Subgroups — C₃×#10588690 #13142715

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63392	C₂	1	2	yes	maximal
4	C₃×#63366	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5007	—	3	1	no	maximal
7	#10588690	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×C₃×Dic₁₈	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Dic₉	—	2	1	no	—
15	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
18	C₃×#869	—	3	1	no	—
19	C₃×#832	—	3	2	no	—
20	C₃×#807	—	3	2	no	—
21	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₄×D₄×C₉	—	2	1	no	—
23	C₄×D₄×C₉	C₂×C₃	1	1	yes	characteristic
24	#63431	C₂×C₃	1	1	yes	characteristic, complemented
25	#63392	C₂×C₃	1	2	yes	—
26	#63366	C₂×C₃	1	2	yes	complemented
27	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×C₉	—	2	1	no	—
29	C₃×D₄×C₉	—	2	1	no	—
30	C₃×D₄×C₉	Q₈	1	2	yes	complemented
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	4	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₉	—	2	2	no	—
37	C₃×Dic₁₈	—	4	2	no	—
38	C₃×C₉⋊C₈	—	2	2	no	—
39	C₃×#474	—	9	1	no	—
40	#5007	—	3	1	no	—
41	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	—	2	1	no	—
43	C₂×C₃×Dic₆	—	6	1	no	—
44	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
45	C₃²×C₂²⋊C₄	—	2	1	no	—
46	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₃×C₄⋊Dic₃	—	3	2	no	—
48	C₃×C₄⋊Dic₃	—	6	1	no	—
49	C₂×C₃×C₃⋊C₈	—	3	2	no	—
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂×Dic₁₈	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₉×C₂²⋊C₄	—	2	2	no	—
58	C₉×C₂²⋊C₄	—	2	1	no	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
61	C₄⋊Dic₉	—	2	1	no	—
62	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
63	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	—	4	1	no	—
66	C₂²×C₃×C₉	—	4	1	no	—
67	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₉	—	2	1	no	—
69	C₃×C₄×C₉	—	4	1	no	—
70	C₃×C₄×C₉	D₈	1	2	yes	—
71	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
72	C₃×C₄×C₉	#41	1	1	yes	characteristic
73	C₃×Dic₉	—	4	2	no	—
74	C₃×C₄⋊Q₈	—	9	1	no	—
75	C₃×C₄×D₄	—	2	1	no	—
76	C₃×C₄×D₄	D₉	1	1	yes	characteristic
77	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
78	#869	—	3	1	no	—
79	C₃×#107	—	9	2	no	—
80	C₃×C₄⋊C₈	—	9	1	no	—
81	C₃×#102	—	9	2	no	—
82	#832	—	3	2	no	—
83	#807	—	3	2	no	—
84	C₄⋊(C₃⋊C₈)	—	3	1	no	—
85	C₂³×C₃²	—	2	1	no	—
86	C₃²×D₄	—	2	1	no	—
87	C₃²×D₄	Dic₆	1	2	yes	—
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	—	4	2	no	—
90	C₂×C₃²×C₄	#67	1	1	yes	characteristic
91	C₂×C₃²×C₄	#67	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₃×Dic₃	—	6	2	no	—
94	C₃×Dic₆	—	12	2	no	—
95	C₂³×C₉	—	2	2	no	—
96	C₂³×C₉	—	2	1	no	—
97	C₃×C₃⋊C₈	—	6	2	no	—
98	D₄×C₉	—	2	2	no	—
99	D₄×C₉	—	2	1	no	—
100	D₄×C₉	—	2	2	no	—
101	D₄×C₉	C₃×Q₈	1	2	yes	complemented
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	—	4	4	no	—
108	C₂×C₄×C₉	—	4	2	no	—
109	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
112	C₂×Dic₉	—	2	2	no	—
113	Dic₁₈	—	4	2	no	—
114	C₉⋊C₈	—	2	2	no	—
115	#474	—	9	1	no	Sylow
116	C₂×C₃×C₉	—	4	2	no	—
117	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₃×C₉	#137	1	1	yes	characteristic
120	C₂×C₃×Q₈	—	18	1	no	—
121	C₂×C₃×D₄	—	2	1	no	—
122	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
123	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	2	2	no	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂×Dic₆	—	6	1	no	—
128	C₂×C₃×C₈	—	9	2	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	—	9	2	no	—
131	C₃×C₄⋊C₄	—	18	1	no	—
132	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
133	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
134	C₃×C₂²⋊C₄	—	2	2	no	—
135	C₃×C₂²⋊C₄	—	2	1	no	—
136	C₃×C₂²⋊C₄	—	2	1	no	—
137	C₃×C₄²	—	2	1	no	—
138	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
139	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
140	C₄⋊Dic₃	—	3	2	no	—
141	C₄⋊Dic₃	—	6	1	no	—
142	C₂×C₃⋊C₈	—	3	2	no	—
143	C₂²×C₃²	—	2	2	no	—
144	C₂²×C₃²	—	4	1	no	—
145	C₂²×C₃²	—	4	1	no	—
146	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
147	C₃²×C₄	—	2	1	no	—
148	C₃²×C₄	—	4	1	no	—
149	C₃²×C₄	#213	1	2	yes	—
150	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
151	C₃²×C₄	#235	1	1	yes	characteristic
152	C₃×Dic₃	—	12	2	no	—
153	C₂²×C₉	—	2	4	no	—
154	C₂²×C₉	—	2	2	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	4	2	no	—
157	C₂²×C₉	—	4	2	no	—
158	C₂²×C₉	—	4	1	no	—
159	C₂²×C₉	—	4	1	no	—
160	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	4	1	no	—
166	C₄×C₉	—	4	2	no	—
167	C₄×C₉	—	4	1	no	—
168	C₄×C₉	C₃×D₈	1	2	yes	—
169	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
170	C₄×C₉	C₃×#41	1	1	yes	characteristic
171	Dic₉	—	4	2	no	—
172	C₄⋊Q₈	—	9	1	no	—
173	C₄×D₄	C₃×D₉	1	1	yes	characteristic
174	#107	—	9	2	no	—
175	C₄⋊C₈	—	9	1	no	—
176	#102	—	9	2	no	—
177	C₃×C₉	#474	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	2	2	no	—
179	C₂³×C₃	—	2	1	no	—
180	C₂³×C₃	—	2	1	no	—
181	C₃×Q₈	—	36	2	no	—
182	C₃×D₄	—	2	2	no	—
183	C₃×D₄	—	2	1	no	—
184	C₃×D₄	—	2	1	no	—
185	C₃×D₄	—	2	2	no	—
186	C₃×D₄	Dic₁₈	1	2	yes	complemented
187	C₃×D₄	C₃×Dic₆	1	2	yes	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	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	—	4	4	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	18	2	no	—
198	C₂×C₃×C₄	#611	1	1	yes	characteristic
199	C₂×C₃×C₄	#611	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
201	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
202	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
204	C₂×Dic₃	—	6	2	no	—
205	Dic₆	—	12	2	no	—
206	C₃×C₈	—	18	2	no	—
207	C₃⋊C₈	—	6	2	no	—
208	C₂×C₃²	—	4	2	no	—
209	C₂×C₃²	#924	1	1	yes	characteristic
210	C₂×C₃²	C₂×#213	1	1	yes	characteristic
211	C₂×C₃²	#951	1	1	yes	characteristic
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	4	4	no	—
216	C₂×C₉	—	4	2	no	—
217	C₂×C₉	C₃×C₂²⋊Q₈	1	1	yes	characteristic
218	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
219	C₂×C₉	C₃×#137	1	1	yes	characteristic
220	C₂×Q₈	—	18	1	no	—
221	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂×C₈	—	9	2	no	—
224	C₄⋊C₄	—	9	2	no	—
225	C₄⋊C₄	—	18	1	no	—
226	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
229	C₂²×C₃	—	2	4	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	4	2	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	—	4	1	no	—
238	C₂²×C₃	—	4	1	no	—
239	C₂²×C₃	C₂×#611	1	1	yes	characteristic
240	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	4	1	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	—	4	1	no	—
249	C₃×C₄	—	4	1	no	—
250	C₃×C₄	—	36	2	no	—
251	C₃×C₄	#3673	1	2	yes	—
252	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
253	C₃×C₄	#3697	1	1	yes	characteristic
254	C₃×C₄	C₃×#213	1	2	yes	—
255	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
256	C₃×C₄	C₃×#235	1	1	yes	characteristic
257	Dic₃	—	12	2	no	—
258	C₃²	#5007	1	1	yes	characteristic
259	C₉	—	2	1	no	—
260	C₉	C₃×#474	1	1	yes	characteristic, complemented
261	C₂³	—	2	1	no	—
262	Q₈	—	36	2	no	—
263	D₄	—	2	1	no	—
264	D₄	C₃×Dic₁₈	1	2	yes	complemented
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	18	2	no	—
268	C₂×C₄	C₃×#611	1	1	yes	characteristic
269	C₂×C₄	C₃×#611	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
271	C₈	—	18	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	4	4	no	—
276	C₂×C₃	—	4	2	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	#63486	1	1	yes	characteristic
279	C₂×C₃	C₂×#3673	1	1	yes	characteristic
280	C₂×C₃	#63511	1	1	yes	characteristic
281	C₂×C₃	C₃×#924	1	1	yes	characteristic
282	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
283	C₂×C₃	C₃×#951	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	C₂×C₃×#611	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	36	2	no	—
291	C₄	C₃×#3673	1	2	yes	—
292	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
293	C₄	C₃×#3697	1	1	yes	characteristic
294	C₃	—	2	1	no	—
295	C₃	#10588690	1	1	yes	characteristic, direct product factor
296	C₃	C₃×#5007	1	1	yes	characteristic
297	C₂	—	4	2	no	—
298	C₂	C₃×#63486	1	1	yes	characteristic
299	C₂	C₂×C₃×#3673	1	1	yes	characteristic
300	C₂	C₃×#63511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃²×D₄	C₂×D₃
hypercenter	C₃×C₄×D₄	D₉
hypocenter	C₉	C₃×#474
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×#63431	C₂	semidirect
4. C₃×#63366	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588690	C₃	direct (N × H)
24. #63431	C₂×C₃	semidirect
26. #63366	C₂×C₃	semidirect
27. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
30. C₃×D₄×C₉	Q₈	semidirect
101. D₄×C₉	C₃×Q₈	semidirect
177. C₃×C₉	#474	semidirect
186. C₃×D₄	Dic₁₈	semidirect
260. C₉	C₃×#474	semidirect
264. D₄	C₃×Dic₁₈	semidirect
295. C₃	#10588690	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×C₄×D₄×C₉	2
C₃×#63431	2
C₃×#63392	2
C₃×#63366	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#5007	3
#10588690	3

Subgroups — C3×#10588690 #13142715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588690 #13142715