Subgroups — C₃×#10596202 #13179482

Order 1728.

1424 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	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64155	C₂	1	2	yes	maximal
3	C₃×Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
4	C₃×#64107	C₂	1	4	yes	maximal
5	C₃×#64098	C₂	1	4	yes	maximal, complemented
6	C₃×#64081	C₂	1	1	yes	maximal, characteristic
7	C₃×#64079	C₂	1	1	yes	maximal, characteristic, complemented
8	#10596202	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#5731	—	3	4	no	maximal
10	C₂×C₃³×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#3752	C₂²	1	2	yes	—
15	C₃×#3752	C₂²	1	4	yes	—
16	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
17	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
18	C₃×#3750	C₂²	1	4	yes	—
19	C₃×#3750	C₂²	1	2	yes	—
20	C₃×#3750	C₂²	1	4	yes	—
21	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₃⋊Dic₃	C₂²	1	4	yes	—
23	C₃²×C₄⋊Q₈	—	2	4	no	—
24	C₃²×C₄⋊Q₈	D₃	1	4	yes	complemented
25	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	#64155	C₂×C₃	1	2	yes	—
27	Q₈×C₃⋊Dic₃	C₂×C₃	1	2	yes	complemented
28	#64107	C₂×C₃	1	4	yes	—
29	#64098	C₂×C₃	1	4	yes	complemented
30	#64081	C₂×C₃	1	1	yes	characteristic
31	#64079	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#946	—	3	8	no	—
33	C₃×Q₈×Dic₃	—	3	8	no	—
34	C₃×#898	—	3	16	no	—
35	C₃×#889	—	3	16	no	—
36	C₃×#872	—	3	4	no	—
37	C₃×#870	—	3	4	no	—
38	C₂²×C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
39	C₃³×Q₈	—	2	4	no	—
40	C₂×C₃³×C₄	C₂³	1	4	yes	—
41	C₂×C₃³×C₄	C₂³	1	2	yes	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
44	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
45	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
46	C₃×C₄×C₃⋊D₃	—	2	2	no	—
47	C₃×#541	—	9	1	no	—
48	#5731	—	3	4	no	—
49	C₂×C₃²×Q₈	—	2	8	no	—
50	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
51	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
52	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄×D₃	—	3	4	no	—
54	C₃²×C₄⋊C₄	—	2	16	no	—
55	C₃²×C₄⋊C₄	C₂×D₃	1	16	yes	—
56	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₃²×C₄²	—	2	4	no	—
58	C₃²×C₄²	C₂×D₃	1	4	yes	—
59	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
60	#3752	C₂²×C₃	1	2	yes	—
61	#3752	C₂²×C₃	1	4	yes	—
62	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
63	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	4	yes	—
64	#3750	C₂²×C₃	1	4	yes	—
65	#3750	C₂²×C₃	1	2	yes	—
66	#3750	C₂²×C₃	1	4	yes	—
67	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
68	C₄×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
69	C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
70	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
71	C₃×C₄⋊Dic₃	—	3	16	no	—
72	C₃×C₄⋊Dic₃	—	3	8	no	—
73	C₃×Dic₃⋊C₄	—	3	16	no	—
74	C₃×Dic₃⋊C₄	—	3	8	no	—
75	C₃×Dic₃⋊C₄	—	3	16	no	—
76	C₃×C₄×Dic₃	—	3	16	no	—
77	C₃×C₄×Dic₃	—	3	4	no	—
78	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
79	C₂×C₃×C₃⋊D₃	—	2	1	no	—
80	C₂×C₃×C₃⋊D₃	—	2	2	no	—
81	C₃³×C₄	—	2	4	no	—
82	C₃³×C₄	—	2	2	no	—
83	C₃³×C₄	#41	1	2	yes	—
84	C₃×C₃⋊Dic₃	—	2	1	no	—
85	C₃×C₃⋊Dic₃	—	2	2	no	—
86	C₃×C₃⋊Dic₃	—	2	4	no	—
87	C₃×C₄⋊Q₈	—	2	4	no	—
88	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
89	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
90	C₃×#126	—	9	4	no	—
91	C₃×#125	—	9	4	no	—
92	C₃×#125	—	9	1	no	—
93	C₃×C₂²⋊Q₈	—	9	2	no	—
94	C₃×C₄×Q₈	—	9	2	no	—
95	C₃×#117	—	9	1	no	—
96	#946	—	3	8	no	—
97	Q₈×Dic₃	—	3	8	no	—
98	#898	—	3	16	no	—
99	#889	—	3	16	no	—
100	#872	—	3	4	no	—
101	#870	—	3	4	no	—
102	C₂²×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
103	C₂²×C₃×D₃	—	3	4	no	—
104	C₃²×Q₈	—	2	4	no	—
105	C₃²×Q₈	—	2	16	no	—
106	C₃²×Q₈	—	4	16	no	—
107	C₂×C₃²×C₄	—	2	8	no	—
108	C₂×C₃²×C₄	—	2	4	no	—
109	C₂×C₃²×C₄	—	2	16	no	—
110	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
111	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
112	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
113	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
114	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
115	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
117	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
118	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
119	C₄×C₃⋊D₃	—	2	2	no	—
120	C₂×C₃×Dic₃	—	3	8	no	—
121	C₂×C₃×Dic₃	—	3	16	no	—
122	C₂×C₃×Dic₃	—	3	4	no	—
123	C₃×C₄×D₃	—	6	8	no	—
124	#541	—	9	1	no	Sylow
125	C₂×C₃³	C₂×#41	1	1	yes	characteristic
126	C₂×C₃³	2₋¹⁺⁴	1	2	yes	—
127	C₃×C₃⋊D₃	—	4	1	no	—
128	C₂×C₃×Q₈	—	2	8	no	—
129	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
130	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
131	C₂²×C₃×C₄	—	9	1	no	—
132	C₂×C₄×D₃	—	3	4	no	—
133	C₃×C₄⋊C₄	—	2	16	no	—
134	C₃×C₄⋊C₄	—	9	2	no	—
135	C₃×C₄⋊C₄	—	9	4	no	—
136	C₃×C₄⋊C₄	—	9	4	no	—
137	C₃×C₄⋊C₄	—	9	2	no	—
138	C₃×C₄⋊C₄	—	9	4	no	—
139	C₃×C₄⋊C₄	C₂×C₃×D₃	1	16	yes	—
140	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
141	C₃×C₂²⋊C₄	—	9	2	no	—
142	C₃×C₂²⋊C₄	—	9	4	no	—
143	C₃×C₄²	—	2	4	no	—
144	C₃×C₄²	—	9	1	no	—
145	C₃×C₄²	—	9	4	no	—
146	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
147	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
148	(C₂×D₃)⋊C₄	—	3	8	no	—
149	(C₂×D₃)⋊C₄	—	3	16	no	—
150	C₄⋊Dic₃	—	3	8	no	—
151	C₄⋊Dic₃	—	3	16	no	—
152	Dic₃⋊C₄	—	3	16	no	—
153	Dic₃⋊C₄	—	3	8	no	—
154	Dic₃⋊C₄	—	3	16	no	—
155	C₄×Dic₃	—	3	4	no	—
156	C₄×Dic₃	—	3	16	no	—
157	C₂²×C₃²	—	2	4	no	—
158	C₂²×C₃²	C₂³×D₃	1	4	yes	—
159	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
160	C₂×C₃⋊D₃	—	2	1	no	—
161	C₂×C₃⋊D₃	—	2	2	no	—
162	C₂×C₃×D₃	—	6	4	no	—
163	C₂×C₃×D₃	—	6	8	no	—
164	C₃²×C₄	—	2	16	no	—
165	C₃²×C₄	—	2	8	no	—
166	C₃²×C₄	—	2	4	no	—
167	C₃²×C₄	—	2	2	no	—
168	C₃²×C₄	—	2	8	no	—
169	C₃²×C₄	—	2	16	no	—
170	C₃²×C₄	—	4	16	no	—
171	C₃²×C₄	#237	1	8	yes	—
172	C₃²×C₄	C₃×#41	1	2	yes	—
173	C₃⋊Dic₃	—	2	1	no	—
174	C₃⋊Dic₃	—	2	2	no	—
175	C₃⋊Dic₃	—	2	4	no	—
176	C₃×Dic₃	—	6	4	no	—
177	C₃×Dic₃	—	6	8	no	—
178	C₃×Dic₃	—	6	16	no	—
179	C₄⋊Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
180	#126	—	9	4	no	—
181	#125	—	9	4	no	—
182	#125	—	9	1	no	—
183	C₂²⋊Q₈	—	9	2	no	—
184	C₄×Q₈	—	9	2	no	—
185	#117	—	9	1	no	—
186	C₃³	#541	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₃	—	9	1	no	—
188	C₂²×D₃	—	3	4	no	—
189	C₃×Q₈	—	2	16	no	—
190	C₃×Q₈	—	2	4	no	—
191	C₃×Q₈	—	4	16	no	—
192	C₂×C₃×C₄	—	2	8	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	16	no	—
195	C₂×C₃×C₄	—	9	2	no	—
196	C₂×C₃×C₄	—	9	4	no	—
197	C₂×C₃×C₄	—	9	1	no	—
198	C₂×C₃×C₄	—	18	2	no	—
199	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
200	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
202	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
203	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
204	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
205	C₂×Dic₃	—	3	8	no	—
206	C₂×Dic₃	—	3	16	no	—
207	C₂×Dic₃	—	3	4	no	—
208	C₄×D₃	—	6	8	no	—
209	C₂×C₃²	—	2	4	no	—
210	C₂×C₃²	—	2	8	no	—
211	C₂×C₃²	C₂×#237	1	4	yes	—
212	C₂×C₃²	#1007	1	8	yes	—
213	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
214	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	—
215	C₃⋊D₃	—	4	1	no	—
216	C₃×D₃	—	12	4	no	—
217	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
218	C₂²×C₄	—	9	1	no	—
219	C₄⋊C₄	—	9	2	no	—
220	C₄⋊C₄	—	9	2	no	—
221	C₄⋊C₄	—	9	4	no	—
222	C₄⋊C₄	—	9	4	no	—
223	C₄⋊C₄	—	9	4	no	—
224	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	4	yes	—
225	C₂²⋊C₄	—	9	2	no	—
226	C₂²⋊C₄	—	9	4	no	—
227	C₄²	—	9	1	no	—
228	C₄²	—	9	4	no	—
229	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	18	1	no	—
232	C₂²×C₃	—	18	2	no	—
233	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
234	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
235	C₂×D₃	—	6	4	no	—
236	C₂×D₃	—	6	8	no	—
237	C₃×C₄	—	2	16	no	—
238	C₃×C₄	—	2	16	no	—
239	C₃×C₄	—	2	8	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	8	no	—
243	C₃×C₄	—	4	16	no	—
244	C₃×C₄	—	18	1	no	—
245	C₃×C₄	—	18	2	no	—
246	C₃×C₄	—	18	4	no	—
247	C₃×C₄	C₃×#237	1	8	yes	—
248	C₃×C₄	#3830	1	2	yes	—
249	Dic₃	—	6	4	no	—
250	Dic₃	—	6	8	no	—
251	Dic₃	—	6	16	no	—
252	C₃²	—	2	4	no	—
253	C₃²	#5731	1	4	yes	complemented
254	C₃²	C₃×#541	1	1	yes	characteristic, complemented
255	C₂³	—	9	1	no	—
256	Q₈	—	2	4	no	—
257	C₂×C₄	—	9	2	no	—
258	C₂×C₄	—	9	4	no	—
259	C₂×C₄	—	9	1	no	—
260	C₂×C₄	—	18	2	no	—
261	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
262	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
263	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
264	C₂×C₃	—	2	4	no	—
265	C₂×C₃	—	2	8	no	—
266	C₂×C₃	—	36	1	no	—
267	C₂×C₃	C₂×C₃×#237	1	4	yes	—
268	C₂×C₃	C₃×#1007	1	8	yes	—
269	C₂×C₃	C₂×#3830	1	1	yes	characteristic
270	C₂×C₃	#64364	1	2	yes	—
271	D₃	—	12	4	no	—
272	C₂²	—	18	1	no	—
273	C₂²	—	18	2	no	—
274	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	18	1	no	—
278	C₄	—	18	2	no	—
279	C₄	—	18	4	no	—
280	C₄	C₃×#3830	1	2	yes	—
281	C₃	—	2	4	no	—
282	C₃	C₃×#5731	1	4	yes	complemented
283	C₃	#10596202	1	1	yes	characteristic, direct product factor
284	C₂	—	36	1	no	—
285	C₂	C₂×C₃×#3830	1	1	yes	characteristic
286	C₂	C₃×#64364	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂²×C₃×C₃⋊D₃	C₂³
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#541
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃×Q₈×C₃⋊Dic₃	C₂	semidirect
5. C₃×#64098	C₂	semidirect
7. C₃×#64079	C₂	semidirect
8. #10596202	C₃	direct (N × H)
24. C₃²×C₄⋊Q₈	D₃	semidirect
25. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
27. Q₈×C₃⋊Dic₃	C₂×C₃	semidirect
29. #64098	C₂×C₃	semidirect
31. #64079	C₂×C₃	semidirect
88. C₃×C₄⋊Q₈	C₃×D₃	semidirect
89. C₃×C₄⋊Q₈	C₃⋊D₃	semidirect
179. C₄⋊Q₈	C₃×C₃⋊D₃	semidirect
186. C₃³	#541	semidirect
253. C₃²	#5731	semidirect
254. C₃²	C₃×#541	semidirect
282. C₃	C₃×#5731	semidirect
283. C₃	#10596202	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₃×#64155	2
C₃×Q₈×C₃⋊Dic₃	2
C₃×#64107	2
C₃×#64098	2
C₃×#64081	2
C₃×#64079	2
#10596202	3
C₃×#5731	3

Subgroups — C3×#10596202 #13179482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596202 #13179482