Subgroups — C₃×#10589679 #13152683

Order 1728.

1006 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3701	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63457	C₂	1	6	yes	maximal, complemented
4	C₃×#63436	C₂	1	3	yes	maximal, complemented
5	C₃×C₄×D₃₆	C₂	1	3	yes	maximal, complemented
6	C₃×#5577	—	3	1	no	maximal
7	#10589679	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₃×#3701	—	2	8	no	—
10	C₂×C₃×D₃₆	C₂²	1	6	yes	—
11	C₂×C₃×D₃₆	C₂²	1	3	yes	—
12	C₂×C₃×D₃₆	C₂²	1	6	yes	—
13	C₂×C₃×C₄×D₉	C₂²	1	6	yes	—
14	C₃×C₉×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×C₄²×C₉	C₂²	1	3	yes	—
16	C₃×(C₂×D₉)⋊C₄	C₂²	1	6	yes	—
17	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₂×C₃×#239	—	3	2	no	—
19	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
20	C₃×#895	—	3	6	no	—
21	C₃×#874	—	3	3	no	—
22	C₃×C₄×D₁₂	—	3	3	no	—
23	C₂×#3701	C₂×C₃	1	2	yes	complemented
24	C₄×Q₈×C₉	—	2	1	no	—
25	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
26	#63457	C₂×C₃	1	6	yes	complemented
27	#63436	C₂×C₃	1	3	yes	complemented
28	C₄×D₃₆	C₂×C₃	1	3	yes	complemented
29	C₂²×C₃×D₉	C₂³	1	6	yes	—
30	C₃×Q₈×C₉	D₄	1	4	yes	complemented
31	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₃×Dic₉	C₂³	1	2	yes	—
35	C₃×D₃₆	—	2	12	no	—
36	C₃×D₃₆	—	2	6	no	—
37	C₃×D₃₆	—	4	6	no	—
38	C₃×C₄×D₉	—	2	12	no	—
39	C₃×#550	—	9	1	no	—
40	#5577	—	3	1	no	—
41	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
42	C₃×#239	—	6	8	no	—
43	C₂×C₃×D₁₂	—	3	6	no	—
44	C₂×C₃×D₁₂	—	3	3	no	—
45	C₂×C₃×D₁₂	—	3	6	no	—
46	C₂×C₃×C₄×D₃	—	3	6	no	—
47	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
48	C₃²×C₄²	C₂×D₃	1	3	yes	—
49	C₃×(C₂×D₃)⋊C₄	—	3	6	no	—
50	C₃×C₄⋊Dic₃	—	3	1	no	—
51	C₂×Q₈×C₉	—	2	1	no	—
52	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
53	#3701	—	2	8	no	—
54	C₂×D₃₆	C₂²×C₃	1	3	yes	—
55	C₂×D₃₆	C₂²×C₃	1	6	yes	—
56	C₂×D₃₆	C₂²×C₃	1	6	yes	—
57	C₂×C₄×D₉	C₂²×C₃	1	6	yes	—
58	C₉×C₄⋊C₄	—	2	3	no	—
59	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
60	C₄²×C₉	—	2	3	no	—
61	C₄²×C₉	C₂²×C₃	1	3	yes	—
62	(C₂×D₉)⋊C₄	C₂²×C₃	1	6	yes	—
63	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₃×D₉	—	2	6	no	—
66	C₂×C₃×D₉	—	2	6	no	—
67	C₂×C₃×D₉	—	2	6	no	—
68	C₃×C₄×C₉	—	2	3	no	—
69	C₃×C₄×C₉	C₂×D₄	1	6	yes	—
70	C₃×C₄×C₉	#41	1	2	yes	complemented
71	C₃×Dic₉	—	2	2	no	—
72	C₂×C₃×#41	—	9	2	no	—
73	C₂×#239	—	3	2	no	—
74	C₃×#127	—	9	3	no	—
75	C₃×#121	—	9	6	no	—
76	C₃×C₄×Q₈	—	2	1	no	—
77	C₃×C₄×Q₈	D₉	1	1	yes	characteristic, complemented
78	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
79	C₃×C₄×D₄	—	9	3	no	—
80	#895	—	3	6	no	—
81	#874	—	3	3	no	—
82	C₄×D₁₂	—	3	3	no	—
83	C₂²×C₃×D₃	—	3	6	no	—
84	C₃²×Q₈	D₁₂	1	4	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
87	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	3	2	no	—
89	C₃×D₁₂	—	6	12	no	—
90	C₃×D₁₂	—	6	6	no	—
91	C₃×D₁₂	—	12	6	no	—
92	C₃×C₄×D₃	—	6	12	no	—
93	C₂²×D₉	C₂³×C₃	1	6	yes	—
94	Q₈×C₉	—	2	4	no	—
95	Q₈×C₉	C₃×D₄	1	4	yes	complemented
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	3	no	—
98	C₂×C₄×C₉	—	2	3	no	—
99	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
100	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
101	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
102	C₂×Dic₉	C₂³×C₃	1	2	yes	—
103	D₃₆	—	2	12	no	—
104	D₃₆	—	2	6	no	—
105	D₃₆	—	4	6	no	—
106	C₄×D₉	—	2	12	no	—
107	#550	—	9	1	no	Sylow
108	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
110	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₃×D₉	—	4	6	no	—
112	C₃×#41	—	18	8	no	—
113	C₂×C₃×Q₈	—	2	1	no	—
114	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
115	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
116	C₂×C₃×D₄	—	9	3	no	—
117	C₂×C₃×D₄	—	9	6	no	—
118	C₂×C₃×D₄	—	9	6	no	—
119	C₂²×C₃×C₄	—	9	6	no	—
120	#239	—	6	8	no	—
121	C₂×D₁₂	—	3	3	no	—
122	C₂×D₁₂	—	3	6	no	—
123	C₂×D₁₂	—	3	6	no	—
124	C₂×C₄×D₃	—	3	6	no	—
125	C₃×C₄⋊C₄	—	2	3	no	—
126	C₃×C₄⋊C₄	—	9	1	no	—
127	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
128	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
129	C₃×C₂²⋊C₄	—	9	6	no	—
130	C₃×C₄²	—	2	3	no	—
131	C₃×C₄²	C₂×D₉	1	3	yes	—
132	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
133	(C₂×D₃)⋊C₄	—	3	6	no	—
134	C₄⋊Dic₃	—	3	1	no	—
135	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
136	C₂×C₃×D₃	—	6	6	no	—
137	C₂×C₃×D₃	—	6	6	no	—
138	C₂×C₃×D₃	—	6	6	no	—
139	C₃²×C₄	—	2	3	no	—
140	C₃²×C₄	C₂×D₁₂	1	6	yes	—
141	C₃²×C₄	#239	1	2	yes	—
142	C₃×Dic₃	—	6	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
145	C₂×D₉	—	2	6	no	—
146	C₂×D₉	—	2	6	no	—
147	C₂×D₉	—	2	6	no	—
148	C₄×C₉	—	2	3	no	—
149	C₄×C₉	—	2	6	no	—
150	C₄×C₉	—	2	2	no	—
151	C₄×C₉	—	4	3	no	—
152	C₄×C₉	C₂×C₃×D₄	1	6	yes	—
153	C₄×C₉	C₃×#41	1	2	yes	complemented
154	Dic₉	—	2	2	no	—
155	C₂×#41	—	9	2	no	—
156	#127	—	9	3	no	—
157	#121	—	9	6	no	—
158	C₄×Q₈	C₃×D₉	1	1	yes	characteristic, complemented
159	C₄×D₄	—	9	3	no	—
160	C₃×C₉	#550	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	9	6	no	—
162	C₂²×D₃	—	3	6	no	—
163	C₃×Q₈	—	2	4	no	—
164	C₃×Q₈	D₃₆	1	4	yes	complemented
165	C₃×Q₈	C₃×D₁₂	1	4	yes	—
166	C₃×D₄	—	18	12	no	—
167	C₃×D₄	—	18	6	no	—
168	C₃×D₄	—	36	6	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	3	no	—
171	C₂×C₃×C₄	—	2	3	no	—
172	C₂×C₃×C₄	—	9	2	no	—
173	C₂×C₃×C₄	—	18	12	no	—
174	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
175	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
176	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
178	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
179	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
180	C₂×Dic₃	—	3	2	no	—
181	D₁₂	—	6	12	no	—
182	D₁₂	—	6	6	no	—
183	D₁₂	—	12	6	no	—
184	C₄×D₃	—	6	12	no	—
185	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
186	C₂×C₃²	C₂×#239	1	1	yes	characteristic
187	C₂×C₃²	#1009	1	1	yes	characteristic
188	C₃×D₃	—	12	6	no	—
189	C₂×C₉	—	2	1	no	—
190	C₂×C₉	—	2	1	no	—
191	C₂×C₉	—	2	1	no	—
192	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
193	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
194	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
195	D₉	—	4	6	no	—
196	#41	—	18	8	no	—
197	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
198	C₂×D₄	—	9	3	no	—
199	C₂×D₄	—	9	6	no	—
200	C₂×D₄	—	9	6	no	—
201	C₂²×C₄	—	9	6	no	—
202	C₄⋊C₄	—	9	1	no	—
203	C₄⋊C₄	C₂×C₃×D₉	1	3	yes	—
204	C₂²⋊C₄	—	9	6	no	—
205	C₄²	C₂×C₃×D₉	1	3	yes	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	18	6	no	—
208	C₂²×C₃	—	18	6	no	—
209	C₂²×C₃	—	18	6	no	—
210	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
211	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
212	C₂×D₃	—	6	6	no	—
213	C₂×D₃	—	6	6	no	—
214	C₂×D₃	—	6	6	no	—
215	C₃×C₄	—	2	3	no	—
216	C₃×C₄	—	2	3	no	—
217	C₃×C₄	—	2	6	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	3	no	—
220	C₃×C₄	—	18	2	no	—
221	C₃×C₄	C₂×D₃₆	1	6	yes	—
222	C₃×C₄	#3701	1	2	yes	complemented
223	C₃×C₄	C₂×C₃×D₁₂	1	6	yes	—
224	C₃×C₄	C₃×#239	1	2	yes	—
225	Dic₃	—	6	2	no	—
226	C₃²	#5577	1	1	yes	characteristic
227	C₉	—	2	1	no	—
228	C₉	C₃×#550	1	1	yes	characteristic, complemented
229	C₂³	—	9	6	no	—
230	Q₈	C₃×D₃₆	1	4	yes	complemented
231	D₄	—	18	12	no	—
232	D₄	—	18	6	no	—
233	D₄	—	36	6	no	—
234	C₂×C₄	—	9	2	no	—
235	C₂×C₄	—	18	12	no	—
236	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
237	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
238	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	36	6	no	—
243	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
244	C₂×C₃	C₂×#3701	1	1	yes	characteristic
245	C₂×C₃	#63715	1	1	yes	characteristic
246	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
247	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
248	C₂×C₃	C₃×#1009	1	1	yes	characteristic
249	D₃	—	12	6	no	—
250	C₂²	—	18	6	no	—
251	C₂²	—	18	6	no	—
252	C₂²	—	18	6	no	—
253	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
254	C₄	—	2	3	no	—
255	C₄	—	18	2	no	—
256	C₄	C₂×C₃×D₃₆	1	6	yes	—
257	C₄	C₃×#3701	1	2	yes	complemented
258	C₃	—	2	1	no	—
259	C₃	#10589679	1	1	yes	characteristic, direct product factor
260	C₃	C₃×#5577	1	1	yes	characteristic
261	C₂	—	36	6	no	—
262	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic
263	C₂	C₂×C₃×#3701	1	1	yes	characteristic
264	C₂	C₃×#63715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₃×C₄×Q₈	D₉
hypocenter	C₉	C₃×#550
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3701	C₂	semidirect
2. C₃×C₄×Q₈×C₉	C₂	semidirect
3. C₃×#63457	C₂	semidirect
4. C₃×#63436	C₂	semidirect
5. C₃×C₄×D₃₆	C₂	semidirect
7. #10589679	C₃	direct (N × H)
23. C₂×#3701	C₂×C₃	semidirect
25. C₄×Q₈×C₉	C₂×C₃	semidirect
26. #63457	C₂×C₃	semidirect
27. #63436	C₂×C₃	semidirect
28. C₄×D₃₆	C₂×C₃	semidirect
30. C₃×Q₈×C₉	D₄	semidirect
70. C₃×C₄×C₉	#41	semidirect
77. C₃×C₄×Q₈	D₉	semidirect
95. Q₈×C₉	C₃×D₄	semidirect
153. C₄×C₉	C₃×#41	semidirect
158. C₄×Q₈	C₃×D₉	semidirect
160. C₃×C₉	#550	semidirect
164. C₃×Q₈	D₃₆	semidirect
222. C₃×C₄	#3701	semidirect
228. C₉	C₃×#550	semidirect
230. Q₈	C₃×D₃₆	semidirect
257. C₄	C₃×#3701	semidirect
259. C₃	#10589679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3701	2
C₃×C₄×Q₈×C₉	2
C₃×#63457	2
C₃×#63436	2
C₃×C₄×D₃₆	2
C₃×#5577	3
#10589679	3

Subgroups — C3×#10589679 #13152683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589679 #13152683