Subgroups — C₃²×#4841 #13162279

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#204	C₂	1	2	yes	maximal
2	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#816	C₂	1	2	yes	maximal, complemented
6	C₃²×#461	—	3	1	no	maximal
7	C₃×#4841	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×D₃	—	2	1	no	—
10	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
11	C₂×C₃³×C₈	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
14	C₃²×C₄⋊Dic₃	—	2	1	no	—
15	C₃²×C₄⋊Dic₃	—	2	1	no	—
16	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₃²×#204	—	2	4	no	—
18	C₂×C₃²×SD₁₆	—	3	2	no	—
19	C₃²×C₄⋊Q₈	—	3	1	no	—
20	C₃²×C₄×D₄	—	3	1	no	—
21	C₂×C₃×#204	C₂×C₃	1	8	yes	—
22	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
23	C₃×#869	C₂×C₃	1	4	yes	complemented
24	C₃²×C₄⋊C₈	—	2	4	no	—
25	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
27	C₃²×#103	—	3	2	no	—
28	C₃×#816	C₂×C₃	1	8	yes	complemented
29	C₂²×C₃²×D₃	—	2	1	no	—
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	—	2	2	no	—
34	C₂×C₃²×Dic₃	—	2	1	no	—
35	C₃²×D₁₂	—	2	1	no	—
36	C₃²×D₁₂	D₄	1	2	yes	—
37	C₃²×C₄×D₃	—	4	2	no	—
38	C₃²×Dic₆	—	2	2	no	—
39	C₃²×Dic₆	—	2	2	no	—
40	C₃³×C₈	—	2	2	no	—
41	C₃×#461	—	3	4	no	—
42	#4841	C₃²	1	1	yes	characteristic, direct product factor
43	C₂×C₃²×Q₈	—	3	2	no	—
44	C₂×C₃²×D₄	—	3	1	no	—
45	C₂²×C₃²×C₄	—	6	1	no	—
46	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
47	C₂×C₃×C₄×D₃	—	2	4	no	—
48	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
49	C₃²×SD₁₆	—	6	4	no	—
50	C₂×C₃²×C₈	—	2	8	no	—
51	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
52	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
53	C₃²×C₄⋊C₄	—	3	1	no	—
54	C₃²×C₄⋊C₄	—	6	1	no	—
55	C₃²×C₄⋊C₄	—	6	1	no	—
56	C₃²×C₂²⋊C₄	—	6	1	no	—
57	C₃²×C₄²	—	2	4	no	—
58	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₃²×C₄²	C₂²×C₃	1	4	yes	—
60	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
61	C₃×C₄⋊Dic₃	—	2	4	no	—
62	C₃×C₄⋊Dic₃	—	2	4	no	—
63	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
64	C₃×#204	—	2	16	no	—
65	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×D₃	—	2	2	no	—
67	C₂×C₃²×D₃	—	4	1	no	—
68	C₂×C₃²×D₃	—	4	1	no	—
69	C₃³×C₄	—	2	1	no	—
70	C₃³×C₄	SD₁₆	1	2	yes	complemented
71	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃³×C₄	#41	1	1	yes	characteristic
73	C₃²×Dic₃	—	4	2	no	—
74	C₃²×Dic₃	—	4	1	no	—
75	C₂×C₃×SD₁₆	—	3	8	no	—
76	C₃×C₄⋊Q₈	—	3	4	no	—
77	C₃×C₄×D₄	—	3	4	no	—
78	C₂×#204	C₂×C₃²	1	2	yes	—
79	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
80	#869	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₃×C₄⋊C₈	—	2	4	no	—
82	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
83	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
84	C₃×#103	—	3	8	no	—
85	#816	C₂×C₃²	1	2	yes	complemented
86	C₂³×C₃²	—	6	1	no	—
87	C₂²×C₃×D₃	—	2	4	no	—
88	C₃²×Q₈	—	6	2	no	—
89	C₃²×Q₈	—	6	2	no	—
90	C₃²×D₄	—	3	2	no	—
91	C₃²×D₄	—	6	1	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	6	1	no	—
96	C₂×C₃²×C₄	—	6	2	no	—
97	C₂×C₃²×C₄	—	12	2	no	—
98	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
99	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
101	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
102	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
104	C₂×C₃×Dic₃	—	2	8	no	—
105	C₂×C₃×Dic₃	—	2	4	no	—
106	C₃×D₁₂	—	2	4	no	—
107	C₃×D₁₂	C₃×D₄	1	8	yes	—
108	C₃×C₄×D₃	—	4	8	no	—
109	C₃×Dic₆	—	2	8	no	—
110	C₃×Dic₆	—	2	8	no	—
111	C₃²×C₈	—	2	2	no	—
112	C₃²×C₈	—	2	8	no	—
113	C₃²×C₈	—	4	8	no	—
114	#461	—	3	1	no	Sylow
115	C₂×C₃³	#121	1	1	yes	characteristic
116	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₃³	#137	1	1	yes	characteristic
118	C₃²×D₃	—	4	2	no	—
119	C₂×C₃×Q₈	—	3	8	no	—
120	C₂×C₃×D₄	—	3	4	no	—
121	C₂²×C₃×C₄	—	6	4	no	—
122	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
123	C₂×C₄×D₃	—	2	1	no	—
124	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
125	C₃×SD₁₆	—	6	16	no	—
126	C₂×C₃×C₈	—	2	8	no	—
127	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
128	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
129	C₃×C₄⋊C₄	—	3	4	no	—
130	C₃×C₄⋊C₄	—	6	4	no	—
131	C₃×C₄⋊C₄	—	6	4	no	—
132	C₃×C₂²⋊C₄	—	6	4	no	—
133	C₃×C₄²	—	2	4	no	—
134	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
135	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	—	2	1	no	—
137	C₄⋊Dic₃	—	2	1	no	—
138	C₄⋊Dic₃	—	2	1	no	—
139	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
140	#204	—	2	4	no	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	—	6	2	no	—
143	C₂²×C₃²	—	12	1	no	—
144	C₂²×C₃²	—	12	1	no	—
145	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
146	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
147	C₂×C₃×D₃	—	2	8	no	—
148	C₂×C₃×D₃	—	4	4	no	—
149	C₂×C₃×D₃	—	4	4	no	—
150	C₃²×C₄	—	2	1	no	—
151	C₃²×C₄	—	2	8	no	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	—	2	4	no	—
154	C₃²×C₄	—	2	4	no	—
155	C₃²×C₄	—	4	4	no	—
156	C₃²×C₄	—	12	2	no	—
157	C₃²×C₄	—	12	1	no	—
158	C₃²×C₄	#204	1	2	yes	complemented
159	C₃²×C₄	C₃×SD₁₆	1	8	yes	complemented
160	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
161	C₃²×C₄	#239	1	1	yes	characteristic
162	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
163	C₃²×C₄	C₃×#41	1	4	yes	—
164	C₃×Dic₃	—	4	8	no	—
165	C₃×Dic₃	—	4	4	no	—
166	C₂×SD₁₆	—	3	2	no	—
167	C₄⋊Q₈	—	3	1	no	—
168	C₄×D₄	—	3	1	no	—
169	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
170	#103	—	3	2	no	—
171	C₃³	#461	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	6	4	no	—
173	C₂²×D₃	—	2	1	no	—
174	C₃×Q₈	—	6	8	no	—
175	C₃×Q₈	—	6	8	no	—
176	C₃×D₄	—	3	8	no	—
177	C₃×D₄	—	6	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	6	4	no	—
182	C₂×C₃×C₄	—	6	8	no	—
183	C₂×C₃×C₄	—	12	8	no	—
184	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
185	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
186	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
189	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
190	C₂×Dic₃	—	2	2	no	—
191	C₂×Dic₃	—	2	1	no	—
192	D₁₂	—	2	1	no	—
193	D₁₂	C₃²×D₄	1	2	yes	—
194	C₄×D₃	—	4	2	no	—
195	Dic₆	—	2	2	no	—
196	Dic₆	—	2	2	no	—
197	C₃×C₈	—	2	8	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	4	8	no	—
200	C₂×C₃²	—	2	4	no	—
201	C₂×C₃²	—	2	4	no	—
202	C₂×C₃²	—	2	4	no	—
203	C₂×C₃²	—	12	2	no	—
204	C₂×C₃²	#895	1	1	yes	characteristic
205	C₂×C₃²	C₂×#204	1	1	yes	characteristic
206	C₂×C₃²	#909	1	1	yes	characteristic
207	C₂×C₃²	C₃×#121	1	4	yes	—
208	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
209	C₂×C₃²	C₃×#137	1	4	yes	—
210	C₃×D₃	—	4	8	no	—
211	C₂×Q₈	—	3	2	no	—
212	C₂×D₄	—	3	1	no	—
213	C₂²×C₄	—	6	1	no	—
214	SD₁₆	—	6	4	no	—
215	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	6	1	no	—
218	C₄⋊C₄	—	6	1	no	—
219	C₂²⋊C₄	—	6	1	no	—
220	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	—	6	8	no	—
223	C₂²×C₃	—	12	4	no	—
224	C₂²×C₃	—	12	4	no	—
225	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
226	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
227	C₂×D₃	—	2	2	no	—
228	C₂×D₃	—	4	1	no	—
229	C₂×D₃	—	4	1	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	4	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	8	no	—
235	C₃×C₄	—	4	4	no	—
236	C₃×C₄	—	12	8	no	—
237	C₃×C₄	—	12	4	no	—
238	C₃×C₄	C₃×#204	1	8	yes	complemented
239	C₃×C₄	C₃²×SD₁₆	1	2	yes	complemented
240	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
241	C₃×C₄	C₃×#239	1	4	yes	—
242	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
243	C₃×C₄	C₃²×#41	1	1	yes	characteristic
244	Dic₃	—	4	2	no	—
245	Dic₃	—	4	1	no	—
246	C₃²	—	2	4	no	—
247	C₃²	#4841	1	1	yes	characteristic, direct product factor
248	C₃²	C₃×#461	1	4	yes	complemented
249	C₂³	—	6	1	no	—
250	Q₈	—	6	2	no	—
251	Q₈	—	6	2	no	—
252	D₄	—	3	2	no	—
253	D₄	—	6	1	no	—
254	C₂×C₄	—	6	1	no	—
255	C₂×C₄	—	6	2	no	—
256	C₂×C₄	—	12	2	no	—
257	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
258	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₂×C₃	—	2	4	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	4	no	—
264	C₂×C₃	—	12	8	no	—
265	C₂×C₃	C₃×#895	1	4	yes	—
266	C₂×C₃	C₂×C₃×#204	1	4	yes	—
267	C₂×C₃	C₃×#909	1	4	yes	—
268	C₂×C₃	C₃²×#121	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
270	C₂×C₃	C₃²×#137	1	1	yes	characteristic
271	D₃	—	4	2	no	—
272	C₂²	—	6	2	no	—
273	C₂²	—	12	1	no	—
274	C₂²	—	12	1	no	—
275	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	12	2	no	—
278	C₄	—	12	1	no	—
279	C₄	C₃²×#204	1	2	yes	complemented
280	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
281	C₄	C₃²×#239	1	1	yes	characteristic
282	C₃	—	2	4	no	—
283	C₃	C₃×#4841	1	4	yes	direct product factor
284	C₃	C₃²×#461	1	1	yes	characteristic, complemented
285	C₂	—	12	2	no	—
286	C₂	C₃²×#895	1	1	yes	characteristic
287	C₂	C₂×C₃²×#204	1	1	yes	characteristic
288	C₂	C₃²×#909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#461
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₃²×#869	C₂	semidirect
4. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃²×#816	C₂	semidirect
7. C₃×#4841	C₃	direct (N × H)
23. C₃×#869	C₂×C₃	semidirect
25. C₃²×C₄⋊C₈	D₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
28. C₃×#816	C₂×C₃	semidirect
42. #4841	C₃²	direct (N × H)
70. C₃³×C₄	SD₁₆	semidirect
80. #869	C₂×C₃²	semidirect
82. C₃×C₄⋊C₈	C₃×D₃	semidirect
83. C₃×C₄⋊C₈	C₂×C₃²	semidirect
85. #816	C₂×C₃²	semidirect
158. C₃²×C₄	#204	semidirect
159. C₃²×C₄	C₃×SD₁₆	semidirect
169. C₄⋊C₈	C₃²×D₃	semidirect
171. C₃³	#461	semidirect
238. C₃×C₄	C₃×#204	semidirect
239. C₃×C₄	C₃²×SD₁₆	semidirect
247. C₃²	#4841	direct (N × H)
248. C₃²	C₃×#461	semidirect
279. C₄	C₃²×#204	semidirect
283. C₃	C₃×#4841	direct (N × H)
284. C₃	C₃²×#461	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#204	2
C₃²×C₄×D₁₂	2
C₃²×#869	2
C₃³×C₄⋊C₈	2
C₃²×#816	2
C₃²×#461	3
C₃×#4841	3

Subgroups — C32×#4841 #13162279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4841 #13162279