Subgroups — C₃²×#5697 #13179074

Order 1728.

1664 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#895	C₂	1	2	yes	maximal, complemented
3	C₃²×#895	C₂	1	4	yes	maximal, complemented
4	C₃²×#894	C₂	1	4	yes	maximal, complemented
5	C₃²×#893	C₂	1	2	yes	maximal, complemented
6	C₃²×#874	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#872	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#540	—	3	1	no	maximal
9	C₃×#5697	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
11	C₂×C₃²×D₁₂	C₂²	1	4	yes	—
12	C₂×C₃²×D₁₂	C₂²	1	4	yes	—
13	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₄×D₃	C₂²	1	4	yes	—
15	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃³×C₄²	C₂²	1	1	yes	characteristic
18	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
19	C₃²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
20	C₃²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
21	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
22	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
23	C₃²×#127	—	3	1	no	—
24	C₃²×#125	—	2	4	no	—
25	C₃²×#125	D₃	1	1	yes	characteristic, complemented
26	C₃²×#125	C₂×C₃	1	4	yes	complemented
27	C₃²×#123	—	3	4	no	—
28	C₃²×#121	—	3	4	no	—
29	C₃²×#121	—	3	2	no	—
30	C₃²×C₄×D₄	—	3	2	no	—
31	C₃²×#117	—	3	1	no	—
32	C₃×#895	C₂×C₃	1	16	yes	complemented
33	C₃×#895	C₂×C₃	1	8	yes	complemented
34	C₃×#894	C₂×C₃	1	16	yes	complemented
35	C₃×#893	C₂×C₃	1	8	yes	complemented
36	C₃×#874	C₂×C₃	1	4	yes	complemented
37	C₃×#872	C₂×C₃	1	4	yes	complemented
38	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
39	C₂²×C₃²×D₃	C₂³	1	4	yes	—
40	C₂×C₃³×C₄	C₂³	1	2	yes	—
41	C₂×C₃³×C₄	C₂³	1	4	yes	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
44	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
45	C₃²×D₁₂	—	2	4	no	—
46	C₃²×D₁₂	—	4	4	no	—
47	C₃²×D₁₂	—	4	4	no	—
48	C₃²×C₄×D₃	—	2	2	no	—
49	C₃²×C₄×D₃	—	4	4	no	—
50	C₃×#540	—	3	4	no	—
51	#5697	C₃²	1	1	yes	characteristic, direct product factor
52	C₂×C₃²×D₄	—	3	2	no	—
53	C₂×C₃²×D₄	—	3	4	no	—
54	C₂×C₃²×D₄	—	3	4	no	—
55	C₂²×C₃²×C₄	—	3	1	no	—
56	C₂²×C₃²×C₄	—	3	4	no	—
57	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
58	C₂×C₃×D₁₂	C₂²×C₃	1	16	yes	—
59	C₂×C₃×D₁₂	C₂²×C₃	1	16	yes	—
60	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
61	C₂×C₃×C₄×D₃	C₂²×C₃	1	16	yes	—
62	C₃²×C₄⋊C₄	—	2	8	no	—
63	C₃²×C₄⋊C₄	—	2	16	no	—
64	C₃²×C₄⋊C₄	—	3	2	no	—
65	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
66	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
67	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
68	C₃²×C₄⋊C₄	C₂²×C₃	1	16	yes	—
69	C₃²×C₂²⋊C₄	—	3	2	no	—
70	C₃²×C₂²⋊C₄	—	3	4	no	—
71	C₃²×C₂²⋊C₄	—	3	4	no	—
72	C₃²×C₄²	—	2	4	no	—
73	C₃²×C₄²	—	3	1	no	—
74	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
75	C₃²×C₄²	C₂²×C₃	1	4	yes	—
76	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	—
77	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	—
78	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
79	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
80	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
81	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
82	C₂×C₃²×D₃	—	2	1	no	—
83	C₂×C₃²×D₃	—	2	4	no	—
84	C₂×C₃²×D₃	—	2	4	no	—
85	C₂×C₃²×D₃	—	2	2	no	—
86	C₂×C₃²×D₃	—	2	4	no	—
87	C₃³×C₄	—	2	4	no	—
88	C₃³×C₄	—	2	2	no	—
89	C₃³×C₄	#41	1	2	yes	—
90	C₃²×Dic₃	—	2	1	no	—
91	C₃²×Dic₃	—	2	2	no	—
92	C₃×#127	—	3	4	no	—
93	C₃×#125	—	2	4	no	—
94	C₃×#125	C₃×D₃	1	4	yes	complemented
95	C₃×#125	C₂×C₃²	1	1	yes	characteristic, complemented
96	C₃×#123	—	3	16	no	—
97	C₃×#121	—	3	16	no	—
98	C₃×#121	—	3	8	no	—
99	C₃×C₄×D₄	—	3	8	no	—
100	C₃×#117	—	3	4	no	—
101	#895	C₂×C₃²	1	4	yes	complemented
102	#895	C₂×C₃²	1	2	yes	complemented
103	#894	C₂×C₃²	1	4	yes	complemented
104	#893	C₂×C₃²	1	2	yes	complemented
105	#874	C₂×C₃²	1	1	yes	characteristic, complemented
106	#872	C₂×C₃²	1	1	yes	characteristic, complemented
107	C₂³×C₃²	—	3	1	no	—
108	C₂³×C₃²	—	3	4	no	—
109	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
110	C₂²×C₃×D₃	C₂³×C₃	1	16	yes	—
111	C₃²×D₄	—	6	4	no	—
112	C₃²×D₄	—	12	4	no	—
113	C₃²×D₄	—	12	4	no	—
114	C₂×C₃²×C₄	—	2	4	no	—
115	C₂×C₃²×C₄	—	2	8	no	—
116	C₂×C₃²×C₄	—	2	16	no	—
117	C₂×C₃²×C₄	—	3	2	no	—
118	C₂×C₃²×C₄	—	3	1	no	—
119	C₂×C₃²×C₄	—	6	2	no	—
120	C₂×C₃²×C₄	—	12	4	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
125	C₂×C₃²×C₄	C₂³×C₃	1	16	yes	—
126	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
127	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
128	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
129	C₃×D₁₂	—	2	16	no	—
130	C₃×D₁₂	—	4	16	no	—
131	C₃×D₁₂	—	4	16	no	—
132	C₃×C₄×D₃	—	2	8	no	—
133	C₃×C₄×D₃	—	4	16	no	—
134	#540	—	3	1	no	Sylow
135	C₂×C₃³	C₂×#41	1	1	yes	characteristic
136	C₂×C₃³	H(5,𝔽₂)	1	2	yes	—
137	C₃²×D₃	—	4	1	no	—
138	C₃²×D₃	—	4	4	no	—
139	C₂×C₃×D₄	—	3	8	no	—
140	C₂×C₃×D₄	—	3	16	no	—
141	C₂×C₃×D₄	—	3	16	no	—
142	C₂²×C₃×C₄	—	3	4	no	—
143	C₂²×C₃×C₄	—	3	16	no	—
144	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
145	C₂×D₁₂	C₂²×C₃²	1	4	yes	—
146	C₂×D₁₂	C₂²×C₃²	1	4	yes	—
147	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
148	C₂×C₄×D₃	C₂²×C₃²	1	4	yes	—
149	C₃×C₄⋊C₄	—	2	8	no	—
150	C₃×C₄⋊C₄	—	2	16	no	—
151	C₃×C₄⋊C₄	—	3	8	no	—
152	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
153	C₃×C₄⋊C₄	C₂×C₃×D₃	1	16	yes	—
154	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
155	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
156	C₃×C₂²⋊C₄	—	3	8	no	—
157	C₃×C₂²⋊C₄	—	3	16	no	—
158	C₃×C₂²⋊C₄	—	3	16	no	—
159	C₃×C₄²	—	2	4	no	—
160	C₃×C₄²	—	3	4	no	—
161	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
162	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
163	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
164	(C₂×D₃)⋊C₄	C₂²×C₃²	1	4	yes	—
165	(C₂×D₃)⋊C₄	C₂²×C₃²	1	4	yes	—
166	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
167	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
168	C₂²×C₃²	—	2	4	no	—
169	C₂²×C₃²	—	6	1	no	—
170	C₂²×C₃²	—	6	4	no	—
171	C₂²×C₃²	—	6	4	no	—
172	C₂²×C₃²	—	6	2	no	—
173	C₂²×C₃²	—	6	4	no	—
174	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
175	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
176	C₂×C₃×D₃	—	2	4	no	—
177	C₂×C₃×D₃	—	2	16	no	—
178	C₂×C₃×D₃	—	2	16	no	—
179	C₂×C₃×D₃	—	2	8	no	—
180	C₂×C₃×D₃	—	2	16	no	—
181	C₃²×C₄	—	2	4	no	—
182	C₃²×C₄	—	2	2	no	—
183	C₃²×C₄	—	2	8	no	—
184	C₃²×C₄	—	2	16	no	—
185	C₃²×C₄	—	2	8	no	—
186	C₃²×C₄	—	4	16	no	—
187	C₃²×C₄	—	4	8	no	—
188	C₃²×C₄	—	6	1	no	—
189	C₃²×C₄	—	6	2	no	—
190	C₃²×C₄	#239	1	2	yes	—
191	C₃²×C₄	C₃×#41	1	8	yes	—
192	C₃×Dic₃	—	2	4	no	—
193	C₃×Dic₃	—	2	8	no	—
194	#127	—	3	1	no	—
195	#125	C₃²×D₃	1	1	yes	characteristic, complemented
196	#123	—	3	4	no	—
197	#121	—	3	4	no	—
198	#121	—	3	2	no	—
199	C₄×D₄	—	3	2	no	—
200	#117	—	3	1	no	—
201	C₃³	#540	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	3	4	no	—
203	C₂³×C₃	—	3	16	no	—
204	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
205	C₂²×D₃	C₂³×C₃²	1	4	yes	—
206	C₃×D₄	—	6	16	no	—
207	C₃×D₄	—	12	16	no	—
208	C₃×D₄	—	12	16	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	8	no	—
211	C₂×C₃×C₄	—	2	16	no	—
212	C₂×C₃×C₄	—	3	8	no	—
213	C₂×C₃×C₄	—	3	4	no	—
214	C₂×C₃×C₄	—	6	8	no	—
215	C₂×C₃×C₄	—	12	16	no	—
216	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
217	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
218	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
219	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
220	C₂×C₃×C₄	C₂³×C₃²	1	4	yes	—
221	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
222	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
223	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
224	D₁₂	—	2	4	no	—
225	D₁₂	—	4	4	no	—
226	D₁₂	—	4	4	no	—
227	C₄×D₃	—	2	2	no	—
228	C₄×D₃	—	4	4	no	—
229	C₂×C₃²	—	2	4	no	—
230	C₂×C₃²	—	2	8	no	—
231	C₂×C₃²	—	12	1	no	—
232	C₂×C₃²	—	12	4	no	—
233	C₂×C₃²	C₂×#239	1	1	yes	characteristic
234	C₂×C₃²	#1009	1	2	yes	—
235	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
236	C₂×C₃²	C₃×H(5,𝔽₂)	1	8	yes	—
237	C₃×D₃	—	4	4	no	—
238	C₃×D₃	—	4	16	no	—
239	C₂×D₄	—	3	2	no	—
240	C₂×D₄	—	3	4	no	—
241	C₂×D₄	—	3	4	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	4	no	—
244	C₄⋊C₄	—	3	2	no	—
245	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
246	C₄⋊C₄	C₂×C₃²×D₃	1	4	yes	—
247	C₂²⋊C₄	—	3	2	no	—
248	C₂²⋊C₄	—	3	4	no	—
249	C₂²⋊C₄	—	3	4	no	—
250	C₄²	—	3	1	no	—
251	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	—	6	4	no	—
254	C₂²×C₃	—	6	16	no	—
255	C₂²×C₃	—	6	16	no	—
256	C₂²×C₃	—	6	8	no	—
257	C₂²×C₃	—	6	16	no	—
258	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
259	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
260	C₂×D₃	—	2	1	no	—
261	C₂×D₃	—	2	4	no	—
262	C₂×D₃	—	2	4	no	—
263	C₂×D₃	—	2	2	no	—
264	C₂×D₃	—	2	4	no	—
265	C₃×C₄	—	2	16	no	—
266	C₃×C₄	—	2	8	no	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	8	no	—
270	C₃×C₄	—	4	16	no	—
271	C₃×C₄	—	4	8	no	—
272	C₃×C₄	—	6	4	no	—
273	C₃×C₄	—	6	8	no	—
274	C₃×C₄	C₃×#239	1	8	yes	—
275	C₃×C₄	C₃²×#41	1	2	yes	—
276	Dic₃	—	2	1	no	—
277	Dic₃	—	2	2	no	—
278	C₃²	—	2	4	no	—
279	C₃²	#5697	1	1	yes	characteristic, direct product factor
280	C₃²	C₃×#540	1	4	yes	complemented
281	C₂³	—	3	1	no	—
282	C₂³	—	3	4	no	—
283	D₄	—	6	4	no	—
284	D₄	—	12	4	no	—
285	D₄	—	12	4	no	—
286	C₂×C₄	—	3	2	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	12	4	no	—
290	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
291	C₂×C₄	C₂²×C₃²×D₃	1	4	yes	—
292	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
293	C₂×C₃	—	2	4	no	—
294	C₂×C₃	—	2	8	no	—
295	C₂×C₃	—	12	4	no	—
296	C₂×C₃	—	12	16	no	—
297	C₂×C₃	C₂×C₃×#239	1	4	yes	—
298	C₂×C₃	C₃×#1009	1	8	yes	—
299	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
300	C₂×C₃	C₃²×H(5,𝔽₂)	1	2	yes	—
301	D₃	—	4	1	no	—
302	D₃	—	4	4	no	—
303	C₂²	—	6	1	no	—
304	C₂²	—	6	4	no	—
305	C₂²	—	6	4	no	—
306	C₂²	—	6	2	no	—
307	C₂²	—	6	4	no	—
308	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	6	1	no	—
312	C₄	—	6	2	no	—
313	C₄	C₃²×#239	1	2	yes	—
314	C₃	—	2	4	no	—
315	C₃	C₃×#5697	1	4	yes	direct product factor
316	C₃	C₃²×#540	1	1	yes	characteristic, complemented
317	C₂	—	12	1	no	—
318	C₂	—	12	4	no	—
319	C₂	C₂×C₃²×#239	1	1	yes	characteristic
320	C₂	C₃²×#1009	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#125	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#125	D₃
hypocenter	C₃	C₃²×#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#125

Complements

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

N	H ≅ G/N	type
1. C₃³×#125	C₂	semidirect
2. C₃²×#895	C₂	semidirect
3. C₃²×#895	C₂	semidirect
4. C₃²×#894	C₂	semidirect
5. C₃²×#893	C₂	semidirect
6. C₃²×#874	C₂	semidirect
7. C₃²×#872	C₂	semidirect
9. C₃×#5697	C₃	direct (N × H)
25. C₃²×#125	D₃	semidirect
26. C₃²×#125	C₂×C₃	semidirect
32. C₃×#895	C₂×C₃	semidirect
33. C₃×#895	C₂×C₃	semidirect
34. C₃×#894	C₂×C₃	semidirect
35. C₃×#893	C₂×C₃	semidirect
36. C₃×#874	C₂×C₃	semidirect
37. C₃×#872	C₂×C₃	semidirect
51. #5697	C₃²	direct (N × H)
94. C₃×#125	C₃×D₃	semidirect
95. C₃×#125	C₂×C₃²	semidirect
101. #895	C₂×C₃²	semidirect
102. #895	C₂×C₃²	semidirect
103. #894	C₂×C₃²	semidirect
104. #893	C₂×C₃²	semidirect
105. #874	C₂×C₃²	semidirect
106. #872	C₂×C₃²	semidirect
195. #125	C₃²×D₃	semidirect
201. C₃³	#540	semidirect
279. C₃²	#5697	direct (N × H)
280. C₃²	C₃×#540	semidirect
315. C₃	C₃×#5697	direct (N × H)
316. C₃	C₃²×#540	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#125	2
C₃²×#895	2
C₃²×#894	2
C₃²×#893	2
C₃²×#874	2
C₃²×#872	2
C₃²×#540	3
C₃×#5697	3

Subgroups — C32×#5697 #13179074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5697 #13179074