Subgroups — C₃×#10588682 #13142677

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63369	C₂	1	4	yes	maximal, complemented
5	C₃×#5001	—	3	1	no	maximal
6	#10588682	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃×D₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₃₆	—	2	1	no	—
10	C₂×C₃×D₃₆	—	2	2	no	—
11	C₂×C₃×D₃₆	C₄	1	4	yes	complemented
12	C₂×C₃×D₃₆	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₃×C₄²×C₉	—	2	1	no	—
16	C₃×#3667	—	2	2	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	complemented
18	C₂²×C₃×D₁₂	—	3	1	no	—
19	C₃²×#117	D₃	1	1	yes	characteristic
20	C₂×C₃×#208	—	3	1	no	—
21	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic
22	C₃×#809	—	3	4	no	—
23	C₉×#117	—	2	1	no	—
24	C₉×#117	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#3667	C₂×C₃	1	1	yes	characteristic, complemented
26	#63369	C₂×C₃	1	4	yes	complemented
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₃×D₉	—	2	4	no	—
29	C₂²×C₃×D₉	—	2	2	no	—
30	C₂²×C₃×D₉	—	4	4	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₃×D₃₆	—	2	4	no	—
37	C₃×D₃₆	—	2	2	no	—
38	C₃×D₃₆	C₂×C₄	1	4	yes	—
39	C₃×C₉⋊C₈	—	2	2	no	—
40	C₃×#418	—	9	1	no	—
41	#5001	—	3	1	no	—
42	C₂³×C₃×D₃	—	6	1	no	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×D₁₂	—	3	4	no	—
45	C₂×C₃×D₁₂	—	3	2	no	—
46	C₂×C₃×D₁₂	—	6	2	no	—
47	C₂×C₃×D₁₂	—	6	1	no	—
48	C₂³×D₉	—	2	1	no	—
49	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₃²×C₂²⋊C₄	—	2	1	no	—
51	C₃²×C₄²	—	2	1	no	—
52	C₃×#208	—	6	2	no	—
53	C₂×C₃×C₃⋊C₈	—	3	2	no	—
54	C₂²×C₄×C₉	—	2	1	no	—
55	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
56	C₂×D₃₆	—	2	1	no	—
57	C₂×D₃₆	—	2	2	no	—
58	C₂×D₃₆	C₃×C₄	1	4	yes	complemented
59	C₂×D₃₆	C₂²×C₃	1	2	yes	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
62	C₉×C₂²⋊C₄	—	2	2	no	—
63	C₉×C₂²⋊C₄	—	2	1	no	—
64	C₄²×C₉	—	2	2	no	—
65	C₄²×C₉	—	2	1	no	—
66	#3667	—	2	2	no	—
67	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	complemented
68	C₂²×C₃×C₉	—	2	2	no	—
69	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×D₉	—	2	4	no	—
72	C₂×C₃×D₉	—	4	8	no	—
73	C₂×C₃×D₉	—	4	4	no	—
74	C₃×C₄×C₉	—	4	2	no	—
75	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
76	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×D₄	—	9	1	no	—
79	C₂²×D₁₂	—	3	1	no	—
80	C₂×C₃×M₁₆	—	9	1	no	—
81	C₃×#117	—	2	1	no	—
82	C₃×#117	D₉	1	1	yes	characteristic, complemented
83	C₃×#117	C₃×D₃	1	1	yes	characteristic
84	C₂×#208	—	3	1	no	—
85	C₃×#102	—	9	4	no	—
86	#809	—	3	4	no	—
87	C₂³×C₃²	#67	1	1	yes	characteristic
88	C₂²×C₃×D₃	—	6	4	no	—
89	C₂²×C₃×D₃	—	6	2	no	—
90	C₂²×C₃×D₃	—	12	4	no	—
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
93	C₂×C₃²×C₄	D₁₂	1	2	yes	—
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₃×D₁₂	—	3	4	no	—
97	C₃×D₁₂	—	6	4	no	—
98	C₃×D₁₂	—	6	2	no	—
99	C₂³×C₉	—	2	1	no	—
100	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
101	C₂²×D₉	—	2	4	no	—
102	C₂²×D₉	—	2	2	no	—
103	C₂²×D₉	—	4	4	no	—
104	C₃×C₃⋊C₈	—	6	2	no	—
105	C₂×C₄×C₉	—	2	4	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
112	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
113	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
115	D₃₆	—	2	4	no	—
116	D₃₆	—	2	2	no	—
117	D₃₆	C₂×C₃×C₄	1	4	yes	—
118	C₉⋊C₈	—	2	2	no	—
119	#418	—	9	1	no	Sylow
120	C₂×C₃×C₉	—	2	2	no	—
121	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₃×D₉	—	4	4	no	—
124	C₂⁴×C₃	—	18	1	no	—
125	C₂³×D₃	—	6	1	no	—
126	C₂×C₃×D₄	—	9	4	no	—
127	C₂×C₃×D₄	—	9	2	no	—
128	C₂×C₃×D₄	—	18	2	no	—
129	C₂×C₃×D₄	—	18	1	no	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
133	C₂×D₁₂	—	3	4	no	—
134	C₂×D₁₂	—	3	2	no	—
135	C₂×D₁₂	—	6	2	no	—
136	C₂×D₁₂	—	6	1	no	—
137	C₃×M₁₆	—	18	2	no	—
138	C₂×C₃×C₈	—	9	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	complemented
141	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
142	C₃×C₂²⋊C₄	—	2	2	no	—
143	C₃×C₂²⋊C₄	—	2	1	no	—
144	C₃×C₂²⋊C₄	—	2	1	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	#208	—	6	2	no	—
149	C₂×C₃⋊C₈	—	3	2	no	—
150	C₂²×C₃²	—	2	2	no	—
151	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
152	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
153	C₂×C₃×D₃	—	6	4	no	—
154	C₂×C₃×D₃	—	12	8	no	—
155	C₂×C₃×D₃	—	12	4	no	—
156	C₃²×C₄	—	4	2	no	—
157	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
158	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
159	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	C₃×C₂²⋊C₄	1	2	yes	—
165	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×D₉	—	2	4	no	—
167	C₂×D₉	—	4	8	no	—
168	C₂×D₉	—	4	4	no	—
169	C₄×C₉	—	2	2	no	—
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	4	4	no	—
173	C₄×C₉	—	4	2	no	—
174	C₄×C₉	C₃×C₂²⋊C₄	1	2	yes	—
175	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂²×D₄	—	9	1	no	—
178	C₂×M₁₆	—	9	1	no	—
179	#117	C₃×D₉	1	1	yes	characteristic, complemented
180	#102	—	9	4	no	—
181	C₃×C₉	#418	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₃	—	2	1	no	—
183	C₂³×C₃	—	18	4	no	—
184	C₂³×C₃	—	18	2	no	—
185	C₂³×C₃	—	36	4	no	—
186	C₂³×C₃	#611	1	1	yes	characteristic
187	C₂³×C₃	C₃×#67	1	1	yes	characteristic
188	C₂²×D₃	—	6	4	no	—
189	C₂²×D₃	—	6	2	no	—
190	C₂²×D₃	—	12	4	no	—
191	C₃×D₄	—	9	4	no	—
192	C₃×D₄	—	18	4	no	—
193	C₃×D₄	—	18	2	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
202	C₂×C₃×C₄	D₃₆	1	2	yes	—
203	C₂×C₃×C₄	#611	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
205	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
206	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
207	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
209	D₁₂	—	3	4	no	—
210	D₁₂	—	6	4	no	—
211	D₁₂	—	6	2	no	—
212	C₃×C₈	—	18	2	no	—
213	C₃⋊C₈	—	6	2	no	—
214	C₂×C₃²	—	2	2	no	—
215	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
216	C₂×C₃²	#949	1	2	yes	—
217	C₃×D₃	—	12	4	no	—
218	C₂×C₉	—	2	2	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	2	2	no	—
221	C₂×C₉	—	4	2	no	—
222	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₉	C₃×AGL(1,ℤ/8)	1	2	yes	—
224	D₉	—	4	4	no	—
225	C₂⁴	—	18	1	no	—
226	C₂×D₄	—	9	4	no	—
227	C₂×D₄	—	9	2	no	—
228	C₂×D₄	—	18	2	no	—
229	C₂×D₄	—	18	1	no	—
230	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
231	M₁₆	—	18	2	no	—
232	C₂×C₈	—	9	2	no	—
233	C₄⋊C₄	C₂×C₃×D₉	1	2	yes	complemented
234	C₂²⋊C₄	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	4	2	no	—
241	C₂²×C₃	—	18	4	no	—
242	C₂²×C₃	—	36	8	no	—
243	C₂²×C₃	—	36	4	no	—
244	C₂²×C₃	(C₂×D₉)⋊C₄	1	2	yes	—
245	C₂²×C₃	C₂×#611	1	1	yes	characteristic
246	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
247	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
248	C₂×D₃	—	6	4	no	—
249	C₂×D₃	—	12	8	no	—
250	C₂×D₃	—	12	4	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	4	4	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
258	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
259	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
260	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
261	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
262	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
263	C₃²	#5001	1	1	yes	characteristic
264	C₉	—	2	1	no	—
265	C₉	C₃×#418	1	1	yes	characteristic, complemented
266	C₂³	—	18	2	no	—
267	C₂³	—	18	4	no	—
268	C₂³	—	36	4	no	—
269	C₂³	C₃×#611	1	1	yes	characteristic
270	D₄	—	9	4	no	—
271	D₄	—	18	2	no	—
272	D₄	—	18	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
275	C₂×C₄	C₃×D₃₆	1	2	yes	—
276	C₂×C₄	C₃×#611	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
278	C₈	—	18	2	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	4	2	no	—
284	C₂×C₃	—	36	4	no	—
285	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₃	#63512	1	2	yes	—
287	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
288	C₂×C₃	C₃×#949	1	2	yes	—
289	D₃	—	12	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	18	4	no	—
292	C₂²	—	36	8	no	—
293	C₂²	—	36	4	no	—
294	C₂²	C₃×(C₂×D₉)⋊C₄	1	2	yes	—
295	C₂²	C₂×C₃×#611	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	C₃×(C₂×D₉)⋊C₄	1	2	yes	—
298	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
299	C₄	C₂×C₃×D₃₆	1	1	yes	characteristic
300	C₃	—	2	1	no	—
301	C₃	#10588682	1	1	yes	characteristic, direct product factor
302	C₃	C₃×#5001	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	36	4	no	—
305	C₂	C₂×C₃×(C₂×D₉)⋊C₄	1	1	yes	characteristic
306	C₂	C₃×#63512	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂²×C₃×D₃₆	C₂
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×#117	D₉
hypocenter	C₉	C₃×#418
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×#117	C₂	semidirect
3. C₂×C₃×#3667	C₂	semidirect
4. C₃×#63369	C₂	semidirect
6. #10588682	C₃	direct (N × H)
11. C₂×C₃×D₃₆	C₄	semidirect
13. C₃×C₉×C₄⋊C₄	C₂²	semidirect
17. C₂×C₃×C₉⋊C₈	C₂²	semidirect
24. C₉×#117	C₂×C₃	semidirect
25. C₂×#3667	C₂×C₃	semidirect
26. #63369	C₂×C₃	semidirect
58. C₂×D₃₆	C₃×C₄	semidirect
61. C₉×C₄⋊C₄	C₂²×C₃	semidirect
67. C₂×C₉⋊C₈	C₂²×C₃	semidirect
82. C₃×#117	D₉	semidirect
140. C₃×C₄⋊C₄	C₂×D₉	semidirect
179. #117	C₃×D₉	semidirect
181. C₃×C₉	#418	semidirect
233. C₄⋊C₄	C₂×C₃×D₉	semidirect
265. C₉	C₃×#418	semidirect
301. C₃	#10588682	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×D₃₆	2
C₃×C₉×#117	2
C₂×C₃×#3667	2
C₃×#63369	2
C₃×#5001	3
#10588682	3

Subgroups — C3×#10588682 #13142677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588682 #13142677