Subgroups — C₃×#10592774 #13163114

Order 1728.

1200 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3754	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×#634	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63659	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592774	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5075	—	3	4	no	maximal
8	C₂×C₃³×D₄	—	2	1	no	—
9	C₂×C₃³×D₄	—	2	1	no	—
10	C₂×C₃³×D₄	C₂²	1	2	yes	—
11	C₂×C₃×#634	C₂²	1	1	yes	characteristic
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#3754	—	2	4	no	—
14	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
15	C₃×#3750	—	2	1	no	—
16	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃²×#127	—	2	4	no	—
19	C₃²×#127	D₃	1	4	yes	—
20	C₃²×#127	C₂×C₃	1	1	yes	characteristic
21	C₂×#3754	C₂×C₃	1	2	yes	complemented
22	C₄×#634	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×#214	—	3	8	no	—
24	C₃×C₄×Dic₆	—	3	4	no	—
25	#63659	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#832	—	3	8	no	—
28	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
29	C₂³×C₃³	—	2	2	no	—
30	C₃³×D₄	—	2	2	no	—
31	C₃³×D₄	—	2	2	no	—
32	C₃³×D₄	—	4	2	no	—
33	C₃³×D₄	—	4	2	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	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
38	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
39	C₃×#634	—	2	1	no	—
40	C₃×#634	D₄	1	2	yes	complemented
41	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₃×#460	—	9	1	no	—
43	#5075	—	3	4	no	—
44	C₂×C₃²×D₄	—	2	8	no	—
45	C₂×C₃²×D₄	—	2	8	no	—
46	C₂×C₃²×D₄	—	2	8	no	—
47	C₂×C₃²×D₄	—	2	4	no	—
48	C₂×C₃²×D₄	—	2	4	no	—
49	C₂×C₃²×D₄	—	2	1	no	—
50	C₂×C₃²×D₄	—	2	1	no	—
51	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
52	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
53	C₂×#634	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×Dic₆	—	3	4	no	—
55	C₃²×C₄²	—	2	4	no	—
56	C₃²×C₄²	C₂×D₃	1	4	yes	—
57	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
58	#3754	—	2	4	no	—
59	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
60	#3750	—	2	1	no	—
61	C₄×C₃⋊Dic₃	—	2	1	no	—
62	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
63	C₃×#214	—	6	16	no	—
64	C₃×C₄⋊Dic₃	—	3	4	no	—
65	C₃×Dic₃⋊C₄	—	6	4	no	—
66	C₃×C₄×Dic₃	—	6	4	no	—
67	C₂×C₃×C₃⋊C₈	—	3	8	no	—
68	C₂²×C₃³	—	4	2	no	—
69	C₂²×C₃³	—	4	2	no	—
70	C₂²×C₃³	—	4	2	no	—
71	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
72	C₃³×C₄	—	2	1	no	—
73	C₃³×C₄	SD₁₆	1	2	yes	complemented
74	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
75	C₃³×C₄	#41	1	1	yes	characteristic
76	C₃×C₃⋊Dic₃	—	2	2	no	—
77	C₃×C₃⋊Dic₃	—	4	1	no	—
78	C₂×C₃×SD₁₆	—	9	2	no	—
79	C₃×#127	—	2	4	no	—
80	C₃×#127	C₃×D₃	1	4	yes	—
81	C₃×#127	C₃⋊D₃	1	1	yes	characteristic
82	C₃×C₄×Q₈	—	9	1	no	—
83	C₂×#214	—	3	8	no	—
84	C₄×Dic₆	—	3	4	no	—
85	C₃×C₄⋊C₈	—	9	1	no	—
86	C₃×#102	—	9	2	no	—
87	#832	—	3	8	no	—
88	C₄⋊(C₃⋊C₈)	—	3	4	no	—
89	C₂³×C₃²	—	2	16	no	—
90	C₂³×C₃²	—	2	8	no	—
91	C₂³×C₃²	—	2	2	no	—
92	C₃²×D₄	—	2	2	no	—
93	C₃²×D₄	—	2	2	no	—
94	C₃²×D₄	—	2	8	no	—
95	C₃²×D₄	—	2	8	no	—
96	C₃²×D₄	—	4	8	no	—
97	C₃²×D₄	—	4	8	no	—
98	C₃²×D₄	—	4	16	no	—
99	C₃²×D₄	—	4	8	no	—
100	C₃²×D₄	—	4	2	no	—
101	C₃²×D₄	—	4	16	no	—
102	C₃²×D₄	—	4	8	no	—
103	C₃²×D₄	—	4	2	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	C₂×C₃⋊Dic₃	—	2	1	no	—
114	C₂×C₃⋊Dic₃	—	2	1	no	—
115	#634	—	2	1	no	—
116	#634	C₃×D₄	1	2	yes	complemented
117	C₂×C₃×Dic₃	—	6	4	no	—
118	C₂×C₃×Dic₃	—	6	4	no	—
119	C₃×Dic₆	—	3	8	no	—
120	C₃×Dic₆	—	6	4	no	—
121	C₃⋊(C₃⋊C₈)	—	2	2	no	—
122	C₃×C₃⋊C₈	—	6	8	no	—
123	#460	—	9	1	no	Sylow
124	C₂×C₃³	—	8	2	no	—
125	C₂×C₃³	#121	1	1	yes	characteristic
126	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₃×Q₈	—	9	1	no	—
129	C₂×C₃×D₄	—	2	8	no	—
130	C₂×C₃×D₄	—	2	8	no	—
131	C₂×C₃×D₄	—	2	4	no	—
132	C₂×C₃×D₄	—	2	4	no	—
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	—	2	1	no	—
135	C₂×C₃×D₄	—	2	8	no	—
136	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
137	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
138	C₂×Dic₆	—	3	4	no	—
139	C₃×SD₁₆	—	18	4	no	—
140	C₂×C₃×C₈	—	9	2	no	—
141	C₃×C₄⋊C₄	—	9	1	no	—
142	C₃×C₄⋊C₄	—	18	1	no	—
143	C₃×C₄²	—	2	4	no	—
144	C₃×C₄²	—	18	1	no	—
145	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
146	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
147	#214	—	6	16	no	—
148	C₄⋊Dic₃	—	3	4	no	—
149	Dic₃⋊C₄	—	6	4	no	—
150	C₄×Dic₃	—	6	4	no	—
151	C₂×C₃⋊C₈	—	3	8	no	—
152	C₂²×C₃²	—	2	4	no	—
153	C₂²×C₃²	—	4	16	no	—
154	C₂²×C₃²	—	4	16	no	—
155	C₂²×C₃²	—	4	16	no	—
156	C₂²×C₃²	—	4	8	no	—
157	C₂²×C₃²	—	4	8	no	—
158	C₂²×C₃²	—	4	8	no	—
159	C₂²×C₃²	—	4	2	no	—
160	C₂²×C₃²	—	4	2	no	—
161	C₂²×C₃²	—	4	2	no	—
162	C₂²×C₃²	C₂×#67	1	4	yes	—
163	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
164	C₃²×C₄	—	2	8	no	—
165	C₃²×C₄	—	2	4	no	—
166	C₃²×C₄	—	2	1	no	—
167	C₃²×C₄	—	2	8	no	—
168	C₃²×C₄	—	2	4	no	—
169	C₃²×C₄	—	2	4	no	—
170	C₃²×C₄	#214	1	8	yes	complemented
171	C₃²×C₄	C₃×SD₁₆	1	2	yes	complemented
172	C₃²×C₄	D₃×D₄	1	4	yes	—
173	C₃²×C₄	#237	1	4	yes	—
174	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₃²×C₄	C₃×#41	1	1	yes	characteristic
176	C₃⋊Dic₃	—	2	2	no	—
177	C₃⋊Dic₃	—	4	1	no	—
178	C₃×Dic₃	—	6	8	no	—
179	C₃×Dic₃	—	12	4	no	—
180	C₂×SD₁₆	—	9	2	no	—
181	#127	C₃×C₃⋊D₃	1	1	yes	characteristic
182	C₄×Q₈	—	9	1	no	—
183	C₄⋊C₈	—	9	1	no	—
184	#102	—	9	2	no	—
185	C₃³	#460	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	2	16	no	—
187	C₂³×C₃	—	2	8	no	—
188	C₂³×C₃	—	2	2	no	—
189	C₃×Q₈	—	9	2	no	—
190	C₃×Q₈	—	18	1	no	—
191	C₃×D₄	—	2	8	no	—
192	C₃×D₄	—	2	8	no	—
193	C₃×D₄	—	2	2	no	—
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	—	4	8	no	—
196	C₃×D₄	—	4	8	no	—
197	C₃×D₄	—	4	16	no	—
198	C₃×D₄	—	4	8	no	—
199	C₃×D₄	—	4	2	no	—
200	C₃×D₄	—	4	16	no	—
201	C₃×D₄	—	4	8	no	—
202	C₃×D₄	—	4	2	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	18	1	no	—
207	C₂×C₃×C₄	—	18	1	no	—
208	C₂×C₃×C₄	C₃×#67	1	4	yes	—
209	C₂×C₃×C₄	C₃×#67	1	4	yes	—
210	C₂×C₃×C₄	#638	1	1	yes	characteristic
211	C₂×C₃×C₄	#638	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
213	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
214	C₂×Dic₃	—	6	4	no	—
215	C₂×Dic₃	—	6	4	no	—
216	Dic₆	—	3	8	no	—
217	Dic₆	—	6	4	no	—
218	C₃×C₈	—	18	2	no	—
219	C₃⋊C₈	—	6	8	no	—
220	C₂×C₃²	—	2	4	no	—
221	C₂×C₃²	—	2	4	no	—
222	C₂×C₃²	—	2	4	no	—
223	C₂×C₃²	—	8	16	no	—
224	C₂×C₃²	—	8	8	no	—
225	C₂×C₃²	—	8	2	no	—
226	C₂×C₃²	#932	1	4	yes	—
227	C₂×C₃²	C₂×#214	1	4	yes	—
228	C₂×C₃²	#938	1	4	yes	—
229	C₂×C₃²	C₃×#121	1	1	yes	characteristic
230	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
231	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂×Q₈	—	9	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
236	SD₁₆	—	18	4	no	—
237	C₂×C₈	—	9	2	no	—
238	C₄⋊C₄	—	9	1	no	—
239	C₄⋊C₄	—	18	1	no	—
240	C₄²	—	18	1	no	—
241	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	—	4	16	no	—
244	C₂²×C₃	—	4	16	no	—
245	C₂²×C₃	—	4	16	no	—
246	C₂²×C₃	—	4	8	no	—
247	C₂²×C₃	—	4	8	no	—
248	C₂²×C₃	—	4	8	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	—	4	2	no	—
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
253	C₂²×C₃	C₂×#638	1	1	yes	characteristic
254	C₃×C₄	—	2	8	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	8	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	18	2	no	—
261	C₃×C₄	—	36	1	no	—
262	C₃×C₄	C₃×#214	1	8	yes	complemented
263	C₃×C₄	#3754	1	2	yes	complemented
264	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
265	C₃×C₄	C₃×#237	1	4	yes	—
266	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
267	C₃×C₄	#3830	1	1	yes	characteristic
268	Dic₃	—	6	8	no	—
269	Dic₃	—	12	4	no	—
270	C₃²	—	2	4	no	—
271	C₃²	#5075	1	4	yes	complemented
272	C₃²	C₃×#460	1	1	yes	characteristic, complemented
273	C₂³	—	2	2	no	—
274	Q₈	—	9	2	no	—
275	Q₈	—	18	1	no	—
276	D₄	—	2	2	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	18	1	no	—
282	C₂×C₄	C₃×#638	1	1	yes	characteristic
283	C₂×C₄	C₃×#638	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
285	C₈	—	18	2	no	—
286	C₂×C₃	—	2	4	no	—
287	C₂×C₃	—	2	4	no	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	8	16	no	—
290	C₂×C₃	—	8	2	no	—
291	C₂×C₃	—	8	8	no	—
292	C₂×C₃	C₃×#932	1	4	yes	—
293	C₂×C₃	C₂×C₃×#214	1	4	yes	—
294	C₂×C₃	C₃×#938	1	4	yes	—
295	C₂×C₃	#64141	1	1	yes	characteristic
296	C₂×C₃	C₂×#3754	1	1	yes	characteristic
297	C₂×C₃	#64147	1	1	yes	characteristic
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₂×C₃×#638	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	18	2	no	—
304	C₄	—	36	1	no	—
305	C₄	C₃×#3754	1	2	yes	complemented
306	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
307	C₄	C₃×#3830	1	1	yes	characteristic
308	C₃	—	2	4	no	—
309	C₃	C₃×#5075	1	4	yes	complemented
310	C₃	#10592774	1	1	yes	characteristic, direct product factor
311	C₂	—	8	2	no	—
312	C₂	C₃×#64141	1	1	yes	characteristic
313	C₂	C₂×C₃×#3754	1	1	yes	characteristic
314	C₂	C₃×#64147	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, prime radical	C₃³×#127	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₃×#127	C₃⋊D₃
hypocenter	C₃²	C₃×#460
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3754	C₂	semidirect
3. C₃×C₄×#634	C₂	semidirect
4. C₃×#63659	C₂	semidirect
5. C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
6. #10592774	C₃	direct (N × H)
21. C₂×#3754	C₂×C₃	semidirect
22. C₄×#634	C₂×C₃	semidirect
25. #63659	C₂×C₃	semidirect
26. C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	semidirect
40. C₃×#634	D₄	semidirect
73. C₃³×C₄	SD₁₆	semidirect
116. #634	C₃×D₄	semidirect
170. C₃²×C₄	#214	semidirect
171. C₃²×C₄	C₃×SD₁₆	semidirect
185. C₃³	#460	semidirect
262. C₃×C₄	C₃×#214	semidirect
263. C₃×C₄	#3754	semidirect
271. C₃²	#5075	semidirect
272. C₃²	C₃×#460	semidirect
305. C₄	C₃×#3754	semidirect
309. C₃	C₃×#5075	semidirect
310. C₃	#10592774	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×#127	2
C₂×C₃×#3754	2
C₃×C₄×#634	2
C₃×#63659	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10592774	3
C₃×#5075	3

Subgroups — C3×#10592774 #13163114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592774 #13163114