Subgroups — C₃×#10596760 #13181747

Order 1728.

5444 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64366	C₂	1	10	yes	maximal, complemented
3	C₃×#41×C₃⋊D₃	C₂	1	10	yes	maximal, complemented
4	C₃×#64364	C₂	1	5	yes	maximal, complemented
5	C₂×C₃×#3832	C₂	1	5	yes	maximal, complemented
6	#10596760	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5968	—	3	4	no	maximal
8	C₃³×#41	C₂²	1	10	yes	complemented
9	C₂×C₃³×Q₈	C₂²	1	5	yes	complemented
10	C₃×#3832	C₂²	1	10	yes	complemented
11	C₃×#3832	C₂²	1	20	yes	complemented
12	C₃×Q₈×C₃⋊D₃	C₂²	1	10	yes	complemented
13	C₃×#3830	C₂²	1	10	yes	complemented
14	C₃×D₄×C₃⋊D₃	C₂²	1	30	yes	complemented
15	C₃×#3828	C₂²	1	30	yes	complemented
16	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	15	yes	complemented
17	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	15	yes	complemented
18	C₃²×2₋¹⁺⁴	—	2	4	no	—
19	C₃²×2₋¹⁺⁴	D₃	1	4	yes	complemented
20	C₃²×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
21	#64366	C₂×C₃	1	10	yes	complemented
22	#41×C₃⋊D₃	C₂×C₃	1	10	yes	complemented
23	#64364	C₂×C₃	1	5	yes	complemented
24	C₂×#3832	C₂×C₃	1	5	yes	complemented
25	C₃×#1009	—	3	40	no	—
26	C₃×D₃×#41	—	3	40	no	—
27	C₃×#1007	—	3	20	no	—
28	C₂×C₃×#239	—	3	20	no	—
29	C₂²×C₃×C₃⋊D₃	C₂³	1	15	yes	—
30	C₃³×Q₈	C₂³	1	10	yes	—
31	C₃³×D₄	C₂³	1	10	yes	—
32	C₂×C₃³×C₄	C₂³	1	15	yes	—
33	C₃×#638	C₂³	1	20	yes	—
34	C₂×C₃×C₃⋊Dic₃	C₂³	1	5	yes	—
35	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	30	yes	—
36	C₃×C₄×C₃⋊D₃	C₂³	1	10	yes	—
37	C₃×C₄×C₃⋊D₃	C₂³	1	30	yes	—
38	C₃×#634	C₂³	1	10	yes	—
39	C₃×#585	—	9	1	no	—
40	#5968	—	3	4	no	—
41	C₃²×#41	—	2	40	no	—
42	C₃²×#41	C₂×D₃	1	40	yes	complemented
43	C₃²×#41	C₂²×C₃	1	10	yes	complemented
44	C₂×C₃²×Q₈	—	2	20	no	—
45	C₂×C₃²×Q₈	C₂×D₃	1	20	yes	complemented
46	C₂×C₃²×Q₈	C₂²×C₃	1	5	yes	complemented
47	#3832	C₂²×C₃	1	10	yes	complemented
48	#3832	C₂²×C₃	1	20	yes	complemented
49	Q₈×C₃⋊D₃	C₂²×C₃	1	10	yes	complemented
50	#3830	C₂²×C₃	1	10	yes	complemented
51	D₄×C₃⋊D₃	C₂²×C₃	1	30	yes	complemented
52	#3828	C₂²×C₃	1	30	yes	complemented
53	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	15	yes	complemented
54	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	15	yes	complemented
55	C₃×#239	—	3	40	no	—
56	C₃×#239	—	3	80	no	—
57	C₃×D₃×Q₈	—	3	40	no	—
58	C₃×#237	—	3	40	no	—
59	C₃×D₃×D₄	—	3	120	no	—
60	C₃×#235	—	3	120	no	—
61	C₂×C₃×D₁₂	—	3	60	no	—
62	C₂×C₃×C₄×D₃	—	3	60	no	—
63	C₂²×C₃³	C₂⁴	1	5	yes	—
64	C₂×C₃×C₃⋊D₃	—	4	15	no	—
65	C₂×C₃×C₃⋊D₃	C₂⁴	1	10	yes	—
66	C₃³×C₄	C₂⁴	1	10	yes	—
67	C₃×C₃⋊Dic₃	C₂⁴	1	5	yes	—
68	C₃×C₃⋊Dic₃	C₂⁴	1	1	yes	characteristic
69	C₃×2₋¹⁺⁴	—	2	4	no	—
70	C₃×2₋¹⁺⁴	—	9	5	no	—
71	C₃×2₋¹⁺⁴	C₃×D₃	1	4	yes	complemented
72	C₃×2₋¹⁺⁴	C₃⋊D₃	1	1	yes	characteristic, complemented
73	C₃×H(5,𝔽₂)	—	9	10	no	—
74	C₂×C₃×#41	—	9	10	no	—
75	C₂×C₃×#41	—	9	5	no	—
76	#1009	—	3	40	no	—
77	D₃×#41	—	3	40	no	—
78	#1007	—	3	20	no	—
79	C₂×#239	—	3	20	no	—
80	C₂²×C₃⋊D₃	C₂³×C₃	1	15	yes	—
81	C₂²×C₃×D₃	—	3	60	no	—
82	C₃²×Q₈	—	2	40	no	—
83	C₃²×Q₈	C₂²×D₃	1	40	yes	—
84	C₃²×Q₈	C₂³×C₃	1	10	yes	—
85	C₃²×D₄	—	2	40	no	—
86	C₃²×D₄	C₂²×D₃	1	40	yes	—
87	C₃²×D₄	C₂³×C₃	1	10	yes	—
88	C₂×C₃²×C₄	—	2	60	no	—
89	C₂×C₃²×C₄	C₂²×D₃	1	60	yes	—
90	C₂×C₃²×C₄	C₂³×C₃	1	15	yes	—
91	#638	C₂³×C₃	1	20	yes	—
92	C₂×C₃⋊Dic₃	C₂³×C₃	1	5	yes	—
93	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	30	yes	—
94	C₄×C₃⋊D₃	C₂³×C₃	1	10	yes	—
95	C₄×C₃⋊D₃	C₂³×C₃	1	30	yes	—
96	#634	C₂³×C₃	1	10	yes	—
97	C₃×#67	—	3	80	no	—
98	C₂×C₃×Dic₃	—	3	20	no	—
99	C₃×D₁₂	—	3	120	no	—
100	C₃×C₄×D₃	—	3	40	no	—
101	C₃×C₄×D₃	—	3	120	no	—
102	C₃×Dic₆	—	3	40	no	—
103	#585	—	9	1	no	Sylow
104	C₂×C₃³	—	2	5	no	—
105	C₂×C₃³	C₂⁵	1	1	yes	characteristic
106	C₃×C₃⋊D₃	—	2	10	no	—
107	C₃×#41	—	2	40	no	—
108	C₃×#41	—	9	10	no	—
109	C₃×#41	—	9	20	no	—
110	C₃×#41	—	9	10	no	—
111	C₃×#41	—	9	30	no	—
112	C₃×#41	C₂×C₃×D₃	1	40	yes	complemented
113	C₃×#41	C₂×C₃⋊D₃	1	10	yes	complemented
114	C₂×C₃×Q₈	—	2	20	no	—
115	C₂×C₃×Q₈	—	9	10	no	—
116	C₂×C₃×Q₈	C₂×C₃×D₃	1	20	yes	complemented
117	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	5	yes	complemented
118	C₂×C₃×D₄	—	9	30	no	—
119	C₂×C₃×D₄	—	9	15	no	—
120	C₂²×C₃×C₄	—	9	15	no	—
121	#239	—	3	40	no	—
122	#239	—	3	80	no	—
123	D₃×Q₈	—	3	40	no	—
124	#237	—	3	40	no	—
125	D₃×D₄	—	3	120	no	—
126	#235	—	3	120	no	—
127	C₂×D₁₂	—	3	60	no	—
128	C₂×C₄×D₃	—	3	60	no	—
129	C₂²×C₃²	—	2	20	no	—
130	C₂²×C₃²	C₂³×D₃	1	20	yes	—
131	C₂²×C₃²	C₂⁴×C₃	1	5	yes	—
132	C₂×C₃⋊D₃	—	4	15	no	—
133	C₂×C₃⋊D₃	C₂⁴×C₃	1	10	yes	—
134	C₂×C₃×D₃	—	3	40	no	—
135	C₂×C₃×D₃	—	12	60	no	—
136	C₃²×C₄	—	2	40	no	—
137	C₃²×C₄	C₂³×D₃	1	40	yes	—
138	C₃²×C₄	C₂⁴×C₃	1	10	yes	—
139	C₃⋊Dic₃	C₂⁴×C₃	1	5	yes	—
140	C₃⋊Dic₃	C₂⁴×C₃	1	1	yes	characteristic
141	C₃×Dic₃	—	3	20	no	—
142	C₃×Dic₃	—	3	4	no	—
143	2₋¹⁺⁴	—	9	5	no	—
144	2₋¹⁺⁴	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
145	H(5,𝔽₂)	—	9	10	no	—
146	C₂×#41	—	9	10	no	—
147	C₂×#41	—	9	5	no	—
148	C₃³	#585	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	9	15	no	—
150	C₂²×D₃	—	3	60	no	—
151	C₃×Q₈	—	2	40	no	—
152	C₃×Q₈	—	9	10	no	—
153	C₃×Q₈	C₂²×C₃×D₃	1	40	yes	—
154	C₃×Q₈	C₂²×C₃⋊D₃	1	10	yes	—
155	C₃×D₄	—	2	40	no	—
156	C₃×D₄	—	9	20	no	—
157	C₃×D₄	—	9	30	no	—
158	C₃×D₄	C₂²×C₃×D₃	1	40	yes	—
159	C₃×D₄	C₂²×C₃⋊D₃	1	10	yes	—
160	C₂×C₃×C₄	—	2	60	no	—
161	C₂×C₃×C₄	—	9	5	no	—
162	C₂×C₃×C₄	—	9	10	no	—
163	C₂×C₃×C₄	—	9	30	no	—
164	C₂×C₃×C₄	C₂²×C₃×D₃	1	60	yes	—
165	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	15	yes	—
166	#67	—	3	80	no	—
167	C₂×Dic₃	—	3	20	no	—
168	D₁₂	—	3	120	no	—
169	C₄×D₃	—	3	40	no	—
170	C₄×D₃	—	3	120	no	—
171	Dic₆	—	3	40	no	—
172	C₂×C₃²	—	2	5	no	—
173	C₂×C₃²	—	2	4	no	—
174	C₂×C₃²	—	2	20	no	—
175	C₂×C₃²	—	4	20	no	—
176	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
177	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
178	C₃⋊D₃	—	2	10	no	—
179	C₃×D₃	—	6	40	no	—
180	#41	—	9	10	no	—
181	#41	—	9	20	no	—
182	#41	—	9	10	no	—
183	#41	—	9	30	no	—
184	#41	C₂×C₃×C₃⋊D₃	1	10	yes	complemented
185	C₂×Q₈	—	9	10	no	—
186	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	5	yes	complemented
187	C₂×D₄	—	9	30	no	—
188	C₂×D₄	—	9	15	no	—
189	C₂²×C₄	—	9	15	no	—
190	C₂²×C₃	—	2	20	no	—
191	C₂²×C₃	—	9	10	no	—
192	C₂²×C₃	—	36	15	no	—
193	C₂²×C₃	C₂³×C₃×D₃	1	20	yes	—
194	C₂²×C₃	C₂³×C₃⋊D₃	1	5	yes	—
195	C₂×D₃	—	3	40	no	—
196	C₂×D₃	—	12	60	no	—
197	C₃×C₄	—	2	40	no	—
198	C₃×C₄	—	9	1	no	—
199	C₃×C₄	—	9	5	no	—
200	C₃×C₄	C₂³×C₃×D₃	1	40	yes	—
201	C₃×C₄	C₂³×C₃⋊D₃	1	10	yes	—
202	Dic₃	—	3	20	no	—
203	Dic₃	—	3	4	no	—
204	C₃²	—	2	4	no	—
205	C₃²	#5968	1	4	yes	complemented
206	C₃²	C₃×#585	1	1	yes	characteristic, complemented
207	C₂³	—	9	15	no	—
208	Q₈	—	9	10	no	—
209	Q₈	C₂²×C₃×C₃⋊D₃	1	10	yes	—
210	D₄	—	9	20	no	—
211	D₄	—	9	30	no	—
212	D₄	C₂²×C₃×C₃⋊D₃	1	10	yes	—
213	C₂×C₄	—	9	5	no	—
214	C₂×C₄	—	9	10	no	—
215	C₂×C₄	—	9	30	no	—
216	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	15	yes	—
217	C₂×C₃	—	2	20	no	—
218	C₂×C₃	—	2	5	no	—
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	—	4	20	no	—
221	C₂×C₃	—	18	10	no	—
222	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
223	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
224	D₃	—	6	40	no	—
225	C₂²	—	9	10	no	—
226	C₂²	—	36	15	no	—
227	C₂²	C₂³×C₃×C₃⋊D₃	1	5	yes	—
228	C₄	—	9	1	no	—
229	C₄	—	9	5	no	—
230	C₄	C₂³×C₃×C₃⋊D₃	1	10	yes	—
231	C₃	—	2	4	no	—
232	C₃	C₃×#5968	1	4	yes	complemented
233	C₃	#10596760	1	1	yes	characteristic, direct product factor
234	C₂	—	2	5	no	—
235	C₂	—	18	10	no	—
236	C₂	C₂⁴×C₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂⁴×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁵×C₃
Frattini	C₂	C₂⁴×C₃×C₃⋊D₃
Fitting	C₃³×2₋¹⁺⁴	C₂
socle	C₂×C₃³	C₂⁵
hypercenter	C₃×2₋¹⁺⁴	C₃⋊D₃
hypocenter	C₃²	C₃×#585
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₃³×2₋¹⁺⁴	C₂	semidirect
2. C₃×#64366	C₂	semidirect
3. C₃×#41×C₃⋊D₃	C₂	semidirect
4. C₃×#64364	C₂	semidirect
5. C₂×C₃×#3832	C₂	semidirect
6. #10596760	C₃	direct (N × H)
8. C₃³×#41	C₂²	semidirect
9. C₂×C₃³×Q₈	C₂²	semidirect
10. C₃×#3832	C₂²	semidirect
11. C₃×#3832	C₂²	semidirect
12. C₃×Q₈×C₃⋊D₃	C₂²	semidirect
13. C₃×#3830	C₂²	semidirect
14. C₃×D₄×C₃⋊D₃	C₂²	semidirect
15. C₃×#3828	C₂²	semidirect
16. C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	semidirect
17. C₂×C₃×C₄×C₃⋊D₃	C₂²	semidirect
19. C₃²×2₋¹⁺⁴	D₃	semidirect
20. C₃²×2₋¹⁺⁴	C₂×C₃	semidirect
21. #64366	C₂×C₃	semidirect
22. #41×C₃⋊D₃	C₂×C₃	semidirect
23. #64364	C₂×C₃	semidirect
24. C₂×#3832	C₂×C₃	semidirect
42. C₃²×#41	C₂×D₃	semidirect
43. C₃²×#41	C₂²×C₃	semidirect
45. C₂×C₃²×Q₈	C₂×D₃	semidirect
46. C₂×C₃²×Q₈	C₂²×C₃	semidirect
47. #3832	C₂²×C₃	semidirect
48. #3832	C₂²×C₃	semidirect
49. Q₈×C₃⋊D₃	C₂²×C₃	semidirect
50. #3830	C₂²×C₃	semidirect
51. D₄×C₃⋊D₃	C₂²×C₃	semidirect
52. #3828	C₂²×C₃	semidirect
53. C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	semidirect
54. C₂×C₄×C₃⋊D₃	C₂²×C₃	semidirect
71. C₃×2₋¹⁺⁴	C₃×D₃	semidirect
72. C₃×2₋¹⁺⁴	C₃⋊D₃	semidirect
112. C₃×#41	C₂×C₃×D₃	semidirect
113. C₃×#41	C₂×C₃⋊D₃	semidirect
116. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
117. C₂×C₃×Q₈	C₂×C₃⋊D₃	semidirect
144. 2₋¹⁺⁴	C₃×C₃⋊D₃	semidirect
148. C₃³	#585	semidirect
184. #41	C₂×C₃×C₃⋊D₃	semidirect
186. C₂×Q₈	C₂×C₃×C₃⋊D₃	semidirect
205. C₃²	#5968	semidirect
206. C₃²	C₃×#585	semidirect
232. C₃	C₃×#5968	semidirect
233. C₃	#10596760	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×2₋¹⁺⁴	2
C₃×#64366	2
C₃×#41×C₃⋊D₃	2
C₃×#64364	2
C₂×C₃×#3832	2
#10596760	3
C₃×#5968	3

Subgroups — C3×#10596760 #13181747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596760 #13181747