Subgroups — C₃²×#5094 #13162530

Order 1728.

848 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃²×#810	C₂	1	2	yes	maximal
4	C₃²×#808	C₂	1	2	yes	maximal
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₃²×#475	—	3	1	no	maximal
7	C₃×#5094	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	—	2	1	no	—
9	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
10	C₃³×C₄⋊C₄	—	2	1	no	—
11	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₃²×Dic₃⋊C₄	—	2	1	no	—
15	C₃²×C₄×Dic₃	—	2	1	no	—
16	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃²×C₄⋊Q₈	—	2	4	no	—
18	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	—
20	C₃²×C₄×Q₈	—	3	1	no	—
21	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
22	C₃²×#106	—	3	2	no	—
23	C₃²×C₄⋊C₈	—	3	1	no	—
24	C₃²×#103	—	3	2	no	—
25	C₃×#810	C₂×C₃	1	8	yes	—
26	C₃×#808	C₂×C₃	1	8	yes	—
27	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	—
28	C₃³×Q₈	—	4	2	no	—
29	C₂×C₃³×C₄	—	2	2	no	—
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	—	2	1	no	—
34	C₂×C₃²×Dic₃	—	2	1	no	—
35	C₃²×Dic₆	—	2	1	no	—
36	C₃²×Dic₆	Q₈	1	2	yes	complemented
37	C₃²×C₃⋊C₈	—	2	2	no	—
38	C₃×#475	—	3	4	no	—
39	#5094	C₃²	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	—	2	8	no	—
41	C₂×C₃²×Q₈	—	2	1	no	—
42	C₂×C₃²×Q₈	—	2	4	no	—
43	C₂×C₃²×Q₈	—	3	1	no	—
44	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
45	C₂×C₃²×C₈	—	3	2	no	—
46	C₃²×C₄⋊C₄	—	2	8	no	—
47	C₃²×C₄⋊C₄	—	2	8	no	—
48	C₃²×C₄⋊C₄	—	2	1	no	—
49	C₃²×C₄⋊C₄	—	2	4	no	—
50	C₃²×C₄⋊C₄	—	3	1	no	—
51	C₃²×C₄⋊C₄	—	6	1	no	—
52	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
54	C₃²×C₄²	—	2	4	no	—
55	C₃²×C₄²	—	6	1	no	—
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₃²×C₄²	C₂²×C₃	1	4	yes	—
58	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
59	C₃×Dic₃⋊C₄	—	2	4	no	—
60	C₃×C₄×Dic₃	—	2	4	no	—
61	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
62	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
63	C₃³×C₄	—	2	1	no	—
64	C₃³×C₄	—	4	2	no	—
65	C₃³×C₄	SD₁₆	1	2	yes	—
66	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₃³×C₄	#41	1	1	yes	characteristic
68	C₃²×Dic₃	—	2	2	no	—
69	C₃²×Dic₃	—	4	1	no	—
70	C₃×C₄⋊Q₈	—	2	4	no	—
71	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	—
72	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
73	C₃×C₄×Q₈	—	3	4	no	—
74	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
75	C₃×#106	—	3	8	no	—
76	C₃×C₄⋊C₈	—	3	4	no	—
77	C₃×#103	—	3	8	no	—
78	#810	C₂×C₃²	1	2	yes	—
79	#808	C₂×C₃²	1	2	yes	—
80	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic
81	C₃²×Q₈	—	3	2	no	—
82	C₃²×Q₈	—	4	16	no	—
83	C₃²×Q₈	—	4	2	no	—
84	C₃²×Q₈	—	4	8	no	—
85	C₃²×Q₈	—	6	1	no	—
86	C₂×C₃²×C₄	—	2	16	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	—	2	8	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	—	6	1	no	—
93	C₂×C₃²×C₄	—	6	1	no	—
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₂×C₃²×C₄	#67	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
98	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
100	C₂×C₃×Dic₃	—	2	4	no	—
101	C₂×C₃×Dic₃	—	2	4	no	—
102	C₃×Dic₆	—	2	4	no	—
103	C₃×Dic₆	C₃×Q₈	1	8	yes	complemented
104	C₃²×C₈	—	6	2	no	—
105	C₃×C₃⋊C₈	—	2	8	no	—
106	#475	—	3	1	no	Sylow
107	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₃³	#137	1	1	yes	characteristic
110	C₂×C₃×Q₈	—	2	8	no	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	2	1	no	—
113	C₂×C₃×Q₈	—	3	4	no	—
114	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
115	C₂×C₃×C₈	—	3	8	no	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₃×C₄⋊C₄	—	2	4	no	—
118	C₃×C₄⋊C₄	—	2	1	no	—
119	C₃×C₄⋊C₄	—	2	8	no	—
120	C₃×C₄⋊C₄	—	3	4	no	—
121	C₃×C₄⋊C₄	—	6	4	no	—
122	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
123	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
124	C₃×C₄²	—	2	4	no	—
125	C₃×C₄²	—	6	4	no	—
126	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
127	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
128	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
129	Dic₃⋊C₄	—	2	1	no	—
130	C₄×Dic₃	—	2	1	no	—
131	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
134	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
135	C₃²×C₄	—	2	8	no	—
136	C₃²×C₄	—	2	1	no	—
137	C₃²×C₄	—	2	4	no	—
138	C₃²×C₄	—	2	8	no	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	2	4	no	—
141	C₃²×C₄	—	4	16	no	—
142	C₃²×C₄	—	4	2	no	—
143	C₃²×C₄	—	4	8	no	—
144	C₃²×C₄	—	6	2	no	—
145	C₃²×C₄	—	12	1	no	—
146	C₃²×C₄	#214	1	2	yes	—
147	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
148	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
149	C₃²×C₄	#239	1	1	yes	characteristic
150	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
151	C₃²×C₄	C₃×#41	1	4	yes	—
152	C₃×Dic₃	—	2	8	no	—
153	C₃×Dic₃	—	4	4	no	—
154	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic
155	C₄×Q₈	—	3	1	no	—
156	#106	—	3	2	no	—
157	C₄⋊C₈	—	3	1	no	—
158	#103	—	3	2	no	—
159	C₃³	#475	1	1	yes	characteristic, complemented, Sylow
160	C₃×Q₈	—	3	8	no	—
161	C₃×Q₈	—	4	16	no	—
162	C₃×Q₈	—	4	8	no	—
163	C₃×Q₈	—	4	2	no	—
164	C₃×Q₈	—	6	4	no	—
165	C₂×C₃×C₄	—	2	16	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	6	4	no	—
172	C₂×C₃×C₄	—	6	4	no	—
173	C₂×C₃×C₄	C₃×#67	1	4	yes	—
174	C₂×C₃×C₄	C₃×#67	1	4	yes	—
175	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
178	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	Dic₆	—	2	1	no	—
182	Dic₆	C₃²×Q₈	1	2	yes	complemented
183	C₃×C₈	—	6	8	no	—
184	C₃⋊C₈	—	2	2	no	—
185	C₂×C₃²	—	2	4	no	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	C₂×#214	1	1	yes	characteristic
189	C₂×C₃²	#942	1	1	yes	characteristic
190	C₂×C₃²	#946	1	1	yes	characteristic
191	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
192	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
193	C₂×C₃²	C₃×#137	1	4	yes	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	3	1	no	—
196	C₂×C₈	—	3	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	3	1	no	—
199	C₄⋊C₄	—	6	1	no	—
200	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
201	C₄²	—	6	1	no	—
202	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
203	C₂²×C₃	—	2	4	no	—
204	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
205	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
206	C₃×C₄	—	2	8	no	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	8	no	—
212	C₃×C₄	—	4	16	no	—
213	C₃×C₄	—	4	8	no	—
214	C₃×C₄	—	4	2	no	—
215	C₃×C₄	—	6	8	no	—
216	C₃×C₄	—	12	4	no	—
217	C₃×C₄	C₃×#214	1	8	yes	—
218	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
219	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
220	C₃×C₄	C₃×#239	1	4	yes	—
221	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
222	C₃×C₄	C₃²×#41	1	1	yes	characteristic
223	Dic₃	—	2	2	no	—
224	Dic₃	—	4	1	no	—
225	C₃²	—	2	4	no	—
226	C₃²	#5094	1	1	yes	characteristic, direct product factor
227	C₃²	C₃×#475	1	4	yes	complemented
228	Q₈	—	3	2	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	6	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	6	1	no	—
233	C₂×C₄	—	6	1	no	—
234	C₂×C₄	C₃²×#67	1	1	yes	characteristic
235	C₂×C₄	C₃²×#67	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
237	C₈	—	6	2	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	2	4	no	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	C₂×C₃×#214	1	4	yes	—
242	C₂×C₃	C₃×#942	1	4	yes	—
243	C₂×C₃	C₃×#946	1	4	yes	—
244	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
246	C₂×C₃	C₃²×#137	1	1	yes	characteristic
247	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	6	2	no	—
251	C₄	—	12	1	no	—
252	C₄	C₃²×#214	1	2	yes	—
253	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
254	C₄	C₃²×#239	1	1	yes	characteristic
255	C₃	—	2	4	no	—
256	C₃	C₃×#5094	1	4	yes	direct product factor
257	C₃	C₃²×#475	1	1	yes	characteristic, complemented
258	C₂	C₂×C₃²×#214	1	1	yes	characteristic
259	C₂	C₃²×#942	1	1	yes	characteristic
260	C₂	C₃²×#946	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×#475
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×#5094	C₃	direct (N × H)
36. C₃²×Dic₆	Q₈	semidirect
39. #5094	C₃²	direct (N × H)
103. C₃×Dic₆	C₃×Q₈	semidirect
159. C₃³	#475	semidirect
182. Dic₆	C₃²×Q₈	semidirect
226. C₃²	#5094	direct (N × H)
227. C₃²	C₃×#475	semidirect
256. C₃	C₃×#5094	direct (N × H)
257. C₃	C₃²×#475	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
C₃²×C₄×Dic₆	2
C₃²×#810	2
C₃²×#808	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#475	3
C₃×#5094	3

Subgroups — C32×#5094 #13162530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5094 #13162530