Subgroups — #13164952

Order 1728.

3001 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#11722639	C₂	1	6	yes	maximal, complemented
3	#41×(C₃⋊D₃)⋊C₃	C₂	1	6	yes	maximal, complemented
4	#64361⋊C₃	C₂	1	9	yes	maximal, complemented
5	C₂×#3830⋊C₃	C₂	1	9	yes	maximal, complemented
6	#10596758	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#5966	—	3	1	no	maximal
8	#41×H(3,𝔽₃)	C₂²	1	6	yes	complemented
9	C₂×D₄×H(3,𝔽₃)	C₂²	1	9	yes	complemented
10	C₂×#638⋊C₃	C₂²	1	18	yes	complemented
11	C₂²×(C₃⋊Dic₃)⋊C₃	C₂²	1	6	yes	complemented
12	#88722	C₂²	1	2	yes	complemented
13	Q₈×(C₃⋊D₃)⋊C₃	C₂²	1	6	yes	complemented
14	#3830⋊C₃	C₂²	1	18	yes	complemented
15	#3830⋊C₃	C₂²	1	36	yes	complemented
16	D₄×(C₃⋊D₃)⋊C₃	C₂²	1	18	yes	complemented
17	#3828⋊C₃	C₂²	1	18	yes	complemented
18	C₂×C₄×(C₃⋊D₃)⋊C₃	C₂²	1	9	yes	complemented
19	C₂×#634⋊C₃	C₂²	1	9	yes	complemented
20	C₃²×H(5,𝔽₂)	—	2	1	no	—
21	C₃²×H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
22	C₃²×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
23	#64367	C₂×C₃	1	6	yes	complemented
24	#41×C₃⋊D₃	C₂×C₃	1	6	yes	complemented
25	#64361	C₂×C₃	1	9	yes	complemented
26	C₂×#3830	C₂×C₃	1	9	yes	complemented
27	C₃×#1010	—	3	6	no	—
28	C₃×D₃×#41	—	3	6	no	—
29	C₃×#1004	—	3	9	no	—
30	C₂×C₃×#237	—	3	9	no	—
31	C₂³×H(3,𝔽₃)	C₂³	1	6	yes	—
32	C₂²×(C₃⋊D₃)⋊C₃	C₂³	1	9	yes	—
33	Q₈×H(3,𝔽₃)	C₂³	1	2	yes	—
34	D₄×H(3,𝔽₃)	C₂³	1	18	yes	—
35	C₂×C₄×H(3,𝔽₃)	C₂³	1	9	yes	—
36	#638⋊C₃	C₂³	1	36	yes	—
37	C₂×(C₃⋊Dic₃)⋊C₃	C₂³	1	18	yes	—
38	C₂×(C₃⋊Dic₃)⋊C₃	C₂³	1	9	yes	—
39	C₄⋊((C₃⋊D₃)⋊C₃)	C₂³	1	6	yes	—
40	C₄×(C₃⋊D₃)⋊C₃	C₂³	1	6	yes	—
41	C₄×(C₃⋊D₃)⋊C₃	C₂³	1	18	yes	—
42	#634⋊C₃	C₂³	1	18	yes	—
43	C₃×#585	—	9	1	no	—
44	#5966	—	3	1	no	—
45	#5966	—	9	1	no	—
46	C₃²×#41	—	2	6	no	—
47	C₃²×#41	C₂×D₃	1	6	yes	complemented
48	C₃²×#41	C₂²×C₃	1	6	yes	complemented
49	C₂×C₃²×D₄	—	2	9	no	—
50	C₂×C₃²×D₄	C₂×D₃	1	9	yes	complemented
51	C₂×C₃²×D₄	C₂²×C₃	1	9	yes	complemented
52	C₂×#638	C₂²×C₃	1	18	yes	complemented
53	C₂²×C₃⋊Dic₃	C₂²×C₃	1	6	yes	complemented
54	#3832	C₂²×C₃	1	2	yes	complemented
55	Q₈×C₃⋊D₃	C₂²×C₃	1	6	yes	complemented
56	#3830	C₂²×C₃	1	18	yes	complemented
57	#3830	C₂²×C₃	1	36	yes	complemented
58	D₄×C₃⋊D₃	C₂²×C₃	1	18	yes	complemented
59	#3828	C₂²×C₃	1	18	yes	complemented
60	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	9	yes	complemented
61	C₂×#634	C₂²×C₃	1	9	yes	complemented
62	C₂×C₃×#67	—	3	18	no	—
63	C₂²×C₃×Dic₃	—	3	6	no	—
64	C₃×#239	—	3	2	no	—
65	C₃×D₃×Q₈	—	3	6	no	—
66	C₃×#237	—	3	18	no	—
67	C₃×#237	—	3	36	no	—
68	C₃×D₃×D₄	—	3	18	no	—
69	C₃×#235	—	3	18	no	—
70	C₂×C₃×C₄×D₃	—	3	9	no	—
71	C₂×C₃×Dic₆	—	3	9	no	—
72	C₂²×H(3,𝔽₃)	—	4	6	no	—
73	C₂²×H(3,𝔽₃)	C₂⁴	1	9	yes	—
74	C₂×(C₃⋊D₃)⋊C₃	—	4	9	no	—
75	C₂×(C₃⋊D₃)⋊C₃	C₂⁴	1	6	yes	—
76	C₄×H(3,𝔽₃)	C₂⁴	1	6	yes	—
77	(C₃⋊Dic₃)⋊C₃	C₂⁴	1	9	yes	—
78	(C₃⋊Dic₃)⋊C₃	C₂⁴	1	1	yes	characteristic
79	C₃×2₋¹⁺⁴	—	9	6	no	—
80	C₃×H(5,𝔽₂)	—	3	1	no	—
81	C₃×H(5,𝔽₂)	—	3	1	no	—
82	C₃×H(5,𝔽₂)	—	6	1	no	—
83	C₃×H(5,𝔽₂)	—	9	9	no	—
84	C₃×H(5,𝔽₂)	C₃×D₃	1	1	yes	characteristic
85	C₂×C₃×#41	—	9	9	no	—
86	C₂×C₃×#41	—	9	6	no	—
87	#1010	—	3	6	no	—
88	#1010	—	9	6	no	—
89	D₃×#41	—	3	6	no	—
90	D₃×#41	—	9	6	no	—
91	#1004	—	3	9	no	—
92	#1004	—	9	9	no	—
93	C₂×#237	—	3	9	no	—
94	C₂×#237	—	9	9	no	—
95	C₂³×C₃²	—	2	6	no	—
96	C₂³×C₃²	C₂²×D₃	1	6	yes	—
97	C₂³×C₃²	C₂³×C₃	1	6	yes	—
98	C₂²×C₃⋊D₃	C₂³×C₃	1	9	yes	—
99	C₂²×C₃×D₃	—	3	9	no	—
100	C₃²×Q₈	—	2	2	no	—
101	C₃²×Q₈	C₂²×D₃	1	2	yes	—
102	C₃²×Q₈	C₂³×C₃	1	2	yes	—
103	C₃²×D₄	—	2	18	no	—
104	C₃²×D₄	C₂²×D₃	1	18	yes	—
105	C₃²×D₄	C₂³×C₃	1	18	yes	—
106	C₂×C₃²×C₄	—	2	9	no	—
107	C₂×C₃²×C₄	C₂²×D₃	1	9	yes	—
108	C₂×C₃²×C₄	C₂³×C₃	1	9	yes	—
109	#638	C₂³×C₃	1	36	yes	—
110	C₂×C₃⋊Dic₃	C₂³×C₃	1	18	yes	—
111	C₂×C₃⋊Dic₃	C₂³×C₃	1	9	yes	—
112	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	6	yes	—
113	C₄×C₃⋊D₃	C₂³×C₃	1	6	yes	—
114	C₄×C₃⋊D₃	C₂³×C₃	1	18	yes	—
115	#634	C₂³×C₃	1	18	yes	—
116	C₃×#67	—	3	36	no	—
117	C₂×C₃×Dic₃	—	3	18	no	—
118	C₂×C₃×Dic₃	—	3	9	no	—
119	C₃×D₁₂	—	3	6	no	—
120	C₃×C₄×D₃	—	3	6	no	—
121	C₃×C₄×D₃	—	3	18	no	—
122	C₃×Dic₆	—	3	18	no	—
123	#585	—	9	1	no	Sylow
124	C₂×H(3,𝔽₃)	—	2	9	no	—
125	C₂×H(3,𝔽₃)	C₂⁵	1	1	yes	characteristic
126	(C₃⋊D₃)⋊C₃	—	2	6	no	—
127	C₃×#41	—	3	6	no	—
128	C₃×#41	—	3	6	no	—
129	C₃×#41	—	6	6	no	—
130	C₃×#41	—	9	18	no	—
131	C₃×#41	—	9	18	no	—
132	C₃×#41	—	9	36	no	—
133	C₃×#41	—	9	2	no	—
134	C₃×#41	C₂×C₃×D₃	1	6	yes	—
135	C₂×C₃×Q₈	—	9	9	no	—
136	C₂×C₃×Q₈	—	9	6	no	—
137	C₂×C₃×D₄	—	3	9	no	—
138	C₂×C₃×D₄	—	3	9	no	—
139	C₂×C₃×D₄	—	6	9	no	—
140	C₂×C₃×D₄	—	9	18	no	—
141	C₂×C₃×D₄	—	9	18	no	—
142	C₂×C₃×D₄	C₂×C₃×D₃	1	9	yes	—
143	C₂²×C₃×C₄	—	9	9	no	—
144	C₂²×C₃×C₄	—	9	6	no	—
145	C₂×#67	—	3	18	no	—
146	C₂×#67	—	9	18	no	—
147	C₂²×Dic₃	—	3	6	no	—
148	C₂²×Dic₃	—	9	6	no	—
149	#239	—	3	2	no	—
150	#239	—	9	2	no	—
151	D₃×Q₈	—	3	6	no	—
152	D₃×Q₈	—	9	6	no	—
153	#237	—	3	18	no	—
154	#237	—	3	36	no	—
155	#237	—	9	18	no	—
156	#237	—	9	36	no	—
157	D₃×D₄	—	3	18	no	—
158	D₃×D₄	—	9	18	no	—
159	#235	—	3	18	no	—
160	#235	—	9	18	no	—
161	C₂×C₄×D₃	—	3	9	no	—
162	C₂×C₄×D₃	—	9	9	no	—
163	C₂×Dic₆	—	3	9	no	—
164	C₂×Dic₆	—	9	9	no	—
165	C₂²×C₃²	—	2	9	no	—
166	C₂²×C₃²	—	4	6	no	—
167	C₂²×C₃²	—	4	6	no	—
168	C₂²×C₃²	—	8	6	no	—
169	C₂²×C₃²	C₂³×D₃	1	9	yes	—
170	C₂²×C₃²	C₂⁴×C₃	1	9	yes	—
171	C₂×C₃⋊D₃	—	4	9	no	—
172	C₂×C₃⋊D₃	C₂⁴×C₃	1	6	yes	—
173	C₂×C₃×D₃	—	3	6	no	—
174	C₂×C₃×D₃	—	12	9	no	—
175	C₃²×C₄	—	2	6	no	—
176	C₃²×C₄	C₂³×D₃	1	6	yes	—
177	C₃²×C₄	C₂⁴×C₃	1	6	yes	—
178	C₃⋊Dic₃	C₂⁴×C₃	1	9	yes	—
179	C₃⋊Dic₃	C₂⁴×C₃	1	1	yes	characteristic
180	C₃×Dic₃	—	3	9	no	—
181	C₃×Dic₃	—	3	1	no	—
182	2₋¹⁺⁴	—	9	6	no	—
183	H(5,𝔽₂)	—	9	9	no	—
184	H(5,𝔽₂)	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
185	C₂×#41	—	9	6	no	—
186	C₂×#41	—	9	9	no	—
187	H(3,𝔽₃)	#585	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₃	—	3	6	no	—
189	C₂³×C₃	—	3	6	no	—
190	C₂³×C₃	—	6	6	no	—
191	C₂³×C₃	—	9	9	no	—
192	C₂³×C₃	C₂²×C₃×D₃	1	6	yes	—
193	C₂²×D₃	—	3	9	no	—
194	C₂²×D₃	—	9	9	no	—
195	C₃×Q₈	—	3	2	no	—
196	C₃×Q₈	—	3	2	no	—
197	C₃×Q₈	—	6	2	no	—
198	C₃×Q₈	—	9	18	no	—
199	C₃×Q₈	C₂²×C₃×D₃	1	2	yes	—
200	C₃×D₄	—	3	18	no	—
201	C₃×D₄	—	3	18	no	—
202	C₃×D₄	—	6	18	no	—
203	C₃×D₄	—	9	6	no	—
204	C₃×D₄	—	9	36	no	—
205	C₃×D₄	C₂²×C₃×D₃	1	18	yes	—
206	C₂×C₃×C₄	—	3	9	no	—
207	C₂×C₃×C₄	—	3	9	no	—
208	C₂×C₃×C₄	—	6	9	no	—
209	C₂×C₃×C₄	—	9	6	no	—
210	C₂×C₃×C₄	—	9	18	no	—
211	C₂×C₃×C₄	—	9	9	no	—
212	C₂×C₃×C₄	—	9	18	no	—
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	9	yes	—
214	#67	—	3	36	no	—
215	#67	—	9	36	no	—
216	C₂×Dic₃	—	3	18	no	—
217	C₂×Dic₃	—	3	9	no	—
218	C₂×Dic₃	—	9	18	no	—
219	C₂×Dic₃	—	9	9	no	—
220	D₁₂	—	3	6	no	—
221	D₁₂	—	9	6	no	—
222	C₄×D₃	—	3	6	no	—
223	C₄×D₃	—	3	18	no	—
224	C₄×D₃	—	9	6	no	—
225	C₄×D₃	—	9	18	no	—
226	Dic₆	—	3	18	no	—
227	Dic₆	—	9	18	no	—
228	C₂×C₃²	—	2	9	no	—
229	C₂×C₃²	—	2	9	no	—
230	C₂×C₃²	—	2	1	no	—
231	C₂×C₃²	—	4	9	no	—
232	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
233	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
234	C₃⋊D₃	—	2	6	no	—
235	C₃×D₃	—	6	6	no	—
236	#41	—	9	2	no	—
237	#41	—	9	18	no	—
238	#41	—	9	36	no	—
239	#41	—	9	18	no	—
240	#41	C₂×(C₃⋊D₃)⋊C₃	1	6	yes	complemented
241	C₂×Q₈	—	9	6	no	—
242	C₂×Q₈	—	9	9	no	—
243	C₂×D₄	—	9	18	no	—
244	C₂×D₄	—	9	18	no	—
245	C₂×D₄	C₂×(C₃⋊D₃)⋊C₃	1	9	yes	complemented
246	C₂²×C₄	—	9	6	no	—
247	C₂²×C₄	—	9	9	no	—
248	C₂²×C₃	—	3	9	no	—
249	C₂²×C₃	—	3	9	no	—
250	C₂²×C₃	—	4	6	no	—
251	C₂²×C₃	—	6	9	no	—
252	C₂²×C₃	—	9	6	no	—
253	C₂²×C₃	—	12	6	no	—
254	C₂²×C₃	—	12	6	no	—
255	C₂²×C₃	—	24	6	no	—
256	C₂²×C₃	—	36	9	no	—
257	C₂²×C₃	C₂³×C₃×D₃	1	9	yes	—
258	C₂×D₃	—	3	6	no	—
259	C₂×D₃	—	9	6	no	—
260	C₂×D₃	—	12	9	no	—
261	C₂×D₃	—	36	9	no	—
262	C₃×C₄	—	3	6	no	—
263	C₃×C₄	—	3	6	no	—
264	C₃×C₄	—	6	6	no	—
265	C₃×C₄	—	9	1	no	—
266	C₃×C₄	—	9	9	no	—
267	C₃×C₄	C₂³×C₃×D₃	1	6	yes	—
268	Dic₃	—	3	9	no	—
269	Dic₃	—	3	1	no	—
270	Dic₃	—	9	9	no	—
271	Dic₃	—	9	1	no	—
272	C₃²	—	2	1	no	—
273	C₃²	#5966	1	1	yes	characteristic, complemented
274	C₃²	C₃×#585	1	1	yes	characteristic, complemented
275	C₂³	—	9	9	no	—
276	C₂³	C₂²×(C₃⋊D₃)⋊C₃	1	6	yes	—
277	Q₈	—	9	18	no	—
278	Q₈	C₂²×(C₃⋊D₃)⋊C₃	1	2	yes	—
279	D₄	—	9	36	no	—
280	D₄	—	9	6	no	—
281	D₄	C₂²×(C₃⋊D₃)⋊C₃	1	18	yes	—
282	C₂×C₄	—	9	9	no	—
283	C₂×C₄	—	9	18	no	—
284	C₂×C₄	—	9	6	no	—
285	C₂×C₄	—	9	18	no	—
286	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	9	yes	—
287	C₂×C₃	—	2	9	no	—
288	C₂×C₃	—	3	1	no	—
289	C₂×C₃	—	3	1	no	—
290	C₂×C₃	—	6	9	no	—
291	C₂×C₃	—	6	9	no	—
292	C₂×C₃	—	6	1	no	—
293	C₂×C₃	—	12	9	no	—
294	C₂×C₃	—	18	6	no	—
295	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
296	D₃	—	6	6	no	—
297	D₃	—	18	6	no	—
298	C₂²	—	4	6	no	—
299	C₂²	—	9	6	no	—
300	C₂²	—	36	9	no	—
301	C₂²	C₂³×(C₃⋊D₃)⋊C₃	1	9	yes	—
302	C₄	—	9	1	no	—
303	C₄	—	9	9	no	—
304	C₄	C₂³×(C₃⋊D₃)⋊C₃	1	6	yes	—
305	C₃	—	3	1	no	—
306	C₃	—	3	1	no	—
307	C₃	—	6	1	no	—
308	C₃	C₃×#5966	1	1	yes	characteristic
309	C₂	—	2	9	no	—
310	C₂	—	18	6	no	—
311	C₂	C₂⁴×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×C₃²	C₂⁵×C₃
Frattini, socle	C₂×C₃	C₂⁴×C₃×D₃
Fitting	H(3,𝔽₃)×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×#585
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×H(5,𝔽₂)	C₂	semidirect
2. #11722639	C₂	semidirect
3. #41×(C₃⋊D₃)⋊C₃	C₂	semidirect
4. #64361⋊C₃	C₂	semidirect
5. C₂×#3830⋊C₃	C₂	semidirect
6. #10596758	C₃	semidirect
8. #41×H(3,𝔽₃)	C₂²	semidirect
9. C₂×D₄×H(3,𝔽₃)	C₂²	semidirect
10. C₂×#638⋊C₃	C₂²	semidirect
11. C₂²×(C₃⋊Dic₃)⋊C₃	C₂²	semidirect
12. #88722	C₂²	semidirect
13. Q₈×(C₃⋊D₃)⋊C₃	C₂²	semidirect
14. #3830⋊C₃	C₂²	semidirect
15. #3830⋊C₃	C₂²	semidirect
16. D₄×(C₃⋊D₃)⋊C₃	C₂²	semidirect
17. #3828⋊C₃	C₂²	semidirect
18. C₂×C₄×(C₃⋊D₃)⋊C₃	C₂²	semidirect
19. C₂×#634⋊C₃	C₂²	semidirect
21. C₃²×H(5,𝔽₂)	D₃	semidirect
22. C₃²×H(5,𝔽₂)	C₂×C₃	semidirect
23. #64367	C₂×C₃	semidirect
24. #41×C₃⋊D₃	C₂×C₃	semidirect
25. #64361	C₂×C₃	semidirect
26. C₂×#3830	C₂×C₃	semidirect
47. C₃²×#41	C₂×D₃	semidirect
48. C₃²×#41	C₂²×C₃	semidirect
50. C₂×C₃²×D₄	C₂×D₃	semidirect
51. C₂×C₃²×D₄	C₂²×C₃	semidirect
52. C₂×#638	C₂²×C₃	semidirect
53. C₂²×C₃⋊Dic₃	C₂²×C₃	semidirect
54. #3832	C₂²×C₃	semidirect
55. Q₈×C₃⋊D₃	C₂²×C₃	semidirect
56. #3830	C₂²×C₃	semidirect
57. #3830	C₂²×C₃	semidirect
58. D₄×C₃⋊D₃	C₂²×C₃	semidirect
59. #3828	C₂²×C₃	semidirect
60. C₂×C₄×C₃⋊D₃	C₂²×C₃	semidirect
61. C₂×#634	C₂²×C₃	semidirect
184. H(5,𝔽₂)	(C₃⋊D₃)⋊C₃	semidirect
187. H(3,𝔽₃)	#585	semidirect
240. #41	C₂×(C₃⋊D₃)⋊C₃	semidirect
245. C₂×D₄	C₂×(C₃⋊D₃)⋊C₃	semidirect
273. C₃²	#5966	semidirect
274. C₃²	C₃×#585	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
H(3,𝔽₃)×H(5,𝔽₂)	2
#11722639	2
#41×(C₃⋊D₃)⋊C₃	2
#64361⋊C₃	2
C₂×#3830⋊C₃	2
#10596758	3
C₃×#5966	3