Subgroups — C₃×#10592830 #13163168

Order 1728.

1412 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63648	C₂	1	4	yes	maximal, complemented
5	#10592830	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃×#5131	—	3	4	no	maximal
7	C₂³×C₃×C₃⋊D₃	—	2	1	no	—
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₄	1	4	yes	complemented
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
13	C₃³×M₁₆	—	2	2	no	—
14	C₂×C₃³×C₈	C₂²	1	2	yes	complemented
15	C₃×#3757	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	complemented
17	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
18	C₂²×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×D₁₂	—	3	4	no	—
20	C₂×C₃²×M₁₆	—	2	4	no	—
21	C₂×C₃²×M₁₆	D₃	1	4	yes	complemented
22	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	#64135	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×#926	—	3	4	no	—
25	#63648	C₂×C₃	1	4	yes	complemented
26	C₃×#821	—	3	16	no	—
27	C₂³×C₃³	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
29	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
30	C₂²×C₃×C₃⋊D₃	—	4	4	no	—
31	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	D₄	1	2	yes	—
34	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
36	C₃×C₄⋊(C₃⋊D₃)	—	2	4	no	—
37	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
38	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	4	yes	—
39	C₃³×C₈	—	2	2	no	—
40	C₃×#418	—	9	1	no	—
41	#5131	—	3	4	no	—
42	C₂³×C₃⋊D₃	—	2	1	no	—
43	C₂³×C₃×D₃	—	6	4	no	—
44	C₂²×C₃²×C₄	—	2	4	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
46	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
48	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
49	C₂×C₄⋊(C₃⋊D₃)	C₃×C₄	1	4	yes	complemented
50	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
51	C₂×C₃×D₁₂	—	3	16	no	—
52	C₂×C₃×D₁₂	—	3	8	no	—
53	C₂×C₃×D₁₂	—	6	4	no	—
54	C₂×C₃×D₁₂	—	6	8	no	—
55	C₃²×M₁₆	—	2	16	no	—
56	C₃²×M₁₆	—	2	8	no	—
57	C₃²×M₁₆	—	2	2	no	—
58	C₂×C₃²×C₈	—	2	8	no	—
59	C₂×C₃²×C₈	C₂×D₃	1	8	yes	complemented
60	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	complemented
61	#3757	—	2	1	no	—
62	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	complemented
63	C₄×C₃⋊Dic₃	—	2	1	no	—
64	C₃×#217	—	6	4	no	—
65	C₃×C₄⋊Dic₃	—	3	8	no	—
66	C₃×C₄×Dic₃	—	6	4	no	—
67	C₂²×C₃³	—	2	2	no	—
68	C₂²×C₃³	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₃⋊D₃	—	2	4	no	—
71	C₂×C₃×C₃⋊D₃	—	4	8	no	—
72	C₂×C₃×C₃⋊D₃	—	4	4	no	—
73	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
74	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
76	C₃×C₃⋊Dic₃	—	4	2	no	—
77	C₂²×C₃×D₄	—	9	1	no	—
78	C₂²×D₁₂	—	3	4	no	—
79	C₂×C₃×M₁₆	—	2	4	no	—
80	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	complemented
81	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
82	C₃×#117	—	9	1	no	—
83	#926	—	3	4	no	—
84	C₃×#102	—	9	4	no	—
85	#821	—	3	16	no	—
86	C₂³×C₃²	—	2	4	no	—
87	C₂³×C₃²	D₁₂	1	4	yes	—
88	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₃⋊D₃	—	2	4	no	—
90	C₂²×C₃⋊D₃	—	2	2	no	—
91	C₂²×C₃⋊D₃	—	4	4	no	—
92	C₂²×C₃×D₃	—	6	16	no	—
93	C₂²×C₃×D₃	—	6	8	no	—
94	C₂²×C₃×D₃	—	12	16	no	—
95	C₂×C₃²×C₄	—	2	8	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	2	8	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
100	C₂×C₃²×C₄	D₁₂	1	4	yes	—
101	C₂×C₃²×C₄	#67	1	8	yes	—
102	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
105	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
106	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₃⋊Dic₃	—	2	2	no	—
108	C₄⋊(C₃⋊D₃)	—	2	4	no	—
109	C₄⋊(C₃⋊D₃)	—	2	2	no	—
110	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	4	yes	—
111	C₂×C₃×Dic₃	—	6	8	no	—
112	C₃×D₁₂	—	3	16	no	—
113	C₃×D₁₂	—	6	16	no	—
114	C₃×D₁₂	—	6	8	no	—
115	C₃²×C₈	—	2	16	no	—
116	C₃²×C₈	—	2	8	no	—
117	C₃²×C₈	—	2	2	no	—
118	#418	—	9	1	no	Sylow
119	C₂×C₃³	—	2	2	no	—
120	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₃³	AGL(1,ℤ/8)	1	2	yes	—
122	C₃×C₃⋊D₃	—	4	4	no	—
123	C₂⁴×C₃	—	18	1	no	—
124	C₂³×D₃	—	6	4	no	—
125	C₂×C₃×D₄	—	9	4	no	—
126	C₂×C₃×D₄	—	9	2	no	—
127	C₂×C₃×D₄	—	18	1	no	—
128	C₂×C₃×D₄	—	18	2	no	—
129	C₂²×C₃×C₄	—	2	4	no	—
130	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
131	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
132	C₂×D₁₂	—	3	16	no	—
133	C₂×D₁₂	—	3	8	no	—
134	C₂×D₁₂	—	6	4	no	—
135	C₂×D₁₂	—	6	8	no	—
136	C₃×M₁₆	—	2	16	no	—
137	C₃×M₁₆	—	2	8	no	—
138	C₃×M₁₆	—	2	2	no	—
139	C₂×C₃×C₈	—	2	8	no	—
140	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	complemented
141	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	complemented
142	C₃×C₄⋊C₄	—	9	2	no	—
143	C₃×C₂²⋊C₄	—	18	1	no	—
144	C₃×C₄²	—	18	1	no	—
145	#217	—	6	4	no	—
146	C₄⋊Dic₃	—	3	8	no	—
147	C₄×Dic₃	—	6	4	no	—
148	C₂²×C₃²	—	2	2	no	—
149	C₂²×C₃²	—	2	8	no	—
150	C₂²×C₃²	—	2	4	no	—
151	C₂²×C₃²	—	2	8	no	—
152	C₂²×C₃²	—	4	8	no	—
153	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
154	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
155	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
156	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₃⋊D₃	—	2	4	no	—
158	C₂×C₃⋊D₃	—	4	8	no	—
159	C₂×C₃⋊D₃	—	4	4	no	—
160	C₂×C₃×D₃	—	6	16	no	—
161	C₂×C₃×D₃	—	12	32	no	—
162	C₂×C₃×D₃	—	12	16	no	—
163	C₃²×C₄	—	2	8	no	—
164	C₃²×C₄	—	2	4	no	—
165	C₃²×C₄	—	2	4	no	—
166	C₃²×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
167	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
168	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
169	C₃²×C₄	C₂×#67	1	4	yes	—
170	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
171	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
172	C₃⋊Dic₃	—	4	2	no	—
173	C₃×Dic₃	—	12	8	no	—
174	C₂²×D₄	—	9	1	no	—
175	C₂×M₁₆	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
176	#117	—	9	1	no	—
177	#102	—	9	4	no	—
178	C₃³	#418	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	4	no	—
180	C₂³×C₃	—	18	4	no	—
181	C₂³×C₃	—	18	2	no	—
182	C₂³×C₃	—	36	4	no	—
183	C₂³×C₃	C₃×D₁₂	1	4	yes	—
184	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
185	C₂²×D₃	—	6	16	no	—
186	C₂²×D₃	—	6	8	no	—
187	C₂²×D₃	—	12	16	no	—
188	C₃×D₄	—	9	4	no	—
189	C₃×D₄	—	18	4	no	—
190	C₃×D₄	—	18	2	no	—
191	C₂×C₃×C₄	—	2	8	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	8	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	18	2	no	—
196	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
197	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
198	C₂×C₃×C₄	C₃×#67	1	8	yes	—
199	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
200	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
201	C₂×C₃×C₄	#638	1	2	yes	—
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
203	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
204	C₂×Dic₃	—	6	8	no	—
205	D₁₂	—	3	16	no	—
206	D₁₂	—	6	16	no	—
207	D₁₂	—	6	8	no	—
208	C₃×C₈	—	2	16	no	—
209	C₃×C₈	—	2	8	no	—
210	C₃×C₈	—	2	2	no	—
211	C₂×C₃²	—	2	2	no	—
212	C₂×C₃²	—	2	8	no	—
213	C₂×C₃²	—	2	4	no	—
214	C₂×C₃²	—	2	8	no	—
215	C₂×C₃²	—	4	8	no	—
216	C₂×C₃²	#908	1	8	yes	—
217	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
218	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
220	C₃⋊D₃	—	4	4	no	—
221	C₃×D₃	—	12	16	no	—
222	C₂⁴	—	18	1	no	—
223	C₂×D₄	—	9	4	no	—
224	C₂×D₄	—	9	2	no	—
225	C₂×D₄	—	18	1	no	—
226	C₂×D₄	—	18	2	no	—
227	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
230	C₄⋊C₄	—	9	2	no	—
231	C₂²⋊C₄	—	18	1	no	—
232	C₄²	—	18	1	no	—
233	C₂²×C₃	—	2	8	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	8	no	—
236	C₂²×C₃	—	2	4	no	—
237	C₂²×C₃	—	4	8	no	—
238	C₂²×C₃	—	18	4	no	—
239	C₂²×C₃	—	36	8	no	—
240	C₂²×C₃	—	36	4	no	—
241	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
242	C₂²×C₃	#3752	1	2	yes	—
243	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
244	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
245	C₂×D₃	—	6	16	no	—
246	C₂×D₃	—	12	32	no	—
247	C₂×D₃	—	12	16	no	—
248	C₃×C₄	—	2	8	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	36	2	no	—
252	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
253	C₃×C₄	#3752	1	2	yes	—
254	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
255	C₃×C₄	C₂×C₃×#67	1	4	yes	—
256	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
257	C₃×C₄	C₂×#638	1	1	yes	characteristic
258	Dic₃	—	12	8	no	—
259	C₃²	—	2	4	no	—
260	C₃²	#5131	1	4	yes	complemented
261	C₃²	C₃×#418	1	1	yes	characteristic, complemented
262	C₂³	—	18	4	no	—
263	C₂³	—	18	2	no	—
264	C₂³	—	36	4	no	—
265	C₂³	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
266	D₄	—	9	4	no	—
267	D₄	—	18	4	no	—
268	D₄	—	18	2	no	—
269	C₂×C₄	—	18	2	no	—
270	C₂×C₄	C₃×C₄×C₃⋊D₃	1	2	yes	—
271	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
272	C₂×C₄	C₃×#638	1	2	yes	—
273	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	8	no	—
277	C₂×C₃	—	2	8	no	—
278	C₂×C₃	—	2	4	no	—
279	C₂×C₃	—	4	8	no	—
280	C₂×C₃	—	36	4	no	—
281	C₂×C₃	C₃×#908	1	8	yes	—
282	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	4	yes	—
283	C₂×C₃	#64117	1	2	yes	—
284	C₂×C₃	C₂×#3752	1	1	yes	characteristic
285	D₃	—	12	16	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	18	4	no	—
288	C₂²	—	36	8	no	—
289	C₂²	—	36	4	no	—
290	C₂²	C₃×#3752	1	2	yes	—
291	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
292	C₄	—	36	2	no	—
293	C₄	C₃×#3752	1	2	yes	—
294	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
295	C₄	C₂×C₃×#638	1	1	yes	characteristic
296	C₃	—	2	4	no	—
297	C₃	C₃×#5131	1	4	yes	complemented
298	C₃	#10592830	1	1	yes	characteristic, direct product factor
299	C₂	—	2	2	no	—
300	C₂	—	36	4	no	—
301	C₂	C₃×#64117	1	2	yes	—
302	C₂	C₂×C₃×#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂²×C₃×C₄⋊(C₃⋊D₃)	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	C₃⋊D₃
hypocenter	C₃²	C₃×#418
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×M₁₆	C₂	semidirect
3. C₃×#64135	C₂	semidirect
4. C₃×#63648	C₂	semidirect
5. #10592830	C₃	direct (N × H)
11. C₂×C₃×C₄⋊(C₃⋊D₃)	C₄	semidirect
14. C₂×C₃³×C₈	C₂²	semidirect
16. C₃×C₄⋊(C₃⋊Dic₃)	C₂²	semidirect
21. C₂×C₃²×M₁₆	D₃	semidirect
22. C₂×C₃²×M₁₆	C₂×C₃	semidirect
23. #64135	C₂×C₃	semidirect
25. #63648	C₂×C₃	semidirect
49. C₂×C₄⋊(C₃⋊D₃)	C₃×C₄	semidirect
59. C₂×C₃²×C₈	C₂×D₃	semidirect
60. C₂×C₃²×C₈	C₂²×C₃	semidirect
62. C₄⋊(C₃⋊Dic₃)	C₂²×C₃	semidirect
80. C₂×C₃×M₁₆	C₃×D₃	semidirect
81. C₂×C₃×M₁₆	C₃⋊D₃	semidirect
140. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
141. C₂×C₃×C₈	C₂×C₃⋊D₃	semidirect
175. C₂×M₁₆	C₃×C₃⋊D₃	semidirect
178. C₃³	#418	semidirect
229. C₂×C₈	C₂×C₃×C₃⋊D₃	semidirect
260. C₃²	#5131	semidirect
261. C₃²	C₃×#418	semidirect
297. C₃	C₃×#5131	semidirect
298. C₃	#10592830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄⋊(C₃⋊D₃)	2
C₂×C₃³×M₁₆	2
C₃×#64135	2
C₃×#63648	2
#10592830	3
C₃×#5131	3

Subgroups — C3×#10592830 #13163168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592830 #13163168