Subgroups — C₃²×#4794 #13162229

Order 1728.

768 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	C₃²×#947	C₂	1	1	yes	maximal, characteristic
2	C₃²×#888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#821	C₂	1	1	yes	maximal, characteristic
6	C₃²×#815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#809	C₂	1	1	yes	maximal, characteristic
8	C₃²×#489	—	3	1	no	maximal
9	C₃×#4794	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Dic₆	—	2	1	no	—
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₃²×Dic₃⋊C₄	—	2	1	no	—
19	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×C₄⋊Q₈	—	3	1	no	—
22	C₃²×#124	—	3	1	no	—
23	C₃×#947	C₂×C₃	1	4	yes	—
24	C₃×#888	C₂×C₃	1	4	yes	complemented
25	C₃²×#103	—	2	4	no	—
26	C₃²×#103	—	3	1	no	—
27	C₃²×#103	D₃	1	1	yes	characteristic, complemented
28	C₃²×#103	C₂×C₃	1	4	yes	complemented
29	C₃²×#102	—	3	1	no	—
30	C₃²×#102	—	3	1	no	—
31	C₃²×#97	—	3	1	no	—
32	C₃×#835	C₂×C₃	1	4	yes	complemented
33	C₃×#821	C₂×C₃	1	4	yes	—
34	C₃×#815	C₂×C₃	1	4	yes	complemented
35	C₃×#809	C₂×C₃	1	4	yes	—
36	C₂²×C₃²×D₃	—	2	1	no	—
37	C₃³×Q₈	—	2	1	no	—
38	C₃³×Q₈	—	2	1	no	—
39	C₂×C₃³×C₄	—	2	1	no	—
40	C₂×C₃³×C₄	—	2	1	no	—
41	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃²×Dic₃	—	2	1	no	—
43	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
44	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
45	C₃²×D₁₂	—	2	1	no	—
46	C₃²×D₁₂	—	2	1	no	—
47	C₃²×Dic₆	—	4	2	no	—
48	C₃³×C₈	—	2	1	no	—
49	C₃²×C₃⋊C₈	—	2	1	no	—
50	C₃×#489	—	3	4	no	—
51	#4794	C₃²	1	1	yes	characteristic, direct product factor
52	C₂×C₃²×Q₈	—	2	4	no	—
53	C₂×C₃²×Q₈	—	6	1	no	—
54	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
56	C₂×C₃²×D₄	—	3	1	no	—
57	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
58	C₂×C₃×Dic₆	—	2	4	no	—
59	C₂×C₃²×C₈	—	2	4	no	—
60	C₂×C₃²×C₈	—	3	1	no	—
61	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
63	C₃²×C₄⋊C₄	—	2	4	no	—
64	C₃²×C₄⋊C₄	—	3	1	no	—
65	C₃²×C₄⋊C₄	—	6	1	no	—
66	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₃²×C₂²⋊C₄	—	6	1	no	—
69	C₃²×C₂²⋊C₄	—	6	1	no	—
70	C₃²×C₄²	—	3	1	no	—
71	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
72	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
73	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
74	C₃×Dic₃⋊C₄	—	2	4	no	—
75	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
76	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
77	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×D₃	—	4	1	no	—
79	C₂×C₃²×D₃	—	4	1	no	—
80	C₂×C₃²×D₃	—	4	1	no	—
81	C₃³×C₄	—	4	1	no	—
82	C₃³×C₄	—	4	1	no	—
83	C₃³×C₄	#41	1	1	yes	characteristic
84	C₃³×C₄	#41	1	1	yes	characteristic
85	C₃²×Dic₃	—	2	1	no	—
86	C₃²×Dic₃	—	2	1	no	—
87	C₃²×Dic₃	—	4	1	no	—
88	C₃×C₄⋊Q₈	—	3	4	no	—
89	C₃×#124	—	3	4	no	—
90	#947	C₂×C₃²	1	1	yes	characteristic
91	#888	C₂×C₃²	1	1	yes	characteristic, complemented
92	C₃×#103	—	2	4	no	—
93	C₃×#103	—	3	4	no	—
94	C₃×#103	C₃×D₃	1	4	yes	complemented
95	C₃×#103	C₂×C₃²	1	1	yes	characteristic, complemented
96	C₃×#102	—	3	4	no	—
97	C₃×#102	—	3	4	no	—
98	C₃×#97	—	3	4	no	—
99	#835	C₂×C₃²	1	1	yes	characteristic, complemented
100	#821	C₂×C₃²	1	1	yes	characteristic
101	#815	C₂×C₃²	1	1	yes	characteristic, complemented
102	#809	C₂×C₃²	1	1	yes	characteristic
103	C₂³×C₃²	—	6	1	no	—
104	C₂²×C₃×D₃	—	2	4	no	—
105	C₃²×Q₈	—	2	8	no	—
106	C₃²×Q₈	—	2	1	no	—
107	C₃²×Q₈	—	2	1	no	—
108	C₃²×Q₈	—	2	4	no	—
109	C₃²×Q₈	—	2	4	no	—
110	C₃²×Q₈	—	4	4	no	—
111	C₃²×Q₈	—	12	2	no	—
112	C₃²×D₄	—	6	1	no	—
113	C₃²×D₄	—	6	1	no	—
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	—	2	1	no	—
116	C₂×C₃²×C₄	—	2	4	no	—
117	C₂×C₃²×C₄	—	2	4	no	—
118	C₂×C₃²×C₄	—	2	4	no	—
119	C₂×C₃²×C₄	—	3	1	no	—
120	C₂×C₃²×C₄	—	3	1	no	—
121	C₂×C₃²×C₄	—	4	4	no	—
122	C₂×C₃²×C₄	—	4	4	no	—
123	C₂×C₃²×C₄	—	6	1	no	—
124	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
126	C₂×C₃×Dic₃	—	2	4	no	—
127	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
128	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
129	C₃×D₁₂	—	2	4	no	—
130	C₃×D₁₂	—	2	4	no	—
131	C₃×Dic₆	—	4	8	no	—
132	C₃²×C₈	—	2	1	no	—
133	C₃²×C₈	—	2	4	no	—
134	C₃²×C₈	—	4	4	no	—
135	C₃²×C₈	—	6	1	no	—
136	C₃×C₃⋊C₈	—	2	4	no	—
137	#489	—	3	1	no	Sylow
138	C₂×C₃³	#124	1	1	yes	characteristic
139	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₃³	#137	1	1	yes	characteristic
141	C₃²×D₃	—	8	1	no	—
142	C₂×C₃×Q₈	—	2	4	no	—
143	C₂×C₃×Q₈	—	6	4	no	—
144	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
145	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
146	C₂×C₃×D₄	—	3	4	no	—
147	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
148	C₂×Dic₆	—	2	1	no	—
149	C₂×C₃×C₈	—	2	4	no	—
150	C₂×C₃×C₈	—	3	4	no	—
151	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
152	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	3	4	no	—
155	C₃×C₄⋊C₄	—	6	4	no	—
156	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
157	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
158	C₃×C₂²⋊C₄	—	6	4	no	—
159	C₃×C₂²⋊C₄	—	6	4	no	—
160	C₃×C₄²	—	3	4	no	—
161	(C₂×D₃)⋊C₄	—	2	1	no	—
162	(C₂×D₃)⋊C₄	—	2	1	no	—
163	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
164	Dic₃⋊C₄	—	2	1	no	—
165	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
167	C₂²×C₃²	—	2	4	no	—
168	C₂²×C₃²	—	12	1	no	—
169	C₂²×C₃²	—	12	1	no	—
170	C₂²×C₃²	—	12	1	no	—
171	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
172	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
173	C₂×C₃×D₃	—	4	4	no	—
174	C₂×C₃×D₃	—	4	4	no	—
175	C₂×C₃×D₃	—	4	4	no	—
176	C₃²×C₄	—	2	4	no	—
177	C₃²×C₄	—	2	4	no	—
178	C₃²×C₄	—	4	8	no	—
179	C₃²×C₄	—	4	1	no	—
180	C₃²×C₄	—	4	1	no	—
181	C₃²×C₄	—	4	4	no	—
182	C₃²×C₄	—	4	4	no	—
183	C₃²×C₄	—	6	1	no	—
184	C₃²×C₄	—	6	1	no	—
185	C₃²×C₄	—	8	4	no	—
186	C₃²×C₄	—	12	1	no	—
187	C₃²×C₄	#235	1	1	yes	characteristic
188	C₃²×C₄	#237	1	1	yes	characteristic
189	C₃²×C₄	C₃×#41	1	4	yes	—
190	C₃²×C₄	C₃×#41	1	4	yes	—
191	C₃×Dic₃	—	2	4	no	—
192	C₃×Dic₃	—	2	4	no	—
193	C₃×Dic₃	—	4	4	no	—
194	C₄⋊Q₈	—	3	1	no	—
195	#124	—	3	1	no	—
196	#103	—	3	1	no	—
197	#103	C₃²×D₃	1	1	yes	characteristic, complemented
198	#102	—	3	1	no	—
199	#102	—	3	1	no	—
200	#97	—	3	1	no	—
201	C₃³	#489	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	6	4	no	—
203	C₂²×D₃	—	2	1	no	—
204	C₃×Q₈	—	2	8	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₃×Q₈	—	2	4	no	—
207	C₃×Q₈	—	2	1	no	—
208	C₃×Q₈	—	2	1	no	—
209	C₃×Q₈	—	4	4	no	—
210	C₃×Q₈	—	12	8	no	—
211	C₃×D₄	—	6	4	no	—
212	C₃×D₄	—	6	4	no	—
213	C₂×C₃×C₄	—	2	4	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	3	4	no	—
219	C₂×C₃×C₄	—	3	4	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	—	4	4	no	—
222	C₂×C₃×C₄	—	6	4	no	—
223	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
224	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
225	C₂×Dic₃	—	2	1	no	—
226	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
227	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
228	D₁₂	—	2	1	no	—
229	D₁₂	—	2	1	no	—
230	Dic₆	—	4	2	no	—
231	C₃×C₈	—	2	4	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	4	4	no	—
234	C₃×C₈	—	6	4	no	—
235	C₃⋊C₈	—	2	1	no	—
236	C₂×C₃²	—	2	4	no	—
237	C₂×C₃²	—	2	4	no	—
238	C₂×C₃²	—	2	4	no	—
239	C₂×C₃²	—	24	1	no	—
240	C₂×C₃²	#885	1	1	yes	characteristic
241	C₂×C₃²	#914	1	1	yes	characteristic
242	C₂×C₃²	#918	1	1	yes	characteristic
243	C₂×C₃²	C₃×#124	1	4	yes	—
244	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
245	C₂×C₃²	C₃×#137	1	4	yes	—
246	C₃×D₃	—	8	4	no	—
247	C₂×Q₈	—	6	1	no	—
248	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
249	C₂×D₄	—	3	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
255	C₂²⋊C₄	—	6	1	no	—
256	C₂²⋊C₄	—	6	1	no	—
257	C₄²	—	3	1	no	—
258	C₂²×C₃	—	2	4	no	—
259	C₂²×C₃	—	12	4	no	—
260	C₂²×C₃	—	12	4	no	—
261	C₂²×C₃	—	12	4	no	—
262	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
263	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
264	C₂×D₃	—	4	1	no	—
265	C₂×D₃	—	4	1	no	—
266	C₂×D₃	—	4	1	no	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	4	no	—
269	C₃×C₄	—	4	8	no	—
270	C₃×C₄	—	4	4	no	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	6	4	no	—
275	C₃×C₄	—	6	4	no	—
276	C₃×C₄	—	8	4	no	—
277	C₃×C₄	—	12	4	no	—
278	C₃×C₄	C₃×#235	1	4	yes	—
279	C₃×C₄	C₃×#237	1	4	yes	—
280	C₃×C₄	C₃²×#41	1	1	yes	characteristic
281	C₃×C₄	C₃²×#41	1	1	yes	characteristic
282	Dic₃	—	2	1	no	—
283	Dic₃	—	2	1	no	—
284	Dic₃	—	4	1	no	—
285	C₃²	—	2	4	no	—
286	C₃²	#4794	1	1	yes	characteristic, direct product factor
287	C₃²	C₃×#489	1	4	yes	complemented
288	C₂³	—	6	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	12	2	no	—
292	D₄	—	6	1	no	—
293	D₄	—	6	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	6	1	no	—
302	C₂×C₃	—	2	4	no	—
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	—	2	4	no	—
305	C₂×C₃	—	24	4	no	—
306	C₂×C₃	C₃×#885	1	4	yes	—
307	C₂×C₃	C₃×#914	1	4	yes	—
308	C₂×C₃	C₃×#918	1	4	yes	—
309	C₂×C₃	C₃²×#124	1	1	yes	characteristic
310	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₂×C₃	C₃²×#137	1	1	yes	characteristic
312	D₃	—	8	1	no	—
313	C₂²	—	12	1	no	—
314	C₂²	—	12	1	no	—
315	C₂²	—	12	1	no	—
316	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	6	1	no	—
320	C₄	—	6	1	no	—
321	C₄	—	12	1	no	—
322	C₄	C₃²×#235	1	1	yes	characteristic
323	C₄	C₃²×#237	1	1	yes	characteristic
324	C₃	—	2	4	no	—
325	C₃	C₃×#4794	1	4	yes	direct product factor
326	C₃	C₃²×#489	1	1	yes	characteristic, complemented
327	C₂	—	24	1	no	—
328	C₂	C₃²×#885	1	1	yes	characteristic
329	C₂	C₃²×#914	1	1	yes	characteristic
330	C₂	C₃²×#918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#103	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×#103	D₃
hypocenter	C₃	C₃²×#489
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#103

Complements

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

N	H ≅ G/N	type
2. C₃²×#888	C₂	semidirect
3. C₃³×#103	C₂	semidirect
4. C₃²×#835	C₂	semidirect
6. C₃²×#815	C₂	semidirect
9. C₃×#4794	C₃	direct (N × H)
24. C₃×#888	C₂×C₃	semidirect
27. C₃²×#103	D₃	semidirect
28. C₃²×#103	C₂×C₃	semidirect
32. C₃×#835	C₂×C₃	semidirect
34. C₃×#815	C₂×C₃	semidirect
51. #4794	C₃²	direct (N × H)
91. #888	C₂×C₃²	semidirect
94. C₃×#103	C₃×D₃	semidirect
95. C₃×#103	C₂×C₃²	semidirect
99. #835	C₂×C₃²	semidirect
101. #815	C₂×C₃²	semidirect
197. #103	C₃²×D₃	semidirect
201. C₃³	#489	semidirect
286. C₃²	#4794	direct (N × H)
287. C₃²	C₃×#489	semidirect
325. C₃	C₃×#4794	direct (N × H)
326. C₃	C₃²×#489	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#947	2
C₃²×#888	2
C₃³×#103	2
C₃²×#835	2
C₃²×#821	2
C₃²×#815	2
C₃²×#809	2
C₃²×#489	3
C₃×#4794	3

Subgroups — C32×#4794 #13162229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4794 #13162229