Subgroups — C₃×#10592828 #13163166

Order 1728.

1220 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	C₂×C₃×#3828	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63647	C₂	1	4	yes	maximal, complemented
5	#10592828	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃×#5129	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×#638	C₄	1	2	yes	—
9	C₃×#3828	—	2	2	no	—
10	C₃×#3828	—	2	2	no	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	2	yes	—
13	C₂×C₃×#634	C₄	1	1	yes	characteristic
14	C₃³×M₁₆	—	2	2	no	—
15	C₂×C₃³×C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₂×#3828	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#235	—	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	C₂²×C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃×C₃⋊C₈	—	3	4	no	—
25	#63647	C₂×C₃	1	4	yes	complemented
26	C₃×(C₂×D₃)⋊C₈	—	3	16	no	—
27	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	D₄	1	2	yes	—
31	C₂×C₃³×C₄	D₄	1	2	yes	—
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₃×#638	—	2	4	no	—
34	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
35	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
36	C₃×C₄×C₃⋊D₃	—	2	2	no	—
37	C₃×C₄×C₃⋊D₃	—	2	2	no	—
38	C₃×#634	—	2	2	no	—
39	C₃³×C₈	—	2	2	no	—
40	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
41	C₃×#408	—	9	1	no	—
42	#5129	—	3	4	no	—
43	C₂²×C₃²×C₄	—	2	4	no	—
44	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
45	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂×#638	C₃×C₄	1	2	yes	—
47	#3828	—	2	2	no	—
48	#3828	—	2	2	no	—
49	C₂×C₄⋊(C₃⋊D₃)	C₃×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
51	C₂×#634	C₃×C₄	1	1	yes	characteristic
52	C₂×C₃×#67	—	3	8	no	—
53	C₃×#235	—	6	8	no	—
54	C₃×#235	—	6	8	no	—
55	C₂×C₃×D₁₂	—	3	4	no	—
56	C₂×C₃×C₄×D₃	—	3	8	no	—
57	C₂×C₃×Dic₆	—	3	4	no	—
58	C₃²×M₁₆	—	2	16	no	—
59	C₃²×M₁₆	—	2	8	no	—
60	C₃²×M₁₆	—	2	2	no	—
61	C₂×C₃²×C₈	—	2	8	no	—
62	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
63	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
64	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
65	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
66	C₂×C₃×C₃⋊C₈	—	3	8	no	—
67	C₂×C₃×C₃⋊C₈	—	6	8	no	—
68	C₂²×C₃³	—	2	2	no	—
69	C₂²×C₃³	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₃⋊D₃	—	2	2	no	—
72	C₂×C₃×C₃⋊D₃	—	2	4	no	—
73	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
74	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
75	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
76	C₃×C₃⋊Dic₃	—	2	2	no	—
77	C₂×C₃×#41	—	9	1	no	—
78	C₂×#235	—	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	C₂²×C₃×C₈	—	9	1	no	—
83	C₂²×C₃⋊C₈	—	3	4	no	—
84	C₃×C₂²⋊C₈	—	9	4	no	—
85	(C₂×D₃)⋊C₈	—	3	16	no	—
86	C₂³×C₃²	—	2	4	no	—
87	C₂³×C₃²	C₄×D₃	1	4	yes	—
88	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₃×D₃	—	3	8	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	—	2	8	no	—
93	C₂×C₃²×C₄	—	2	8	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
96	C₂×C₃²×C₄	D₁₂	1	8	yes	—
97	C₂×C₃²×C₄	#67	1	8	yes	—
98	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
100	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
101	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
102	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
103	#638	—	2	4	no	—
104	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
105	C₄⋊(C₃⋊D₃)	—	2	2	no	—
106	C₄×C₃⋊D₃	—	2	2	no	—
107	C₄×C₃⋊D₃	—	2	2	no	—
108	#634	—	2	2	no	—
109	C₃×#67	—	6	16	no	—
110	C₂×C₃×Dic₃	—	3	8	no	—
111	C₃×D₁₂	—	6	8	no	—
112	C₃×C₄×D₃	—	6	8	no	—
113	C₃×C₄×D₃	—	6	8	no	—
114	C₃×Dic₆	—	6	8	no	—
115	C₃²×C₈	—	2	16	no	—
116	C₃²×C₈	—	2	8	no	—
117	C₃²×C₈	—	2	2	no	—
118	C₃⋊(C₃⋊C₈)	—	2	2	no	—
119	C₃×C₃⋊C₈	—	6	8	no	—
120	#408	—	9	1	no	Sylow
121	C₂×C₃³	—	2	2	no	—
122	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₃³	#131	1	2	yes	—
124	C₃×C₃⋊D₃	—	4	2	no	—
125	C₃×#41	—	18	2	no	—
126	C₃×#41	—	18	2	no	—
127	C₂×C₃×Q₈	—	9	1	no	—
128	C₂×C₃×D₄	—	9	2	no	—
129	C₂×C₃×D₄	—	9	1	no	—
130	C₂²×C₃×C₄	—	2	4	no	—
131	C₂²×C₃×C₄	—	9	2	no	—
132	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
133	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
134	C₂×#67	—	3	8	no	—
135	#235	—	6	8	no	—
136	#235	—	6	8	no	—
137	C₂×D₁₂	—	3	4	no	—
138	C₂×C₄×D₃	—	3	8	no	—
139	C₂×Dic₆	—	3	4	no	—
140	C₃×M₁₆	—	2	16	no	—
141	C₃×M₁₆	—	2	8	no	—
142	C₃×M₁₆	—	2	2	no	—
143	C₂×C₃×C₈	—	2	8	no	—
144	C₂×C₃×C₈	—	9	2	no	—
145	C₂×C₃×C₈	—	18	2	no	—
146	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
147	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
148	C₂×C₃⋊C₈	—	3	8	no	—
149	C₂×C₃⋊C₈	—	6	8	no	—
150	C₂²×C₃²	—	2	2	no	—
151	C₂²×C₃²	—	2	8	no	—
152	C₂²×C₃²	—	2	4	no	—
153	C₂²×C₃²	—	2	8	no	—
154	C₂²×C₃²	—	4	8	no	—
155	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
156	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
157	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
158	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₂×C₃⋊D₃	—	2	2	no	—
160	C₂×C₃⋊D₃	—	2	4	no	—
161	C₂×C₃×D₃	—	6	16	no	—
162	C₂×C₃×D₃	—	6	8	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×D₁₂	1	4	yes	—
169	C₃²×C₄	C₂×#67	1	4	yes	—
170	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
172	C₃⋊Dic₃	—	2	2	no	—
173	C₃×Dic₃	—	6	8	no	—
174	C₂×#41	—	9	1	no	—
175	C₂×M₁₆	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
176	C₂²×C₈	—	9	1	no	—
177	C₂²⋊C₈	—	9	4	no	—
178	C₃³	#408	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	4	no	—
180	C₂³×C₃	—	9	2	no	—
181	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
182	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
183	C₂²×D₃	—	3	8	no	—
184	C₃×Q₈	—	18	2	no	—
185	C₃×D₄	—	18	4	no	—
186	C₃×D₄	—	18	2	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	8	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	9	2	no	—
192	C₂×C₃×C₄	—	18	2	no	—
193	C₂×C₃×C₄	—	18	2	no	—
194	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
195	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
196	C₂×C₃×C₄	C₃×#67	1	8	yes	—
197	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
199	C₂×C₃×C₄	#638	1	2	yes	—
200	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
201	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
202	#67	—	6	16	no	—
203	C₂×Dic₃	—	3	8	no	—
204	D₁₂	—	6	8	no	—
205	C₄×D₃	—	6	8	no	—
206	C₄×D₃	—	6	8	no	—
207	Dic₆	—	6	8	no	—
208	C₃×C₈	—	2	16	no	—
209	C₃×C₈	—	2	8	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	—	18	2	no	—
212	C₃⋊C₈	—	6	8	no	—
213	C₂×C₃²	—	2	2	no	—
214	C₂×C₃²	—	2	8	no	—
215	C₂×C₃²	—	2	4	no	—
216	C₂×C₃²	—	2	8	no	—
217	C₂×C₃²	—	4	8	no	—
218	C₂×C₃²	#907	1	8	yes	—
219	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
220	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₃²	C₃×#131	1	2	yes	—
222	C₃⋊D₃	—	4	2	no	—
223	C₃×D₃	—	12	8	no	—
224	#41	—	18	2	no	—
225	#41	—	18	2	no	—
226	C₂×Q₈	—	9	1	no	—
227	C₂×D₄	—	9	2	no	—
228	C₂×D₄	—	9	1	no	—
229	C₂²×C₄	—	9	2	no	—
230	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
231	M₁₆	—	2	2	no	—
232	C₂×C₈	—	9	2	no	—
233	C₂×C₈	—	18	2	no	—
234	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
235	C₂²×C₃	—	2	8	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	4	no	—
239	C₂²×C₃	—	4	8	no	—
240	C₂²×C₃	—	18	4	no	—
241	C₂²×C₃	—	18	2	no	—
242	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
243	C₂²×C₃	#3752	1	2	yes	—
244	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
245	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
246	C₂×D₃	—	6	16	no	—
247	C₂×D₃	—	6	8	no	—
248	C₃×C₄	—	2	8	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	18	2	no	—
252	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
253	C₃×C₄	#3752	1	2	yes	—
254	C₃×C₄	C₂×C₃×D₁₂	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	—	6	8	no	—
259	C₃²	—	2	4	no	—
260	C₃²	#5129	1	4	yes	complemented
261	C₃²	C₃×#408	1	1	yes	characteristic, complemented
262	C₂³	—	9	2	no	—
263	C₂³	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
264	Q₈	—	18	2	no	—
265	D₄	—	18	4	no	—
266	D₄	—	18	2	no	—
267	C₂×C₄	—	9	2	no	—
268	C₂×C₄	—	18	2	no	—
269	C₂×C₄	—	18	2	no	—
270	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
271	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	2	yes	—
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	C₈	—	18	2	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	—	2	8	no	—
278	C₂×C₃	—	2	8	no	—
279	C₂×C₃	—	2	4	no	—
280	C₂×C₃	—	4	8	no	—
281	C₂×C₃	—	36	2	no	—
282	C₂×C₃	C₃×#907	1	8	yes	—
283	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	4	yes	—
284	C₂×C₃	#64116	1	2	yes	—
285	C₂×C₃	C₂×#3752	1	1	yes	characteristic
286	D₃	—	12	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	18	4	no	—
289	C₂²	—	18	2	no	—
290	C₂²	C₃×#3752	1	2	yes	—
291	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
292	C₄	—	18	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×#5129	1	4	yes	complemented
298	C₃	#10592828	1	1	yes	characteristic, direct product factor
299	C₂	—	2	2	no	—
300	C₂	—	36	2	no	—
301	C₂	C₃×#64116	1	2	yes	—
302	C₂	C₂×C₃×#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂×C₃×#3828	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	C₃⋊D₃
hypocenter	C₃²	C₃×#408
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×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. C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
4. C₃×#63647	C₂	semidirect
5. #10592828	C₃	direct (N × H)
21. C₂×C₃²×M₁₆	D₃	semidirect
22. C₂×C₃²×M₁₆	C₂×C₃	semidirect
23. C₂²×C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
25. #63647	C₂×C₃	semidirect
80. C₂×C₃×M₁₆	C₃×D₃	semidirect
81. C₂×C₃×M₁₆	C₃⋊D₃	semidirect
175. C₂×M₁₆	C₃×C₃⋊D₃	semidirect
178. C₃³	#408	semidirect
260. C₃²	#5129	semidirect
261. C₃²	C₃×#408	semidirect
297. C₃	C₃×#5129	semidirect
298. C₃	#10592828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3828	2
C₂×C₃³×M₁₆	2
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₃×#63647	2
#10592828	3
C₃×#5129	3

Subgroups — C3×#10592828 #13163166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592828 #13163166