Subgroups — C₃×#10592726 #13163065

Order 1728.

1048 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₂×C₃×#3755	C₂	1	2	yes	maximal, complemented
3	C₃×#64083	C₂	1	1	yes	maximal, characteristic
4	C₃×#63636	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592726	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5027	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
12	C₃³×C₄⋊C₄	—	2	1	no	—
13	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃³×C₄²	—	2	1	no	—
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#3755	—	2	4	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃²×C₄×Q₈	—	2	4	no	—
19	C₃²×C₄×Q₈	D₃	1	4	yes	complemented
20	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#3755	C₂×C₃	1	2	yes	complemented
22	#64083	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#215	—	3	8	no	—
24	C₃×#874	—	3	4	no	—
25	#63636	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#809	—	3	8	no	—
28	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
29	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
30	C₃³×Q₈	—	2	1	no	—
31	C₃³×Q₈	D₄	1	2	yes	complemented
32	C₂×C₃³×C₄	—	2	1	no	—
33	C₂×C₃³×C₄	—	2	1	no	—
34	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
35	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
36	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
37	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
38	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
39	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
40	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
41	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₃×#460	—	9	1	no	—
43	#5027	—	3	4	no	—
44	C₂×C₃²×Q₈	—	2	4	no	—
45	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
46	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
48	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
49	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
50	C₂×C₃×D₁₂	—	3	8	no	—
51	C₂×C₃×D₁₂	—	6	4	no	—
52	C₂×C₃×D₁₂	—	6	4	no	—
53	C₃²×C₄⋊C₄	—	2	8	no	—
54	C₃²×C₄⋊C₄	—	2	4	no	—
55	C₃²×C₄⋊C₄	—	2	4	no	—
56	C₃²×C₄⋊C₄	—	2	1	no	—
57	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
58	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₃²×C₄²	—	2	8	no	—
60	C₃²×C₄²	—	2	4	no	—
61	C₃²×C₄²	—	2	4	no	—
62	C₃²×C₄²	—	2	1	no	—
63	C₃²×C₄²	C₂×D₃	1	4	yes	—
64	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
65	#3755	—	2	4	no	—
66	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
67	C₃×#215	—	6	16	no	—
68	C₂×C₃×C₃⋊C₈	—	3	8	no	—
69	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₃⋊D₃	—	4	2	no	—
71	C₂×C₃×C₃⋊D₃	—	4	2	no	—
72	C₂×C₃×C₃⋊D₃	—	4	2	no	—
73	C₃³×C₄	—	2	2	no	—
74	C₃³×C₄	—	2	1	no	—
75	C₃³×C₄	—	4	1	no	—
76	C₃³×C₄	SD₁₆	1	2	yes	complemented
77	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₃³×C₄	#41	1	1	yes	characteristic
79	C₂×C₃×SD₁₆	—	9	2	no	—
80	C₃×#127	—	9	1	no	—
81	C₃×C₄×Q₈	—	2	4	no	—
82	C₃×C₄×Q₈	C₃×D₃	1	4	yes	complemented
83	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
84	C₂×#215	—	3	8	no	—
85	#874	—	3	4	no	—
86	C₃×C₄⋊C₈	—	9	1	no	—
87	C₃×#102	—	9	2	no	—
88	#809	—	3	8	no	—
89	C₄⋊(C₃⋊C₈)	—	3	4	no	—
90	C₂²×C₃⋊D₃	—	2	2	no	—
91	C₂²×C₃×D₃	—	6	8	no	—
92	C₃²×Q₈	—	2	8	no	—
93	C₃²×Q₈	—	2	4	no	—
94	C₃²×Q₈	—	2	1	no	—
95	C₃²×Q₈	—	2	8	no	—
96	C₃²×Q₈	D₁₂	1	8	yes	complemented
97	C₃²×Q₈	C₃×D₄	1	2	yes	complemented
98	C₂×C₃²×C₄	—	2	8	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	—	2	1	no	—
101	C₂×C₃²×C₄	—	2	8	no	—
102	C₂×C₃²×C₄	—	2	4	no	—
103	C₂×C₃²×C₄	—	2	1	no	—
104	C₂×C₃²×C₄	—	2	4	no	—
105	C₂×C₃²×C₄	—	2	4	no	—
106	C₂×C₃²×C₄	—	2	4	no	—
107	C₂×C₃²×C₄	#67	1	4	yes	—
108	C₂×C₃²×C₄	#67	1	4	yes	—
109	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
112	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₄⋊(C₃⋊D₃)	—	2	2	no	—
114	C₄⋊(C₃⋊D₃)	—	2	2	no	—
115	C₄⋊(C₃⋊D₃)	—	4	2	no	—
116	C₄⋊(C₃⋊D₃)	—	4	2	no	—
117	C₃×D₁₂	—	6	8	no	—
118	C₃×D₁₂	—	6	8	no	—
119	C₃×D₁₂	—	12	8	no	—
120	C₃×D₁₂	—	12	8	no	—
121	C₃⋊(C₃⋊C₈)	—	2	2	no	—
122	C₃×C₃⋊C₈	—	6	8	no	—
123	#460	—	9	1	no	Sylow
124	C₂×C₃³	#121	1	1	yes	characteristic
125	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₃×C₃⋊D₃	—	8	2	no	—
128	C₂×C₃×Q₈	—	2	4	no	—
129	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
130	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
131	C₂×C₃×D₄	—	9	2	no	—
132	C₂×C₃×D₄	—	18	1	no	—
133	C₂×C₃×D₄	—	18	1	no	—
134	C₂×D₁₂	—	3	8	no	—
135	C₂×D₁₂	—	6	4	no	—
136	C₂×D₁₂	—	6	4	no	—
137	C₃×SD₁₆	—	18	4	no	—
138	C₂×C₃×C₈	—	9	2	no	—
139	C₃×C₄⋊C₄	—	2	8	no	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	4	no	—
143	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
144	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
145	C₃×C₄²	—	2	8	no	—
146	C₃×C₄²	—	2	4	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	2	4	no	—
149	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
150	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
151	#215	—	6	16	no	—
152	C₂×C₃⋊C₈	—	3	8	no	—
153	C₂²×C₃²	—	2	4	no	—
154	C₂²×C₃²	C₂×#67	1	4	yes	—
155	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₃⋊D₃	—	4	2	no	—
157	C₂×C₃⋊D₃	—	4	2	no	—
158	C₂×C₃⋊D₃	—	4	2	no	—
159	C₂×C₃×D₃	—	12	8	no	—
160	C₂×C₃×D₃	—	12	8	no	—
161	C₂×C₃×D₃	—	12	8	no	—
162	C₃²×C₄	—	2	16	no	—
163	C₃²×C₄	—	2	8	no	—
164	C₃²×C₄	—	2	2	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	—	2	8	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃²×C₄	—	2	4	no	—
169	C₃²×C₄	—	2	4	no	—
170	C₃²×C₄	—	4	4	no	—
171	C₃²×C₄	—	4	8	no	—
172	C₃²×C₄	—	4	4	no	—
173	C₃²×C₄	—	4	1	no	—
174	C₃²×C₄	#215	1	8	yes	complemented
175	C₃²×C₄	C₃×SD₁₆	1	2	yes	complemented
176	C₃²×C₄	C₂×D₁₂	1	4	yes	—
177	C₃²×C₄	#235	1	4	yes	—
178	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
179	C₃²×C₄	C₃×#41	1	1	yes	characteristic
180	C₂×SD₁₆	—	9	2	no	—
181	#127	—	9	1	no	—
182	C₄×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
183	C₄⋊C₈	—	9	1	no	—
184	#102	—	9	2	no	—
185	C₃³	#460	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	18	2	no	—
187	C₂²×D₃	—	6	8	no	—
188	C₃×Q₈	—	2	8	no	—
189	C₃×Q₈	—	2	4	no	—
190	C₃×Q₈	—	2	1	no	—
191	C₃×Q₈	—	2	8	no	—
192	C₃×Q₈	C₃×D₁₂	1	8	yes	complemented
193	C₃×Q₈	C₄⋊(C₃⋊D₃)	1	2	yes	complemented
194	C₃×D₄	—	18	2	no	—
195	C₃×D₄	—	18	2	no	—
196	C₃×D₄	—	36	2	no	—
197	C₃×D₄	—	36	2	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	8	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	C₃×#67	1	4	yes	—
208	C₂×C₃×C₄	C₃×#67	1	4	yes	—
209	C₂×C₃×C₄	#638	1	1	yes	characteristic
210	C₂×C₃×C₄	#638	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
212	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
213	D₁₂	—	6	8	no	—
214	D₁₂	—	6	8	no	—
215	D₁₂	—	12	8	no	—
216	D₁₂	—	12	8	no	—
217	C₃×C₈	—	18	2	no	—
218	C₃⋊C₈	—	6	8	no	—
219	C₂×C₃²	—	2	4	no	—
220	C₂×C₃²	—	2	4	no	—
221	C₂×C₃²	—	2	4	no	—
222	C₂×C₃²	#930	1	4	yes	—
223	C₂×C₃²	C₂×#215	1	4	yes	—
224	C₂×C₃²	#949	1	4	yes	—
225	C₂×C₃²	C₃×#121	1	1	yes	characteristic
226	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
227	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₃⋊D₃	—	8	2	no	—
229	C₃×D₃	—	24	8	no	—
230	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
231	C₂×D₄	—	9	2	no	—
232	C₂×D₄	—	18	1	no	—
233	C₂×D₄	—	18	1	no	—
234	SD₁₆	—	18	4	no	—
235	C₂×C₈	—	9	2	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
238	C₄²	—	2	1	no	—
239	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	36	2	no	—
242	C₂²×C₃	—	36	2	no	—
243	C₂²×C₃	—	36	2	no	—
244	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
245	C₂²×C₃	C₂×#638	1	1	yes	characteristic
246	C₂×D₃	—	12	8	no	—
247	C₂×D₃	—	12	8	no	—
248	C₂×D₃	—	12	8	no	—
249	C₃×C₄	—	2	16	no	—
250	C₃×C₄	—	2	8	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	8	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	—	4	8	no	—
259	C₃×C₄	—	4	4	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	C₃×#215	1	8	yes	complemented
262	C₃×C₄	#3755	1	2	yes	complemented
263	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
264	C₃×C₄	C₃×#235	1	4	yes	—
265	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
266	C₃×C₄	#3828	1	1	yes	characteristic
267	C₃²	—	2	4	no	—
268	C₃²	#5027	1	4	yes	complemented
269	C₃²	C₃×#460	1	1	yes	characteristic, complemented
270	C₂³	—	18	2	no	—
271	Q₈	—	2	1	no	—
272	Q₈	C₃×C₄⋊(C₃⋊D₃)	1	2	yes	complemented
273	D₄	—	18	2	no	—
274	D₄	—	18	2	no	—
275	D₄	—	36	2	no	—
276	D₄	—	36	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	C₃×#638	1	1	yes	characteristic
280	C₂×C₄	C₃×#638	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
282	C₈	—	18	2	no	—
283	C₂×C₃	—	2	4	no	—
284	C₂×C₃	—	2	4	no	—
285	C₂×C₃	—	2	4	no	—
286	C₂×C₃	—	72	2	no	—
287	C₂×C₃	C₃×#930	1	4	yes	—
288	C₂×C₃	C₂×C₃×#215	1	4	yes	—
289	C₂×C₃	C₃×#949	1	4	yes	—
290	C₂×C₃	#64139	1	1	yes	characteristic
291	C₂×C₃	C₂×#3755	1	1	yes	characteristic
292	C₂×C₃	#64158	1	1	yes	characteristic
293	D₃	—	24	8	no	—
294	C₂²	—	36	2	no	—
295	C₂²	—	36	2	no	—
296	C₂²	—	36	2	no	—
297	C₂²	C₂×C₃×#638	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₃×#3755	1	2	yes	complemented
302	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
303	C₄	C₃×#3828	1	1	yes	characteristic
304	C₃	—	2	4	no	—
305	C₃	C₃×#5027	1	4	yes	complemented
306	C₃	#10592726	1	1	yes	characteristic, direct product factor
307	C₂	—	72	2	no	—
308	C₂	C₃×#64139	1	1	yes	characteristic
309	C₂	C₂×C₃×#3755	1	1	yes	characteristic
310	C₂	C₃×#64158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×Q₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃×#64083	C₂
hypercenter	C₃×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#460
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄×Q₈	C₂	semidirect
2. C₂×C₃×#3755	C₂	semidirect
4. C₃×#63636	C₂	semidirect
5. C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
6. #10592726	C₃	direct (N × H)
19. C₃²×C₄×Q₈	D₃	semidirect
20. C₃²×C₄×Q₈	C₂×C₃	semidirect
21. C₂×#3755	C₂×C₃	semidirect
25. #63636	C₂×C₃	semidirect
26. C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	semidirect
31. C₃³×Q₈	D₄	semidirect
76. C₃³×C₄	SD₁₆	semidirect
82. C₃×C₄×Q₈	C₃×D₃	semidirect
83. C₃×C₄×Q₈	C₃⋊D₃	semidirect
96. C₃²×Q₈	D₁₂	semidirect
97. C₃²×Q₈	C₃×D₄	semidirect
174. C₃²×C₄	#215	semidirect
175. C₃²×C₄	C₃×SD₁₆	semidirect
182. C₄×Q₈	C₃×C₃⋊D₃	semidirect
185. C₃³	#460	semidirect
192. C₃×Q₈	C₃×D₁₂	semidirect
193. C₃×Q₈	C₄⋊(C₃⋊D₃)	semidirect
261. C₃×C₄	C₃×#215	semidirect
262. C₃×C₄	#3755	semidirect
268. C₃²	#5027	semidirect
269. C₃²	C₃×#460	semidirect
272. Q₈	C₃×C₄⋊(C₃⋊D₃)	semidirect
301. C₄	C₃×#3755	semidirect
305. C₃	C₃×#5027	semidirect
306. C₃	#10592726	direct (N × H)

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×C₃×#3755	2
C₃×#64083	2
C₃×#63636	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10592726	3
C₃×#5027	3

Subgroups — C3×#10592726 #13163065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592726 #13163065