Subgroups — C₃×#10588655 #13142549

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×Dic₉⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
5	C₃×#4975	—	3	1	no	maximal
6	#10588655	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	—	2	2	no	—
12	C₃×C₄⋊Dic₉	—	2	2	no	—
13	C₃×Dic₉⋊C₄	—	2	8	no	—
14	C₃×C₄×Dic₉	C₄	1	4	yes	complemented
15	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
17	C₂×C₃×Dic₃⋊C₄	—	3	4	no	—
18	C₂×C₃×C₄×Dic₃	—	3	1	no	—
19	C₂×C₉×C₄⋊C₄	—	2	1	no	—
20	C₂×C₉×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄⋊Dic₉	C₂×C₃	1	1	yes	characteristic
22	C₂×Dic₉⋊C₄	C₂×C₃	1	4	yes	—
23	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
24	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	2	4	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
30	C₂×C₃×Dic₉	—	2	4	no	—
31	C₂×C₃×Dic₉	—	2	2	no	—
32	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
33	C₂×C₃×Dic₉	D₄	1	4	yes	—
34	C₂×C₃×Dic₉	Q₈	1	4	yes	—
35	C₃×#384	—	9	1	no	—
36	#4975	—	3	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	3	2	no	—
40	C₂²×C₃×Dic₃	—	3	2	no	—
41	C₃²×C₄⋊C₄	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	6	2	no	—
43	C₃×Dic₃⋊C₄	—	6	8	no	—
44	C₃×C₄×Dic₃	—	3	4	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	—	2	2	no	—
47	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
48	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
50	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
51	C₉×C₄⋊C₄	—	2	4	no	—
52	C₉×C₄⋊C₄	—	2	2	no	—
53	C₄⋊Dic₉	—	2	2	no	—
54	Dic₉⋊C₄	—	2	8	no	—
55	C₄×Dic₉	C₃×C₄	1	4	yes	complemented
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
61	C₃×C₄×C₉	—	4	2	no	—
62	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
63	C₃×Dic₉	—	4	2	no	—
64	C₃×Dic₉	C₄⋊C₄	1	8	yes	complemented
65	C₂×C₃×C₄⋊C₄	—	2	1	no	—
66	C₂×C₃×C₄⋊C₄	—	9	4	no	—
67	C₂×C₃×C₄⋊C₄	—	9	1	no	—
68	C₂×C₃×C₄⋊C₄	D₉	1	1	yes	characteristic
69	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
70	C₂×C₃×C₄²	—	9	1	no	—
71	C₂×C₄⋊Dic₃	—	3	1	no	—
72	C₂×Dic₃⋊C₄	—	3	4	no	—
73	C₂×C₄×Dic₃	—	3	1	no	—
74	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	Dic₆	1	2	yes	—
78	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
79	C₂×C₃²×C₄	#67	1	2	yes	—
80	C₂×C₃×Dic₃	—	3	4	no	—
81	C₂×C₃×Dic₃	—	3	4	no	—
82	C₂×C₃×Dic₃	—	3	4	no	—
83	C₂×C₃×Dic₃	—	6	4	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	4	4	no	—
94	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
97	C₂×Dic₉	—	2	4	no	—
98	C₂×Dic₉	—	2	2	no	—
99	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
100	C₂×Dic₉	C₃×D₄	1	4	yes	—
101	C₂×Dic₉	C₃×Q₈	1	4	yes	—
102	#384	—	9	1	no	Sylow
103	C₂×C₃×C₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₉	#127	1	1	yes	characteristic
106	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
107	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	—	9	2	no	—
111	C₂²×C₃×C₄	—	9	2	no	—
112	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
113	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
116	C₂²×Dic₃	—	3	2	no	—
117	C₂²×Dic₃	—	3	2	no	—
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	—	18	8	no	—
122	C₃×C₄⋊C₄	—	18	2	no	—
123	C₃×C₄²	—	9	4	no	—
124	C₄⋊Dic₃	—	6	2	no	—
125	Dic₃⋊C₄	—	6	8	no	—
126	C₄×Dic₃	—	3	4	no	—
127	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
128	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂²×C₃²	D₃×D₄	1	2	yes	—
130	C₂²×C₃²	D₃×Q₈	1	2	yes	—
131	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
132	C₃²×C₄	—	4	2	no	—
133	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
134	C₃×Dic₃	—	3	8	no	—
135	C₃×Dic₃	—	12	2	no	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	2	1	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂²×C₉	C₂×C₃×D₄	1	2	yes	—
143	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
145	C₂²×C₉	C₂×C₃×Q₈	1	2	yes	—
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	—	4	4	no	—
148	C₄×C₉	—	4	2	no	—
149	C₄×C₉	C₃×C₄⋊C₄	1	4	yes	complemented
150	Dic₉	—	4	2	no	—
151	Dic₉	C₃×C₄⋊C₄	1	8	yes	complemented
152	C₂×C₄⋊C₄	—	9	4	no	—
153	C₂×C₄⋊C₄	—	9	1	no	—
154	C₂×C₄⋊C₄	C₃×D₉	1	1	yes	characteristic
155	C₂×C₄²	—	9	1	no	—
156	C₃×C₉	#384	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
159	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	4	4	no	—
169	C₂×C₃×C₄	—	9	4	no	—
170	C₂×C₃×C₄	—	9	4	no	—
171	C₂×C₃×C₄	—	9	4	no	—
172	C₂×C₃×C₄	—	18	4	no	—
173	C₂×C₃×C₄	—	18	2	no	—
174	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
175	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
176	C₂×C₃×C₄	#611	1	2	yes	—
177	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
178	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
179	C₂×C₃×C₄	C₃×#67	1	2	yes	—
180	C₂×Dic₃	—	3	4	no	—
181	C₂×Dic₃	—	3	4	no	—
182	C₂×Dic₃	—	3	4	no	—
183	C₂×Dic₃	—	6	4	no	—
184	C₂×Dic₃	—	6	2	no	—
185	C₂×C₃²	#888	1	2	yes	—
186	C₂×C₃²	D₃×C₄⋊C₄	1	2	yes	—
187	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
188	C₂×C₃²	#940	1	1	yes	characteristic
189	C₂×C₃²	#944	1	1	yes	characteristic
190	C₂×C₉	—	2	2	no	—
191	C₂×C₉	—	2	2	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	C₂×C₃×C₄⋊C₄	1	2	yes	—
196	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₉	C₃×#127	1	1	yes	characteristic
198	C₂×C₉	C₃×C₄⋊Q₈	1	2	yes	—
199	C₂×C₉	C₃×C₄⋊Q₈	1	1	yes	characteristic
200	C₂²×C₄	—	9	2	no	—
201	C₂²×C₄	—	9	2	no	—
202	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
203	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	18	8	no	—
206	C₄⋊C₄	—	18	2	no	—
207	C₄²	—	9	4	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
214	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
215	C₂²×C₃	D₄×D₉	1	2	yes	—
216	C₂²×C₃	Q₈×D₉	1	2	yes	—
217	C₂²×C₃	C₂×#611	1	1	yes	characteristic
218	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
220	C₂²×C₃	C₃×D₃×D₄	1	2	yes	—
221	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
222	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	4	4	no	—
225	C₃×C₄	—	4	2	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	9	8	no	—
228	C₃×C₄	—	36	2	no	—
229	C₃×C₄	Dic₉⋊C₄	1	4	yes	complemented
230	C₃×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
231	Dic₃	—	3	8	no	—
232	Dic₃	—	12	2	no	—
233	C₃²	#4975	1	1	yes	characteristic
234	C₉	—	2	1	no	—
235	C₉	C₃×#384	1	1	yes	characteristic, complemented
236	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	9	4	no	—
240	C₂×C₄	—	9	4	no	—
241	C₂×C₄	—	9	4	no	—
242	C₂×C₄	—	18	4	no	—
243	C₂×C₄	—	18	2	no	—
244	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
245	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
246	C₂×C₄	C₃×#611	1	2	yes	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	#63450	1	2	yes	—
253	C₂×C₃	C₄⋊C₄×D₉	1	2	yes	—
254	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
255	C₂×C₃	#63502	1	1	yes	characteristic
256	C₂×C₃	#63506	1	1	yes	characteristic
257	C₂×C₃	C₃×#888	1	2	yes	—
258	C₂×C₃	C₃×D₃×C₄⋊C₄	1	2	yes	—
259	C₂×C₃	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
260	C₂×C₃	C₃×#940	1	1	yes	characteristic
261	C₂×C₃	C₃×#944	1	1	yes	characteristic
262	C₂²	C₂×C₃×Dic₁₈	1	1	yes	characteristic
263	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
264	C₂²	C₃×D₄×D₉	1	2	yes	—
265	C₂²	C₃×Q₈×D₉	1	2	yes	—
266	C₂²	C₂×C₃×#611	1	1	yes	characteristic
267	C₄	—	4	2	no	—
268	C₄	—	9	8	no	—
269	C₄	—	36	2	no	—
270	C₄	C₃×Dic₉⋊C₄	1	4	yes	complemented
271	C₃	—	2	1	no	—
272	C₃	#10588655	1	1	yes	characteristic, direct product factor
273	C₃	C₃×#4975	1	1	yes	characteristic
274	C₂	C₃×#63450	1	2	yes	—
275	C₂	C₃×C₄⋊C₄×D₉	1	2	yes	—
276	C₂	C₂×C₃×Dic₉⋊C₄	1	1	yes	characteristic
277	C₂	C₃×#63502	1	1	yes	characteristic
278	C₂	C₃×#63506	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₄	D₉
hypocenter	C₉	C₃×#384
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #10588655	C₃	direct (N × H)
14. C₃×C₄×Dic₉	C₄	semidirect
55. C₄×Dic₉	C₃×C₄	semidirect
62. C₃×C₄×C₉	C₄⋊C₄	semidirect
64. C₃×Dic₉	C₄⋊C₄	semidirect
149. C₄×C₉	C₃×C₄⋊C₄	semidirect
151. Dic₉	C₃×C₄⋊C₄	semidirect
156. C₃×C₉	#384	semidirect
229. C₃×C₄	Dic₉⋊C₄	semidirect
235. C₉	C₃×#384	semidirect
270. C₄	C₃×Dic₉⋊C₄	semidirect
272. C₃	#10588655	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×C₉×C₄⋊C₄	2
C₂×C₃×C₄⋊Dic₉	2
C₂×C₃×Dic₉⋊C₄	2
C₂×C₃×C₄×Dic₉	2
C₃×#4975	3
#10588655	3

Subgroups — C3×#10588655 #13142549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588655 #13142549