Subgroups — C₃²×#5116 #13162552

Order 1728.

988 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#924	C₂	1	2	yes	maximal
3	C₂×C₃²×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#818	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#816	C₂	1	2	yes	maximal, complemented
6	C₃²×#467	—	3	1	no	maximal
7	C₃×#5116	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	—	2	2	no	—
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃²×#217	—	2	2	no	—
14	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₃²×Dic₃⋊C₄	—	2	2	no	—
16	C₃²×Dic₁₂	—	2	2	no	—
17	C₂×C₃²×Q₁₆	—	3	1	no	—
18	C₂²×C₃²×C₈	—	2	4	no	—
19	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
20	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	—
21	C₃²×C₂²⋊Q₈	—	3	2	no	—
22	C₃×#924	C₂×C₃	1	8	yes	—
23	C₂×C₃×Dic₁₂	C₂×C₃	1	4	yes	complemented
24	C₃²×#107	—	3	1	no	—
25	C₃²×#103	—	3	2	no	—
26	C₃×#818	C₂×C₃	1	4	yes	complemented
27	C₃×#816	C₂×C₃	1	8	yes	complemented
28	C₂³×C₃³	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	—	2	1	no	—
30	C₂×C₃³×C₄	—	2	1	no	—
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	2	no	—
35	C₃²×Dic₆	—	2	2	no	—
36	C₃²×Dic₆	—	2	2	no	—
37	C₃³×C₈	—	2	1	no	—
38	C₃³×C₈	D₄	1	2	yes	—
39	C₃×#467	—	3	4	no	—
40	#5116	C₃²	1	1	yes	characteristic, direct product factor
41	C₂×C₃²×Q₈	—	3	2	no	—
42	C₂²×C₃²×C₄	—	2	4	no	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
46	C₃²×Q₁₆	—	6	2	no	—
47	C₂×C₃²×C₈	—	2	16	no	—
48	C₂×C₃²×C₈	—	2	4	no	—
49	C₂×C₃²×C₈	—	2	4	no	—
50	C₂×C₃²×C₈	—	2	2	no	—
51	C₂×C₃²×C₈	—	2	8	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
55	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
56	C₃²×C₄⋊C₄	—	3	2	no	—
57	C₃²×C₄⋊C₄	—	6	2	no	—
58	C₃²×C₂²⋊C₄	—	6	2	no	—
59	C₃×#217	—	2	8	no	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
61	C₃×Dic₃⋊C₄	—	2	8	no	—
62	C₃×Dic₁₂	—	2	8	no	—
63	C₂²×C₃³	—	2	1	no	—
64	C₂²×C₃³	—	2	1	no	—
65	C₂²×C₃³	Q₁₆	1	2	yes	complemented
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
69	C₃³×C₄	#41	1	1	yes	characteristic
70	C₃²×Dic₃	—	4	2	no	—
71	C₃²×Dic₃	—	4	2	no	—
72	C₂×C₃×Q₁₆	—	3	4	no	—
73	C₂²×C₃×C₈	—	2	4	no	—
74	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
75	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
76	C₃×C₂²⋊Q₈	—	3	8	no	—
77	#924	C₂×C₃²	1	2	yes	—
78	C₂×Dic₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
79	C₃×#107	—	3	4	no	—
80	C₃×#103	—	3	8	no	—
81	#818	C₂×C₃²	1	1	yes	characteristic, complemented
82	#816	C₂×C₃²	1	2	yes	complemented
83	C₂³×C₃²	—	2	4	no	—
84	C₂³×C₃²	D₁₂	1	1	yes	characteristic
85	C₂³×C₃²	C₃×D₄	1	4	yes	—
86	C₃²×Q₈	—	6	2	no	—
87	C₃²×Q₈	—	6	2	no	—
88	C₂×C₃²×C₄	—	2	8	no	—
89	C₂×C₃²×C₄	—	2	8	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	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	—	2	4	no	—
96	C₂×C₃²×C₄	—	6	2	no	—
97	C₂×C₃²×C₄	—	6	2	no	—
98	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
100	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
102	C₂×C₃×Dic₃	—	2	8	no	—
103	C₂×C₃×Dic₃	—	2	8	no	—
104	C₃×Dic₆	—	2	8	no	—
105	C₃×Dic₆	—	2	8	no	—
106	C₃²×C₈	—	2	8	no	—
107	C₃²×C₈	—	2	1	no	—
108	C₃²×C₈	—	2	4	no	—
109	C₃²×C₈	—	2	8	no	—
110	C₃²×C₈	#67	1	2	yes	—
111	C₃²×C₈	C₃×D₄	1	8	yes	—
112	#467	—	3	1	no	Sylow
113	C₂×C₃³	—	2	2	no	—
114	C₂×C₃³	#121	1	1	yes	characteristic
115	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₃³	#135	1	1	yes	characteristic
117	C₂×C₃×Q₈	—	3	8	no	—
118	C₂²×C₃×C₄	—	2	4	no	—
119	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
120	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
121	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
122	C₃×Q₁₆	—	6	8	no	—
123	C₂×C₃×C₈	—	2	16	no	—
124	C₂×C₃×C₈	—	2	8	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	4	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
129	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
130	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
132	C₃×C₄⋊C₄	—	3	8	no	—
133	C₃×C₄⋊C₄	—	6	8	no	—
134	C₃×C₂²⋊C₄	—	6	8	no	—
135	#217	—	2	2	no	—
136	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
137	Dic₃⋊C₄	—	2	2	no	—
138	Dic₁₂	—	2	2	no	—
139	C₂²×C₃²	—	2	8	no	—
140	C₂²×C₃²	—	2	8	no	—
141	C₂²×C₃²	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	—	2	4	no	—
144	C₂²×C₃²	—	2	4	no	—
145	C₂²×C₃²	—	2	8	no	—
146	C₂²×C₃²	—	2	4	no	—
147	C₂²×C₃²	Dic₁₂	1	2	yes	complemented
148	C₂²×C₃²	C₃×Q₁₆	1	8	yes	complemented
149	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
150	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
151	C₃²×C₄	—	2	8	no	—
152	C₃²×C₄	—	2	1	no	—
153	C₃²×C₄	—	2	4	no	—
154	C₃²×C₄	—	2	4	no	—
155	C₃²×C₄	—	2	4	no	—
156	C₃²×C₄	—	12	2	no	—
157	C₃²×C₄	—	12	2	no	—
158	C₃²×C₄	#235	1	1	yes	characteristic
159	C₃²×C₄	C₂×#67	1	1	yes	characteristic
160	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
161	C₃²×C₄	C₃×#41	1	4	yes	—
162	C₃×Dic₃	—	4	8	no	—
163	C₃×Dic₃	—	4	8	no	—
164	C₂×Q₁₆	—	3	1	no	—
165	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
166	C₂²⋊Q₈	—	3	2	no	—
167	#107	—	3	1	no	—
168	#103	—	3	2	no	—
169	C₃³	#467	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	4	no	—
171	C₂³×C₃	C₃×D₁₂	1	4	yes	—
172	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
173	C₃×Q₈	—	6	8	no	—
174	C₃×Q₈	—	6	8	no	—
175	C₂×C₃×C₄	—	2	8	no	—
176	C₂×C₃×C₄	—	2	8	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	6	8	no	—
184	C₂×C₃×C₄	—	6	8	no	—
185	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
186	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
188	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
189	C₂×Dic₃	—	2	2	no	—
190	C₂×Dic₃	—	2	2	no	—
191	Dic₆	—	2	2	no	—
192	Dic₆	—	2	2	no	—
193	C₃×C₈	—	2	8	no	—
194	C₃×C₈	—	2	4	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	8	no	—
197	C₃×C₈	C₃×#67	1	8	yes	—
198	C₃×C₈	C₃²×D₄	1	2	yes	—
199	C₂×C₃²	—	2	16	no	—
200	C₂×C₃²	—	2	8	no	—
201	C₂×C₃²	—	2	2	no	—
202	C₂×C₃²	—	2	4	no	—
203	C₂×C₃²	—	2	4	no	—
204	C₂×C₃²	—	2	4	no	—
205	C₂×C₃²	#904	1	1	yes	characteristic
206	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
207	C₂×C₃²	#930	1	1	yes	characteristic
208	C₂×C₃²	C₃×#121	1	4	yes	—
209	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
210	C₂×C₃²	C₃×#135	1	4	yes	—
211	C₂×Q₈	—	3	2	no	—
212	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
213	Q₁₆	—	6	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	6	2	no	—
219	C₂²⋊C₄	—	6	2	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	2	8	no	—
222	C₂²×C₃	—	2	4	no	—
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	2	8	no	—
228	C₂²×C₃	C₃×Dic₁₂	1	8	yes	complemented
229	C₂²×C₃	C₃²×Q₁₆	1	2	yes	complemented
230	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
231	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
232	C₃×C₄	—	2	8	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	4	no	—
237	C₃×C₄	—	12	8	no	—
238	C₃×C₄	—	12	8	no	—
239	C₃×C₄	C₃×#235	1	4	yes	—
240	C₃×C₄	C₂×C₃×#67	1	4	yes	—
241	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
242	C₃×C₄	C₃²×#41	1	1	yes	characteristic
243	Dic₃	—	4	2	no	—
244	Dic₃	—	4	2	no	—
245	C₃²	—	2	4	no	—
246	C₃²	#5116	1	1	yes	characteristic, direct product factor
247	C₃²	C₃×#467	1	4	yes	complemented
248	C₂³	C₃²×D₁₂	1	1	yes	characteristic
249	Q₈	—	6	2	no	—
250	Q₈	—	6	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	C₃²×#67	1	2	yes	—
259	C₂×C₃	—	2	16	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	8	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	4	no	—
264	C₂×C₃	—	2	4	no	—
265	C₂×C₃	C₃×#904	1	4	yes	—
266	C₂×C₃	C₂×C₃×Dic₁₂	1	4	yes	—
267	C₂×C₃	C₃×#930	1	4	yes	—
268	C₂×C₃	C₃²×#121	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
270	C₂×C₃	C₃²×#135	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₃²×Dic₁₂	1	2	yes	complemented
274	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	12	2	no	—
277	C₄	—	12	2	no	—
278	C₄	C₃²×#235	1	1	yes	characteristic
279	C₄	C₂×C₃²×#67	1	1	yes	characteristic
280	C₃	—	2	4	no	—
281	C₃	C₃×#5116	1	4	yes	direct product factor
282	C₃	C₃²×#467	1	1	yes	characteristic, complemented
283	C₂	—	2	2	no	—
284	C₂	C₃²×#904	1	1	yes	characteristic
285	C₂	C₂×C₃²×Dic₁₂	1	1	yes	characteristic
286	C₂	C₃²×#930	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	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂³×C₃³	D₄
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₃²×#467
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₃²×Dic₁₂	C₂	semidirect
4. C₃²×#818	C₂	semidirect
5. C₃²×#816	C₂	semidirect
7. C₃×#5116	C₃	direct (N × H)
23. C₂×C₃×Dic₁₂	C₂×C₃	semidirect
26. C₃×#818	C₂×C₃	semidirect
27. C₃×#816	C₂×C₃	semidirect
40. #5116	C₃²	direct (N × H)
65. C₂²×C₃³	Q₁₆	semidirect
78. C₂×Dic₁₂	C₂×C₃²	semidirect
81. #818	C₂×C₃²	semidirect
82. #816	C₂×C₃²	semidirect
147. C₂²×C₃²	Dic₁₂	semidirect
148. C₂²×C₃²	C₃×Q₁₆	semidirect
169. C₃³	#467	semidirect
228. C₂²×C₃	C₃×Dic₁₂	semidirect
229. C₂²×C₃	C₃²×Q₁₆	semidirect
246. C₃²	#5116	direct (N × H)
247. C₃²	C₃×#467	semidirect
273. C₂²	C₃²×Dic₁₂	semidirect
281. C₃	C₃×#5116	direct (N × H)
282. C₃	C₃²×#467	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃³×C₈	2
C₃²×#924	2
C₂×C₃²×Dic₁₂	2
C₃²×#818	2
C₃²×#816	2
C₃²×#467	3
C₃×#5116	3

Subgroups — C32×#5116 #13162552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5116 #13162552