Subgroups — C₃²×#4823 #13162259

Order 1728.

800 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	C₃²×#950	C₂	1	1	yes	maximal, characteristic
2	C₃²×#948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#907	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#822	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#803	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#444	—	3	1	no	maximal
9	C₃×#4823	C₃	1	4	yes	maximal, direct product factor
10	C₃³×#41	C₂²	1	1	yes	characteristic
11	C₃²×#235	C₂²	1	1	yes	characteristic
12	C₃³×M₁₆	C₂²	1	1	yes	characteristic
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃²×#216	C₄	1	1	yes	characteristic, complemented
15	C₃²×#215	C₄	1	1	yes	characteristic, complemented
16	C₃²×#214	C₄	1	1	yes	characteristic, complemented
17	C₃²×#213	C₄	1	1	yes	characteristic, complemented
18	C₃²×#208	—	2	1	no	—
19	C₃²×#208	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	—	2	1	no	—
21	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃²×ΓL(1,𝔽₂₅)	—	2	1	no	—
24	C₃²×D₃×C₈	—	2	1	no	—
25	C₃²×#135	—	3	1	no	—
26	C₃²×#131	—	3	1	no	—
27	C₃²×#131	—	3	1	no	—
28	C₃×#950	C₂×C₃	1	4	yes	—
29	C₃×#948	C₂×C₃	1	4	yes	complemented
30	C₃×#907	C₂×C₃	1	4	yes	complemented
31	C₃²×#108	—	3	1	no	—
32	C₃²×#104	—	2	4	no	—
33	C₃²×#104	—	3	1	no	—
34	C₃²×#104	D₃	1	1	yes	characteristic, complemented
35	C₃²×#104	C₂×C₃	1	4	yes	complemented
36	C₃²×#97	—	3	1	no	—
37	C₃×#822	C₂×C₃	1	4	yes	complemented
38	C₃×#805	C₂×C₃	1	4	yes	complemented
39	C₃×#803	C₂×C₃	1	4	yes	complemented
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	—	2	1	no	—
45	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
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	C₃³×C₈	—	2	1	no	—
51	C₃²×C₃⋊C₈	—	2	1	no	—
52	C₃²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₃×#444	—	3	4	no	—
57	#4823	C₃²	1	1	yes	characteristic, direct product factor
58	C₃²×#41	—	2	4	no	—
59	C₃²×#41	—	3	1	no	—
60	C₃²×#41	C₂×D₃	1	1	yes	characteristic
61	C₃²×#41	C₂²×C₃	1	4	yes	—
62	C₃×#235	C₂²×C₃	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	C₃²×M₁₆	—	2	4	no	—
68	C₃²×M₁₆	—	3	1	no	—
69	C₃²×M₁₆	—	6	1	no	—
70	C₃²×M₁₆	—	6	1	no	—
71	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
72	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
73	C₂×C₃²×C₈	—	3	1	no	—
74	C₂×C₃²×C₈	—	3	1	no	—
75	C₂×C₃²×C₈	—	6	1	no	—
76	C₂×C₃²×C₈	—	6	1	no	—
77	C₃²×C₄²	—	2	4	no	—
78	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
79	C₃²×C₄²	C₂²×C₃	1	4	yes	—
80	C₃×#216	C₃×C₄	1	4	yes	complemented
81	C₃×#215	C₃×C₄	1	4	yes	complemented
82	C₃×#214	C₃×C₄	1	4	yes	complemented
83	C₃×#213	C₃×C₄	1	4	yes	complemented
84	C₃×#208	—	2	4	no	—
85	C₃×#208	C₂²×C₃	1	4	yes	—
86	C₂×C₃×C₃⋊C₈	—	2	4	no	—
87	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
88	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
89	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
90	C₃×D₃×C₈	—	2	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	—	4	1	no	—
96	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
98	C₃²×Dic₃	—	2	1	no	—
99	C₃×#135	—	3	4	no	—
100	C₃×#131	—	3	4	no	—
101	C₃×#131	—	3	4	no	—
102	#950	C₂×C₃²	1	1	yes	characteristic
103	#948	C₂×C₃²	1	1	yes	characteristic, complemented
104	#907	C₂×C₃²	1	1	yes	characteristic, complemented
105	C₃×#108	—	3	4	no	—
106	C₃×#104	—	2	4	no	—
107	C₃×#104	—	3	4	no	—
108	C₃×#104	C₃×D₃	1	4	yes	complemented
109	C₃×#104	C₂×C₃²	1	1	yes	characteristic, complemented
110	C₃×#97	—	3	4	no	—
111	#822	C₂×C₃²	1	1	yes	characteristic, complemented
112	#805	C₂×C₃²	1	1	yes	characteristic, complemented
113	#803	C₂×C₃²	1	1	yes	characteristic, complemented
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×C₃×C₄	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×C₃×C₄	1	4	yes	—
126	C₂×C₃²×C₄	—	2	1	no	—
127	C₂×C₃²×C₄	—	2	1	no	—
128	C₂×C₃²×C₄	—	2	4	no	—
129	C₂×C₃²×C₄	—	2	4	no	—
130	C₂×C₃²×C₄	—	2	4	no	—
131	C₂×C₃²×C₄	—	4	4	no	—
132	C₂×C₃²×C₄	—	4	4	no	—
133	C₂×C₃²×C₄	—	6	1	no	—
134	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
136	C₃×#67	—	2	4	no	—
137	C₃×D₁₂	C₂×C₃×C₄	1	4	yes	—
138	C₃×C₄×D₃	—	2	4	no	—
139	C₃×Dic₆	C₂×C₃×C₄	1	4	yes	—
140	C₃²×C₈	—	2	1	no	—
141	C₃²×C₈	—	2	4	no	—
142	C₃²×C₈	—	3	1	no	—
143	C₃²×C₈	—	3	1	no	—
144	C₃²×C₈	—	3	1	no	—
145	C₃²×C₈	—	3	1	no	—
146	C₃²×C₈	—	4	4	no	—
147	C₃²×C₈	—	6	1	no	—
148	C₃×C₃⋊C₈	—	2	4	no	—
149	C₃×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
150	C₃×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
151	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
152	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
153	#444	—	3	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	C₂×C₃×D₃	1	4	yes	—
161	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
162	#235	C₂²×C₃²	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	—	2	4	no	—
168	C₃×M₁₆	—	3	4	no	—
169	C₃×M₁₆	—	6	4	no	—
170	C₃×M₁₆	—	6	4	no	—
171	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
172	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
173	C₂×C₃×C₈	—	3	4	no	—
174	C₂×C₃×C₈	—	3	4	no	—
175	C₂×C₃×C₈	—	6	4	no	—
176	C₂×C₃×C₈	—	6	4	no	—
177	C₃×C₄²	—	2	4	no	—
178	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
179	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
180	#216	C₃²×C₄	1	1	yes	characteristic, complemented
181	#215	C₃²×C₄	1	1	yes	characteristic, complemented
182	#214	C₃²×C₄	1	1	yes	characteristic, complemented
183	#213	C₃²×C₄	1	1	yes	characteristic, complemented
184	#208	—	2	1	no	—
185	#208	C₂²×C₃²	1	1	yes	characteristic
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
188	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
189	ΓL(1,𝔽₂₅)	—	2	1	no	—
190	D₃×C₈	—	2	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	#237	1	1	yes	characteristic
197	C₂²×C₃²	C₃×#41	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	—	4	1	no	—
205	C₃²×C₄	—	4	4	no	—
206	C₃²×C₄	—	4	8	no	—
207	C₃²×C₄	—	6	1	no	—
208	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
209	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
210	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
211	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
212	C₃×Dic₃	—	2	4	no	—
213	#135	—	3	1	no	—
214	#131	—	3	1	no	—
215	#131	—	3	1	no	—
216	#108	—	3	1	no	—
217	#104	—	3	1	no	—
218	#104	C₃²×D₃	1	1	yes	characteristic, complemented
219	#97	—	3	1	no	—
220	C₃³	#444	1	1	yes	characteristic, complemented, Sylow
221	C₃×Q₈	—	2	4	no	—
222	C₃×Q₈	—	3	4	no	—
223	C₃×Q₈	C₃×C₄×D₃	1	4	yes	—
224	C₃×Q₈	C₂×C₃²×C₄	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	C₃×C₄×D₃	1	4	yes	—
232	C₃×D₄	C₂×C₃²×C₄	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	—	2	1	no	—
237	C₂×C₃×C₄	—	2	4	no	—
238	C₂×C₃×C₄	—	4	4	no	—
239	C₂×C₃×C₄	—	4	4	no	—
240	C₂×C₃×C₄	—	6	4	no	—
241	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
242	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
243	#67	—	2	1	no	—
244	D₁₂	C₂×C₃²×C₄	1	1	yes	characteristic
245	C₄×D₃	—	2	1	no	—
246	Dic₆	C₂×C₃²×C₄	1	1	yes	characteristic
247	C₃×C₈	—	2	4	no	—
248	C₃×C₈	—	2	1	no	—
249	C₃×C₈	—	3	4	no	—
250	C₃×C₈	—	3	4	no	—
251	C₃×C₈	—	3	4	no	—
252	C₃×C₈	—	3	4	no	—
253	C₃×C₈	—	4	4	no	—
254	C₃×C₈	—	6	4	no	—
255	C₃⋊C₈	—	2	1	no	—
256	C₃⋊C₈	C₂×C₃²×C₄	1	1	yes	characteristic
257	C₃⋊C₈	C₂×C₃²×C₄	1	1	yes	characteristic
258	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
259	C₃⋊C₈	C₃²×D₄	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	#67⋊C₄	1	1	yes	characteristic
269	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
270	C₃×D₃	—	4	4	no	—
271	#41	—	3	1	no	—
272	#41	C₂×C₃²×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	—	3	1	no	—
278	M₁₆	—	6	1	no	—
279	M₁₆	—	6	1	no	—
280	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic
281	C₂×C₈	—	3	1	no	—
282	C₂×C₈	—	3	1	no	—
283	C₂×C₈	—	6	1	no	—
284	C₂×C₈	—	6	1	no	—
285	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
286	C₂²×C₃	—	2	4	no	—
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	2	4	no	—
289	C₂²×C₃	—	4	4	no	—
290	C₂²×C₃	—	6	4	no	—
291	C₂²×C₃	C₃×#237	1	4	yes	—
292	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
293	C₂×D₃	—	2	1	no	—
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	4	no	—
300	C₃×C₄	—	4	1	no	—
301	C₃×C₄	—	4	8	no	—
302	C₃×C₄	—	6	4	no	—
303	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
304	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
305	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
306	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
307	Dic₃	—	2	1	no	—
308	C₃²	—	2	4	no	—
309	C₃²	#4823	1	1	yes	characteristic, direct product factor
310	C₃²	C₃×#444	1	4	yes	complemented
311	Q₈	—	3	1	no	—
312	Q₈	C₃²×C₄×D₃	1	1	yes	characteristic
313	D₄	—	2	1	no	—
314	D₄	—	3	1	no	—
315	D₄	—	6	1	no	—
316	D₄	C₃²×C₄×D₃	1	1	yes	characteristic
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	6	1	no	—
320	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	3	1	no	—
323	C₈	—	3	1	no	—
324	C₈	—	3	1	no	—
325	C₈	—	3	1	no	—
326	C₈	—	6	1	no	—
327	C₂×C₃	—	2	4	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	4	no	—
330	C₂×C₃	—	4	4	no	—
331	C₂×C₃	—	4	4	no	—
332	C₂×C₃	—	4	1	no	—
333	C₂×C₃	—	8	4	no	—
334	C₂×C₃	—	12	4	no	—
335	C₂×C₃	C₃×#67⋊C₄	1	4	yes	—
336	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
337	D₃	—	4	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	6	1	no	—
340	C₂²	C₃²×#237	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	6	1	no	—
344	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
345	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
346	C₃	—	2	4	no	—
347	C₃	C₃×#4823	1	4	yes	direct product factor
348	C₃	C₃²×#444	1	1	yes	characteristic, complemented
349	C₂	—	2	1	no	—
350	C₂	—	4	1	no	—
351	C₂	—	12	1	no	—
352	C₂	C₃²×#67⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#104	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃²×#950	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#104	D₃
hypocenter	C₃	C₃²×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#104

Complements

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

N	H ≅ G/N	type
2. C₃²×#948	C₂	semidirect
3. C₃²×#907	C₂	semidirect
4. C₃³×#104	C₂	semidirect
5. C₃²×#822	C₂	semidirect
6. C₃²×#805	C₂	semidirect
7. C₃²×#803	C₂	semidirect
9. C₃×#4823	C₃	direct (N × H)
14. C₃²×#216	C₄	semidirect
15. C₃²×#215	C₄	semidirect
16. C₃²×#214	C₄	semidirect
17. C₃²×#213	C₄	semidirect
29. C₃×#948	C₂×C₃	semidirect
30. C₃×#907	C₂×C₃	semidirect
34. C₃²×#104	D₃	semidirect
35. C₃²×#104	C₂×C₃	semidirect
37. C₃×#822	C₂×C₃	semidirect
38. C₃×#805	C₂×C₃	semidirect
39. C₃×#803	C₂×C₃	semidirect
57. #4823	C₃²	direct (N × H)
80. C₃×#216	C₃×C₄	semidirect
81. C₃×#215	C₃×C₄	semidirect
82. C₃×#214	C₃×C₄	semidirect
83. C₃×#213	C₃×C₄	semidirect
103. #948	C₂×C₃²	semidirect
104. #907	C₂×C₃²	semidirect
108. C₃×#104	C₃×D₃	semidirect
109. C₃×#104	C₂×C₃²	semidirect
111. #822	C₂×C₃²	semidirect
112. #805	C₂×C₃²	semidirect
113. #803	C₂×C₃²	semidirect
180. #216	C₃²×C₄	semidirect
181. #215	C₃²×C₄	semidirect
182. #214	C₃²×C₄	semidirect
183. #213	C₃²×C₄	semidirect
218. #104	C₃²×D₃	semidirect
220. C₃³	#444	semidirect
309. C₃²	#4823	direct (N × H)
310. C₃²	C₃×#444	semidirect
347. C₃	C₃×#4823	direct (N × H)
348. C₃	C₃²×#444	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#950	2
C₃²×#948	2
C₃²×#907	2
C₃³×#104	2
C₃²×#822	2
C₃²×#805	2
C₃²×#803	2
C₃²×#444	3
C₃×#4823	3

Subgroups — C32×#4823 #13162259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4823 #13162259