Subgroups — C₃²×#5204 #13162638

Order 1728.

840 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#137	C₂	1	1	yes	maximal, characteristic
2	C₃²×#951	C₂	1	1	yes	maximal, characteristic
3	C₃²×#948	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃²×#216	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#836	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#824	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#822	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#473	—	3	1	no	maximal
9	C₃×#5204	C₃	1	4	yes	maximal, direct product factor
10	C₃³×#41	C₂²	1	1	yes	characteristic
11	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
13	C₃³×Q₁₆	—	2	1	no	—
14	C₃³×SD₁₆	—	2	1	no	—
15	C₃³×M₁₆	C₂²	1	1	yes	characteristic
16	C₃²×#216	—	2	1	no	—
17	C₃²×#216	—	2	1	no	—
18	C₃²×#216	—	2	1	no	—
19	C₃²×#214	—	2	1	no	—
20	C₃²×#208	—	2	1	no	—
21	C₃²×#208	C₂²	1	1	yes	characteristic
22	C₃²×#208	C₂²	1	1	yes	characteristic
23	C₂×C₃²×C₃⋊C₈	—	2	1	no	—
24	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₃²×#137	—	2	4	no	—
26	C₃²×#137	—	3	1	no	—
27	C₃²×#137	D₃	1	1	yes	characteristic
28	C₃²×#137	C₂×C₃	1	4	yes	—
29	C₂×C₃²×Q₁₆	—	3	1	no	—
30	C₃²×#131	—	3	1	no	—
31	C₃×#951	C₂×C₃	1	4	yes	—
32	C₃×#948	C₂×C₃	1	4	yes	—
33	C₂×C₃×#216	C₂×C₃	1	4	yes	complemented
34	C₃²×#108	—	3	1	no	—
35	C₃²×#101	—	3	1	no	—
36	C₃²×#101	—	3	1	no	—
37	C₃×#836	C₂×C₃	1	4	yes	complemented
38	C₃×#824	C₂×C₃	1	4	yes	complemented
39	C₃×#822	C₂×C₃	1	4	yes	complemented
40	C₃³×Q₈	—	2	1	no	—
41	C₃³×Q₈	—	2	1	no	—
42	C₃³×Q₈	D₄	1	1	yes	characteristic
43	C₃³×D₄	—	2	1	no	—
44	C₃³×D₄	D₄	1	1	yes	characteristic
45	C₂×C₃³×C₄	—	2	1	no	—
46	C₂×C₃³×C₄	—	2	1	no	—
47	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃²×Dic₃	—	2	1	no	—
49	C₃²×Dic₆	—	2	1	no	—
50	C₃²×Dic₆	—	2	1	no	—
51	C₃³×C₈	—	2	1	no	—
52	C₃²×C₃⋊C₈	—	2	1	no	—
53	C₃²×C₃⋊C₈	—	2	1	no	—
54	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₃×#473	—	3	4	no	—
57	#5204	C₃²	1	1	yes	characteristic, direct product factor
58	C₃²×#41	—	2	4	no	—
59	C₃²×#41	C₂×D₃	1	1	yes	characteristic
60	C₃²×#41	C₂²×C₃	1	4	yes	—
61	C₂×C₃²×Q₈	—	2	4	no	—
62	C₂×C₃²×Q₈	—	3	1	no	—
63	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
65	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
66	C₃²×Q₁₆	—	2	8	no	—
67	C₃²×Q₁₆	—	2	1	no	—
68	C₃²×Q₁₆	—	2	4	no	—
69	C₃²×Q₁₆	—	6	1	no	—
70	C₃²×Q₁₆	—	6	1	no	—
71	C₃²×Q₁₆	—	6	1	no	—
72	C₃²×SD₁₆	—	2	8	no	—
73	C₃²×SD₁₆	—	2	1	no	—
74	C₃²×SD₁₆	—	2	4	no	—
75	C₃²×SD₁₆	—	6	1	no	—
76	C₃²×M₁₆	—	2	4	no	—
77	C₃²×M₁₆	—	3	1	no	—
78	C₃²×M₁₆	—	3	1	no	—
79	C₃²×M₁₆	—	6	1	no	—
80	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
81	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
82	C₂×C₃²×C₈	—	3	1	no	—
83	C₂×C₃²×C₈	—	6	1	no	—
84	C₃×#216	—	2	4	no	—
85	C₃×#216	—	2	4	no	—
86	C₃×#216	—	2	4	no	—
87	C₃×#214	—	2	4	no	—
88	C₃×#208	—	2	4	no	—
89	C₃×#208	C₂²×C₃	1	4	yes	—
90	C₃×#208	C₂²×C₃	1	4	yes	—
91	C₂×C₃×C₃⋊C₈	—	2	4	no	—
92	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
93	C₂²×C₃³	—	2	1	no	—
94	C₂²×C₃³	#41	1	1	yes	characteristic
95	C₃³×C₄	—	2	1	no	—
96	C₃³×C₄	—	4	1	no	—
97	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
98	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
99	C₃²×Dic₃	—	4	1	no	—
100	C₃×#137	—	2	4	no	—
101	C₃×#137	—	3	4	no	—
102	C₃×#137	C₃×D₃	1	4	yes	—
103	C₃×#137	C₂×C₃²	1	1	yes	characteristic
104	C₂×C₃×Q₁₆	—	3	4	no	—
105	C₃×#131	—	3	4	no	—
106	#951	C₂×C₃²	1	1	yes	characteristic
107	#948	C₂×C₃²	1	1	yes	characteristic
108	C₂×#216	C₂×C₃²	1	1	yes	characteristic, complemented
109	C₃×#108	—	3	4	no	—
110	C₃×#101	—	3	4	no	—
111	C₃×#101	—	3	4	no	—
112	#836	C₂×C₃²	1	1	yes	characteristic, complemented
113	#824	C₂×C₃²	1	1	yes	characteristic, complemented
114	#822	C₂×C₃²	1	1	yes	characteristic, complemented
115	C₃²×Q₈	—	2	8	no	—
116	C₃²×Q₈	—	2	1	no	—
117	C₃²×Q₈	—	2	1	no	—
118	C₃²×Q₈	—	2	4	no	—
119	C₃²×Q₈	—	2	4	no	—
120	C₃²×Q₈	—	2	4	no	—
121	C₃²×Q₈	—	4	4	no	—
122	C₃²×Q₈	—	6	1	no	—
123	C₃²×Q₈	—	6	1	no	—
124	C₃²×Q₈	#67	1	1	yes	characteristic
125	C₃²×Q₈	C₃×D₄	1	4	yes	—
126	C₃²×D₄	—	2	1	no	—
127	C₃²×D₄	—	2	4	no	—
128	C₃²×D₄	—	2	4	no	—
129	C₃²×D₄	—	4	4	no	—
130	C₃²×D₄	#67	1	1	yes	characteristic
131	C₃²×D₄	C₃×D₄	1	4	yes	—
132	C₂×C₃²×C₄	—	2	1	no	—
133	C₂×C₃²×C₄	—	2	1	no	—
134	C₂×C₃²×C₄	—	2	4	no	—
135	C₂×C₃²×C₄	—	2	4	no	—
136	C₂×C₃²×C₄	—	2	4	no	—
137	C₂×C₃²×C₄	—	4	4	no	—
138	C₂×C₃²×C₄	—	4	4	no	—
139	C₂×C₃²×C₄	—	6	1	no	—
140	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
142	C₂×C₃×Dic₃	—	2	4	no	—
143	C₃×Dic₆	—	2	4	no	—
144	C₃×Dic₆	—	2	4	no	—
145	C₃²×C₈	—	2	8	no	—
146	C₃²×C₈	—	2	1	no	—
147	C₃²×C₈	—	2	4	no	—
148	C₃²×C₈	—	3	1	no	—
149	C₃²×C₈	—	3	1	no	—
150	C₃²×C₈	—	6	1	no	—
151	C₃²×C₈	—	6	1	no	—
152	C₃×C₃⋊C₈	—	2	4	no	—
153	C₃×C₃⋊C₈	—	2	4	no	—
154	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
155	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
156	#473	—	3	1	no	Sylow
157	C₂×C₃³	—	2	1	no	—
158	C₂×C₃³	—	4	1	no	—
159	C₂×C₃³	#121	1	1	yes	characteristic
160	C₃×#41	—	2	4	no	—
161	C₃×#41	C₂×C₃×D₃	1	4	yes	—
162	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
163	C₂×C₃×Q₈	—	2	4	no	—
164	C₂×C₃×Q₈	—	3	4	no	—
165	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
166	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
167	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
168	C₃×Q₁₆	—	2	8	no	—
169	C₃×Q₁₆	—	2	4	no	—
170	C₃×Q₁₆	—	2	1	no	—
171	C₃×Q₁₆	—	6	4	no	—
172	C₃×Q₁₆	—	6	4	no	—
173	C₃×Q₁₆	—	6	4	no	—
174	C₃×SD₁₆	—	2	8	no	—
175	C₃×SD₁₆	—	2	4	no	—
176	C₃×SD₁₆	—	2	1	no	—
177	C₃×SD₁₆	—	6	4	no	—
178	C₃×M₁₆	—	2	4	no	—
179	C₃×M₁₆	—	3	4	no	—
180	C₃×M₁₆	—	3	4	no	—
181	C₃×M₁₆	—	6	4	no	—
182	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
183	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
184	C₂×C₃×C₈	—	3	4	no	—
185	C₂×C₃×C₈	—	6	4	no	—
186	#216	—	2	1	no	—
187	#216	—	2	1	no	—
188	#216	—	2	1	no	—
189	#214	—	2	1	no	—
190	#208	—	2	1	no	—
191	#208	C₂²×C₃²	1	1	yes	characteristic
192	#208	C₂²×C₃²	1	1	yes	characteristic
193	C₂×C₃⋊C₈	—	2	1	no	—
194	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
195	C₂²×C₃²	—	2	1	no	—
196	C₂²×C₃²	—	2	4	no	—
197	C₂²×C₃²	—	2	4	no	—
198	C₂²×C₃²	—	4	4	no	—
199	C₂²×C₃²	#237	1	1	yes	characteristic
200	C₂²×C₃²	C₃×#41	1	4	yes	—
201	C₃²×C₄	—	2	1	no	—
202	C₃²×C₄	—	2	4	no	—
203	C₃²×C₄	—	2	4	no	—
204	C₃²×C₄	—	2	4	no	—
205	C₃²×C₄	—	4	4	no	—
206	C₃²×C₄	—	4	8	no	—
207	C₃²×C₄	—	4	1	no	—
208	C₃²×C₄	—	4	4	no	—
209	C₃²×C₄	—	12	1	no	—
210	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
211	C₃²×C₄	C₂×#67	1	1	yes	characteristic
212	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
213	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
214	C₃×Dic₃	—	4	4	no	—
215	#137	—	3	1	no	—
216	#137	C₃²×D₃	1	1	yes	characteristic
217	C₂×Q₁₆	—	3	1	no	—
218	#131	—	3	1	no	—
219	#108	—	3	1	no	—
220	#101	—	3	1	no	—
221	#101	—	3	1	no	—
222	C₃³	#473	1	1	yes	characteristic, complemented, Sylow
223	C₃×Q₈	—	2	8	no	—
224	C₃×Q₈	—	2	4	no	—
225	C₃×Q₈	—	2	4	no	—
226	C₃×Q₈	—	2	1	no	—
227	C₃×Q₈	—	2	1	no	—
228	C₃×Q₈	—	2	4	no	—
229	C₃×Q₈	—	4	4	no	—
230	C₃×Q₈	—	6	4	no	—
231	C₃×Q₈	—	6	4	no	—
232	C₃×Q₈	C₃×#67	1	4	yes	—
233	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
234	C₃×D₄	—	2	4	no	—
235	C₃×D₄	—	2	1	no	—
236	C₃×D₄	—	2	4	no	—
237	C₃×D₄	—	4	4	no	—
238	C₃×D₄	C₃×#67	1	4	yes	—
239	C₃×D₄	C₃²×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	—	2	4	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	4	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	2	4	no	—
245	C₂×C₃×C₄	—	4	4	no	—
246	C₂×C₃×C₄	—	4	4	no	—
247	C₂×C₃×C₄	—	6	4	no	—
248	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
249	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
250	C₂×Dic₃	—	2	1	no	—
251	Dic₆	—	2	1	no	—
252	Dic₆	—	2	1	no	—
253	C₃×C₈	—	2	8	no	—
254	C₃×C₈	—	2	4	no	—
255	C₃×C₈	—	2	1	no	—
256	C₃×C₈	—	3	4	no	—
257	C₃×C₈	—	3	4	no	—
258	C₃×C₈	—	6	4	no	—
259	C₃×C₈	—	6	4	no	—
260	C₃⋊C₈	—	2	1	no	—
261	C₃⋊C₈	—	2	1	no	—
262	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
263	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
264	C₂×C₃²	—	2	1	no	—
265	C₂×C₃²	—	2	4	no	—
266	C₂×C₃²	—	2	4	no	—
267	C₂×C₃²	—	4	4	no	—
268	C₂×C₃²	—	4	1	no	—
269	C₂×C₃²	—	4	4	no	—
270	C₂×C₃²	—	8	4	no	—
271	C₂×C₃²	#939	1	1	yes	characteristic
272	C₂×C₃²	C₃×#121	1	4	yes	—
273	#41	C₂×C₃²×D₃	1	1	yes	characteristic
274	C₂×Q₈	—	3	1	no	—
275	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	6	1	no	—
278	Q₁₆	—	6	1	no	—
279	Q₁₆	—	6	1	no	—
280	SD₁₆	—	2	1	no	—
281	SD₁₆	—	6	1	no	—
282	M₁₆	—	3	1	no	—
283	M₁₆	—	3	1	no	—
284	M₁₆	—	6	1	no	—
285	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
286	C₂×C₈	—	3	1	no	—
287	C₂×C₈	—	6	1	no	—
288	C₂²×C₃	—	2	4	no	—
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	—	2	4	no	—
291	C₂²×C₃	—	4	4	no	—
292	C₂²×C₃	C₃×#237	1	4	yes	—
293	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
294	C₃×C₄	—	2	4	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	2	4	no	—
297	C₃×C₄	—	2	4	no	—
298	C₃×C₄	—	4	4	no	—
299	C₃×C₄	—	4	8	no	—
300	C₃×C₄	—	4	4	no	—
301	C₃×C₄	—	4	1	no	—
302	C₃×C₄	—	12	4	no	—
303	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
304	C₃×C₄	C₂×C₃×#67	1	4	yes	—
305	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
306	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
307	Dic₃	—	4	1	no	—
308	C₃²	—	2	4	no	—
309	C₃²	#5204	1	1	yes	characteristic, direct product factor
310	C₃²	C₃×#473	1	4	yes	complemented
311	Q₈	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	6	1	no	—
314	Q₈	—	6	1	no	—
315	Q₈	C₃²×#67	1	1	yes	characteristic
316	D₄	—	2	1	no	—
317	D₄	C₃²×#67	1	1	yes	characteristic
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	6	1	no	—
321	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	3	1	no	—
324	C₈	—	3	1	no	—
325	C₈	—	6	1	no	—
326	C₈	—	6	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	C₃×#939	1	4	yes	—
335	C₂×C₃	C₃²×#121	1	1	yes	characteristic
336	C₂²	—	2	1	no	—
337	C₂²	C₃²×#237	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	12	1	no	—
341	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
342	C₄	C₂×C₃²×#67	1	1	yes	characteristic
343	C₃	—	2	4	no	—
344	C₃	C₃×#5204	1	4	yes	direct product factor
345	C₃	C₃²×#473	1	1	yes	characteristic, complemented
346	C₂	—	2	1	no	—
347	C₂	—	4	1	no	—
348	C₂	C₃²×#939	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#137	C₂
socle	C₂×C₃³	#121
prime radical	C₃³×#41	C₂²
hypercenter	C₃²×#137	D₃
hypocenter	C₃	C₃²×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#137

Complements

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

N	H ≅ G/N	type
4. C₂×C₃²×#216	C₂	semidirect
5. C₃²×#836	C₂	semidirect
6. C₃²×#824	C₂	semidirect
7. C₃²×#822	C₂	semidirect
9. C₃×#5204	C₃	direct (N × H)
33. C₂×C₃×#216	C₂×C₃	semidirect
37. C₃×#836	C₂×C₃	semidirect
38. C₃×#824	C₂×C₃	semidirect
39. C₃×#822	C₂×C₃	semidirect
57. #5204	C₃²	direct (N × H)
108. C₂×#216	C₂×C₃²	semidirect
112. #836	C₂×C₃²	semidirect
113. #824	C₂×C₃²	semidirect
114. #822	C₂×C₃²	semidirect
222. C₃³	#473	semidirect
309. C₃²	#5204	direct (N × H)
310. C₃²	C₃×#473	semidirect
344. C₃	C₃×#5204	direct (N × H)
345. C₃	C₃²×#473	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#137	2
C₃²×#951	2
C₃²×#948	2
C₂×C₃²×#216	2
C₃²×#836	2
C₃²×#824	2
C₃²×#822	2
C₃²×#473	3
C₃×#5204	3

Subgroups — C32×#5204 #13162638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5204 #13162638