Subgroups — C₃×D₃×#1712 #72244943

Order 2304.

2824 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×#493	C₂	1	6	yes	maximal, complemented
2	C₃×#68062	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#71276	C₂	1	6	yes	maximal, complemented
4	C₃×D₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#1712	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×#1712	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#1712	—	3	1	no	maximal
8	C₂×C₃×D₃×D₈	—	2	12	no	—
9	C₃×D₃×#127	C₂²	1	4	yes	—
10	C₃²×#493	C₂²	1	6	yes	complemented
11	C₃×#5151	—	2	12	no	—
12	C₃×#5073	C₂²	1	12	yes	complemented
13	C₃×#4695	C₂²	1	6	yes	complemented
14	C₃×C₄×D₃×C₈	C₂²	1	3	yes	—
15	C₃²×C₈²	C₂²	1	1	yes	characteristic, complemented
16	C₃×C₈×C₃⋊C₈	C₂²	1	3	yes	complemented
17	C₂×C₃×#493	—	3	6	no	—
18	D₃×#493	C₂×C₃	1	6	yes	complemented
19	C₃×#1712	—	2	1	no	—
20	C₃×#1712	—	3	6	no	—
21	C₃×#1712	—	3	1	no	—
22	C₃×#1712	D₃	1	1	yes	characteristic, direct product factor
23	C₃×#1712	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₈²	—	3	1	no	—
25	#71276	C₂×C₃	1	6	yes	complemented
26	#68062	C₂×C₃	1	1	yes	characteristic, complemented
27	D₃×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×D₃×D₄	—	2	12	no	—
29	C₂×C₃²×D₈	—	2	12	no	—
30	C₃²×#127	C₂³	1	4	yes	—
31	C₃×#940	—	2	12	no	—
32	C₂×C₃×#213	—	2	24	no	—
33	C₃×D₃×D₈	—	4	48	no	—
34	C₂×C₃×D₂₄	—	2	12	no	—
35	C₂×C₃×D₃×C₈	D₄	1	6	yes	—
36	C₃×#874	C₂³	1	4	yes	—
37	C₃×C₄²×D₃	C₂³	1	1	yes	characteristic
38	C₃²×C₄×C₈	C₂³	1	3	yes	—
39	C₃×C₈×Dic₃	D₄	1	6	yes	—
40	C₃×C₄×C₃⋊C₈	C₂³	1	3	yes	—
41	C₂×#1712	—	3	1	no	Sylow
42	C₂²×C₃×D₈	—	6	12	no	—
43	C₂×C₃×#127	—	3	4	no	—
44	C₂×D₃×D₈	—	2	12	no	—
45	D₃×#127	C₂²×C₃	1	4	yes	—
46	C₃×#493	—	2	6	no	—
47	C₃×#493	—	3	12	no	—
48	C₃×#493	—	3	6	no	—
49	C₃×#493	—	6	12	no	—
50	C₃×#493	C₂×D₃	1	6	yes	complemented
51	C₃×#493	C₂²×C₃	1	6	yes	complemented
52	C₂×C₃×C₄×C₈	—	3	3	no	—
53	#5151	—	2	12	no	—
54	#5073	C₂²×C₃	1	12	yes	complemented
55	#4695	C₂²×C₃	1	6	yes	complemented
56	C₄×D₃×C₈	C₂²×C₃	1	3	yes	—
57	C₃×C₈²	—	2	1	no	—
58	C₃×C₈²	—	3	3	no	—
59	C₃×C₈²	C₂×D₃	1	1	yes	characteristic, complemented
60	C₃×C₈²	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₈×C₃⋊C₈	C₂²×C₃	1	3	yes	complemented
62	C₂³×C₃×D₃	—	4	4	no	—
63	C₂×C₃²×D₄	—	2	12	no	—
64	C₂×C₃×#67	—	4	12	no	—
65	C₃×D₃×D₄	—	4	48	no	—
66	C₂×C₃×D₁₂	—	2	12	no	—
67	C₂×C₃×C₄×D₃	C₂×D₄	1	3	yes	—
68	C₃²×D₈	—	4	24	no	—
69	C₂×C₃²×C₈	C₂×D₄	1	6	yes	—
70	C₃²×C₄²	C₂⁴	1	1	yes	characteristic
71	C₃×#213	—	4	48	no	—
72	C₃×C₄×Dic₃	C₂×D₄	1	3	yes	—
73	C₂×C₃×C₃⋊C₈	C₂×D₄	1	6	yes	—
74	C₃×D₂₄	—	4	24	no	—
75	C₃×D₃×C₈	D₈	1	24	yes	complemented
76	C₂×#493	—	3	6	no	—
77	#1712	—	3	6	no	—
78	#1712	—	3	1	no	—
79	#1712	C₃×D₃	1	1	yes	characteristic, direct product factor
80	C₂×C₈²	—	3	1	no	—
81	C₂²×C₃×D₄	—	6	12	no	—
82	C₂×D₃×D₄	—	2	12	no	—
83	C₂×C₃×D₈	—	2	12	no	—
84	C₂×C₃×D₈	—	2	12	no	—
85	C₂×C₃×D₈	—	4	12	no	—
86	C₂×C₃×D₈	—	6	24	no	—
87	C₂×C₃×D₈	—	6	12	no	—
88	C₂×C₃×D₈	—	12	48	no	—
89	C₂²×C₃×C₈	—	3	6	no	—
90	C₃×#127	—	2	4	no	—
91	C₃×#127	—	3	4	no	—
92	C₃×#127	—	6	12	no	—
93	C₃×#127	C₂²×D₃	1	4	yes	—
94	C₃×#127	C₂³×C₃	1	4	yes	—
95	C₂×C₃×C₄²	—	3	1	no	—
96	#940	—	2	12	no	—
97	C₂×#213	—	2	24	no	—
98	D₃×D₈	—	4	48	no	—
99	C₂×D₂₄	—	2	12	no	—
100	C₂×D₃×C₈	C₃×D₄	1	6	yes	—
101	#874	C₂³×C₃	1	4	yes	—
102	C₄²×D₃	C₂³×C₃	1	1	yes	characteristic
103	C₃×C₄×C₈	—	2	3	no	—
104	C₃×C₄×C₈	—	3	6	no	—
105	C₃×C₄×C₈	—	3	3	no	—
106	C₃×C₄×C₈	C₂²×D₃	1	3	yes	—
107	C₃×C₄×C₈	C₂³×C₃	1	3	yes	—
108	C₈×Dic₃	C₃×D₄	1	6	yes	—
109	C₄×C₃⋊C₈	C₂³×C₃	1	3	yes	—
110	C₂³×C₃²	—	4	4	no	—
111	C₂²×C₃×D₃	—	4	4	no	—
112	C₂²×C₃×D₃	—	8	24	no	—
113	C₂²×C₃×D₃	#127	1	1	yes	characteristic
114	C₃²×D₄	—	4	24	no	—
115	C₂×C₃²×C₄	C₂²×D₄	1	3	yes	—
116	C₃×#67	—	8	24	no	—
117	C₂×C₃×Dic₃	#127	1	3	yes	—
118	C₃×D₁₂	—	4	24	no	—
119	C₃×C₄×D₃	C₂×D₈	1	12	yes	—
120	C₃²×C₈	C₂×D₈	1	12	yes	complemented
121	C₃×C₃⋊C₈	C₂×D₈	1	12	yes	complemented
122	C₂²×D₈	—	6	12	no	—
123	C₂×#127	—	3	4	no	—
124	#493	—	3	12	no	—
125	#493	—	3	6	no	—
126	#493	—	6	12	no	—
127	#493	C₂×C₃×D₃	1	6	yes	complemented
128	C₂×C₄×C₈	—	3	3	no	—
129	C₈²	—	3	3	no	—
130	C₈²	C₂×C₃×D₃	1	1	yes	characteristic, complemented
131	C₂⁴×C₃	—	12	4	no	—
132	C₂³×D₃	—	4	4	no	—
133	C₂×C₃×D₄	—	2	12	no	—
134	C₂×C₃×D₄	—	2	12	no	—
135	C₂×C₃×D₄	—	4	12	no	—
136	C₂×C₃×D₄	—	6	12	no	—
137	C₂×C₃×D₄	—	12	48	no	—
138	C₂×C₃×D₄	—	12	12	no	—
139	C₂²×C₃×C₄	—	3	3	no	—
140	C₂×#67	—	4	12	no	—
141	D₃×D₄	—	4	48	no	—
142	C₂×D₁₂	—	2	12	no	—
143	C₂×C₄×D₃	C₂×C₃×D₄	1	3	yes	—
144	C₃×D₈	—	4	24	no	—
145	C₃×D₈	—	4	24	no	—
146	C₃×D₈	—	8	24	no	—
147	C₃×D₈	—	12	48	no	—
148	C₃×D₈	—	12	24	no	—
149	C₂×C₃×C₈	—	2	6	no	—
150	C₂×C₃×C₈	—	3	6	no	—
151	C₂×C₃×C₈	—	3	24	no	—
152	C₂×C₃×C₈	D₃×D₄	1	6	yes	—
153	C₂×C₃×C₈	C₂×C₃×D₄	1	6	yes	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	3	3	no	—
156	C₃×C₄²	C₂³×D₃	1	1	yes	characteristic
157	C₃×C₄²	C₂⁴×C₃	1	1	yes	characteristic
158	#213	—	4	48	no	—
159	C₄×Dic₃	C₂×C₃×D₄	1	3	yes	—
160	C₂×C₃⋊C₈	C₂×C₃×D₄	1	6	yes	—
161	D₂₄	—	4	24	no	—
162	D₃×C₈	C₃×D₈	1	24	yes	complemented
163	C₂²×C₃²	—	8	12	no	—
164	C₂²×C₃²	C₂×#127	1	1	yes	characteristic
165	C₂×C₃×D₃	—	8	12	no	—
166	C₂×C₃×D₃	—	16	16	no	—
167	C₂×C₃×D₃	#493	1	6	yes	—
168	C₃²×C₄	C₂²×D₈	1	6	yes	—
169	C₃×Dic₃	#493	1	6	yes	—
170	C₂²×D₄	—	6	12	no	—
171	C₂×D₈	—	2	12	no	—
172	C₂×D₈	—	6	24	no	—
173	C₂×D₈	—	6	12	no	—
174	C₂×D₈	—	12	48	no	—
175	C₂²×C₈	—	3	6	no	—
176	#127	—	3	4	no	—
177	#127	—	6	12	no	—
178	#127	C₂²×C₃×D₃	1	4	yes	—
179	C₂×C₄²	—	3	1	no	—
180	C₄×C₈	—	3	6	no	—
181	C₄×C₈	—	3	3	no	—
182	C₄×C₈	C₂²×C₃×D₃	1	3	yes	—
183	C₂³×C₃	—	3	1	no	—
184	C₂³×C₃	—	4	4	no	—
185	C₂³×C₃	—	4	4	no	—
186	C₂³×C₃	—	8	4	no	—
187	C₂³×C₃	—	12	4	no	—
188	C₂³×C₃	—	24	24	no	—
189	C₂²×D₃	—	4	4	no	—
190	C₂²×D₃	—	8	24	no	—
191	C₂²×D₃	C₃×#127	1	1	yes	characteristic
192	C₃×D₄	—	4	24	no	—
193	C₃×D₄	—	4	24	no	—
194	C₃×D₄	—	8	24	no	—
195	C₃×D₄	—	12	24	no	—
196	C₃×D₄	—	24	24	no	—
197	C₂×C₃×C₄	—	2	3	no	—
198	C₂×C₃×C₄	—	3	3	no	—
199	C₂×C₃×C₄	—	3	12	no	—
200	C₂×C₃×C₄	C₂×D₃×D₄	1	3	yes	—
201	C₂×C₃×C₄	C₂²×C₃×D₄	1	3	yes	—
202	#67	—	8	24	no	—
203	C₂×Dic₃	C₃×#127	1	3	yes	—
204	D₁₂	—	4	24	no	—
205	C₄×D₃	C₂×C₃×D₈	1	12	yes	—
206	C₃×C₈	—	2	12	no	—
207	C₃×C₈	—	3	12	no	—
208	C₃×C₈	D₃×D₈	1	12	yes	complemented
209	C₃×C₈	C₂×C₃×D₈	1	12	yes	complemented
210	C₃⋊C₈	C₂×C₃×D₈	1	12	yes	complemented
211	C₂×C₃²	—	16	4	no	—
212	C₂×C₃²	C₂×#493	1	3	yes	—
213	C₃×D₃	—	16	4	no	—
214	C₃×D₃	#1712	1	4	yes	direct product factor
215	C₂⁴	—	12	4	no	—
216	C₂×D₄	—	2	12	no	—
217	C₂×D₄	—	6	12	no	—
218	C₂×D₄	—	12	48	no	—
219	C₂×D₄	—	12	12	no	—
220	C₂²×C₄	—	3	3	no	—
221	D₈	—	4	24	no	—
222	D₈	—	12	48	no	—
223	D₈	—	12	24	no	—
224	C₂×C₈	—	3	6	no	—
225	C₂×C₈	—	3	24	no	—
226	C₂×C₈	C₃×D₃×D₄	1	6	yes	—
227	C₄²	—	3	3	no	—
228	C₄²	C₂³×C₃×D₃	1	1	yes	characteristic
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	3	6	no	—
231	C₂²×C₃	—	8	12	no	—
232	C₂²×C₃	—	8	12	no	—
233	C₂²×C₃	—	16	12	no	—
234	C₂²×C₃	—	24	12	no	—
235	C₂²×C₃	—	48	16	no	—
236	C₂²×C₃	D₃×#127	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₃×#127	1	1	yes	characteristic
238	C₂×D₃	—	8	12	no	—
239	C₂×D₃	—	16	16	no	—
240	C₂×D₃	C₃×#493	1	6	yes	—
241	C₃×C₄	—	2	6	no	—
242	C₃×C₄	—	3	6	no	—
243	C₃×C₄	C₂×D₃×D₈	1	6	yes	—
244	C₃×C₄	C₂²×C₃×D₈	1	6	yes	—
245	Dic₃	C₃×#493	1	6	yes	—
246	C₃²	C₂×#1712	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	3	1	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	12	4	no	—
250	C₂³	—	24	24	no	—
251	D₄	—	4	24	no	—
252	D₄	—	12	24	no	—
253	D₄	—	24	24	no	—
254	C₂×C₄	—	3	3	no	—
255	C₂×C₄	—	3	12	no	—
256	C₂×C₄	C₂×C₃×D₃×D₄	1	3	yes	—
257	C₈	—	3	12	no	—
258	C₈	C₃×D₃×D₈	1	12	yes	complemented
259	C₂×C₃	—	2	3	no	—
260	C₂×C₃	—	3	4	no	—
261	C₂×C₃	—	16	4	no	—
262	C₂×C₃	—	16	4	no	—
263	C₂×C₃	—	32	4	no	—
264	C₂×C₃	—	48	4	no	—
265	C₂×C₃	D₃×#493	1	3	yes	—
266	C₂×C₃	C₂×C₃×#493	1	3	yes	—
267	D₃	—	16	4	no	—
268	D₃	C₃×#1712	1	4	yes	direct product factor
269	C₂²	—	3	6	no	—
270	C₂²	—	8	12	no	—
271	C₂²	—	24	12	no	—
272	C₂²	—	48	16	no	—
273	C₂²	C₃×D₃×#127	1	1	yes	characteristic
274	C₄	—	3	6	no	—
275	C₄	C₂×C₃×D₃×D₈	1	6	yes	—
276	C₃	—	2	1	no	—
277	C₃	C₂×C₃×#1712	1	1	yes	characteristic, complemented
278	C₃	D₃×#1712	1	1	yes	characteristic, direct product factor
279	C₂	—	3	4	no	—
280	C₂	—	16	4	no	—
281	C₂	—	48	4	no	—
282	C₂	C₃×D₃×#493	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃	D₃×#127
commutator, cosocle	C₃×C₄²	C₂⁴×C₃
Frattini	C₄²	C₂³×C₃×D₃
Fitting	C₃²×#1712	C₂
socle	C₂²×C₃²	C₂×#127
hypercenter	C₃×#1712	D₃
hypocenter	C₃	C₂×C₃×#1712
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#1712

Complements

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

N	H ≅ G/N	type
1. C₃×D₃×#493	C₂	semidirect
2. C₃×#68062	C₂	semidirect
3. C₃×#71276	C₂	semidirect
4. C₃×D₃×C₈²	C₂	semidirect
5. C₃²×#1712	C₂	semidirect
6. D₃×#1712	C₃	direct (N × H)
10. C₃²×#493	C₂²	semidirect
12. C₃×#5073	C₂²	semidirect
13. C₃×#4695	C₂²	semidirect
15. C₃²×C₈²	C₂²	semidirect
16. C₃×C₈×C₃⋊C₈	C₂²	semidirect
18. D₃×#493	C₂×C₃	semidirect
22. C₃×#1712	D₃	direct (N × H)
23. C₃×#1712	C₂×C₃	semidirect
25. #71276	C₂×C₃	semidirect
26. #68062	C₂×C₃	semidirect
27. D₃×C₈²	C₂×C₃	semidirect
50. C₃×#493	C₂×D₃	semidirect
51. C₃×#493	C₂²×C₃	semidirect
54. #5073	C₂²×C₃	semidirect
55. #4695	C₂²×C₃	semidirect
59. C₃×C₈²	C₂×D₃	semidirect
60. C₃×C₈²	C₂²×C₃	semidirect
61. C₈×C₃⋊C₈	C₂²×C₃	semidirect
75. C₃×D₃×C₈	D₈	semidirect
79. #1712	C₃×D₃	direct (N × H)
120. C₃²×C₈	C₂×D₈	semidirect
121. C₃×C₃⋊C₈	C₂×D₈	semidirect
127. #493	C₂×C₃×D₃	semidirect
130. C₈²	C₂×C₃×D₃	semidirect
162. D₃×C₈	C₃×D₈	semidirect
208. C₃×C₈	D₃×D₈	semidirect
209. C₃×C₈	C₂×C₃×D₈	semidirect
210. C₃⋊C₈	C₂×C₃×D₈	semidirect
214. C₃×D₃	#1712	direct (N × H)
246. C₃²	C₂×#1712	semidirect
258. C₈	C₃×D₃×D₈	semidirect
268. D₃	C₃×#1712	direct (N × H)
277. C₃	C₂×C₃×#1712	semidirect
278. C₃	D₃×#1712	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₃×#493	2
C₃×#68062	2
C₃×#71276	2
C₃×D₃×C₈²	2
C₃²×#1712	2
D₃×#1712	3
C₂×C₃×#1712	3

Subgroups — C3×D3×#1712 #72244943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×#1712 #72244943