Subgroups — C₃×#10588521 #13141997

Order 1728.

416 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	C₂×C₃×#3663	C₂	1	2	yes	maximal
2	C₃×C₄²×D₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×Dic₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4837	—	3	1	no	maximal
7	#10588521	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×C₄×D₉	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×D₉	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
14	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
15	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₃×#3663	—	2	4	no	—
18	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	2	no	—
19	C₃×C₄²×D₃	—	3	1	no	—
20	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
21	C₃×Dic₃⋊C₈	—	3	2	no	—
22	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×#3663	C₂×C₃	1	2	yes	—
24	C₄²×D₉	C₂×C₃	1	1	yes	characteristic
25	C₉×C₄⋊C₈	—	2	1	no	—
26	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	Dic₉⋊C₈	C₂×C₃	1	2	yes	complemented
28	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×D₉	—	2	2	no	—
37	C₃×C₄×D₉	—	2	2	no	—
38	C₃×C₄×D₉	D₄	1	2	yes	—
39	C₃×C₄×D₉	Q₈	1	2	yes	—
40	C₃×C₈×C₉	—	2	2	no	—
41	C₃×C₉⋊C₈	—	2	2	no	—
42	C₃×#423	—	9	1	no	—
43	#4837	—	3	1	no	—
44	C₂×C₃×C₄×D₃	—	3	1	no	—
45	C₂×C₃×C₄×D₃	—	3	1	no	—
46	C₂×C₃×C₄×D₃	—	3	1	no	—
47	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
48	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₃×C₄×Dic₃	—	3	1	no	—
50	C₃×C₄×Dic₃	—	3	1	no	—
51	C₃×C₄×Dic₃	—	3	1	no	—
52	C₂×C₃×C₃⋊C₈	—	3	2	no	—
53	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
54	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
56	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₈×C₉	—	2	2	no	—
58	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
61	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
62	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
63	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
65	#3663	—	2	4	no	—
66	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×D₉	—	2	1	no	—
68	C₂×C₃×D₉	—	2	1	no	—
69	C₂×C₃×D₉	C₄⋊C₄	1	2	yes	—
70	C₃×C₄×C₉	—	2	1	no	—
71	C₃×C₄×C₉	M₁₆	1	2	yes	complemented
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
74	C₃×Dic₉	—	2	1	no	—
75	C₃×Dic₉	C₄⋊C₄	1	2	yes	—
76	C₃×Dic₉	M₁₆	1	2	yes	complemented
77	C₂×C₃×M₁₆	—	9	2	no	—
78	C₂×C₃×C₄²	—	9	1	no	—
79	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
80	C₄²×D₃	—	3	1	no	—
81	C₃×C₄⋊C₈	—	2	1	no	—
82	C₃×C₄⋊C₈	—	9	2	no	—
83	C₃×C₄⋊C₈	—	9	1	no	—
84	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
85	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
86	Dic₃⋊C₈	—	3	2	no	—
87	C₄⋊(C₃⋊C₈)	—	3	1	no	—
88	C₂²×C₃×D₃	—	3	1	no	—
89	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₃×Dic₃	—	3	1	no	—
93	C₂×C₃×Dic₃	—	3	1	no	—
94	C₂×C₃×Dic₃	—	3	1	no	—
95	C₃×C₄×D₃	—	3	2	no	—
96	C₃×C₄×D₃	—	3	2	no	—
97	C₃×C₄×D₃	—	6	2	no	—
98	C₃×C₄×D₃	—	6	2	no	—
99	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
100	C₃²×C₈	—	2	2	no	—
101	C₃×C₃⋊C₈	—	6	2	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
108	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
111	C₄×D₉	—	2	2	no	—
112	C₄×D₉	—	2	2	no	—
113	C₄×D₉	C₃×D₄	1	2	yes	—
114	C₄×D₉	C₃×Q₈	1	2	yes	—
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	4	2	no	—
117	C₉⋊C₈	—	2	2	no	—
118	#423	—	9	1	no	Sylow
119	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
122	C₃×D₉	—	2	2	no	—
123	C₂²×C₃×C₄	—	9	1	no	—
124	C₂²×C₃×C₄	—	9	1	no	—
125	C₂²×C₃×C₄	—	9	1	no	—
126	C₂×C₄×D₃	—	3	1	no	—
127	C₂×C₄×D₃	—	3	1	no	—
128	C₂×C₄×D₃	—	3	1	no	—
129	C₃×M₁₆	—	18	4	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	9	2	no	—
132	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
133	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
134	C₃×C₄²	—	2	1	no	—
135	C₃×C₄²	—	9	1	no	—
136	C₃×C₄²	—	9	1	no	—
137	C₃×C₄²	—	9	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	1	no	—
141	C₄×Dic₃	—	3	1	no	—
142	C₄×Dic₃	—	3	1	no	—
143	C₂×C₃⋊C₈	—	3	2	no	—
144	ΓL(1,𝔽₂₅)	—	6	4	no	—
145	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂×C₃×D₃	—	3	2	no	—
147	C₂×C₃×D₃	—	6	1	no	—
148	C₂×C₃×D₃	—	6	1	no	—
149	C₃²×C₄	—	2	1	no	—
150	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
151	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
152	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
153	C₃×Dic₃	—	3	2	no	—
154	C₃×Dic₃	—	3	2	no	—
155	C₃×Dic₃	—	6	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₂×D₉	—	2	1	no	—
159	C₂×D₉	—	2	1	no	—
160	C₂×D₉	C₃×C₄⋊C₄	1	2	yes	—
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	C₃×M₁₆	1	2	yes	complemented
167	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
168	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
169	Dic₉	—	2	1	no	—
170	Dic₉	C₃×C₄⋊C₄	1	2	yes	—
171	Dic₉	C₃×M₁₆	1	2	yes	complemented
172	C₂×M₁₆	—	9	2	no	—
173	C₂×C₄²	—	9	1	no	—
174	C₄⋊C₈	—	9	2	no	—
175	C₄⋊C₈	—	9	1	no	—
176	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
177	C₃×C₉	#423	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	9	1	no	—
179	C₂²×D₃	—	3	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	9	1	no	—
184	C₂×C₃×C₄	—	9	1	no	—
185	C₂×C₃×C₄	—	9	2	no	—
186	C₂×C₃×C₄	—	9	2	no	—
187	C₂×C₃×C₄	—	9	1	no	—
188	C₂×C₃×C₄	—	18	2	no	—
189	C₂×C₃×C₄	—	18	2	no	—
190	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
196	C₂×Dic₃	—	3	1	no	—
197	C₂×Dic₃	—	3	1	no	—
198	C₂×Dic₃	—	3	1	no	—
199	C₄×D₃	—	3	2	no	—
200	C₄×D₃	—	3	2	no	—
201	C₄×D₃	—	6	2	no	—
202	C₄×D₃	—	6	2	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	4	2	no	—
206	C₃×C₈	—	18	2	no	—
207	C₃⋊C₈	—	6	2	no	—
208	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
210	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
211	C₃×D₃	—	6	2	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
216	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
217	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
218	D₉	—	2	2	no	—
219	C₂²×C₄	—	9	1	no	—
220	C₂²×C₄	—	9	1	no	—
221	C₂²×C₄	—	9	1	no	—
222	M₁₆	—	18	4	no	—
223	C₂×C₈	—	9	2	no	—
224	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
225	C₄²	—	9	1	no	—
226	C₄²	—	9	1	no	—
227	C₄²	—	9	1	no	—
228	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	9	2	no	—
231	C₂²×C₃	—	18	1	no	—
232	C₂²×C₃	—	18	1	no	—
233	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
234	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂×D₃	—	3	2	no	—
236	C₂×D₃	—	6	1	no	—
237	C₂×D₃	—	6	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	4	1	no	—
244	C₃×C₄	—	9	2	no	—
245	C₃×C₄	—	9	2	no	—
246	C₃×C₄	—	18	1	no	—
247	C₃×C₄	#3663	1	2	yes	complemented
248	C₃×C₄	D₄×D₉	1	1	yes	characteristic
249	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
250	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
251	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
252	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
253	Dic₃	—	3	2	no	—
254	Dic₃	—	3	2	no	—
255	Dic₃	—	6	1	no	—
256	C₃²	#4837	1	1	yes	characteristic
257	C₉	—	2	1	no	—
258	C₉	C₃×#423	1	1	yes	characteristic, complemented
259	C₂³	—	9	1	no	—
260	C₂×C₄	—	9	1	no	—
261	C₂×C₄	—	9	1	no	—
262	C₂×C₄	—	9	2	no	—
263	C₂×C₄	—	9	2	no	—
264	C₂×C₄	—	9	1	no	—
265	C₂×C₄	—	18	2	no	—
266	C₂×C₄	—	18	2	no	—
267	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
268	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
270	C₈	—	2	2	no	—
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	—	18	2	no	—
276	C₂×C₃	C₄⋊C₄×D₉	1	1	yes	characteristic
277	C₂×C₃	C₂×#3663	1	1	yes	characteristic
278	C₂×C₃	M₁₆×D₉	1	1	yes	characteristic
279	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
281	C₂×C₃	C₃×D₃×M₁₆	1	1	yes	characteristic
282	D₃	—	6	2	no	—
283	C₂²	—	9	2	no	—
284	C₂²	—	18	1	no	—
285	C₂²	—	18	1	no	—
286	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	9	2	no	—
289	C₄	—	9	2	no	—
290	C₄	—	18	1	no	—
291	C₄	C₃×#3663	1	2	yes	complemented
292	C₄	C₃×D₄×D₉	1	1	yes	characteristic
293	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
294	C₃	—	2	1	no	—
295	C₃	#10588521	1	1	yes	characteristic, direct product factor
296	C₃	C₃×#4837	1	1	yes	characteristic
297	C₂	—	18	2	no	—
298	C₂	C₃×C₄⋊C₄×D₉	1	1	yes	characteristic
299	C₂	C₂×C₃×#3663	1	1	yes	characteristic
300	C₂	C₃×M₁₆×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₉	C₂×C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#423
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
4. C₃×Dic₉⋊C₈	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588521	C₃	direct (N × H)
26. C₉×C₄⋊C₈	C₂×C₃	semidirect
27. Dic₉⋊C₈	C₂×C₃	semidirect
28. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
71. C₃×C₄×C₉	M₁₆	semidirect
76. C₃×Dic₉	M₁₆	semidirect
84. C₃×C₄⋊C₈	D₉	semidirect
166. C₄×C₉	C₃×M₁₆	semidirect
171. Dic₉	C₃×M₁₆	semidirect
176. C₄⋊C₈	C₃×D₉	semidirect
177. C₃×C₉	#423	semidirect
247. C₃×C₄	#3663	semidirect
258. C₉	C₃×#423	semidirect
291. C₄	C₃×#3663	semidirect
295. C₃	#10588521	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×#3663	2
C₃×C₄²×D₉	2
C₃×C₉×C₄⋊C₈	2
C₃×Dic₉⋊C₈	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#4837	3
#10588521	3

Subgroups — C3×#10588521 #13141997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588521 #13141997