Subgroups — C₃×#10596190 #13179470

Order 1728.

1872 conjugacy classes of subgroups.

The subgroup lattice has 332 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, complemented
2	C₃×#64148	C₂	1	4	yes	maximal, complemented
3	C₃×#64147	C₂	1	2	yes	maximal, complemented
4	C₃×#64144	C₂	1	4	yes	maximal, complemented
5	C₃×D₄×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
6	C₃×#64081	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64079	C₂	1	1	yes	maximal, characteristic, complemented
8	#10596190	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#5719	—	3	4	no	maximal
10	C₂×C₃³×D₄	C₂²	1	2	yes	—
11	C₂×C₃³×D₄	C₂²	1	4	yes	—
12	C₂×C₃×#638	C₂²	1	4	yes	—
13	C₂²×C₃×C₃⋊Dic₃	C₂²	1	4	yes	—
14	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#3757	C₂²	1	4	yes	—
17	C₃×#3757	C₂²	1	4	yes	—
18	C₃×#3752	C₂²	1	2	yes	—
19	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
20	C₃×#3750	C₂²	1	4	yes	—
21	C₃×#3750	C₂²	1	2	yes	—
22	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₃²×#127	—	2	4	no	—
24	C₃²×#127	D₃	1	4	yes	complemented
25	C₃²×#127	C₂×C₃	1	1	yes	characteristic, complemented
26	#64148	C₂×C₃	1	4	yes	complemented
27	#64147	C₂×C₃	1	2	yes	complemented
28	#64144	C₂×C₃	1	4	yes	complemented
29	D₄×C₃⋊Dic₃	C₂×C₃	1	2	yes	complemented
30	#64081	C₂×C₃	1	1	yes	characteristic, complemented
31	#64079	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#939	—	3	16	no	—
33	C₃×#938	—	3	8	no	—
34	C₃×#935	—	3	16	no	—
35	C₃×D₄×Dic₃	—	3	8	no	—
36	C₃×#872	—	3	4	no	—
37	C₃×#870	—	3	4	no	—
38	C₂³×C₃³	C₂³	1	4	yes	—
39	C₂²×C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
40	C₃³×D₄	—	2	4	no	—
41	C₃³×D₄	—	4	4	no	—
42	C₂×C₃³×C₄	C₂³	1	2	yes	—
43	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
44	C₃×#638	—	4	4	no	—
45	C₂×C₃×C₃⋊Dic₃	—	4	4	no	—
46	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
47	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
48	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
49	C₃×C₄×C₃⋊D₃	—	2	2	no	—
50	C₃×#540	—	9	1	no	—
51	#5719	—	3	4	no	—
52	C₂×C₃²×D₄	—	2	8	no	—
53	C₂×C₃²×D₄	—	2	16	no	—
54	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
55	C₂×C₃²×D₄	C₂×D₃	1	16	yes	—
56	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
57	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
58	C₂×#638	C₂²×C₃	1	4	yes	—
59	C₂²×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
60	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×#67	—	3	16	no	—
62	C₂²×C₃×Dic₃	—	3	16	no	—
63	C₂×C₃×C₄×D₃	—	3	4	no	—
64	C₃²×C₄²	—	2	4	no	—
65	C₃²×C₄²	C₂×D₃	1	4	yes	—
66	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
67	#3757	C₂²×C₃	1	4	yes	—
68	#3757	C₂²×C₃	1	4	yes	—
69	#3752	C₂²×C₃	1	2	yes	—
70	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
71	#3750	C₂²×C₃	1	4	yes	—
72	#3750	C₂²×C₃	1	2	yes	—
73	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
74	C₃×#217	—	3	16	no	—
75	C₃×#217	—	3	16	no	—
76	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
77	C₃×C₄⋊Dic₃	—	3	8	no	—
78	C₃×Dic₃⋊C₄	—	3	8	no	—
79	C₃×Dic₃⋊C₄	—	3	16	no	—
80	C₃×C₄×Dic₃	—	3	4	no	—
81	C₂²×C₃³	—	2	4	no	—
82	C₂²×C₃³	—	2	4	no	—
83	C₂²×C₃³	—	2	4	no	—
84	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
85	C₂×C₃×C₃⋊D₃	—	2	1	no	—
86	C₂×C₃×C₃⋊D₃	—	2	2	no	—
87	C₃³×C₄	—	2	2	no	—
88	C₃³×C₄	#41	1	2	yes	—
89	C₃×C₃⋊Dic₃	—	2	1	no	—
90	C₃×C₃⋊Dic₃	—	2	2	no	—
91	C₃×C₃⋊Dic₃	—	2	4	no	—
92	C₃×#127	—	2	4	no	—
93	C₃×#127	C₃×D₃	1	4	yes	complemented
94	C₃×#127	C₃⋊D₃	1	1	yes	characteristic, complemented
95	C₃×#125	—	9	1	no	—
96	C₃×#123	—	9	4	no	—
97	C₃×#121	—	9	2	no	—
98	C₃×#121	—	9	4	no	—
99	C₃×C₄×D₄	—	9	2	no	—
100	C₃×#117	—	9	1	no	—
101	#939	—	3	16	no	—
102	#938	—	3	8	no	—
103	#935	—	3	16	no	—
104	D₄×Dic₃	—	3	8	no	—
105	#872	—	3	4	no	—
106	#870	—	3	4	no	—
107	C₂³×C₃²	—	2	16	no	—
108	C₂³×C₃²	C₂²×D₃	1	16	yes	—
109	C₂³×C₃²	C₂³×C₃	1	4	yes	—
110	C₂²×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₃×D₃	—	3	4	no	—
112	C₃²×D₄	—	2	4	no	—
113	C₃²×D₄	—	2	16	no	—
114	C₃²×D₄	—	4	16	no	—
115	C₃²×D₄	—	4	16	no	—
116	C₃²×D₄	—	4	4	no	—
117	C₃²×D₄	—	4	32	no	—
118	C₂×C₃²×C₄	—	2	8	no	—
119	C₂×C₃²×C₄	—	2	4	no	—
120	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
122	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
123	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
124	#638	—	4	4	no	—
125	C₂×C₃⋊Dic₃	—	4	4	no	—
126	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
127	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
128	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
129	C₄×C₃⋊D₃	—	2	2	no	—
130	C₃×#67	—	12	16	no	—
131	C₂×C₃×Dic₃	—	3	8	no	—
132	C₂×C₃×Dic₃	—	3	16	no	—
133	C₂×C₃×Dic₃	—	3	4	no	—
134	C₂×C₃×Dic₃	—	12	16	no	—
135	C₃×C₄×D₃	—	6	8	no	—
136	#540	—	9	1	no	Sylow
137	C₂×C₃³	—	4	4	no	—
138	C₂×C₃³	C₂×#41	1	1	yes	characteristic
139	C₂×C₃³	H(5,𝔽₂)	1	2	yes	—
140	C₃×C₃⋊D₃	—	4	1	no	—
141	C₂×C₃×D₄	—	2	8	no	—
142	C₂×C₃×D₄	—	2	16	no	—
143	C₂×C₃×D₄	—	9	4	no	—
144	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
145	C₂×C₃×D₄	C₂×C₃×D₃	1	16	yes	—
146	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
147	C₂×C₃×D₄	C₂×C₃⋊D₃	1	4	yes	—
148	C₂²×C₃×C₄	—	9	4	no	—
149	C₂²×C₃×C₄	—	9	1	no	—
150	C₂×#67	—	3	16	no	—
151	C₂²×Dic₃	—	3	16	no	—
152	C₂×C₄×D₃	—	3	4	no	—
153	C₃×C₄⋊C₄	—	9	2	no	—
154	C₃×C₄⋊C₄	—	9	4	no	—
155	C₃×C₄⋊C₄	—	9	2	no	—
156	C₃×C₂²⋊C₄	—	9	4	no	—
157	C₃×C₂²⋊C₄	—	9	4	no	—
158	C₃×C₂²⋊C₄	—	9	2	no	—
159	C₃×C₄²	—	2	4	no	—
160	C₃×C₄²	—	9	1	no	—
161	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
162	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
163	#217	—	3	16	no	—
164	#217	—	3	16	no	—
165	(C₂×D₃)⋊C₄	—	3	8	no	—
166	C₄⋊Dic₃	—	3	8	no	—
167	Dic₃⋊C₄	—	3	16	no	—
168	Dic₃⋊C₄	—	3	8	no	—
169	C₄×Dic₃	—	3	4	no	—
170	C₂²×C₃²	—	2	4	no	—
171	C₂²×C₃²	—	2	4	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	—	2	4	no	—
174	C₂²×C₃²	—	2	16	no	—
175	C₂²×C₃²	—	2	16	no	—
176	C₂²×C₃²	—	2	16	no	—
177	C₂²×C₃²	—	4	16	no	—
178	C₂²×C₃²	—	4	16	no	—
179	C₂²×C₃²	—	4	16	no	—
180	C₂²×C₃²	C₂³×D₃	1	4	yes	—
181	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
182	C₂×C₃⋊D₃	—	2	1	no	—
183	C₂×C₃⋊D₃	—	2	2	no	—
184	C₂×C₃×D₃	—	6	4	no	—
185	C₂×C₃×D₃	—	6	8	no	—
186	C₃²×C₄	—	2	16	no	—
187	C₃²×C₄	—	2	8	no	—
188	C₃²×C₄	—	2	2	no	—
189	C₃²×C₄	—	2	8	no	—
190	C₃²×C₄	#237	1	8	yes	—
191	C₃²×C₄	C₃×#41	1	2	yes	—
192	C₃⋊Dic₃	—	2	1	no	—
193	C₃⋊Dic₃	—	2	2	no	—
194	C₃⋊Dic₃	—	2	4	no	—
195	C₃×Dic₃	—	6	4	no	—
196	C₃×Dic₃	—	6	8	no	—
197	C₃×Dic₃	—	6	16	no	—
198	#127	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
199	#125	—	9	1	no	—
200	#123	—	9	4	no	—
201	#121	—	9	2	no	—
202	#121	—	9	4	no	—
203	C₄×D₄	—	9	2	no	—
204	#117	—	9	1	no	—
205	C₃³	#540	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	2	16	no	—
207	C₂³×C₃	—	9	1	no	—
208	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
209	C₂³×C₃	C₂²×C₃⋊D₃	1	4	yes	—
210	C₂²×D₃	—	3	4	no	—
211	C₃×D₄	—	2	16	no	—
212	C₃×D₄	—	2	4	no	—
213	C₃×D₄	—	4	16	no	—
214	C₃×D₄	—	4	16	no	—
215	C₃×D₄	—	4	4	no	—
216	C₃×D₄	—	4	32	no	—
217	C₃×D₄	—	36	4	no	—
218	C₂×C₃×C₄	—	2	8	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	9	2	no	—
221	C₂×C₃×C₄	—	9	1	no	—
222	C₂×C₃×C₄	—	9	4	no	—
223	C₂×C₃×C₄	—	18	2	no	—
224	C₂×C₃×C₄	—	36	4	no	—
225	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
226	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
227	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
228	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
229	#67	—	12	16	no	—
230	C₂×Dic₃	—	3	8	no	—
231	C₂×Dic₃	—	3	16	no	—
232	C₂×Dic₃	—	3	4	no	—
233	C₂×Dic₃	—	12	16	no	—
234	C₄×D₃	—	6	8	no	—
235	C₂×C₃²	—	2	4	no	—
236	C₂×C₃²	—	2	8	no	—
237	C₂×C₃²	—	4	4	no	—
238	C₂×C₃²	—	4	16	no	—
239	C₂×C₃²	—	8	16	no	—
240	C₂×C₃²	C₂×#237	1	4	yes	—
241	C₂×C₃²	#1004	1	8	yes	—
242	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
243	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
244	C₃⋊D₃	—	4	1	no	—
245	C₃×D₃	—	12	4	no	—
246	C₂×D₄	—	9	4	no	—
247	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
248	C₂×D₄	C₂×C₃×C₃⋊D₃	1	4	yes	—
249	C₂²×C₄	—	9	4	no	—
250	C₂²×C₄	—	9	1	no	—
251	C₄⋊C₄	—	9	2	no	—
252	C₄⋊C₄	—	9	2	no	—
253	C₄⋊C₄	—	9	4	no	—
254	C₂²⋊C₄	—	9	4	no	—
255	C₂²⋊C₄	—	9	4	no	—
256	C₂²⋊C₄	—	9	2	no	—
257	C₄²	—	9	1	no	—
258	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
259	C₂²×C₃	—	2	16	no	—
260	C₂²×C₃	—	2	16	no	—
261	C₂²×C₃	—	2	16	no	—
262	C₂²×C₃	—	2	4	no	—
263	C₂²×C₃	—	2	4	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	—	2	4	no	—
266	C₂²×C₃	—	4	16	no	—
267	C₂²×C₃	—	4	16	no	—
268	C₂²×C₃	—	4	16	no	—
269	C₂²×C₃	—	18	1	no	—
270	C₂²×C₃	—	18	2	no	—
271	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
272	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
273	C₂×D₃	—	6	4	no	—
274	C₂×D₃	—	6	8	no	—
275	C₃×C₄	—	2	16	no	—
276	C₃×C₄	—	2	8	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	2	8	no	—
279	C₃×C₄	—	18	1	no	—
280	C₃×C₄	—	18	2	no	—
281	C₃×C₄	—	18	4	no	—
282	C₃×C₄	C₃×#237	1	8	yes	—
283	C₃×C₄	#3830	1	2	yes	—
284	Dic₃	—	6	4	no	—
285	Dic₃	—	6	8	no	—
286	Dic₃	—	6	16	no	—
287	C₃²	—	2	4	no	—
288	C₃²	#5719	1	4	yes	complemented
289	C₃²	C₃×#540	1	1	yes	characteristic, complemented
290	C₂³	—	9	1	no	—
291	C₂³	C₂²×C₃×C₃⋊D₃	1	4	yes	—
292	D₄	—	2	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	36	4	no	—
295	C₂×C₄	—	9	2	no	—
296	C₂×C₄	—	9	1	no	—
297	C₂×C₄	—	9	4	no	—
298	C₂×C₄	—	18	2	no	—
299	C₂×C₄	—	36	4	no	—
300	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
301	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
302	C₂×C₃	—	2	4	no	—
303	C₂×C₃	—	2	8	no	—
304	C₂×C₃	—	4	4	no	—
305	C₂×C₃	—	4	16	no	—
306	C₂×C₃	—	8	16	no	—
307	C₂×C₃	—	36	1	no	—
308	C₂×C₃	C₂×C₃×#237	1	4	yes	—
309	C₂×C₃	C₃×#1004	1	8	yes	—
310	C₂×C₃	C₂×#3830	1	1	yes	characteristic
311	C₂×C₃	#64361	1	2	yes	—
312	D₃	—	12	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	18	1	no	—
317	C₂²	—	18	2	no	—
318	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	18	1	no	—
321	C₄	—	18	2	no	—
322	C₄	—	18	4	no	—
323	C₄	C₃×#3830	1	2	yes	—
324	C₃	—	2	4	no	—
325	C₃	C₃×#5719	1	4	yes	complemented
326	C₃	#10596190	1	1	yes	characteristic, direct product factor
327	C₂	—	4	4	no	—
328	C₂	—	36	1	no	—
329	C₂	C₂×C₃×#3830	1	1	yes	characteristic
330	C₂	C₃×#64361	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃×#127	C₃⋊D₃
hypocenter	C₃²	C₃×#540
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₃³×#127	C₂	semidirect
2. C₃×#64148	C₂	semidirect
3. C₃×#64147	C₂	semidirect
4. C₃×#64144	C₂	semidirect
5. C₃×D₄×C₃⋊Dic₃	C₂	semidirect
6. C₃×#64081	C₂	semidirect
7. C₃×#64079	C₂	semidirect
8. #10596190	C₃	direct (N × H)
24. C₃²×#127	D₃	semidirect
25. C₃²×#127	C₂×C₃	semidirect
26. #64148	C₂×C₃	semidirect
27. #64147	C₂×C₃	semidirect
28. #64144	C₂×C₃	semidirect
29. D₄×C₃⋊Dic₃	C₂×C₃	semidirect
30. #64081	C₂×C₃	semidirect
31. #64079	C₂×C₃	semidirect
93. C₃×#127	C₃×D₃	semidirect
94. C₃×#127	C₃⋊D₃	semidirect
198. #127	C₃×C₃⋊D₃	semidirect
205. C₃³	#540	semidirect
288. C₃²	#5719	semidirect
289. C₃²	C₃×#540	semidirect
325. C₃	C₃×#5719	semidirect
326. C₃	#10596190	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
C₃×#64148	2
C₃×#64147	2
C₃×#64144	2
C₃×D₄×C₃⋊Dic₃	2
C₃×#64081	2
C₃×#64079	2
#10596190	3
C₃×#5719	3

Subgroups — C3×#10596190 #13179470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596190 #13179470