Subgroups — C₃×#10588796 #13143099

Order 1728.

478 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63492	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×D₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63380	C₂	1	2	yes	maximal, complemented
5	C₃×#63378	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5115	—	3	1	no	maximal
7	#10588796	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×#611	—	2	2	no	—
10	C₂×C₃×D₃₆	C₂²	1	2	yes	—
11	C₂×C₃×C₈×C₉	—	2	2	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×(C₂×D₉)⋊C₄	—	2	2	no	—
15	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
16	C₃×D₇₂	—	2	2	no	—
17	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
18	C₃×#930	—	3	2	no	—
19	C₂×C₃×D₂₄	—	3	1	no	—
20	C₃×#821	—	3	2	no	—
21	C₃×#818	—	3	1	no	—
22	C₂²×C₈×C₉	—	2	1	no	—
23	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
24	#63492	C₂×C₃	1	2	yes	complemented
25	C₂×D₇₂	C₂×C₃	1	1	yes	characteristic, complemented
26	#63380	C₂×C₃	1	2	yes	complemented
27	#63378	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₃×D₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₃×#611	—	4	4	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₃×D₃₆	—	2	2	no	—
37	C₃×D₃₆	—	2	2	no	—
38	C₃×C₈×C₉	—	2	1	no	—
39	C₃×C₈×C₉	D₄	1	2	yes	complemented
40	C₃×#466	—	9	1	no	—
41	#5115	—	3	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×#67	—	6	2	no	—
44	C₂×C₃×D₁₂	—	3	2	no	—
45	C₂×C₃²×C₈	—	2	2	no	—
46	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
48	C₃×(C₂×D₃)⋊C₄	—	6	2	no	—
49	C₃×C₄⋊Dic₃	—	3	2	no	—
50	C₃×D₂₄	—	6	2	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
53	C₂×#611	—	2	2	no	—
54	C₂×D₃₆	C₂²×C₃	1	2	yes	—
55	C₂×C₈×C₉	—	2	4	no	—
56	C₂×C₈×C₉	—	2	1	no	—
57	C₂×C₈×C₉	—	2	1	no	—
58	C₂×C₈×C₉	—	2	2	no	—
59	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
61	(C₂×D₉)⋊C₄	—	2	2	no	—
62	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
63	D₇₂	—	2	2	no	—
64	C₂²×C₃×C₉	—	2	1	no	—
65	C₂²×C₃×C₉	—	2	1	no	—
66	C₂²×C₃×C₉	D₈	1	2	yes	complemented
67	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×D₉	—	4	2	no	—
69	C₂×C₃×D₉	—	4	2	no	—
70	C₂×C₃×D₉	—	4	2	no	—
71	C₃×C₄×C₉	—	2	1	no	—
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	#41	1	1	yes	characteristic
74	C₃×Dic₉	—	4	2	no	—
75	C₂×C₃×D₈	—	9	1	no	—
76	C₂²×C₃×C₈	—	2	1	no	—
77	C₂²×C₃×C₈	D₉	1	1	yes	characteristic, complemented
78	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
79	C₃×#121	—	9	2	no	—
80	#930	—	3	2	no	—
81	C₂×D₂₄	—	3	1	no	—
82	C₃×#107	—	9	1	no	—
83	C₃×#102	—	9	2	no	—
84	#821	—	3	2	no	—
85	#818	—	3	1	no	—
86	C₂³×C₃²	D₁₂	1	1	yes	characteristic
87	C₂²×C₃×D₃	—	6	2	no	—
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₃×#67	—	12	4	no	—
93	C₂×C₃×Dic₃	—	6	2	no	—
94	C₃×D₁₂	—	6	2	no	—
95	C₃×D₁₂	—	6	2	no	—
96	C₂³×C₉	—	2	1	no	—
97	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
98	C₂²×D₉	—	2	2	no	—
99	C₃²×C₈	—	2	1	no	—
100	C₃²×C₈	#67	1	2	yes	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	2	1	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
109	#611	—	4	4	no	—
110	C₂×Dic₉	—	2	2	no	—
111	D₃₆	—	2	2	no	—
112	D₃₆	—	2	2	no	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	—	2	1	no	—
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	C₃×D₄	1	2	yes	complemented
117	#466	—	9	1	no	Sylow
118	C₂×C₃×C₉	—	2	2	no	—
119	C₂×C₃×C₉	#121	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₃×C₉	#135	1	1	yes	characteristic
122	C₃×D₉	—	8	2	no	—
123	C₂×C₃×D₄	—	9	2	no	—
124	C₂×C₃×D₄	—	18	2	no	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂×#67	—	6	2	no	—
129	C₂×D₁₂	—	3	2	no	—
130	C₃×D₈	—	18	2	no	—
131	C₂×C₃×C₈	—	2	4	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
140	C₃×C₄⋊C₄	—	9	2	no	—
141	C₃×C₂²⋊C₄	—	18	2	no	—
142	(C₂×D₃)⋊C₄	—	6	2	no	—
143	C₄⋊Dic₃	—	3	2	no	—
144	D₂₄	—	6	2	no	—
145	C₂²×C₃²	—	2	1	no	—
146	C₂²×C₃²	—	2	1	no	—
147	C₂²×C₃²	D₂₄	1	2	yes	—
148	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
149	C₂×C₃×D₃	—	12	2	no	—
150	C₂×C₃×D₃	—	12	2	no	—
151	C₂×C₃×D₃	—	12	2	no	—
152	C₃²×C₄	—	2	1	no	—
153	C₃²×C₄	#235	1	1	yes	characteristic
154	C₃²×C₄	C₂×#67	1	1	yes	characteristic
155	C₃×Dic₃	—	12	2	no	—
156	C₂²×C₉	—	2	2	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	1	no	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	C₃×D₈	1	2	yes	complemented
163	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×D₉	—	4	2	no	—
165	C₂×D₉	—	4	2	no	—
166	C₂×D₉	—	4	2	no	—
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
172	C₄×C₉	C₃×#41	1	1	yes	characteristic
173	Dic₉	—	4	2	no	—
174	C₂×D₈	—	9	1	no	—
175	C₂²×C₈	C₃×D₉	1	1	yes	characteristic, complemented
176	#121	—	9	2	no	—
177	#107	—	9	1	no	—
178	#102	—	9	2	no	—
179	C₃×C₉	#466	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	—	18	2	no	—
182	C₂³×C₃	D₃₆	1	1	yes	characteristic
183	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
184	C₂²×D₃	—	6	2	no	—
185	C₃×D₄	—	18	2	no	—
186	C₃×D₄	—	18	2	no	—
187	C₃×D₄	—	36	4	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	18	2	no	—
197	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
199	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
201	#67	—	12	4	no	—
202	C₂×Dic₃	—	6	2	no	—
203	D₁₂	—	6	2	no	—
204	D₁₂	—	6	2	no	—
205	C₃×C₈	—	2	2	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	2	2	no	—
209	C₃×C₈	#611	1	2	yes	complemented
210	C₃×C₈	C₃×#67	1	2	yes	—
211	C₂×C₃²	—	2	2	no	—
212	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
213	C₂×C₃²	#904	1	1	yes	characteristic
214	C₂×C₃²	#930	1	1	yes	characteristic
215	C₃×D₃	—	24	2	no	—
216	C₂×C₉	—	2	4	no	—
217	C₂×C₉	—	2	2	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	C₃×#121	1	1	yes	characteristic
222	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
223	C₂×C₉	C₃×#135	1	1	yes	characteristic
224	D₉	—	8	2	no	—
225	C₂×D₄	—	9	2	no	—
226	C₂×D₄	—	18	2	no	—
227	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
228	D₈	—	18	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
231	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
232	C₄⋊C₄	—	9	2	no	—
233	C₂²⋊C₄	—	18	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	36	2	no	—
243	C₂²×C₃	—	36	2	no	—
244	C₂²×C₃	—	36	2	no	—
245	C₂²×C₃	D₇₂	1	2	yes	complemented
246	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
247	C₂²×C₃	C₃×D₂₄	1	2	yes	—
248	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
249	C₂×D₃	—	12	2	no	—
250	C₂×D₃	—	12	2	no	—
251	C₂×D₃	—	12	2	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	36	2	no	—
258	C₃×C₄	#3697	1	1	yes	characteristic
259	C₃×C₄	C₂×#611	1	1	yes	characteristic
260	C₃×C₄	C₃×#235	1	1	yes	characteristic
261	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
262	Dic₃	—	12	2	no	—
263	C₃²	#5115	1	1	yes	characteristic
264	C₉	—	2	1	no	—
265	C₉	C₃×#466	1	1	yes	characteristic, complemented
266	C₂³	—	18	2	no	—
267	C₂³	C₃×D₃₆	1	1	yes	characteristic
268	D₄	—	18	2	no	—
269	D₄	—	18	2	no	—
270	D₄	—	36	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	18	2	no	—
274	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	C₃×#611	1	2	yes	complemented
278	C₂×C₃	—	2	4	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	72	2	no	—
285	C₂×C₃	C₂×D₇₂	1	1	yes	characteristic
286	C₂×C₃	#63467	1	1	yes	characteristic
287	C₂×C₃	#63492	1	1	yes	characteristic
288	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
289	C₂×C₃	C₃×#904	1	1	yes	characteristic
290	C₂×C₃	C₃×#930	1	1	yes	characteristic
291	D₃	—	24	2	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	36	2	no	—
295	C₂²	—	36	2	no	—
296	C₂²	—	36	2	no	—
297	C₂²	C₃×D₇₂	1	2	yes	complemented
298	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	36	2	no	—
301	C₄	C₃×#3697	1	1	yes	characteristic
302	C₄	C₂×C₃×#611	1	1	yes	characteristic
303	C₃	—	2	1	no	—
304	C₃	#10588796	1	1	yes	characteristic, direct product factor
305	C₃	C₃×#5115	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	72	2	no	—
308	C₂	C₂×C₃×D₇₂	1	1	yes	characteristic
309	C₂	C₃×#63467	1	1	yes	characteristic
310	C₂	C₃×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×D₁₂
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₃×#466
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₈×C₉	C₂	semidirect
2. C₃×#63492	C₂	semidirect
3. C₂×C₃×D₇₂	C₂	semidirect
4. C₃×#63380	C₂	semidirect
5. C₃×#63378	C₂	semidirect
7. #10588796	C₃	direct (N × H)
23. C₂²×C₈×C₉	C₂×C₃	semidirect
24. #63492	C₂×C₃	semidirect
25. C₂×D₇₂	C₂×C₃	semidirect
26. #63380	C₂×C₃	semidirect
27. #63378	C₂×C₃	semidirect
39. C₃×C₈×C₉	D₄	semidirect
66. C₂²×C₃×C₉	D₈	semidirect
77. C₂²×C₃×C₈	D₉	semidirect
116. C₈×C₉	C₃×D₄	semidirect
162. C₂²×C₉	C₃×D₈	semidirect
175. C₂²×C₈	C₃×D₉	semidirect
179. C₃×C₉	#466	semidirect
209. C₃×C₈	#611	semidirect
245. C₂²×C₃	D₇₂	semidirect
265. C₉	C₃×#466	semidirect
277. C₈	C₃×#611	semidirect
297. C₂²	C₃×D₇₂	semidirect
304. C₃	#10588796	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₃×#63492	2
C₂×C₃×D₇₂	2
C₃×#63380	2
C₃×#63378	2
C₃×#5115	3
#10588796	3

Subgroups — C3×#10588796 #13143099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588796 #13143099