Subgroups — C₃×#10589854 #13153402

Order 1728.

866 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×#63715	C₂	1	2	yes	maximal, complemented
2	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63470	C₂	1	6	yes	maximal, complemented
4	C₃×#63467	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×D₇₂	C₂	1	3	yes	maximal, complemented
6	C₃×#5752	—	3	1	no	maximal
7	#10589854	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×C₉×#41	C₂²	1	1	yes	characteristic
9	C₃×#3701	—	2	2	no	—
10	C₃×D₄×D₉	—	2	6	no	—
11	C₃×#3697	C₂²	1	6	yes	complemented
12	C₂×C₃×D₃₆	C₂²	1	6	yes	complemented
13	C₃×C₉×M₁₆	C₂²	1	3	yes	complemented
14	C₂×C₃×C₈×C₉	C₂²	1	3	yes	complemented
15	C₃×D₇₂	C₂²	1	3	yes	complemented
16	C₃×D₇₂	C₂²	1	6	yes	complemented
17	C₃×#3664	C₂²	1	6	yes	complemented
18	C₃×Dic₃₆	C₂²	1	1	yes	characteristic, complemented
19	C₃×#1009	—	3	2	no	—
20	C₃²×#131	D₃	1	1	yes	characteristic
21	C₃×#908	—	3	6	no	—
22	C₃×#904	—	3	3	no	—
23	C₂×C₃×D₂₄	—	3	3	no	—
24	#63715	C₂×C₃	1	2	yes	complemented
25	C₉×#131	—	2	1	no	—
26	C₉×#131	C₂×C₃	1	1	yes	characteristic, complemented
27	#63470	C₂×C₃	1	6	yes	complemented
28	#63467	C₂×C₃	1	3	yes	complemented
29	C₂×D₇₂	C₂×C₃	1	3	yes	complemented
30	C₂²×C₃×D₉	—	2	6	no	—
31	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
32	C₃×D₄×C₉	D₄	1	3	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
34	C₃×#611	—	2	6	no	—
35	C₃×D₃₆	—	2	6	no	—
36	C₃×D₃₆	C₂³	1	6	yes	—
37	C₃×C₄×D₉	—	2	6	no	—
38	C₃×Dic₁₈	C₂³	1	2	yes	—
39	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
40	C₃×C₈×C₉	C₂³	1	3	yes	—
41	C₃×#576	—	9	1	no	—
42	#5752	—	3	1	no	—
43	C₃²×#41	C₂×D₃	1	1	yes	characteristic
44	C₃×#239	—	6	2	no	—
45	C₃×D₃×D₄	—	6	6	no	—
46	C₃×#235	—	3	6	no	—
47	C₂×C₃×D₁₂	—	3	6	no	—
48	C₃²×M₁₆	C₂×D₃	1	3	yes	—
49	C₂×C₃²×C₈	C₂×D₃	1	3	yes	—
50	C₃×Dic₁₂	—	3	1	no	—
51	C₃×D₂₄	—	3	3	no	—
52	C₃×D₂₄	—	3	6	no	—
53	C₃×#204	—	3	6	no	—
54	C₉×#41	—	2	1	no	—
55	C₉×#41	C₂²×C₃	1	1	yes	characteristic
56	#3701	—	2	2	no	—
57	D₄×D₉	—	2	6	no	—
58	#3697	C₂²×C₃	1	6	yes	complemented
59	C₂×D₃₆	C₂²×C₃	1	6	yes	complemented
60	C₉×M₁₆	—	2	3	no	—
61	C₉×M₁₆	C₂²×C₃	1	3	yes	complemented
62	C₂×C₈×C₉	—	2	3	no	—
63	C₂×C₈×C₉	C₂²×C₃	1	3	yes	complemented
64	D₇₂	C₂²×C₃	1	3	yes	complemented
65	D₇₂	C₂²×C₃	1	6	yes	complemented
66	#3664	C₂²×C₃	1	6	yes	complemented
67	Dic₃₆	C₂²×C₃	1	1	yes	characteristic, complemented
68	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
69	C₂×C₃×D₉	—	2	6	no	—
70	C₂×C₃×D₉	—	8	6	no	—
71	C₃×C₄×C₉	C₂×D₄	1	3	yes	—
72	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
73	C₃×Dic₉	—	2	2	no	—
74	C₃×H(5,𝔽₂)	—	9	2	no	—
75	#1009	—	3	2	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	1	no	—
80	C₃×#131	D₉	1	1	yes	characteristic, complemented
81	C₃×#131	C₃×D₃	1	1	yes	characteristic
82	#908	—	3	6	no	—
83	#904	—	3	3	no	—
84	C₂×D₂₄	—	3	3	no	—
85	C₂²×C₃×D₃	—	6	6	no	—
86	C₃²×Q₈	D₁₂	1	1	yes	characteristic
87	C₃²×D₄	D₁₂	1	3	yes	—
88	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
89	C₃×#67	—	6	6	no	—
90	C₃×D₁₂	—	3	6	no	—
91	C₃×D₁₂	—	6	6	no	—
92	C₃×C₄×D₃	—	6	6	no	—
93	C₃×Dic₆	—	3	2	no	—
94	C₂²×D₉	—	2	6	no	—
95	C₃²×C₈	C₂²×D₃	1	3	yes	—
96	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
97	Q₈×C₉	—	2	1	no	—
98	Q₈×C₉	C₃×D₄	1	1	yes	characteristic
99	D₄×C₉	—	2	3	no	—
100	D₄×C₉	C₃×D₄	1	3	yes	—
101	C₂×C₄×C₉	—	2	3	no	—
102	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
103	#611	—	2	6	no	—
104	D₃₆	—	2	6	no	—
105	D₃₆	C₂³×C₃	1	6	yes	—
106	C₄×D₉	—	2	6	no	—
107	Dic₁₈	C₂³×C₃	1	2	yes	—
108	C₈×C₉	—	2	1	no	—
109	C₈×C₉	—	2	3	no	—
110	C₈×C₉	C₂³×C₃	1	1	yes	characteristic
111	C₈×C₉	C₂³×C₃	1	3	yes	—
112	#576	—	9	1	no	Sylow
113	C₂×C₃×C₉	—	2	3	no	—
114	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
115	C₃×D₉	—	4	6	no	—
116	C₃×#41	—	2	1	no	—
117	C₃×#41	—	9	6	no	—
118	C₃×#41	—	18	2	no	—
119	C₃×#41	C₂×D₉	1	1	yes	characteristic
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	2	no	—
124	D₃×D₄	—	6	6	no	—
125	#235	—	3	6	no	—
126	C₂×D₁₂	—	3	6	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	3	no	—
132	C₃×M₁₆	C₂×D₉	1	3	yes	complemented
133	C₃×M₁₆	C₂×C₃×D₃	1	3	yes	—
134	C₂×C₃×C₈	—	2	3	no	—
135	C₂×C₃×C₈	C₂×D₉	1	3	yes	complemented
136	C₂×C₃×C₈	C₂×C₃×D₃	1	3	yes	—
137	Dic₁₂	—	3	1	no	—
138	D₂₄	—	3	3	no	—
139	D₂₄	—	3	6	no	—
140	#204	—	3	6	no	—
141	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
142	C₂×C₃×D₃	—	6	6	no	—
143	C₂×C₃×D₃	—	24	6	no	—
144	C₃²×C₄	C₂×D₁₂	1	3	yes	—
145	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
146	C₃×Dic₃	—	6	2	no	—
147	C₂²×C₉	—	2	3	no	—
148	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
149	C₂×D₉	—	2	6	no	—
150	C₂×D₉	—	8	6	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	3	no	—
153	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
154	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
155	Dic₉	—	2	2	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×D₉	1	1	yes	characteristic, complemented
161	C₃×C₉	#576	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	18	6	no	—
163	C₂²×D₃	—	6	6	no	—
164	C₃×Q₈	—	2	1	no	—
165	C₃×Q₈	—	9	2	no	—
166	C₃×Q₈	D₃₆	1	1	yes	characteristic
167	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
168	C₃×D₄	—	2	3	no	—
169	C₃×D₄	—	9	6	no	—
170	C₃×D₄	—	18	6	no	—
171	C₃×D₄	—	18	6	no	—
172	C₃×D₄	D₃₆	1	3	yes	—
173	C₃×D₄	C₃×D₁₂	1	3	yes	—
174	C₂×C₃×C₄	—	2	3	no	—
175	C₂×C₃×C₄	—	18	6	no	—
176	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
178	#67	—	6	6	no	—
179	D₁₂	—	3	6	no	—
180	D₁₂	—	6	6	no	—
181	C₄×D₃	—	6	6	no	—
182	Dic₆	—	3	2	no	—
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	2	3	no	—
185	C₃×C₈	C₂²×D₉	1	1	yes	characteristic
186	C₃×C₈	C₂²×D₉	1	3	yes	—
187	C₃×C₈	C₂²×C₃×D₃	1	3	yes	—
188	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
189	C₂×C₃²	—	2	3	no	—
190	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
191	C₃×D₃	—	12	6	no	—
192	C₂×C₉	—	2	3	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	4	3	no	—
195	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
196	D₉	—	4	6	no	—
197	#41	—	9	6	no	—
198	#41	—	18	2	no	—
199	#41	C₂×C₃×D₉	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×D₉	1	3	yes	complemented
207	C₂×C₈	C₂×C₃×D₉	1	3	yes	complemented
208	C₂²×C₃	—	2	3	no	—
209	C₂²×C₃	—	18	6	no	—
210	C₂²×C₃	—	72	6	no	—
211	C₂²×C₃	C₂×D₃₆	1	3	yes	—
212	C₂²×C₃	C₂×C₃×D₁₂	1	3	yes	—
213	C₂×D₃	—	6	6	no	—
214	C₂×D₃	—	24	6	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	3	no	—
217	C₃×C₄	—	18	2	no	—
218	C₃×C₄	C₂×D₃₆	1	3	yes	—
219	C₃×C₄	C₂³×D₉	1	1	yes	characteristic
220	C₃×C₄	C₂×C₃×D₁₂	1	3	yes	—
221	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
222	Dic₃	—	6	2	no	—
223	C₃²	#5752	1	1	yes	characteristic
224	C₉	—	2	1	no	—
225	C₉	C₃×#576	1	1	yes	characteristic, complemented
226	C₂³	—	18	6	no	—
227	Q₈	—	9	2	no	—
228	Q₈	C₃×D₃₆	1	1	yes	characteristic
229	D₄	—	9	6	no	—
230	D₄	—	18	6	no	—
231	D₄	—	18	6	no	—
232	D₄	C₃×D₃₆	1	3	yes	—
233	C₂×C₄	—	18	6	no	—
234	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
235	C₈	C₂²×C₃×D₉	1	1	yes	characteristic
236	C₈	C₂²×C₃×D₉	1	3	yes	—
237	C₂×C₃	—	2	3	no	—
238	C₂×C₃	—	2	3	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	4	3	no	—
241	C₂×C₃	—	36	6	no	—
242	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
243	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
244	D₃	—	12	6	no	—
245	C₂²	—	18	6	no	—
246	C₂²	—	72	6	no	—
247	C₂²	C₂×C₃×D₃₆	1	3	yes	—
248	C₄	—	18	2	no	—
249	C₄	C₂×C₃×D₃₆	1	3	yes	—
250	C₄	C₂³×C₃×D₉	1	1	yes	characteristic
251	C₃	—	2	1	no	—
252	C₃	#10589854	1	1	yes	characteristic, direct product factor
253	C₃	C₃×#5752	1	1	yes	characteristic
254	C₂	—	2	3	no	—
255	C₂	—	36	6	no	—
256	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₃₆
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₃×C₉×#131	C₂
socle	C₂×C₃²	C₂²×D₁₂
hypercenter	C₃×#131	D₉
hypocenter	C₉	C₃×#576
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
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×#63715	C₂	semidirect
2. C₃×C₉×#131	C₂	semidirect
3. C₃×#63470	C₂	semidirect
4. C₃×#63467	C₂	semidirect
5. C₂×C₃×D₇₂	C₂	semidirect
7. #10589854	C₃	direct (N × H)
11. C₃×#3697	C₂²	semidirect
12. C₂×C₃×D₃₆	C₂²	semidirect
13. C₃×C₉×M₁₆	C₂²	semidirect
14. C₂×C₃×C₈×C₉	C₂²	semidirect
15. C₃×D₇₂	C₂²	semidirect
16. C₃×D₇₂	C₂²	semidirect
17. C₃×#3664	C₂²	semidirect
18. C₃×Dic₃₆	C₂²	semidirect
24. #63715	C₂×C₃	semidirect
26. C₉×#131	C₂×C₃	semidirect
27. #63470	C₂×C₃	semidirect
28. #63467	C₂×C₃	semidirect
29. C₂×D₇₂	C₂×C₃	semidirect
58. #3697	C₂²×C₃	semidirect
59. C₂×D₃₆	C₂²×C₃	semidirect
61. C₉×M₁₆	C₂²×C₃	semidirect
63. C₂×C₈×C₉	C₂²×C₃	semidirect
64. D₇₂	C₂²×C₃	semidirect
65. D₇₂	C₂²×C₃	semidirect
66. #3664	C₂²×C₃	semidirect
67. Dic₃₆	C₂²×C₃	semidirect
80. C₃×#131	D₉	semidirect
132. C₃×M₁₆	C₂×D₉	semidirect
135. C₂×C₃×C₈	C₂×D₉	semidirect
160. #131	C₃×D₉	semidirect
161. C₃×C₉	#576	semidirect
206. M₁₆	C₂×C₃×D₉	semidirect
207. C₂×C₈	C₂×C₃×D₉	semidirect
225. C₉	C₃×#576	semidirect
252. C₃	#10589854	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×#63715	2
C₃×C₉×#131	2
C₃×#63470	2
C₃×#63467	2
C₂×C₃×D₇₂	2
C₃×#5752	3
#10589854	3

Subgroups — C3×#10589854 #13153402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589854 #13153402