Subgroups — C₃²×#5203 #13162639

Order 1728.

864 conjugacy classes of subgroups.

The subgroup lattice has 362 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, complemented
2	C₃²×#949	C₂	1	1	yes	maximal, characteristic
3	C₃²×#948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×#215	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#836	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#823	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#822	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#471	—	3	1	no	maximal
9	C₃×#5203	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×D₁₂	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×#215	—	2	1	no	—
17	C₃²×#215	—	2	1	no	—
18	C₃²×#215	—	2	1	no	—
19	C₃²×#213	—	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	D₃	1	1	yes	characteristic, complemented
27	C₃²×#137	C₂×C₃	1	4	yes	complemented
28	C₃²×AGL(1,ℤ/8)	—	3	1	no	—
29	C₂×C₃²×SD₁₆	—	3	1	no	—
30	C₃²×#131	—	3	1	no	—
31	C₃×#949	C₂×C₃	1	4	yes	—
32	C₃×#948	C₂×C₃	1	4	yes	complemented
33	C₂×C₃×#215	C₂×C₃	1	4	yes	complemented
34	C₃²×#108	—	3	1	no	—
35	C₃²×#101	—	3	1	no	—
36	C₃²×#100	—	3	1	no	—
37	C₃×#836	C₂×C₃	1	4	yes	complemented
38	C₃×#823	C₂×C₃	1	4	yes	complemented
39	C₃×#822	C₂×C₃	1	4	yes	complemented
40	C₂²×C₃²×D₃	—	2	1	no	—
41	C₃³×Q₈	—	2	1	no	—
42	C₃³×Q₈	—	2	1	no	—
43	C₃³×Q₈	D₄	1	1	yes	characteristic
44	C₃³×D₄	—	2	1	no	—
45	C₃³×D₄	D₄	1	1	yes	characteristic
46	C₂×C₃³×C₄	—	2	1	no	—
47	C₂×C₃³×C₄	—	2	1	no	—
48	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
49	C₃²×D₁₂	—	2	1	no	—
50	C₃²×D₁₂	—	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×#471	—	3	4	no	—
57	#5203	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	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
64	C₂×C₃²×D₄	—	3	1	no	—
65	C₂×C₃×D₁₂	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×SD₁₆	—	2	8	no	—
70	C₃²×SD₁₆	—	2	1	no	—
71	C₃²×SD₁₆	—	2	4	no	—
72	C₃²×SD₁₆	—	6	1	no	—
73	C₃²×SD₁₆	—	6	1	no	—
74	C₃²×SD₁₆	—	6	1	no	—
75	C₃²×D₈	—	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×#215	—	2	4	no	—
85	C₃×#215	—	2	4	no	—
86	C₃×#215	—	2	4	no	—
87	C₃×#213	—	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	C₂×C₃²×D₃	—	4	1	no	—
96	C₂×C₃²×D₃	—	8	1	no	—
97	C₃³×C₄	—	2	1	no	—
98	C₃³×C₄	—	4	1	no	—
99	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
100	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
101	C₃×#137	—	2	4	no	—
102	C₃×#137	C₃×D₃	1	4	yes	complemented
103	C₃×#137	C₂×C₃²	1	1	yes	characteristic, complemented
104	C₃×AGL(1,ℤ/8)	—	3	4	no	—
105	C₂×C₃×SD₁₆	—	3	4	no	—
106	C₃×#131	—	3	4	no	—
107	#949	C₂×C₃²	1	1	yes	characteristic
108	#948	C₂×C₃²	1	1	yes	characteristic, complemented
109	C₂×#215	C₂×C₃²	1	1	yes	characteristic, complemented
110	C₃×#108	—	3	4	no	—
111	C₃×#101	—	3	4	no	—
112	C₃×#100	—	3	4	no	—
113	#836	C₂×C₃²	1	1	yes	characteristic, complemented
114	#823	C₂×C₃²	1	1	yes	characteristic, complemented
115	#822	C₂×C₃²	1	1	yes	characteristic, complemented
116	C₂³×C₃²	—	6	1	no	—
117	C₂²×C₃×D₃	—	2	4	no	—
118	C₃²×Q₈	—	2	8	no	—
119	C₃²×Q₈	—	2	1	no	—
120	C₃²×Q₈	—	2	1	no	—
121	C₃²×Q₈	—	2	4	no	—
122	C₃²×Q₈	—	2	4	no	—
123	C₃²×Q₈	—	2	4	no	—
124	C₃²×Q₈	—	4	4	no	—
125	C₃²×Q₈	#67	1	1	yes	characteristic
126	C₃²×Q₈	C₃×D₄	1	4	yes	—
127	C₃²×D₄	—	2	1	no	—
128	C₃²×D₄	—	2	4	no	—
129	C₃²×D₄	—	2	4	no	—
130	C₃²×D₄	—	4	4	no	—
131	C₃²×D₄	—	6	1	no	—
132	C₃²×D₄	—	6	1	no	—
133	C₃²×D₄	#67	1	1	yes	characteristic
134	C₃²×D₄	C₃×D₄	1	4	yes	—
135	C₂×C₃²×C₄	—	2	1	no	—
136	C₂×C₃²×C₄	—	2	1	no	—
137	C₂×C₃²×C₄	—	2	4	no	—
138	C₂×C₃²×C₄	—	2	4	no	—
139	C₂×C₃²×C₄	—	2	4	no	—
140	C₂×C₃²×C₄	—	4	4	no	—
141	C₂×C₃²×C₄	—	4	4	no	—
142	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
144	C₃×D₁₂	—	2	4	no	—
145	C₃×D₁₂	—	2	4	no	—
146	C₃²×C₈	—	2	8	no	—
147	C₃²×C₈	—	2	1	no	—
148	C₃²×C₈	—	2	4	no	—
149	C₃²×C₈	—	3	1	no	—
150	C₃²×C₈	—	3	1	no	—
151	C₃²×C₈	—	6	1	no	—
152	C₃²×C₈	—	6	1	no	—
153	C₃×C₃⋊C₈	—	2	4	no	—
154	C₃×C₃⋊C₈	—	2	4	no	—
155	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
156	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
157	#471	—	3	1	no	Sylow
158	C₂×C₃³	—	2	1	no	—
159	C₂×C₃³	—	4	1	no	—
160	C₂×C₃³	#121	1	1	yes	characteristic
161	C₃²×D₃	—	8	1	no	—
162	C₃×#41	—	2	4	no	—
163	C₃×#41	C₂×C₃×D₃	1	4	yes	—
164	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
165	C₂×C₃×Q₈	—	2	4	no	—
166	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
167	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
168	C₂×C₃×D₄	—	3	4	no	—
169	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
170	C₃×Q₁₆	—	2	8	no	—
171	C₃×Q₁₆	—	2	4	no	—
172	C₃×Q₁₆	—	2	1	no	—
173	C₃×SD₁₆	—	2	8	no	—
174	C₃×SD₁₆	—	2	4	no	—
175	C₃×SD₁₆	—	2	1	no	—
176	C₃×SD₁₆	—	6	4	no	—
177	C₃×SD₁₆	—	6	4	no	—
178	C₃×SD₁₆	—	6	4	no	—
179	C₃×D₈	—	6	4	no	—
180	C₃×M₁₆	—	2	4	no	—
181	C₃×M₁₆	—	3	4	no	—
182	C₃×M₁₆	—	3	4	no	—
183	C₃×M₁₆	—	6	4	no	—
184	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
185	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
186	C₂×C₃×C₈	—	3	4	no	—
187	C₂×C₃×C₈	—	6	4	no	—
188	#215	—	2	1	no	—
189	#215	—	2	1	no	—
190	#215	—	2	1	no	—
191	#213	—	2	1	no	—
192	#208	—	2	1	no	—
193	#208	C₂²×C₃²	1	1	yes	characteristic
194	#208	C₂²×C₃²	1	1	yes	characteristic
195	C₂×C₃⋊C₈	—	2	1	no	—
196	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
197	C₂²×C₃²	—	2	1	no	—
198	C₂²×C₃²	—	2	4	no	—
199	C₂²×C₃²	—	2	4	no	—
200	C₂²×C₃²	—	4	4	no	—
201	C₂²×C₃²	—	12	1	no	—
202	C₂²×C₃²	—	24	1	no	—
203	C₂²×C₃²	#237	1	1	yes	characteristic
204	C₂²×C₃²	C₃×#41	1	4	yes	—
205	C₂×C₃×D₃	—	4	4	no	—
206	C₂×C₃×D₃	—	8	4	no	—
207	C₃²×C₄	—	2	1	no	—
208	C₃²×C₄	—	2	4	no	—
209	C₃²×C₄	—	2	4	no	—
210	C₃²×C₄	—	2	4	no	—
211	C₃²×C₄	—	4	4	no	—
212	C₃²×C₄	—	4	8	no	—
213	C₃²×C₄	—	4	1	no	—
214	C₃²×C₄	—	4	4	no	—
215	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
216	C₃²×C₄	C₂×#67	1	1	yes	characteristic
217	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
218	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
219	#137	C₃²×D₃	1	1	yes	characteristic, complemented
220	AGL(1,ℤ/8)	—	3	1	no	—
221	C₂×SD₁₆	—	3	1	no	—
222	#131	—	3	1	no	—
223	#108	—	3	1	no	—
224	#101	—	3	1	no	—
225	#100	—	3	1	no	—
226	C₃³	#471	1	1	yes	characteristic, complemented, Sylow
227	C₂³×C₃	—	6	4	no	—
228	C₂²×D₃	—	2	1	no	—
229	C₃×Q₈	—	2	8	no	—
230	C₃×Q₈	—	2	4	no	—
231	C₃×Q₈	—	2	4	no	—
232	C₃×Q₈	—	2	1	no	—
233	C₃×Q₈	—	2	1	no	—
234	C₃×Q₈	—	2	4	no	—
235	C₃×Q₈	—	4	4	no	—
236	C₃×Q₈	C₃×#67	1	4	yes	—
237	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
238	C₃×D₄	—	2	4	no	—
239	C₃×D₄	—	2	1	no	—
240	C₃×D₄	—	2	4	no	—
241	C₃×D₄	—	4	4	no	—
242	C₃×D₄	—	6	4	no	—
243	C₃×D₄	—	6	4	no	—
244	C₃×D₄	C₃×#67	1	4	yes	—
245	C₃×D₄	C₃²×D₄	1	1	yes	characteristic
246	C₂×C₃×C₄	—	2	4	no	—
247	C₂×C₃×C₄	—	2	1	no	—
248	C₂×C₃×C₄	—	2	4	no	—
249	C₂×C₃×C₄	—	2	1	no	—
250	C₂×C₃×C₄	—	2	4	no	—
251	C₂×C₃×C₄	—	4	4	no	—
252	C₂×C₃×C₄	—	4	4	no	—
253	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
254	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
255	D₁₂	—	2	1	no	—
256	D₁₂	—	2	1	no	—
257	C₃×C₈	—	2	8	no	—
258	C₃×C₈	—	2	4	no	—
259	C₃×C₈	—	2	1	no	—
260	C₃×C₈	—	3	4	no	—
261	C₃×C₈	—	3	4	no	—
262	C₃×C₈	—	6	4	no	—
263	C₃×C₈	—	6	4	no	—
264	C₃⋊C₈	—	2	1	no	—
265	C₃⋊C₈	—	2	1	no	—
266	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
267	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
268	C₂×C₃²	—	2	1	no	—
269	C₂×C₃²	—	2	4	no	—
270	C₂×C₃²	—	2	4	no	—
271	C₂×C₃²	—	4	4	no	—
272	C₂×C₃²	—	4	1	no	—
273	C₂×C₃²	—	4	4	no	—
274	C₂×C₃²	—	8	4	no	—
275	C₂×C₃²	—	24	1	no	—
276	C₂×C₃²	#939	1	1	yes	characteristic
277	C₂×C₃²	C₃×#121	1	4	yes	—
278	C₃×D₃	—	8	4	no	—
279	#41	C₂×C₃²×D₃	1	1	yes	characteristic
280	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
281	C₂×D₄	—	3	1	no	—
282	Q₁₆	—	2	1	no	—
283	SD₁₆	—	2	1	no	—
284	SD₁₆	—	6	1	no	—
285	SD₁₆	—	6	1	no	—
286	SD₁₆	—	6	1	no	—
287	D₈	—	6	1	no	—
288	M₁₆	—	3	1	no	—
289	M₁₆	—	3	1	no	—
290	M₁₆	—	6	1	no	—
291	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
292	C₂×C₈	—	3	1	no	—
293	C₂×C₈	—	6	1	no	—
294	C₂²×C₃	—	2	4	no	—
295	C₂²×C₃	—	2	1	no	—
296	C₂²×C₃	—	2	4	no	—
297	C₂²×C₃	—	4	4	no	—
298	C₂²×C₃	—	12	4	no	—
299	C₂²×C₃	—	24	4	no	—
300	C₂²×C₃	C₃×#237	1	4	yes	—
301	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
302	C₂×D₃	—	4	1	no	—
303	C₂×D₃	—	8	1	no	—
304	C₃×C₄	—	2	4	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	4	no	—
307	C₃×C₄	—	2	4	no	—
308	C₃×C₄	—	4	4	no	—
309	C₃×C₄	—	4	8	no	—
310	C₃×C₄	—	4	4	no	—
311	C₃×C₄	—	4	1	no	—
312	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
313	C₃×C₄	C₂×C₃×#67	1	4	yes	—
314	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
315	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
316	C₃²	—	2	4	no	—
317	C₃²	#5203	1	1	yes	characteristic, direct product factor
318	C₃²	C₃×#471	1	4	yes	complemented
319	C₂³	—	6	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	C₃²×#67	1	1	yes	characteristic
323	D₄	—	2	1	no	—
324	D₄	—	6	1	no	—
325	D₄	—	6	1	no	—
326	D₄	C₃²×#67	1	1	yes	characteristic
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	3	1	no	—
332	C₈	—	3	1	no	—
333	C₈	—	6	1	no	—
334	C₈	—	6	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	2	4	no	—
337	C₂×C₃	—	2	4	no	—
338	C₂×C₃	—	4	4	no	—
339	C₂×C₃	—	4	1	no	—
340	C₂×C₃	—	4	4	no	—
341	C₂×C₃	—	8	4	no	—
342	C₂×C₃	—	24	4	no	—
343	C₂×C₃	C₃×#939	1	4	yes	—
344	C₂×C₃	C₃²×#121	1	1	yes	characteristic
345	D₃	—	8	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	12	1	no	—
348	C₂²	—	24	1	no	—
349	C₂²	C₃²×#237	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
353	C₄	C₂×C₃²×#67	1	1	yes	characteristic
354	C₃	—	2	4	no	—
355	C₃	C₃×#5203	1	4	yes	direct product factor
356	C₃	C₃²×#471	1	1	yes	characteristic, complemented
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	24	1	no	—
360	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	C₃²×#949	C₂
hypercenter	C₃²×#137	D₃
hypocenter	C₃	C₃²×#471
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
1. C₃³×#137	C₂	semidirect
3. C₃²×#948	C₂	semidirect
4. C₂×C₃²×#215	C₂	semidirect
5. C₃²×#836	C₂	semidirect
6. C₃²×#823	C₂	semidirect
7. C₃²×#822	C₂	semidirect
9. C₃×#5203	C₃	direct (N × H)
26. C₃²×#137	D₃	semidirect
27. C₃²×#137	C₂×C₃	semidirect
32. C₃×#948	C₂×C₃	semidirect
33. C₂×C₃×#215	C₂×C₃	semidirect
37. C₃×#836	C₂×C₃	semidirect
38. C₃×#823	C₂×C₃	semidirect
39. C₃×#822	C₂×C₃	semidirect
57. #5203	C₃²	direct (N × H)
102. C₃×#137	C₃×D₃	semidirect
103. C₃×#137	C₂×C₃²	semidirect
108. #948	C₂×C₃²	semidirect
109. C₂×#215	C₂×C₃²	semidirect
113. #836	C₂×C₃²	semidirect
114. #823	C₂×C₃²	semidirect
115. #822	C₂×C₃²	semidirect
219. #137	C₃²×D₃	semidirect
226. C₃³	#471	semidirect
317. C₃²	#5203	direct (N × H)
318. C₃²	C₃×#471	semidirect
355. C₃	C₃×#5203	direct (N × H)
356. C₃	C₃²×#471	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×#949	2
C₃²×#948	2
C₂×C₃²×#215	2
C₃²×#836	2
C₃²×#823	2
C₃²×#822	2
C₃²×#471	3
C₃×#5203	3

Subgroups — C32×#5203 #13162639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5203 #13162639