Subgroups — C₃×#10589595 #13152340

Order 1728.

1170 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#611	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63492	C₂	1	4	yes	maximal, complemented
4	C₃×#63448	C₂	1	4	yes	maximal, complemented
5	C₃×#63444	C₂	1	4	yes	maximal, complemented
6	C₃×#5493	—	3	1	no	maximal
7	#10589595	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₃×D₉	C₂²	1	2	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	complemented
11	C₂×C₃×#611	—	2	8	no	—
12	C₂×C₃×#611	C₂²	1	4	yes	—
13	C₂×C₃×#611	C₂²	1	4	yes	complemented
14	C₂²×C₃×Dic₉	C₂²	1	2	yes	complemented
15	C₂×C₃×D₃₆	C₂²	1	4	yes	complemented
16	C₃×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₃×(C₂×D₉)⋊C₄	C₂²	1	8	yes	complemented
18	C₃×C₄⋊Dic₉	C₂²	1	4	yes	complemented
19	C₂²×C₃×#67	—	3	2	no	—
20	C₂×C₃²×C₂²⋊C₄	D₃	1	1	yes	characteristic
21	C₃×#930	—	3	4	no	—
22	C₃×#886	—	3	4	no	—
23	C₃×#882	—	3	4	no	—
24	C₂²×#611	C₂×C₃	1	2	yes	complemented
25	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
26	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
27	#63492	C₂×C₃	1	4	yes	complemented
28	#63448	C₂×C₃	1	4	yes	complemented
29	#63444	C₂×C₃	1	4	yes	complemented
30	C₂³×C₃×C₉	—	4	2	no	—
31	C₂³×C₃×C₉	D₄	1	4	yes	complemented
32	C₂³×C₃×C₉	C₂³	1	2	yes	—
33	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
34	C₂²×C₃×D₉	—	2	4	no	—
35	C₂²×C₃×D₉	—	4	4	no	—
36	C₂²×C₃×D₉	C₂³	1	4	yes	—
37	C₂×C₃×C₄×C₉	—	4	2	no	—
38	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
39	C₃×#611	—	2	8	no	—
40	C₃×#611	—	2	8	no	—
41	C₂×C₃×Dic₉	—	2	4	no	—
42	C₂×C₃×Dic₉	C₂³	1	4	yes	—
43	C₃×D₃₆	—	4	4	no	—
44	C₃×#534	—	9	1	no	—
45	#5493	—	3	1	no	—
46	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
47	C₂³×C₃×D₃	—	3	2	no	—
48	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
49	C₂×C₃×#67	—	3	4	no	—
50	C₂×C₃×#67	—	3	4	no	—
51	C₂×C₃×#67	—	6	8	no	—
52	C₂²×C₃×Dic₃	—	3	2	no	—
53	C₂×C₃×D₁₂	—	3	4	no	—
54	C₂⁴×C₉	—	2	1	no	—
55	C₂⁴×C₉	C₂²×C₃	1	1	yes	characteristic
56	C₂³×D₉	C₂²×C₃	1	2	yes	complemented
57	C₃²×C₂²⋊C₄	C₂×D₃	1	4	yes	—
58	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
59	C₃×C₄⋊Dic₃	—	3	4	no	—
60	C₂²×C₄×C₉	—	2	2	no	—
61	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	complemented
62	C₂×#611	—	2	8	no	—
63	C₂×#611	C₂²×C₃	1	4	yes	—
64	C₂×#611	C₂²×C₃	1	4	yes	complemented
65	C₂²×Dic₉	C₂²×C₃	1	2	yes	complemented
66	C₂×D₃₆	C₂²×C₃	1	4	yes	complemented
67	C₉×C₂²⋊C₄	—	2	4	no	—
68	C₉×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
69	(C₂×D₉)⋊C₄	C₂²×C₃	1	8	yes	complemented
70	C₄⋊Dic₉	C₂²×C₃	1	4	yes	complemented
71	C₂²×C₃×C₉	—	2	2	no	—
72	C₂²×C₃×C₉	—	2	4	no	—
73	C₂²×C₃×C₉	—	4	4	no	—
74	C₂²×C₃×C₉	C₂×D₄	1	4	yes	—
75	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
76	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₃×D₉	—	2	4	no	—
78	C₂×C₃×D₉	—	2	8	no	—
79	C₂×C₃×D₉	—	4	8	no	—
80	C₃×C₄×C₉	—	2	4	no	—
81	C₃×Dic₉	—	2	4	no	—
82	C₂²×C₃×D₄	—	9	2	no	—
83	C₂²×#67	—	3	2	no	—
84	C₃×#123	—	9	4	no	—
85	C₃×#121	—	9	4	no	—
86	C₃×#120	—	9	4	no	—
87	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
88	C₂×C₃×C₂²⋊C₄	D₉	1	1	yes	characteristic, complemented
89	C₂×C₃×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic
90	#930	—	3	4	no	—
91	#886	—	3	4	no	—
92	#882	—	3	4	no	—
93	C₂³×C₃²	—	4	2	no	—
94	C₂³×C₃²	D₁₂	1	4	yes	—
95	C₂³×C₃²	C₂²×D₃	1	2	yes	—
96	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×D₃	—	3	4	no	—
98	C₂²×C₃×D₃	—	6	4	no	—
99	C₂²×C₃×D₃	—	12	4	no	—
100	C₂×C₃²×C₄	—	4	2	no	—
101	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
102	C₃×#67	—	6	8	no	—
103	C₃×#67	—	6	8	no	—
104	C₂×C₃×Dic₃	—	3	4	no	—
105	C₂×C₃×Dic₃	—	6	4	no	—
106	C₃×D₁₂	—	12	4	no	—
107	C₂³×C₉	—	2	4	no	—
108	C₂³×C₉	—	2	1	no	—
109	C₂³×C₉	—	2	2	no	—
110	C₂³×C₉	—	4	2	no	—
111	C₂³×C₉	—	4	4	no	—
112	C₂³×C₉	C₃×D₄	1	4	yes	complemented
113	C₂³×C₉	C₂³×C₃	1	2	yes	—
114	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
115	C₂²×D₉	—	2	4	no	—
116	C₂²×D₉	—	4	4	no	—
117	C₂²×D₉	C₂³×C₃	1	4	yes	—
118	C₂×C₄×C₉	—	2	4	no	—
119	C₂×C₄×C₉	—	4	2	no	—
120	C₂×C₄×C₉	—	4	4	no	—
121	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
122	#611	—	2	8	no	—
123	#611	—	2	8	no	—
124	C₂×Dic₉	—	2	4	no	—
125	C₂×Dic₉	C₂³×C₃	1	4	yes	—
126	D₃₆	—	4	4	no	—
127	#534	—	9	1	no	Sylow
128	C₂×C₃×C₉	—	2	2	no	—
129	C₂×C₃×C₉	—	2	4	no	—
130	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
132	C₃×D₉	—	4	4	no	—
133	C₂⁴×C₃	—	2	1	no	—
134	C₂⁴×C₃	—	9	2	no	—
135	C₂⁴×C₃	C₂×D₉	1	1	yes	characteristic
136	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂³×D₃	—	3	2	no	—
138	C₂×C₃×D₄	—	9	4	no	—
139	C₂×C₃×D₄	—	9	4	no	—
140	C₂×C₃×D₄	—	9	4	no	—
141	C₂×C₃×D₄	—	18	8	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	—	9	2	no	—
144	C₂²×C₃×C₄	C₂×D₉	1	2	yes	complemented
145	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
146	C₂×#67	—	3	4	no	—
147	C₂×#67	—	3	4	no	—
148	C₂×#67	—	6	8	no	—
149	C₂²×Dic₃	—	3	2	no	—
150	C₂×D₁₂	—	3	4	no	—
151	C₃×C₄⋊C₄	—	9	4	no	—
152	C₃×C₂²⋊C₄	—	2	4	no	—
153	C₃×C₂²⋊C₄	—	9	8	no	—
154	C₃×C₂²⋊C₄	C₂×D₉	1	4	yes	complemented
155	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	4	yes	—
156	(C₂×D₃)⋊C₄	—	3	8	no	—
157	C₄⋊Dic₃	—	3	4	no	—
158	C₂²×C₃²	—	2	2	no	—
159	C₂²×C₃²	—	2	4	no	—
160	C₂²×C₃²	—	4	4	no	—
161	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
162	C₂²×C₃²	C₂×D₁₂	1	2	yes	—
163	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
164	C₂×C₃×D₃	—	6	4	no	—
165	C₂×C₃×D₃	—	6	8	no	—
166	C₂×C₃×D₃	—	12	8	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃×Dic₃	—	6	4	no	—
169	C₂²×C₉	—	2	4	no	—
170	C₂²×C₉	—	2	2	no	—
171	C₂²×C₉	—	2	4	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	4	no	—
175	C₂²×C₉	—	4	4	no	—
176	C₂²×C₉	—	4	4	no	—
177	C₂²×C₉	—	4	8	no	—
178	C₂²×C₉	C₂×C₃×D₄	1	4	yes	—
179	C₂²×C₉	C₂×C₃×D₄	1	2	yes	—
180	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
181	C₂×D₉	—	2	4	no	—
182	C₂×D₉	—	2	8	no	—
183	C₂×D₉	—	4	8	no	—
184	C₄×C₉	—	2	4	no	—
185	C₄×C₉	—	4	4	no	—
186	Dic₉	—	2	4	no	—
187	C₂²×D₄	—	9	2	no	—
188	#123	—	9	4	no	—
189	#121	—	9	4	no	—
190	#120	—	9	4	no	—
191	C₂×C₂²⋊C₄	C₃×D₉	1	1	yes	characteristic, complemented
192	C₃×C₉	#534	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₃	—	2	4	no	—
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	—	4	2	no	—
197	C₂³×C₃	—	4	2	no	—
198	C₂³×C₃	—	4	4	no	—
199	C₂³×C₃	—	9	4	no	—
200	C₂³×C₃	—	18	4	no	—
201	C₂³×C₃	—	36	4	no	—
202	C₂³×C₃	D₃₆	1	4	yes	complemented
203	C₂³×C₃	C₂²×D₉	1	2	yes	—
204	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
205	C₂³×C₃	C₃×D₁₂	1	4	yes	—
206	C₂³×C₃	C₂²×C₃×D₃	1	2	yes	—
207	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
208	C₂²×D₃	—	3	4	no	—
209	C₂²×D₃	—	6	4	no	—
210	C₂²×D₃	—	12	4	no	—
211	C₃×D₄	—	18	8	no	—
212	C₃×D₄	—	18	8	no	—
213	C₃×D₄	—	36	4	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	—	4	4	no	—
218	C₂×C₃×C₄	—	9	4	no	—
219	C₂×C₃×C₄	—	18	4	no	—
220	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
221	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
222	#67	—	6	8	no	—
223	#67	—	6	8	no	—
224	C₂×Dic₃	—	3	4	no	—
225	C₂×Dic₃	—	6	4	no	—
226	D₁₂	—	12	4	no	—
227	C₂×C₃²	—	2	2	no	—
228	C₂×C₃²	—	2	4	no	—
229	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
230	C₂×C₃²	#1004	1	2	yes	—
231	C₃×D₃	—	12	4	no	—
232	C₂×C₉	—	2	4	no	—
233	C₂×C₉	—	2	4	no	—
234	C₂×C₉	—	2	2	no	—
235	C₂×C₉	—	2	1	no	—
236	C₂×C₉	—	2	2	no	—
237	C₂×C₉	—	4	4	no	—
238	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₂×C₉	C₃×H(5,𝔽₂)	1	2	yes	—
240	D₉	—	4	4	no	—
241	C₂⁴	—	9	2	no	—
242	C₂⁴	C₂×C₃×D₉	1	1	yes	characteristic
243	C₂×D₄	—	9	4	no	—
244	C₂×D₄	—	9	4	no	—
245	C₂×D₄	—	9	4	no	—
246	C₂×D₄	—	18	8	no	—
247	C₂²×C₄	—	9	2	no	—
248	C₂²×C₄	C₂×C₃×D₉	1	2	yes	complemented
249	C₄⋊C₄	—	9	4	no	—
250	C₂²⋊C₄	—	9	8	no	—
251	C₂²⋊C₄	C₂×C₃×D₉	1	4	yes	complemented
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	2	4	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	4	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	4	4	no	—
261	C₂²×C₃	—	4	4	no	—
262	C₂²×C₃	—	4	4	no	—
263	C₂²×C₃	—	4	8	no	—
264	C₂²×C₃	—	18	4	no	—
265	C₂²×C₃	—	18	8	no	—
266	C₂²×C₃	—	36	8	no	—
267	C₂²×C₃	C₂×D₃₆	1	4	yes	—
268	C₂²×C₃	C₂×D₃₆	1	2	yes	—
269	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
270	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
271	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
272	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
273	C₂×D₃	—	6	4	no	—
274	C₂×D₃	—	6	8	no	—
275	C₂×D₃	—	12	8	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	2	4	no	—
278	C₃×C₄	—	4	4	no	—
279	C₃×C₄	—	18	4	no	—
280	Dic₃	—	6	4	no	—
281	C₃²	#5493	1	1	yes	characteristic
282	C₉	—	2	1	no	—
283	C₉	C₃×#534	1	1	yes	characteristic, complemented
284	C₂³	—	4	2	no	—
285	C₂³	—	9	4	no	—
286	C₂³	—	18	4	no	—
287	C₂³	—	36	4	no	—
288	C₂³	C₃×D₃₆	1	4	yes	complemented
289	C₂³	C₂²×C₃×D₉	1	2	yes	—
290	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
291	D₄	—	18	8	no	—
292	D₄	—	18	8	no	—
293	D₄	—	36	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	9	4	no	—
296	C₂×C₄	—	18	4	no	—
297	C₂×C₄	C₂²×C₃×D₉	1	4	yes	—
298	C₂×C₃	—	2	4	no	—
299	C₂×C₃	—	2	4	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	2	4	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	4	4	no	—
306	C₂×C₃	—	36	4	no	—
307	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
308	C₂×C₃	#63710	1	2	yes	—
309	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
310	C₂×C₃	C₃×#1004	1	2	yes	—
311	D₃	—	12	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	18	4	no	—
316	C₂²	—	18	8	no	—
317	C₂²	—	36	8	no	—
318	C₂²	C₂×C₃×D₃₆	1	4	yes	—
319	C₂²	C₂×C₃×D₃₆	1	2	yes	—
320	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	18	4	no	—
323	C₃	—	2	1	no	—
324	C₃	#10589595	1	1	yes	characteristic, direct product factor
325	C₃	C₃×#5493	1	1	yes	characteristic
326	C₂	—	2	4	no	—
327	C₂	—	2	2	no	—
328	C₂	—	36	4	no	—
329	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic
330	C₂	C₃×#63710	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×C₂²⋊C₄	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₂×C₃×C₂²⋊C₄	D₉
hypocenter	C₉	C₃×#534
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#611	C₂	semidirect
2. C₂×C₃×C₉×C₂²⋊C₄	C₂	semidirect
3. C₃×#63492	C₂	semidirect
4. C₃×#63448	C₂	semidirect
5. C₃×#63444	C₂	semidirect
7. #10589595	C₃	direct (N × H)
9. C₂³×C₃×D₉	C₂²	semidirect
10. C₂²×C₃×C₄×C₉	C₂²	semidirect
13. C₂×C₃×#611	C₂²	semidirect
14. C₂²×C₃×Dic₉	C₂²	semidirect
15. C₂×C₃×D₃₆	C₂²	semidirect
16. C₃×C₉×C₂²⋊C₄	C₂²	semidirect
17. C₃×(C₂×D₉)⋊C₄	C₂²	semidirect
18. C₃×C₄⋊Dic₉	C₂²	semidirect
24. C₂²×#611	C₂×C₃	semidirect
26. C₂×C₉×C₂²⋊C₄	C₂×C₃	semidirect
27. #63492	C₂×C₃	semidirect
28. #63448	C₂×C₃	semidirect
29. #63444	C₂×C₃	semidirect
31. C₂³×C₃×C₉	D₄	semidirect
56. C₂³×D₉	C₂²×C₃	semidirect
61. C₂²×C₄×C₉	C₂²×C₃	semidirect
64. C₂×#611	C₂²×C₃	semidirect
65. C₂²×Dic₉	C₂²×C₃	semidirect
66. C₂×D₃₆	C₂²×C₃	semidirect
68. C₉×C₂²⋊C₄	C₂²×C₃	semidirect
69. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
70. C₄⋊Dic₉	C₂²×C₃	semidirect
88. C₂×C₃×C₂²⋊C₄	D₉	semidirect
112. C₂³×C₉	C₃×D₄	semidirect
144. C₂²×C₃×C₄	C₂×D₉	semidirect
154. C₃×C₂²⋊C₄	C₂×D₉	semidirect
191. C₂×C₂²⋊C₄	C₃×D₉	semidirect
192. C₃×C₉	#534	semidirect
202. C₂³×C₃	D₃₆	semidirect
248. C₂²×C₄	C₂×C₃×D₉	semidirect
251. C₂²⋊C₄	C₂×C₃×D₉	semidirect
283. C₉	C₃×#534	semidirect
288. C₂³	C₃×D₃₆	semidirect
324. C₃	#10589595	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×#611	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₃×#63492	2
C₃×#63448	2
C₃×#63444	2
C₃×#5493	3
#10589595	3

Subgroups — C3×#10589595 #13152340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589595 #13152340