Subgroups — C₃²×#5087 #13162524

Order 1728.

920 conjugacy classes of subgroups.

The subgroup lattice has 286 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, complemented
2	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×#809	C₂	1	2	yes	maximal
4	C₃²×#807	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#474	—	3	1	no	maximal
7	C₃×#5087	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	—	2	1	no	—
9	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×D₃	—	2	1	no	—
11	C₃³×C₄⋊C₄	—	2	1	no	—
12	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
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, complemented
19	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
20	C₃²×C₄×D₄	—	3	1	no	—
21	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
22	C₃²×#107	—	3	2	no	—
23	C₃²×C₄⋊C₈	—	3	1	no	—
24	C₃²×#102	—	3	2	no	—
25	C₃×#809	C₂×C₃	1	8	yes	—
26	C₃×#807	C₂×C₃	1	8	yes	complemented
27	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	complemented
28	C₂²×C₃²×D₃	—	2	1	no	—
29	C₃³×Q₈	—	4	2	no	—
30	C₂×C₃³×C₄	—	2	2	no	—
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃²×Dic₃	—	2	1	no	—
35	C₃²×D₁₂	—	2	1	no	—
36	C₃²×D₁₂	Q₈	1	2	yes	complemented
37	C₃²×C₄×D₃	—	4	2	no	—
38	C₃²×C₃⋊C₈	—	2	2	no	—
39	C₃×#474	—	3	4	no	—
40	#5087	C₃²	1	1	yes	characteristic, direct product factor
41	C₂×C₃²×Q₈	—	2	8	no	—
42	C₂×C₃²×Q₈	—	2	1	no	—
43	C₂×C₃²×Q₈	—	2	4	no	—
44	C₂×C₃²×D₄	—	3	1	no	—
45	C₂²×C₃²×C₄	—	6	1	no	—
46	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
47	C₂×C₃×C₄×D₃	—	2	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	4	no	—
53	C₃²×C₄⋊C₄	—	3	1	no	—
54	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
55	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
56	C₃²×C₂²⋊C₄	—	6	1	no	—
57	C₃²×C₄²	—	2	4	no	—
58	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₃²×C₄²	C₂²×C₃	1	4	yes	—
60	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
62	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
63	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×D₃	—	2	2	no	—
65	C₂×C₃²×D₃	—	4	1	no	—
66	C₂×C₃²×D₃	—	4	1	no	—
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	—	4	2	no	—
69	C₃³×C₄	D₈	1	2	yes	—
70	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
71	C₃³×C₄	#41	1	1	yes	characteristic
72	C₃²×Dic₃	—	4	1	no	—
73	C₃×C₄⋊Q₈	—	2	4	no	—
74	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
75	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
76	C₃×C₄×D₄	—	3	4	no	—
77	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
78	C₃×#107	—	3	8	no	—
79	C₃×C₄⋊C₈	—	3	4	no	—
80	C₃×#102	—	3	8	no	—
81	#809	C₂×C₃²	1	2	yes	—
82	#807	C₂×C₃²	1	2	yes	complemented
83	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic, complemented
84	C₂³×C₃²	—	6	1	no	—
85	C₂²×C₃×D₃	—	2	4	no	—
86	C₃²×Q₈	—	4	16	no	—
87	C₃²×Q₈	—	4	2	no	—
88	C₃²×Q₈	—	4	8	no	—
89	C₃²×D₄	—	3	2	no	—
90	C₃²×D₄	—	6	1	no	—
91	C₂×C₃²×C₄	—	2	16	no	—
92	C₂×C₃²×C₄	—	2	2	no	—
93	C₂×C₃²×C₄	—	2	8	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	2	4	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	6	1	no	—
98	C₂×C₃²×C₄	—	12	2	no	—
99	C₂×C₃²×C₄	#67	1	1	yes	characteristic
100	C₂×C₃²×C₄	#67	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
102	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
103	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
105	C₂×C₃×Dic₃	—	2	4	no	—
106	C₃×D₁₂	—	2	4	no	—
107	C₃×D₁₂	C₃×Q₈	1	8	yes	complemented
108	C₃×C₄×D₃	—	4	8	no	—
109	C₃²×C₈	—	6	2	no	—
110	C₃×C₃⋊C₈	—	2	8	no	—
111	#474	—	3	1	no	Sylow
112	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
114	C₂×C₃³	#137	1	1	yes	characteristic
115	C₃²×D₃	—	4	2	no	—
116	C₂×C₃×Q₈	—	2	8	no	—
117	C₂×C₃×Q₈	—	2	4	no	—
118	C₂×C₃×Q₈	—	2	1	no	—
119	C₂×C₃×D₄	—	3	4	no	—
120	C₂²×C₃×C₄	—	6	4	no	—
121	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
122	C₂×C₄×D₃	—	2	1	no	—
123	C₂×C₃×C₈	—	3	8	no	—
124	C₃×C₄⋊C₄	—	2	8	no	—
125	C₃×C₄⋊C₄	—	2	4	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	—	2	8	no	—
128	C₃×C₄⋊C₄	—	3	4	no	—
129	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
130	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
131	C₃×C₂²⋊C₄	—	6	4	no	—
132	C₃×C₄²	—	2	4	no	—
133	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
134	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
135	(C₂×D₃)⋊C₄	—	2	1	no	—
136	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
137	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
138	C₂²×C₃²	—	2	4	no	—
139	C₂²×C₃²	—	6	2	no	—
140	C₂²×C₃²	—	12	1	no	—
141	C₂²×C₃²	—	12	1	no	—
142	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
143	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
144	C₂×C₃×D₃	—	2	8	no	—
145	C₂×C₃×D₃	—	4	4	no	—
146	C₂×C₃×D₃	—	4	4	no	—
147	C₃²×C₄	—	2	8	no	—
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	2	8	no	—
151	C₃²×C₄	—	2	4	no	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	—	4	16	no	—
154	C₃²×C₄	—	4	2	no	—
155	C₃²×C₄	—	4	8	no	—
156	C₃²×C₄	—	12	1	no	—
157	C₃²×C₄	#213	1	2	yes	—
158	C₃²×C₄	C₃×D₈	1	8	yes	—
159	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
160	C₃²×C₄	#239	1	1	yes	characteristic
161	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
162	C₃²×C₄	C₃×#41	1	4	yes	—
163	C₃×Dic₃	—	4	4	no	—
164	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
165	C₄×D₄	—	3	1	no	—
166	#107	—	3	2	no	—
167	C₄⋊C₈	—	3	1	no	—
168	#102	—	3	2	no	—
169	C₃³	#474	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	6	4	no	—
171	C₂²×D₃	—	2	1	no	—
172	C₃×Q₈	—	4	16	no	—
173	C₃×Q₈	—	4	8	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₃×D₄	—	3	8	no	—
176	C₃×D₄	—	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	—	12	8	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	D₁₂	—	2	1	no	—
193	D₁₂	C₃²×Q₈	1	2	yes	complemented
194	C₄×D₃	—	4	2	no	—
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	—	12	2	no	—
201	C₂×C₃²	C₂×#213	1	1	yes	characteristic
202	C₂×C₃²	#942	1	1	yes	characteristic
203	C₂×C₃²	#946	1	1	yes	characteristic
204	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
205	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
206	C₂×C₃²	C₃×#137	1	4	yes	—
207	C₃×D₃	—	4	8	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×D₄	—	3	1	no	—
210	C₂²×C₄	—	6	1	no	—
211	C₂×C₈	—	3	2	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	3	1	no	—
214	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
215	C₂²⋊C₄	—	6	1	no	—
216	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	6	8	no	—
219	C₂²×C₃	—	12	4	no	—
220	C₂²×C₃	—	12	4	no	—
221	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
222	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
223	C₂×D₃	—	2	2	no	—
224	C₂×D₃	—	4	1	no	—
225	C₂×D₃	—	4	1	no	—
226	C₃×C₄	—	2	8	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	4	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	8	no	—
232	C₃×C₄	—	4	16	no	—
233	C₃×C₄	—	4	8	no	—
234	C₃×C₄	—	4	2	no	—
235	C₃×C₄	—	12	4	no	—
236	C₃×C₄	C₃×#213	1	8	yes	—
237	C₃×C₄	C₃²×D₈	1	2	yes	—
238	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
239	C₃×C₄	C₃×#239	1	4	yes	—
240	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
241	C₃×C₄	C₃²×#41	1	1	yes	characteristic
242	Dic₃	—	4	1	no	—
243	C₃²	—	2	4	no	—
244	C₃²	#5087	1	1	yes	characteristic, direct product factor
245	C₃²	C₃×#474	1	4	yes	complemented
246	C₂³	—	6	1	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	3	2	no	—
249	D₄	—	6	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	6	1	no	—
252	C₂×C₄	—	12	2	no	—
253	C₂×C₄	C₃²×#67	1	1	yes	characteristic
254	C₂×C₄	C₃²×#67	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
256	C₈	—	6	2	no	—
257	C₂×C₃	—	2	4	no	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	—	2	4	no	—
260	C₂×C₃	—	12	8	no	—
261	C₂×C₃	C₂×C₃×#213	1	4	yes	—
262	C₂×C₃	C₃×#942	1	4	yes	—
263	C₂×C₃	C₃×#946	1	4	yes	—
264	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
266	C₂×C₃	C₃²×#137	1	1	yes	characteristic
267	D₃	—	4	2	no	—
268	C₂²	—	6	2	no	—
269	C₂²	—	12	1	no	—
270	C₂²	—	12	1	no	—
271	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	12	1	no	—
275	C₄	C₃²×#213	1	2	yes	—
276	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
277	C₄	C₃²×#239	1	1	yes	characteristic
278	C₃	—	2	4	no	—
279	C₃	C₃×#5087	1	4	yes	direct product factor
280	C₃	C₃²×#474	1	1	yes	characteristic, complemented
281	C₂	—	12	2	no	—
282	C₂	C₂×C₃²×#213	1	1	yes	characteristic
283	C₂	C₃²×#942	1	1	yes	characteristic
284	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	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#474
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
1. C₃³×C₄⋊Q₈	C₂	semidirect
4. C₃²×#807	C₂	semidirect
5. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
7. C₃×#5087	C₃	direct (N × H)
18. C₃²×C₄⋊Q₈	D₃	semidirect
19. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
26. C₃×#807	C₂×C₃	semidirect
27. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
36. C₃²×D₁₂	Q₈	semidirect
40. #5087	C₃²	direct (N × H)
74. C₃×C₄⋊Q₈	C₃×D₃	semidirect
75. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
82. #807	C₂×C₃²	semidirect
83. C₄⋊(C₃⋊C₈)	C₂×C₃²	semidirect
107. C₃×D₁₂	C₃×Q₈	semidirect
164. C₄⋊Q₈	C₃²×D₃	semidirect
169. C₃³	#474	semidirect
193. D₁₂	C₃²×Q₈	semidirect
244. C₃²	#5087	direct (N × H)
245. C₃²	C₃×#474	semidirect
279. C₃	C₃×#5087	direct (N × H)
280. C₃	C₃²×#474	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×D₁₂	2
C₃²×#809	2
C₃²×#807	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#474	3
C₃×#5087	3

Subgroups — C32×#5087 #13162524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5087 #13162524