Subgroups — #13161365

Order 1728.

812 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64157	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#950	C₂	1	1	yes	maximal, characteristic
3	#11723349	C₂	1	1	yes	maximal, characteristic, complemented
4	#11722207	C₂	1	1	yes	maximal, characteristic, complemented
5	#11722058	C₂	1	1	yes	maximal, characteristic, complemented
6	#11722043	C₂	1	1	yes	maximal, characteristic, complemented
7	#11722018	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592895	—	3	1	no	maximal
9	#10591673	—	3	4	no	maximal
10	C₃³×#41	C₂²	1	1	yes	characteristic
11	C₃²×#235	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	#88855	C₂²	1	1	yes	characteristic
14	C₃×#3748	—	2	1	no	—
15	C₃×#3748	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃²×#216	C₄	1	1	yes	characteristic, complemented
19	C₃²×#215	C₄	1	1	yes	characteristic, complemented
20	C₃²×#214	C₄	1	1	yes	characteristic, complemented
21	C₃²×#213	C₄	1	1	yes	characteristic, complemented
22	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	#88786	—	2	1	no	—
24	D₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
25	C₃²×#135	—	3	1	no	—
26	#64157	—	3	1	no	—
27	#64157	D₃	1	1	yes	characteristic, complemented
28	C₃×#950	D₃	1	4	yes	—
29	C₃×#948	—	3	4	no	—
30	#63814	—	3	4	no	—
31	#63664	—	3	1	no	—
32	#63664	—	3	1	no	—
33	#63664	—	3	4	no	—
34	#63664	—	6	4	no	—
35	#63646	—	3	1	no	—
36	C₈×C₃⋊Dic₃	—	3	1	no	—
37	#63577	—	3	4	no	—
38	#63568	—	3	4	no	—
39	#63553	—	3	4	no	—
40	C₃³×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₃³×D₄	—	2	1	no	—
42	C₃³×D₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃³×C₄	—	2	1	no	—
44	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
46	C₃²×#67	—	2	1	no	—
47	C₃²×D₁₂	C₂×C₄	1	1	yes	characteristic
48	C₃²×C₄×D₃	—	2	1	no	—
49	C₃²×Dic₆	C₂×C₄	1	1	yes	characteristic
50	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
51	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
52	C₃×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
53	C₃×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
54	C₃²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	#5196	—	9	4	no	—
57	#5144	—	9	1	no	—
58	C₃²×#41	—	2	4	no	—
59	C₃²×#41	—	3	1	no	—
60	C₃²×#41	C₂×D₃	1	1	yes	characteristic
61	C₃²×#41	C₂×D₃	1	4	yes	—
62	C₃×#235	C₂×D₃	1	4	yes	—
63	C₃²×Q₁₆	—	3	1	no	—
64	C₃²×SD₁₆	—	3	1	no	—
65	C₃²×SD₁₆	—	3	1	no	—
66	C₃²×D₈	—	3	1	no	—
67	C₂×C₃²×C₈	—	3	1	no	—
68	C₄×C₃⋊Dic₃	—	3	1	no	—
69	C₄×C₃⋊Dic₃	—	3	4	no	—
70	C₄×C₃⋊Dic₃	—	6	4	no	—
71	#3748	—	2	1	no	—
72	#3748	—	3	1	no	—
73	#3748	—	3	4	no	—
74	#3748	—	6	1	no	—
75	#3748	—	6	4	no	—
76	#3748	C₂×D₃	1	1	yes	characteristic
77	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
78	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
79	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
80	C₃×#216	Dic₃	1	4	yes	complemented
81	C₃×#215	Dic₃	1	4	yes	complemented
82	C₃×#214	Dic₃	1	4	yes	complemented
83	C₃×#213	Dic₃	1	4	yes	complemented
84	C₃×#208	—	3	4	no	—
85	C₃×#208	—	6	4	no	—
86	C₂×C₃×C₃⋊C₈	—	3	4	no	—
87	C₂×C₃×C₃⋊C₈	—	6	4	no	—
88	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
89	#3711	—	6	4	no	—
90	D₃×C₃⋊C₈	—	6	4	no	—
91	C₂²×C₃³	—	2	1	no	—
92	C₂²×C₃³	#41	1	1	yes	characteristic
93	C₂×C₃²×D₃	—	2	1	no	—
94	C₃³×C₄	—	2	1	no	—
95	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
97	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
98	C₃²×Dic₃	—	2	1	no	—
99	C₃×#135	—	3	4	no	—
100	C₃×#131	—	9	1	no	—
101	#950	C₃⋊D₃	1	1	yes	characteristic
102	#948	—	3	4	no	—
103	#948	—	9	4	no	—
104	#901	—	9	1	no	—
105	#837	—	9	4	no	—
106	#837	—	9	4	no	—
107	#837	—	9	1	no	—
108	#837	—	18	4	no	—
109	#825	—	9	1	no	—
110	#822	—	9	1	no	—
111	#819	—	9	4	no	—
112	C₈×Dic₃	—	9	4	no	—
113	C₈×Dic₃	—	9	1	no	—
114	C₃²×Q₈	—	2	4	no	—
115	C₃²×Q₈	—	3	1	no	—
116	C₃²×Q₈	C₄×D₃	1	1	yes	characteristic
117	C₃²×Q₈	C₂×Dic₃	1	4	yes	—
118	C₃²×D₄	—	2	1	no	—
119	C₃²×D₄	—	2	4	no	—
120	C₃²×D₄	—	2	4	no	—
121	C₃²×D₄	—	3	1	no	—
122	C₃²×D₄	—	4	4	no	—
123	C₃²×D₄	—	6	1	no	—
124	C₃²×D₄	C₄×D₃	1	1	yes	characteristic
125	C₃²×D₄	C₂×Dic₃	1	4	yes	—
126	C₂×C₃²×C₄	—	2	1	no	—
127	C₂×C₃²×C₄	—	2	4	no	—
128	C₂×C₃²×C₄	—	2	4	no	—
129	C₂×C₃²×C₄	—	4	4	no	—
130	C₂×C₃²×C₄	—	6	1	no	—
131	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
132	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₃⋊Dic₃	—	6	1	no	—
134	C₂×C₃⋊Dic₃	—	6	4	no	—
135	C₂×C₃⋊Dic₃	—	12	4	no	—
136	C₃×#67	—	2	4	no	—
137	C₃×D₁₂	C₂×Dic₃	1	4	yes	—
138	C₃×C₄×D₃	—	2	4	no	—
139	C₃×Dic₆	C₂×Dic₃	1	4	yes	—
140	C₃²×C₈	—	3	1	no	—
141	C₃²×C₈	—	3	1	no	—
142	C₃⋊(C₃⋊C₈)	—	2	1	no	—
143	C₃⋊(C₃⋊C₈)	—	6	1	no	—
144	C₃⋊(C₃⋊C₈)	—	6	4	no	—
145	C₃⋊(C₃⋊C₈)	—	12	4	no	—
146	C₃⋊(C₃⋊C₈)	#67	1	1	yes	characteristic
147	C₃⋊(C₃⋊C₈)	#67	1	1	yes	characteristic
148	C₃×C₃⋊C₈	—	3	4	no	—
149	C₃×C₃⋊C₈	—	3	4	no	—
150	C₃×C₃⋊C₈	—	6	4	no	—
151	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
152	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
153	#443	—	27	1	no	Sylow
154	C₂×C₃³	—	2	1	no	—
155	C₂×C₃³	—	4	1	no	—
156	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
157	C₃²×D₃	—	4	1	no	—
158	C₃×#41	—	2	4	no	—
159	C₃×#41	—	3	4	no	—
160	C₃×#41	D₃²	1	4	yes	—
161	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
162	#235	C₂×C₃⋊D₃	1	1	yes	characteristic
163	C₃×Q₁₆	—	3	4	no	—
164	C₃×SD₁₆	—	3	4	no	—
165	C₃×SD₁₆	—	3	4	no	—
166	C₃×D₈	—	3	4	no	—
167	C₃×M₁₆	—	9	1	no	—
168	C₃×M₁₆	—	18	1	no	—
169	C₂×C₃×C₈	—	3	4	no	—
170	C₂×C₃×C₈	—	9	1	no	—
171	C₂×C₃×C₈	—	18	1	no	—
172	#216	C₃⋊Dic₃	1	1	yes	characteristic, complemented
173	#215	C₃⋊Dic₃	1	1	yes	characteristic, complemented
174	#214	C₃⋊Dic₃	1	1	yes	characteristic, complemented
175	#213	C₃⋊Dic₃	1	1	yes	characteristic, complemented
176	C₄×Dic₃	—	9	4	no	—
177	C₄×Dic₃	—	9	1	no	—
178	C₄×Dic₃	—	18	4	no	—
179	#208	—	3	4	no	—
180	#208	—	6	4	no	—
181	#208	—	9	4	no	—
182	#208	—	9	1	no	—
183	#208	—	18	4	no	—
184	#208	—	18	4	no	—
185	C₂×C₃⋊C₈	—	3	4	no	—
186	C₂×C₃⋊C₈	—	6	4	no	—
187	C₂×C₃⋊C₈	—	18	4	no	—
188	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
189	ΓL(1,𝔽₂₅)	—	18	1	no	—
190	D₃×C₈	—	18	1	no	—
191	C₂²×C₃²	—	2	1	no	—
192	C₂²×C₃²	—	2	4	no	—
193	C₂²×C₃²	—	2	4	no	—
194	C₂²×C₃²	—	4	4	no	—
195	C₂²×C₃²	—	6	1	no	—
196	C₂²×C₃²	#235	1	1	yes	characteristic
197	C₂²×C₃²	#237	1	4	yes	—
198	C₂×C₃×D₃	—	2	4	no	—
199	C₃²×C₄	—	2	1	no	—
200	C₃²×C₄	—	2	4	no	—
201	C₃²×C₄	—	2	4	no	—
202	C₃²×C₄	—	2	4	no	—
203	C₃²×C₄	—	4	4	no	—
204	C₃²×C₄	—	6	1	no	—
205	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₃²×C₄	D₃×D₄	1	4	yes	—
207	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
208	C₃²×C₄	C₂×#67	1	1	yes	characteristic
209	C₃⋊Dic₃	—	6	2	no	—
210	C₃⋊Dic₃	—	6	8	no	—
211	C₃⋊Dic₃	—	12	8	no	—
212	C₃×Dic₃	—	2	4	no	—
213	#135	—	3	1	no	—
214	#131	—	9	1	no	—
215	#131	—	27	1	no	—
216	#108	—	27	1	no	—
217	#104	—	27	1	no	—
218	#104	—	27	1	no	—
219	C₄×C₈	—	27	1	no	—
220	C₃³	#443	1	1	yes	characteristic, complemented, Sylow
221	C₃×Q₈	—	2	4	no	—
222	C₃×Q₈	—	3	4	no	—
223	C₃×Q₈	D₃×Dic₃	1	4	yes	—
224	C₃×Q₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
225	C₃×D₄	—	2	4	no	—
226	C₃×D₄	—	2	1	no	—
227	C₃×D₄	—	2	4	no	—
228	C₃×D₄	—	3	4	no	—
229	C₃×D₄	—	4	4	no	—
230	C₃×D₄	—	6	4	no	—
231	C₃×D₄	D₃×Dic₃	1	4	yes	—
232	C₃×D₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
233	C₂×C₃×C₄	—	2	4	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	4	no	—
236	C₂×C₃×C₄	—	4	4	no	—
237	C₂×C₃×C₄	—	6	4	no	—
238	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
239	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
240	#67	—	2	1	no	—
241	C₂×Dic₃	—	18	4	no	—
242	C₂×Dic₃	—	18	1	no	—
243	C₂×Dic₃	—	36	4	no	—
244	D₁₂	C₂×C₃⋊Dic₃	1	1	yes	characteristic
245	C₄×D₃	—	2	1	no	—
246	Dic₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
247	C₃×C₈	—	3	4	no	—
248	C₃×C₈	—	3	4	no	—
249	C₃×C₈	—	9	1	no	—
250	C₃×C₈	—	9	1	no	—
251	C₃×C₈	—	18	1	no	—
252	C₃⋊C₈	—	3	4	no	—
253	C₃⋊C₈	—	3	4	no	—
254	C₃⋊C₈	—	6	4	no	—
255	C₃⋊C₈	—	18	4	no	—
256	C₃⋊C₈	—	18	1	no	—
257	C₃⋊C₈	—	36	4	no	—
258	C₃⋊C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
259	C₃⋊C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
260	C₂×C₃²	—	2	1	no	—
261	C₂×C₃²	—	2	4	no	—
262	C₂×C₃²	—	2	4	no	—
263	C₂×C₃²	—	4	4	no	—
264	C₂×C₃²	—	4	1	no	—
265	C₂×C₃²	—	4	4	no	—
266	C₂×C₃²	—	8	4	no	—
267	C₂×C₃²	—	12	1	no	—
268	C₂×C₃²	C₄×#67	1	1	yes	characteristic
269	C₂×C₃²	D₄×Dic₃	1	4	yes	—
270	C₃×D₃	—	4	4	no	—
271	#41	—	3	1	no	—
272	#41	D₃×C₃⋊D₃	1	1	yes	characteristic
273	Q₁₆	—	3	1	no	—
274	SD₁₆	—	3	1	no	—
275	SD₁₆	—	3	1	no	—
276	D₈	—	3	1	no	—
277	M₁₆	—	9	1	no	—
278	M₁₆	—	18	1	no	—
279	M₁₆	—	27	1	no	—
280	M₁₆	—	54	1	no	—
281	C₂×C₈	—	3	1	no	—
282	C₂×C₈	—	9	1	no	—
283	C₂×C₈	—	18	1	no	—
284	C₂×C₈	—	54	1	no	—
285	C₄²	—	27	1	no	—
286	C₂²×C₃	—	2	1	no	—
287	C₂²×C₃	—	2	4	no	—
288	C₂²×C₃	—	2	4	no	—
289	C₂²×C₃	—	4	4	no	—
290	C₂²×C₃	—	6	4	no	—
291	C₂²×C₃	#3804	1	4	yes	—
292	C₂²×C₃	#3830	1	1	yes	characteristic
293	C₂×D₃	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	4	no	—
296	C₃×C₄	—	2	4	no	—
297	C₃×C₄	—	2	4	no	—
298	C₃×C₄	—	4	4	no	—
299	C₃×C₄	—	6	4	no	—
300	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
301	C₃×C₄	D₃×#67	1	4	yes	—
302	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
303	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
304	Dic₃	—	2	1	no	—
305	Dic₃	—	18	2	no	—
306	Dic₃	—	18	8	no	—
307	Dic₃	—	36	8	no	—
308	C₃²	—	2	4	no	—
309	C₃²	#5144	1	1	yes	characteristic, complemented
310	C₃²	#5196	1	4	yes	complemented
311	Q₈	—	3	1	no	—
312	Q₈	D₃×C₃⋊Dic₃	1	1	yes	characteristic
313	D₄	—	2	1	no	—
314	D₄	—	3	1	no	—
315	D₄	—	6	1	no	—
316	D₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	6	1	no	—
319	C₂×C₄	—	54	1	no	—
320	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
321	C₈	—	3	1	no	—
322	C₈	—	3	1	no	—
323	C₈	—	9	1	no	—
324	C₈	—	9	1	no	—
325	C₈	—	18	1	no	—
326	C₈	—	54	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	4	no	—
329	C₂×C₃	—	2	4	no	—
330	C₂×C₃	—	4	4	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	—	4	4	no	—
333	C₂×C₃	—	8	4	no	—
334	C₂×C₃	—	12	4	no	—
335	C₂×C₃	Dic₃×#67	1	4	yes	—
336	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
337	D₃	—	4	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	6	1	no	—
340	C₂²	#89028	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	6	1	no	—
343	C₄	—	54	2	no	—
344	C₄	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
345	C₄	C₃⋊D₃×#67	1	1	yes	characteristic
346	C₃	—	2	4	no	—
347	C₃	#10591673	1	4	yes	complemented
348	C₃	#10592895	1	1	yes	characteristic, complemented
349	C₂	—	2	1	no	—
350	C₂	—	4	1	no	—
351	C₂	—	12	1	no	—
352	C₂	#67×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₃⋊D₃×#67
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₃³×#41	C₂²
socle	C₂×C₃³	C₄×D₄
prime radical	C₃²×#950	C₂
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	#41	D₃×C₃⋊D₃
hypocenter	C₃³	#443
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₃×#64157	C₂	semidirect
3. #11723349	C₂	semidirect
4. #11722207	C₂	semidirect
5. #11722058	C₂	semidirect
6. #11722043	C₂	semidirect
7. #11722018	C₂	semidirect
18. C₃²×#216	C₄	semidirect
19. C₃²×#215	C₄	semidirect
20. C₃²×#214	C₄	semidirect
21. C₃²×#213	C₄	semidirect
27. #64157	D₃	semidirect
80. C₃×#216	Dic₃	semidirect
81. C₃×#215	Dic₃	semidirect
82. C₃×#214	Dic₃	semidirect
83. C₃×#213	Dic₃	semidirect
172. #216	C₃⋊Dic₃	semidirect
173. #215	C₃⋊Dic₃	semidirect
174. #214	C₃⋊Dic₃	semidirect
175. #213	C₃⋊Dic₃	semidirect
220. C₃³	#443	semidirect
309. C₃²	#5144	semidirect
310. C₃²	#5196	semidirect
347. C₃	#10591673	semidirect
348. C₃	#10592895	semidirect

Maximal subgroups

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

subgroup	index
C₃×#64157	2
C₃²×#950	2
#11723349	2
#11722207	2
#11722058	2
#11722043	2
#11722018	2
#10592895	3
#10591673	3