Subgroups — C₃×#10589827 #13153295

Order 1728.

998 conjugacy classes of subgroups.

The subgroup lattice has 330 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×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63509	C₂	1	2	yes	maximal, complemented
4	C₃×#63457	C₂	1	4	yes	maximal, complemented
5	C₃×#63455	C₂	1	4	yes	maximal, complemented
6	C₃×#63436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄²×D₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5725	—	3	1	no	maximal
9	#10589827	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
11	C₃×#3701	—	2	8	no	—
12	C₂×C₃×D₃₆	C₂²	1	2	yes	—
13	C₂×C₃×D₃₆	C₂²	1	4	yes	—
14	C₂×C₃×D₃₆	C₂²	1	4	yes	—
15	C₂×C₃×C₄×D₉	C₂²	1	4	yes	—
16	C₂×C₃×C₄×D₉	C₂²	1	2	yes	—
17	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
18	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
19	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₃×(C₂×D₉)⋊C₄	C₂²	1	8	yes	—
21	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
22	C₃×C₄×Dic₉	C₂²	1	2	yes	—
23	C₂×C₃×#239	—	3	2	no	—
24	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
25	C₃×#947	—	3	2	no	—
26	C₃×#895	—	3	4	no	—
27	C₃×#893	—	3	4	no	—
28	C₃×#874	—	3	1	no	—
29	C₃×C₄²×D₃	—	3	1	no	—
30	C₂×#3701	C₂×C₃	1	2	yes	complemented
31	C₉×C₄⋊Q₈	—	2	1	no	—
32	C₉×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#63509	C₂×C₃	1	2	yes	complemented
34	#63457	C₂×C₃	1	4	yes	complemented
35	#63455	C₂×C₃	1	4	yes	complemented
36	#63436	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄²×D₉	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃×D₉	C₂³	1	4	yes	—
39	C₂²×C₃×D₉	C₂³	1	1	yes	characteristic
40	C₃×Q₈×C₉	—	2	4	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
44	C₂×C₃×Dic₉	C₂³	1	2	yes	—
45	C₂×C₃×Dic₉	C₂³	1	1	yes	characteristic
46	C₃×D₃₆	—	2	8	no	—
47	C₃×D₃₆	—	2	4	no	—
48	C₃×D₃₆	—	2	8	no	—
49	C₃×C₄×D₉	—	2	4	no	—
50	C₃×C₄×D₉	—	2	8	no	—
51	C₃×C₄×D₉	D₄	1	4	yes	complemented
52	C₃×#532	—	9	1	no	—
53	#5725	—	3	1	no	—
54	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
55	C₃×#239	—	6	8	no	—
56	C₂×C₃×D₁₂	—	3	4	no	—
57	C₂×C₃×D₁₂	—	3	4	no	—
58	C₂×C₃×D₁₂	—	3	2	no	—
59	C₂×C₃×C₄×D₃	—	3	1	no	—
60	C₂×C₃×C₄×D₃	—	3	4	no	—
61	C₂×C₃×C₄×D₃	—	3	2	no	—
62	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
63	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
65	C₃×C₄×Dic₃	—	3	2	no	—
66	C₃×C₄×Dic₃	—	3	1	no	—
67	C₂×Q₈×C₉	—	2	2	no	—
68	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
69	#3701	—	2	8	no	—
70	C₂×D₃₆	C₂²×C₃	1	2	yes	—
71	C₂×D₃₆	C₂²×C₃	1	4	yes	—
72	C₂×D₃₆	C₂²×C₃	1	4	yes	—
73	C₂×C₄×D₉	C₂²×C₃	1	2	yes	—
74	C₂×C₄×D₉	C₂²×C₃	1	4	yes	—
75	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
76	C₉×C₄⋊C₄	—	2	4	no	—
77	C₉×C₄⋊C₄	C₂²×C₃	1	4	yes	—
78	C₄²×C₉	—	2	1	no	—
79	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
80	(C₂×D₉)⋊C₄	C₂²×C₃	1	8	yes	—
81	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
82	C₄×Dic₉	C₂²×C₃	1	2	yes	—
83	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₃×D₉	—	2	2	no	—
85	C₂×C₃×D₉	—	2	4	no	—
86	C₂×C₃×D₉	—	2	8	no	—
87	C₂×C₃×D₉	C₂×D₄	1	2	yes	—
88	C₃×C₄×C₉	—	2	4	no	—
89	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
90	C₃×C₄×C₉	#41	1	4	yes	complemented
91	C₃×Dic₉	—	2	2	no	—
92	C₃×Dic₉	C₂×D₄	1	2	yes	—
93	C₂×C₃×#41	—	9	2	no	—
94	C₂×#239	—	3	2	no	—
95	C₃×C₄⋊Q₈	—	2	1	no	—
96	C₃×C₄⋊Q₈	D₉	1	1	yes	characteristic, complemented
97	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
98	C₃×#127	—	9	1	no	—
99	C₃×#124	—	9	2	no	—
100	C₃×#121	—	9	4	no	—
101	C₃×C₄×D₄	—	9	4	no	—
102	C₂×C₃×C₄²	—	9	1	no	—
103	#947	—	3	2	no	—
104	#895	—	3	4	no	—
105	#893	—	3	4	no	—
106	#874	—	3	1	no	—
107	C₄²×D₃	—	3	1	no	—
108	C₂²×C₃×D₃	—	3	1	no	—
109	C₂²×C₃×D₃	—	3	4	no	—
110	C₃²×Q₈	—	2	4	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
114	C₂×C₃×Dic₃	—	3	1	no	—
115	C₂×C₃×Dic₃	—	3	2	no	—
116	C₃×D₁₂	—	6	8	no	—
117	C₃×D₁₂	—	6	4	no	—
118	C₃×D₁₂	—	6	8	no	—
119	C₃×C₄×D₃	—	3	4	no	—
120	C₃×C₄×D₃	—	6	8	no	—
121	C₃×C₄×D₃	—	6	4	no	—
122	C₂²×D₉	C₂³×C₃	1	4	yes	—
123	C₂²×D₉	C₂³×C₃	1	1	yes	characteristic
124	Q₈×C₉	—	2	4	no	—
125	Q₈×C₉	—	4	4	no	—
126	C₂×C₄×C₉	—	2	2	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	4	no	—
129	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
130	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
132	C₂×Dic₉	C₂³×C₃	1	2	yes	—
133	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
134	D₃₆	—	2	8	no	—
135	D₃₆	—	2	4	no	—
136	D₃₆	—	2	8	no	—
137	C₄×D₉	—	2	4	no	—
138	C₄×D₉	—	2	8	no	—
139	C₄×D₉	C₃×D₄	1	4	yes	complemented
140	#532	—	9	1	no	Sylow
141	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₃×C₉	C₂×#41	1	2	yes	—
143	C₃×D₉	—	2	2	no	—
144	C₃×D₉	—	4	4	no	—
145	C₃×#41	—	18	8	no	—
146	C₂×C₃×Q₈	—	2	2	no	—
147	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
148	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
149	C₂×C₃×D₄	—	9	4	no	—
150	C₂×C₃×D₄	—	9	4	no	—
151	C₂×C₃×D₄	—	9	2	no	—
152	C₂²×C₃×C₄	—	9	1	no	—
153	C₂²×C₃×C₄	—	9	2	no	—
154	C₂²×C₃×C₄	—	9	4	no	—
155	#239	—	6	8	no	—
156	C₂×D₁₂	—	3	4	no	—
157	C₂×D₁₂	—	3	2	no	—
158	C₂×D₁₂	—	3	4	no	—
159	C₂×C₄×D₃	—	3	1	no	—
160	C₂×C₄×D₃	—	3	2	no	—
161	C₂×C₄×D₃	—	3	4	no	—
162	C₃×C₄⋊C₄	—	2	4	no	—
163	C₃×C₄⋊C₄	C₂×D₉	1	4	yes	—
164	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
165	C₃×C₂²⋊C₄	—	9	8	no	—
166	C₃×C₄²	—	2	1	no	—
167	C₃×C₄²	—	9	2	no	—
168	C₃×C₄²	—	9	1	no	—
169	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
170	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
171	(C₂×D₃)⋊C₄	—	3	8	no	—
172	C₄×Dic₃	—	3	2	no	—
173	C₄×Dic₃	—	3	1	no	—
174	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
175	C₂×C₃×D₃	—	3	2	no	—
176	C₂×C₃×D₃	—	6	2	no	—
177	C₂×C₃×D₃	—	6	4	no	—
178	C₂×C₃×D₃	—	6	8	no	—
179	C₃²×C₄	—	2	4	no	—
180	C₃²×C₄	D₃×D₄	1	2	yes	—
181	C₃²×C₄	#239	1	4	yes	—
182	C₃×Dic₃	—	3	2	no	—
183	C₃×Dic₃	—	6	2	no	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
186	C₂×D₉	—	2	2	no	—
187	C₂×D₉	—	2	4	no	—
188	C₂×D₉	—	2	8	no	—
189	C₂×D₉	C₂×C₃×D₄	1	2	yes	—
190	C₄×C₉	—	2	4	no	—
191	C₄×C₉	—	2	2	no	—
192	C₄×C₉	—	2	4	no	—
193	C₄×C₉	—	4	4	no	—
194	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
195	C₄×C₉	C₃×#41	1	4	yes	complemented
196	Dic₉	—	2	2	no	—
197	Dic₉	C₂×C₃×D₄	1	2	yes	—
198	C₂×#41	—	9	2	no	—
199	C₄⋊Q₈	C₃×D₉	1	1	yes	characteristic, complemented
200	#127	—	9	1	no	—
201	#124	—	9	2	no	—
202	#121	—	9	4	no	—
203	C₄×D₄	—	9	4	no	—
204	C₂×C₄²	—	9	1	no	—
205	C₃×C₉	#532	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	9	1	no	—
207	C₂³×C₃	—	9	4	no	—
208	C₂²×D₃	—	3	1	no	—
209	C₂²×D₃	—	3	4	no	—
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	4	4	no	—
213	C₃×D₄	—	18	8	no	—
214	C₃×D₄	—	18	4	no	—
215	C₃×D₄	—	18	8	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	9	1	no	—
220	C₂×C₃×C₄	—	9	4	no	—
221	C₂×C₃×C₄	—	9	2	no	—
222	C₂×C₃×C₄	—	18	8	no	—
223	C₂×C₃×C₄	—	18	4	no	—
224	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
225	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
227	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
228	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
230	C₂×Dic₃	—	3	1	no	—
231	C₂×Dic₃	—	3	2	no	—
232	D₁₂	—	6	8	no	—
233	D₁₂	—	6	4	no	—
234	D₁₂	—	6	8	no	—
235	C₄×D₃	—	3	4	no	—
236	C₄×D₃	—	6	8	no	—
237	C₄×D₃	—	6	4	no	—
238	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
239	C₂×C₃²	C₂×#239	1	2	yes	—
240	C₃×D₃	—	6	2	no	—
241	C₃×D₃	—	12	4	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	2	2	no	—
244	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
245	C₂×C₉	C₂×C₃×#41	1	2	yes	—
246	D₉	—	2	2	no	—
247	D₉	—	4	4	no	—
248	#41	—	18	8	no	—
249	C₂×Q₈	C₂×C₃×D₉	1	2	yes	—
250	C₂×D₄	—	9	4	no	—
251	C₂×D₄	—	9	4	no	—
252	C₂×D₄	—	9	2	no	—
253	C₂²×C₄	—	9	1	no	—
254	C₂²×C₄	—	9	2	no	—
255	C₂²×C₄	—	9	4	no	—
256	C₄⋊C₄	C₂×C₃×D₉	1	4	yes	—
257	C₂²⋊C₄	—	9	8	no	—
258	C₄²	—	9	2	no	—
259	C₄²	—	9	1	no	—
260	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	9	2	no	—
263	C₂²×C₃	—	18	2	no	—
264	C₂²×C₃	—	18	4	no	—
265	C₂²×C₃	—	18	8	no	—
266	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
267	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
268	C₂×D₃	—	3	2	no	—
269	C₂×D₃	—	6	2	no	—
270	C₂×D₃	—	6	4	no	—
271	C₂×D₃	—	6	8	no	—
272	C₃×C₄	—	2	4	no	—
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	4	4	no	—
277	C₃×C₄	—	9	2	no	—
278	C₃×C₄	—	18	2	no	—
279	C₃×C₄	D₄×D₉	1	2	yes	—
280	C₃×C₄	#3701	1	4	yes	complemented
281	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
282	C₃×C₄	C₃×#239	1	4	yes	—
283	Dic₃	—	3	2	no	—
284	Dic₃	—	6	2	no	—
285	C₃²	#5725	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	C₃×#532	1	1	yes	characteristic, complemented
288	C₂³	—	9	1	no	—
289	C₂³	—	9	4	no	—
290	Q₈	—	2	4	no	—
291	D₄	—	18	8	no	—
292	D₄	—	18	4	no	—
293	D₄	—	18	8	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	9	4	no	—
296	C₂×C₄	—	9	2	no	—
297	C₂×C₄	—	18	8	no	—
298	C₂×C₄	—	18	4	no	—
299	C₂×C₄	C₂²×C₃×D₉	1	4	yes	—
300	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	18	2	no	—
305	C₂×C₃	—	36	4	no	—
306	C₂×C₃	C₂×D₄×D₉	1	1	yes	characteristic
307	C₂×C₃	C₂×#3701	1	2	yes	—
308	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
309	C₂×C₃	C₂×C₃×#239	1	2	yes	—
310	D₃	—	6	2	no	—
311	D₃	—	12	4	no	—
312	C₂²	—	9	2	no	—
313	C₂²	—	18	2	no	—
314	C₂²	—	18	4	no	—
315	C₂²	—	18	8	no	—
316	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	9	2	no	—
319	C₄	—	18	2	no	—
320	C₄	C₃×D₄×D₉	1	2	yes	—
321	C₄	C₃×#3701	1	4	yes	complemented
322	C₃	—	2	1	no	—
323	C₃	#10589827	1	1	yes	characteristic, direct product factor
324	C₃	C₃×#5725	1	1	yes	characteristic
325	C₂	—	18	2	no	—
326	C₂	—	36	4	no	—
327	C₂	C₂×C₃×D₄×D₉	1	1	yes	characteristic
328	C₂	C₂×C₃×#3701	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₃×C₄⋊Q₈	D₉
hypocenter	C₉	C₃×#532
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×C₉×C₄⋊Q₈	C₂	semidirect
3. C₃×#63509	C₂	semidirect
4. C₃×#63457	C₂	semidirect
5. C₃×#63455	C₂	semidirect
6. C₃×#63436	C₂	semidirect
7. C₃×C₄²×D₉	C₂	semidirect
9. #10589827	C₃	direct (N × H)
30. C₂×#3701	C₂×C₃	semidirect
32. C₉×C₄⋊Q₈	C₂×C₃	semidirect
33. #63509	C₂×C₃	semidirect
34. #63457	C₂×C₃	semidirect
35. #63455	C₂×C₃	semidirect
36. #63436	C₂×C₃	semidirect
37. C₄²×D₉	C₂×C₃	semidirect
51. C₃×C₄×D₉	D₄	semidirect
90. C₃×C₄×C₉	#41	semidirect
96. C₃×C₄⋊Q₈	D₉	semidirect
139. C₄×D₉	C₃×D₄	semidirect
195. C₄×C₉	C₃×#41	semidirect
199. C₄⋊Q₈	C₃×D₉	semidirect
205. C₃×C₉	#532	semidirect
280. C₃×C₄	#3701	semidirect
287. C₉	C₃×#532	semidirect
321. C₄	C₃×#3701	semidirect
323. C₃	#10589827	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×C₉×C₄⋊Q₈	2
C₃×#63509	2
C₃×#63457	2
C₃×#63455	2
C₃×#63436	2
C₃×C₄²×D₉	2
C₃×#5725	3
#10589827	3

Subgroups — C3×#10589827 #13153295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589827 #13153295