Subgroups — C₃×#10593765 #13173236

Order 1728.

456 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64286	C₂	1	1	yes	maximal, characteristic
2	C₃×#64283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#63767	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#157	C₂	1	2	yes	maximal, complemented
6	#10593765	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₃×C₂²⋊#157	C₄	1	2	yes	—
9	C₃×#3792	C₄	1	2	yes	complemented
10	C₃×C₄⋊#157	C₄	1	2	yes	—
11	C₃×C₄×#157	C₂²	1	2	yes	—
12	C₃×#3788	C₄	1	2	yes	complemented
13	C₃×#3787	C₂²	1	2	yes	—
14	C₂×C₃³×C₈	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic, complemented
16	C₃×C₈×C₃⋊D₃	C₄	1	1	yes	characteristic, complemented
17	C₃×C₈×C₃⋊D₃	C₄	1	1	yes	characteristic, complemented
18	C₃×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
19	C₃×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
20	#64286	C₂×C₃	1	1	yes	characteristic
21	#64283	C₂×C₃	1	1	yes	characteristic
22	C₂×C₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×D₃×C₈	—	6	2	no	—
24	#63767	C₂×C₃	1	2	yes	complemented
25	C₈×#157	C₂×C₃	1	2	yes	complemented
26	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×#157	C₂×C₄	1	4	yes	—
28	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
34	C₃×#622	C₂×C₄	1	4	yes	—
35	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
38	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
39	C₃×#405	—	9	1	no	maximal
40	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	—	6	2	no	—
42	C₂²⋊#157	C₃×C₄	1	2	yes	—
43	#3792	C₃×C₄	1	2	yes	complemented
44	C₄⋊#157	C₃×C₄	1	2	yes	—
45	C₄×#157	C₂²×C₃	1	2	yes	—
46	#3788	C₃×C₄	1	2	yes	complemented
47	#3787	C₂²×C₃	1	2	yes	—
48	C₂×C₃²×C₈	—	2	2	no	—
49	C₂×C₃²×C₈	—	4	2	no	—
50	C₂×C₃²×C₈	C₃×C₄	1	1	yes	characteristic, complemented
51	C₂×C₃⋊(C₃⋊C₈)	C₃×C₄	1	1	yes	characteristic, complemented
52	C₈×C₃⋊D₃	C₃×C₄	1	1	yes	characteristic, complemented
53	C₈×C₃⋊D₃	C₃×C₄	1	1	yes	characteristic, complemented
54	C₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
55	C₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	—	6	2	no	—
57	C₃×D₃×C₈	—	6	2	no	—
58	C₃×D₃×C₈	—	6	2	no	—
59	C₃×D₃×C₈	—	6	2	no	—
60	C₃×D₃×C₈	—	6	2	no	—
61	C₂²×C₃³	C₄²	1	1	yes	characteristic
62	C₂×C₃×C₃⋊D₃	—	2	2	no	—
63	C₂×C₃×C₃⋊D₃	C₄²	1	1	yes	characteristic
64	C₂×C₃×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
65	C₃×#157	—	2	4	no	—
66	C₃³×C₄	C₄²	1	1	yes	characteristic
67	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₃⋊Dic₃	C₄²	1	1	yes	characteristic
69	C₃×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×M₁₆	—	9	1	no	—
71	C₂²×C₃×C₈	—	9	1	no	—
72	C₃×#117	—	9	1	no	—
73	C₂×D₃×C₈	—	6	2	no	—
74	C₃×#97	—	9	2	no	—
75	C₃×C₄×C₈	—	9	2	no	—
76	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	6	2	no	—
78	C₂×#157	C₂×C₃×C₄	1	4	yes	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	4	2	no	—
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
83	C₄×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
84	C₄×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
85	C₄×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
86	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	6	2	no	—
88	C₃×C₄×D₃	—	6	2	no	—
89	C₃×C₄×D₃	—	6	2	no	—
90	C₃×C₄×D₃	—	6	2	no	—
91	C₃×C₄×D₃	—	6	2	no	—
92	#622	C₂×C₃×C₄	1	4	yes	—
93	C₃²×C₈	—	2	2	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃²×C₈	—	4	2	no	—
96	C₃²×C₈	—	4	2	no	—
97	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	1	yes	characteristic
100	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	1	yes	characteristic
101	C₃×C₃⋊C₈	—	6	2	no	—
102	C₃×C₃⋊C₈	—	6	2	no	—
103	#405	—	9	1	no	Sylow
104	C₂×C₃³	—	2	1	no	—
105	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
106	C₃×C₃⋊D₃	—	2	1	no	—
107	C₃×C₃⋊D₃	#131	1	2	yes	—
108	C₂²×C₃×C₄	—	9	1	no	—
109	C₂×C₄×D₃	—	6	2	no	—
110	C₃×M₁₆	—	9	2	no	—
111	C₃×M₁₆	—	9	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	4	2	no	—
114	C₂×C₃×C₈	—	9	2	no	—
115	C₂×C₃×C₈	—	9	1	no	—
116	C₂×C₃×C₈	—	9	1	no	—
117	C₂×C₃×C₈	—	9	1	no	—
118	C₂×C₃×C₈	—	9	1	no	—
119	C₂×C₃×C₈	—	9	1	no	—
120	C₂×C₃×C₈	#157	1	1	yes	characteristic, complemented
121	C₃×C₄⋊C₄	—	9	2	no	—
122	C₃×C₂²⋊C₄	—	9	2	no	—
123	C₃×C₄²	—	9	2	no	—
124	C₂×C₃⋊C₈	—	6	2	no	—
125	D₃×C₈	—	6	2	no	—
126	D₃×C₈	—	6	2	no	—
127	D₃×C₈	—	6	2	no	—
128	D₃×C₈	—	6	2	no	—
129	C₂²×C₃²	—	2	2	no	—
130	C₂²×C₃²	—	4	2	no	—
131	C₂²×C₃²	C₃×C₄²	1	1	yes	characteristic
132	C₂×C₃⋊D₃	—	2	2	no	—
133	C₂×C₃⋊D₃	C₃×C₄²	1	1	yes	characteristic
134	C₂×C₃⋊D₃	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₃×D₃	—	6	2	no	—
136	C₂×C₃×D₃	—	6	2	no	—
137	C₂×C₃×D₃	—	6	8	no	—
138	#157	—	2	4	no	—
139	C₃²×C₄	—	2	2	no	—
140	C₃²×C₄	—	2	2	no	—
141	C₃²×C₄	—	4	2	no	—
142	C₃²×C₄	—	4	2	no	—
143	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
144	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₃⋊Dic₃	C₃×C₄²	1	1	yes	characteristic
146	C₃⋊Dic₃	C₂²×C₃×C₄	1	1	yes	characteristic
147	C₃×Dic₃	—	6	2	no	—
148	C₃×Dic₃	—	6	2	no	—
149	C₂×M₁₆	—	9	1	no	—
150	C₂²×C₈	—	9	1	no	—
151	#117	—	9	1	no	—
152	#97	—	9	2	no	—
153	C₄×C₈	—	9	2	no	—
154	C₃³	#405	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₃	—	9	1	no	—
156	C₂²×D₃	—	6	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	9	1	no	—
160	C₂×C₃×C₄	—	9	1	no	—
161	C₂×C₃×C₄	—	9	1	no	—
162	C₂×C₃×C₄	—	9	1	no	—
163	C₂×C₃×C₄	—	9	1	no	—
164	C₂×C₃×C₄	—	9	4	no	—
165	C₂×C₃×C₄	C₂×#157	1	1	yes	characteristic
166	C₂×Dic₃	—	6	2	no	—
167	C₄×D₃	—	6	2	no	—
168	C₄×D₃	—	6	2	no	—
169	C₄×D₃	—	6	2	no	—
170	C₄×D₃	—	6	2	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	4	2	no	—
174	C₃×C₈	—	4	2	no	—
175	C₃×C₈	—	9	4	no	—
176	C₃×C₈	—	9	1	no	—
177	C₃×C₈	—	9	1	no	—
178	C₃×C₈	C₂×#157	1	1	yes	characteristic
179	C₃×C₈	C₂×#157	1	1	yes	characteristic
180	C₃⋊C₈	—	6	2	no	—
181	C₃⋊C₈	—	6	2	no	—
182	C₂×C₃²	—	2	1	no	—
183	C₂×C₃²	—	2	4	no	—
184	C₂×C₃²	—	2	2	no	—
185	C₂×C₃²	—	4	4	no	—
186	C₂×C₃²	—	4	2	no	—
187	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
188	C₃⋊D₃	—	2	1	no	—
189	C₃⋊D₃	C₃×#131	1	2	yes	—
190	C₃×D₃	—	6	4	no	—
191	C₃×D₃	—	6	4	no	—
192	C₂²×C₄	—	9	1	no	—
193	M₁₆	—	9	2	no	—
194	M₁₆	—	9	2	no	—
195	C₂×C₈	—	9	2	no	—
196	C₂×C₈	—	9	1	no	—
197	C₂×C₈	—	9	1	no	—
198	C₂×C₈	—	9	1	no	—
199	C₂×C₈	—	9	1	no	—
200	C₂×C₈	—	9	1	no	—
201	C₂×C₈	C₃×#157	1	1	yes	characteristic, complemented
202	C₄⋊C₄	—	9	2	no	—
203	C₂²⋊C₄	—	9	2	no	—
204	C₄²	—	9	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	9	1	no	—
208	C₂²×C₃	—	9	1	no	—
209	C₂²×C₃	—	18	2	no	—
210	C₂²×C₃	C₄×#157	1	1	yes	characteristic
211	C₂×D₃	—	6	2	no	—
212	C₂×D₃	—	6	2	no	—
213	C₂×D₃	—	6	8	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	—	4	2	no	—
218	C₃×C₄	—	9	1	no	—
219	C₃×C₄	—	9	1	no	—
220	C₃×C₄	—	18	4	no	—
221	C₃×C₄	C₄×#157	1	1	yes	characteristic
222	C₃×C₄	C₂²×#157	1	1	yes	characteristic
223	Dic₃	—	6	2	no	—
224	Dic₃	—	6	2	no	—
225	C₃²	—	2	2	no	—
226	C₃²	—	4	2	no	—
227	C₃²	C₃×#405	1	1	yes	characteristic, complemented
228	C₂³	—	9	1	no	—
229	C₂×C₄	—	9	1	no	—
230	C₂×C₄	—	9	1	no	—
231	C₂×C₄	—	9	1	no	—
232	C₂×C₄	—	9	1	no	—
233	C₂×C₄	—	9	1	no	—
234	C₂×C₄	—	9	4	no	—
235	C₂×C₄	C₂×C₃×#157	1	1	yes	characteristic
236	C₈	—	9	4	no	—
237	C₈	—	9	1	no	—
238	C₈	—	9	1	no	—
239	C₈	C₂×C₃×#157	1	1	yes	characteristic
240	C₈	C₂×C₃×#157	1	1	yes	characteristic
241	C₂×C₃	—	2	4	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	4	4	no	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	—	9	2	no	—
247	C₂×C₃	—	18	1	no	—
248	C₂×C₃	C₂×C₄×#157	1	1	yes	characteristic
249	D₃	—	6	4	no	—
250	D₃	—	6	4	no	—
251	C₂²	—	9	1	no	—
252	C₂²	—	9	1	no	—
253	C₂²	—	18	2	no	—
254	C₂²	C₃×C₄×#157	1	1	yes	characteristic
255	C₄	—	9	1	no	—
256	C₄	—	9	1	no	—
257	C₄	—	18	4	no	—
258	C₄	C₃×C₄×#157	1	1	yes	characteristic
259	C₄	C₂²×C₃×#157	1	1	yes	characteristic
260	C₃	—	2	2	no	—
261	C₃	—	4	2	no	—
262	C₃	#10593765	1	1	yes	characteristic, direct product factor
263	C₂	—	2	1	no	—
264	C₂	—	9	2	no	—
265	C₂	—	18	1	no	—
266	C₂	C₂×C₃×C₄×#157	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂×#157
commutator	C₂×C₃²	C₂×C₃×C₄²
Frattini	C₄	C₂²×C₃×#157
Fitting	C₂×C₃³×C₈	C₄
socle	C₂×C₃³	C₂×C₄²
prime radical	C₂²×C₃×C₃⋊D₃	C₂×C₄
cosocle	C₄×C₃⋊D₃	C₂³×C₃
hypercenter	C₂×C₃×C₈	#157
hypocenter	C₃²	C₃×#405
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#63767	C₂	semidirect
5. C₃×C₈×#157	C₂	semidirect
6. #10593765	C₃	direct (N × H)
9. C₃×#3792	C₄	semidirect
12. C₃×#3788	C₄	semidirect
14. C₂×C₃³×C₈	C₄	semidirect
15. C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	semidirect
16. C₃×C₈×C₃⋊D₃	C₄	semidirect
17. C₃×C₈×C₃⋊D₃	C₄	semidirect
24. #63767	C₂×C₃	semidirect
25. C₈×#157	C₂×C₃	semidirect
43. #3792	C₃×C₄	semidirect
46. #3788	C₃×C₄	semidirect
50. C₂×C₃²×C₈	C₃×C₄	semidirect
51. C₂×C₃⋊(C₃⋊C₈)	C₃×C₄	semidirect
52. C₈×C₃⋊D₃	C₃×C₄	semidirect
53. C₈×C₃⋊D₃	C₃×C₄	semidirect
120. C₂×C₃×C₈	#157	semidirect
154. C₃³	#405	semidirect
201. C₂×C₈	C₃×#157	semidirect
227. C₃²	C₃×#405	semidirect
262. C₃	#10593765	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×#64286	2
C₃×#64283	2
C₂×C₃×C₈×C₃⋊D₃	2
C₃×#63767	2
C₃×C₈×#157	2
#10593765	3
C₃×#405	9

Subgroups — C3×#10593765 #13173236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10593765 #13173236