226	C₃×C₄	—	4	8	no	—
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	6	8	no	—
229	C₃×C₄	—	12	4	no	—
230	C₃×C₄	C₃×#216	1	8	yes	complemented
231	C₃×C₄	C₃²×Q₁₆	1	2	yes	complemented
232	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
233	C₃×C₄	C₃×#237	1	4	yes	—
234	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
235	C₃×C₄	C₃²×#41	1	1	yes	characteristic
236	Dic₃	—	2	2	no	—
237	Dic₃	—	4	1	no	—
238	C₃²	—	2	4	no	—
239	C₃²	#5090	1	1	yes	characteristic, direct product factor
240	C₃²	C₃×#462	1	4	yes	complemented
241	Q₈	—	2	2	no	—
242	Q₈	—	2	2	no	—
243	Q₈	—	3	2	no	—
244	Q₈	—	6	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	6	1	no	—
247	C₂×C₄	—	6	1	no	—
248	C₂×C₄	C₃²×#67	1	1	yes	characteristic
249	C₂×C₄	C₃²×#67	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
251	C₈	—	6	2	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	C₃×#938	1	4	yes	—
256	C₂×C₃	C₃×#942	1	4	yes	—
257	C₂×C₃	C₂×C₃×#216	1	4	yes	—
258	C₂×C₃	C₃²×#121	1	1	yes	characteristic
259	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
260	C₂×C₃	C₃²×#137	1	1	yes	characteristic
261	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	6	2	no	—
265	C₄	—	12	1	no	—
266	C₄	C₃²×#216	1	2	yes	complemented
267	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
268	C₄	C₃²×#237	1	1	yes	characteristic
269	C₃	—	2	4	no	—
270	C₃	C₃×#5090	1	4	yes	direct product factor
271	C₃	C₃²×#462	1	1	yes	characteristic, complemented
272	C₂	C₃²×#938	1	1	yes	characteristic
273	C₂	C₃²×#942	1	1	yes	characteristic
274	C₂	C₂×C₃²×#216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#462
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
7. C₃×#5090	C₃	direct (N × H)
41. #5090	C₃²	direct (N × H)
69. C₃³×C₄	Q₁₆	semidirect
155. C₃²×C₄	#216	semidirect
156. C₃²×C₄	C₃×Q₁₆	semidirect
168. C₃³	#462	semidirect
230. C₃×C₄	C₃×#216	semidirect
231. C₃×C₄	C₃²×Q₁₆	semidirect
239. C₃²	#5090	direct (N × H)
240. C₃²	C₃×#462	semidirect
266. C₄	C₃²×#216	semidirect
270. C₃	C₃×#5090	direct (N × H)
271. C₃	C₃²×#462	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₂×C₃²×#216	2
C₃²×C₄×Dic₆	2
C₃²×#835	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#462	3
C₃×#5090	3

Subgroups — C32×#5090 #13162528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5090 #13162528