Subgroups — C₃×#10596223 #13179503

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64366	C₂	1	2	yes	maximal, complemented
2	C₃³×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64117	C₂	1	6	yes	maximal, complemented
4	C₃×#64113	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#3745	C₂	1	3	yes	maximal, complemented
6	#10596223	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5752	—	3	4	no	maximal
8	C₃³×#41	C₂²	1	1	yes	characteristic
9	C₃×#3832	—	2	2	no	—
10	C₃×D₄×C₃⋊D₃	—	2	6	no	—
11	C₃×#3828	C₂²	1	6	yes	complemented
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	6	yes	complemented
13	C₃³×M₁₆	C₂²	1	3	yes	complemented
14	C₂×C₃³×C₈	C₂²	1	3	yes	complemented
15	C₃×#3746	C₂²	1	1	yes	characteristic, complemented
16	C₃×#3745	C₂²	1	3	yes	complemented
17	C₃×#3745	C₂²	1	6	yes	complemented
18	C₃×#3744	C₂²	1	6	yes	complemented
19	#64366	C₂×C₃	1	2	yes	complemented
20	C₃×#1009	—	3	8	no	—
21	C₃²×#131	—	2	4	no	—
22	C₃²×#131	D₃	1	4	yes	complemented
23	C₃²×#131	C₂×C₃	1	1	yes	characteristic, complemented
24	#64117	C₂×C₃	1	6	yes	complemented
25	#64113	C₂×C₃	1	3	yes	complemented
26	C₂×#3745	C₂×C₃	1	3	yes	complemented
27	C₃×#908	—	3	24	no	—
28	C₃×#904	—	3	12	no	—
29	C₂×C₃×D₂₄	—	3	12	no	—
30	C₂²×C₃×C₃⋊D₃	—	2	6	no	—
31	C₃³×Q₈	D₄	1	1	yes	characteristic
32	C₃³×D₄	D₄	1	3	yes	—
33	C₂×C₃³×C₄	C₂³	1	3	yes	—
34	C₃×#638	—	2	6	no	—
35	C₃×C₄⋊(C₃⋊D₃)	—	2	6	no	—
36	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	6	yes	—
37	C₃×C₄×C₃⋊D₃	—	2	6	no	—
38	C₃×#634	C₂³	1	2	yes	—
39	C₃³×C₈	C₂³	1	3	yes	—
40	C₃³×C₈	C₂³	1	1	yes	characteristic
41	C₃×#576	—	9	1	no	—
42	#5752	—	3	4	no	—
43	C₃²×#41	—	2	4	no	—
44	C₃²×#41	C₂×D₃	1	4	yes	—
45	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
46	#3832	—	2	2	no	—
47	D₄×C₃⋊D₃	—	2	6	no	—
48	#3828	C₂²×C₃	1	6	yes	complemented
49	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	6	yes	complemented
50	C₃×#239	—	6	8	no	—
51	C₃×D₃×D₄	—	6	24	no	—
52	C₃×#235	—	3	24	no	—
53	C₂×C₃×D₁₂	—	3	24	no	—
54	C₃²×M₁₆	—	2	12	no	—
55	C₃²×M₁₆	C₂×D₃	1	12	yes	complemented
56	C₃²×M₁₆	C₂²×C₃	1	3	yes	complemented
57	C₂×C₃²×C₈	—	2	12	no	—
58	C₂×C₃²×C₈	C₂×D₃	1	12	yes	complemented
59	C₂×C₃²×C₈	C₂²×C₃	1	3	yes	complemented
60	#3746	C₂²×C₃	1	1	yes	characteristic, complemented
61	#3745	C₂²×C₃	1	3	yes	complemented
62	#3745	C₂²×C₃	1	6	yes	complemented
63	#3744	C₂²×C₃	1	6	yes	complemented
64	C₃×Dic₁₂	—	3	4	no	—
65	C₃×D₂₄	—	3	12	no	—
66	C₃×D₂₄	—	3	24	no	—
67	C₃×#204	—	3	24	no	—
68	C₂²×C₃³	C₂×D₄	1	3	yes	—
69	C₂×C₃×C₃⋊D₃	—	2	6	no	—
70	C₂×C₃×C₃⋊D₃	—	8	6	no	—
71	C₃³×C₄	C₂×D₄	1	3	yes	—
72	C₃³×C₄	C₂⁴	1	1	yes	characteristic
73	C₃×C₃⋊Dic₃	—	2	2	no	—
74	C₃×H(5,𝔽₂)	—	9	2	no	—
75	#1009	—	3	8	no	—
76	C₃×AGL(1,ℤ/8)	—	9	6	no	—
77	C₃×#135	—	9	3	no	—
78	C₂×C₃×D₈	—	9	3	no	—
79	C₃×#131	—	2	4	no	—
80	C₃×#131	C₃×D₃	1	4	yes	complemented
81	C₃×#131	C₃⋊D₃	1	1	yes	characteristic, complemented
82	#908	—	3	24	no	—
83	#904	—	3	12	no	—
84	C₂×D₂₄	—	3	12	no	—
85	C₂²×C₃⋊D₃	—	2	6	no	—
86	C₂²×C₃×D₃	—	6	24	no	—
87	C₃²×Q₈	—	2	4	no	—
88	C₃²×Q₈	D₁₂	1	4	yes	—
89	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
90	C₃²×D₄	—	2	12	no	—
91	C₃²×D₄	D₁₂	1	12	yes	—
92	C₃²×D₄	C₃×D₄	1	3	yes	—
93	C₂×C₃²×C₄	—	2	12	no	—
94	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
95	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
96	#638	—	2	6	no	—
97	C₄⋊(C₃⋊D₃)	—	2	6	no	—
98	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	6	yes	—
99	C₄×C₃⋊D₃	—	2	6	no	—
100	#634	C₂³×C₃	1	2	yes	—
101	C₃×#67	—	6	24	no	—
102	C₃×D₁₂	—	3	24	no	—
103	C₃×D₁₂	—	6	24	no	—
104	C₃×C₄×D₃	—	6	24	no	—
105	C₃×Dic₆	—	3	8	no	—
106	C₃²×C₈	—	2	4	no	—
107	C₃²×C₈	—	2	12	no	—
108	C₃²×C₈	C₂²×D₃	1	12	yes	—
109	C₃²×C₈	C₂²×D₃	1	4	yes	—
110	C₃²×C₈	C₂³×C₃	1	3	yes	—
111	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
112	#576	—	9	1	no	Sylow
113	C₂×C₃³	—	2	3	no	—
114	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
115	C₃×C₃⋊D₃	—	4	6	no	—
116	C₃×#41	—	2	4	no	—
117	C₃×#41	—	9	6	no	—
118	C₃×#41	—	18	2	no	—
119	C₃×#41	C₂×C₃×D₃	1	4	yes	—
120	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
121	C₂×C₃×D₄	—	9	6	no	—
122	C₂×C₃×D₄	—	18	6	no	—
123	#239	—	6	8	no	—
124	D₃×D₄	—	6	24	no	—
125	#235	—	3	24	no	—
126	C₂×D₁₂	—	3	24	no	—
127	C₃×Q₁₆	—	9	1	no	—
128	C₃×SD₁₆	—	9	6	no	—
129	C₃×D₈	—	9	3	no	—
130	C₃×D₈	—	9	6	no	—
131	C₃×M₁₆	—	2	12	no	—
132	C₃×M₁₆	C₂×C₃×D₃	1	12	yes	complemented
133	C₃×M₁₆	C₂×C₃⋊D₃	1	3	yes	complemented
134	C₂×C₃×C₈	—	2	12	no	—
135	C₂×C₃×C₈	C₂×C₃×D₃	1	12	yes	complemented
136	C₂×C₃×C₈	C₂×C₃⋊D₃	1	3	yes	complemented
137	Dic₁₂	—	3	4	no	—
138	D₂₄	—	3	12	no	—
139	D₂₄	—	3	24	no	—
140	#204	—	3	24	no	—
141	C₂²×C₃²	—	2	12	no	—
142	C₂²×C₃²	C₂×D₁₂	1	12	yes	—
143	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
144	C₂×C₃⋊D₃	—	2	6	no	—
145	C₂×C₃⋊D₃	—	8	6	no	—
146	C₂×C₃×D₃	—	6	24	no	—
147	C₂×C₃×D₃	—	24	24	no	—
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	—	2	12	no	—
150	C₃²×C₄	C₂×D₁₂	1	12	yes	—
151	C₃²×C₄	C₂×C₃×D₄	1	3	yes	—
152	C₃²×C₄	C₂³×D₃	1	4	yes	—
153	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
154	C₃⋊Dic₃	—	2	2	no	—
155	C₃×Dic₃	—	6	8	no	—
156	H(5,𝔽₂)	—	9	2	no	—
157	AGL(1,ℤ/8)	—	9	6	no	—
158	#135	—	9	3	no	—
159	C₂×D₈	—	9	3	no	—
160	#131	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
161	C₃³	#576	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	18	6	no	—
163	C₂²×D₃	—	6	24	no	—
164	C₃×Q₈	—	2	4	no	—
165	C₃×Q₈	—	9	2	no	—
166	C₃×Q₈	C₃×D₁₂	1	4	yes	—
167	C₃×Q₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
168	C₃×D₄	—	2	12	no	—
169	C₃×D₄	—	9	6	no	—
170	C₃×D₄	—	18	6	no	—
171	C₃×D₄	—	18	6	no	—
172	C₃×D₄	C₃×D₁₂	1	12	yes	—
173	C₃×D₄	C₄⋊(C₃⋊D₃)	1	3	yes	—
174	C₂×C₃×C₄	—	2	12	no	—
175	C₂×C₃×C₄	—	18	6	no	—
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
177	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
178	#67	—	6	24	no	—
179	D₁₂	—	3	24	no	—
180	D₁₂	—	6	24	no	—
181	C₄×D₃	—	6	24	no	—
182	Dic₆	—	3	8	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	—	2	12	no	—
185	C₃×C₈	C₂²×C₃×D₃	1	12	yes	—
186	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
187	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
188	C₃×C₈	C₂²×C₃⋊D₃	1	3	yes	—
189	C₂×C₃²	—	2	3	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	—	2	12	no	—
192	C₂×C₃²	—	4	12	no	—
193	C₂×C₃²	C₂²×D₁₂	1	4	yes	—
194	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
195	C₃⋊D₃	—	4	6	no	—
196	C₃×D₃	—	12	24	no	—
197	#41	—	9	6	no	—
198	#41	—	18	2	no	—
199	#41	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
200	C₂×D₄	—	9	6	no	—
201	C₂×D₄	—	18	6	no	—
202	Q₁₆	—	9	1	no	—
203	SD₁₆	—	9	6	no	—
204	D₈	—	9	3	no	—
205	D₈	—	9	6	no	—
206	M₁₆	C₂×C₃×C₃⋊D₃	1	3	yes	complemented
207	C₂×C₈	C₂×C₃×C₃⋊D₃	1	3	yes	complemented
208	C₂²×C₃	—	2	12	no	—
209	C₂²×C₃	—	18	6	no	—
210	C₂²×C₃	—	72	6	no	—
211	C₂²×C₃	C₂×C₃×D₁₂	1	12	yes	—
212	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
213	C₂×D₃	—	6	24	no	—
214	C₂×D₃	—	24	24	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	12	no	—
217	C₃×C₄	—	18	2	no	—
218	C₃×C₄	C₂×C₃×D₁₂	1	12	yes	—
219	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
220	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
221	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
222	Dic₃	—	6	8	no	—
223	C₃²	—	2	4	no	—
224	C₃²	#5752	1	4	yes	complemented
225	C₃²	C₃×#576	1	1	yes	characteristic, complemented
226	C₂³	—	18	6	no	—
227	Q₈	—	9	2	no	—
228	Q₈	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
229	D₄	—	9	6	no	—
230	D₄	—	18	6	no	—
231	D₄	—	18	6	no	—
232	D₄	C₃×C₄⋊(C₃⋊D₃)	1	3	yes	—
233	C₂×C₄	—	18	6	no	—
234	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
235	C₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
236	C₈	C₂²×C₃×C₃⋊D₃	1	3	yes	—
237	C₂×C₃	—	2	12	no	—
238	C₂×C₃	—	2	3	no	—
239	C₂×C₃	—	2	4	no	—
240	C₂×C₃	—	4	12	no	—
241	C₂×C₃	—	36	6	no	—
242	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
243	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
244	D₃	—	12	24	no	—
245	C₂²	—	18	6	no	—
246	C₂²	—	72	6	no	—
247	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	3	yes	—
248	C₄	—	18	2	no	—
249	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	3	yes	—
250	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
251	C₃	—	2	4	no	—
252	C₃	C₃×#5752	1	4	yes	complemented
253	C₃	#10596223	1	1	yes	characteristic, direct product factor
254	C₂	—	2	3	no	—
255	C₂	—	36	6	no	—
256	C₂	C₂²×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×#131	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃×#131	C₃⋊D₃
hypocenter	C₃²	C₃×#576
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×#131

Complements

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

N	H ≅ G/N	type
1. C₃×#64366	C₂	semidirect
2. C₃³×#131	C₂	semidirect
3. C₃×#64117	C₂	semidirect
4. C₃×#64113	C₂	semidirect
5. C₂×C₃×#3745	C₂	semidirect
6. #10596223	C₃	direct (N × H)
11. C₃×#3828	C₂²	semidirect
12. C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	semidirect
13. C₃³×M₁₆	C₂²	semidirect
14. C₂×C₃³×C₈	C₂²	semidirect
15. C₃×#3746	C₂²	semidirect
16. C₃×#3745	C₂²	semidirect
17. C₃×#3745	C₂²	semidirect
18. C₃×#3744	C₂²	semidirect
19. #64366	C₂×C₃	semidirect
22. C₃²×#131	D₃	semidirect
23. C₃²×#131	C₂×C₃	semidirect
24. #64117	C₂×C₃	semidirect
25. #64113	C₂×C₃	semidirect
26. C₂×#3745	C₂×C₃	semidirect
48. #3828	C₂²×C₃	semidirect
49. C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	semidirect
55. C₃²×M₁₆	C₂×D₃	semidirect
56. C₃²×M₁₆	C₂²×C₃	semidirect
58. C₂×C₃²×C₈	C₂×D₃	semidirect
59. C₂×C₃²×C₈	C₂²×C₃	semidirect
60. #3746	C₂²×C₃	semidirect
61. #3745	C₂²×C₃	semidirect
62. #3745	C₂²×C₃	semidirect
63. #3744	C₂²×C₃	semidirect
80. C₃×#131	C₃×D₃	semidirect
81. C₃×#131	C₃⋊D₃	semidirect
132. C₃×M₁₆	C₂×C₃×D₃	semidirect
133. C₃×M₁₆	C₂×C₃⋊D₃	semidirect
135. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
136. C₂×C₃×C₈	C₂×C₃⋊D₃	semidirect
160. #131	C₃×C₃⋊D₃	semidirect
161. C₃³	#576	semidirect
206. M₁₆	C₂×C₃×C₃⋊D₃	semidirect
207. C₂×C₈	C₂×C₃×C₃⋊D₃	semidirect
224. C₃²	#5752	semidirect
225. C₃²	C₃×#576	semidirect
252. C₃	C₃×#5752	semidirect
253. C₃	#10596223	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×#64366	2
C₃³×#131	2
C₃×#64117	2
C₃×#64113	2
C₂×C₃×#3745	2
#10596223	3
C₃×#5752	3

Subgroups — C3×#10596223 #13179503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596223 #13179503