Subgroups — #13160404

Order 1728.

988 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11723967	C₂	1	1	yes	maximal, characteristic
2	C₃×#64135	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11722055	C₂	1	2	yes	maximal, complemented
5	#11722052	C₂	1	2	yes	maximal, complemented
6	#10592805	—	3	1	no	maximal
7	#10591122	—	3	4	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	#88864	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
11	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₃×#3757	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
14	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
15	C₂×C₃²×C₃⋊C₈	C₄	1	4	yes	complemented
16	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃²×C₈	—	3	1	no	—
19	#64135	—	3	1	no	—
20	#64135	—	3	4	no	—
21	#64135	—	6	4	no	—
22	#64135	D₃	1	1	yes	characteristic
23	C₃×#926	—	3	4	no	—
24	C₂²×C₃×C₃⋊C₈	D₃	1	4	yes	—
25	#63645	—	3	2	no	—
26	#63644	—	3	2	no	—
27	#63575	—	3	8	no	—
28	#63573	—	3	8	no	—
29	C₂³×C₃³	D₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	Q₈	1	2	yes	—
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
35	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
36	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	#5106	—	9	4	no	—
39	#4998	—	9	1	no	—
40	C₂²×C₃²×C₄	—	2	4	no	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
43	C₂×C₃²×C₈	—	3	4	no	—
44	C₂×C₃²×C₈	—	3	1	no	—
45	C₂×C₃²×C₈	—	3	1	no	—
46	#3757	—	2	1	no	—
47	#3757	—	6	8	no	—
48	#3757	—	6	4	no	—
49	#3757	—	6	1	no	—
50	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
51	C₄⋊(C₃⋊Dic₃)	—	3	8	no	—
52	C₄⋊(C₃⋊Dic₃)	—	6	8	no	—
53	C₄⋊(C₃⋊Dic₃)	C₂×D₃	1	2	yes	—
54	C₄×C₃⋊Dic₃	—	2	1	no	—
55	C₄×C₃⋊Dic₃	—	6	8	no	—
56	C₄×C₃⋊Dic₃	—	6	4	no	—
57	C₄×C₃⋊Dic₃	—	6	1	no	—
58	C₃×#217	—	6	4	no	—
59	C₃×C₄⋊Dic₃	—	3	8	no	—
60	C₃×C₄×Dic₃	—	6	4	no	—
61	C₂×C₃×C₃⋊C₈	Dic₃	1	16	yes	complemented
62	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
63	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
64	C₂²×C₃³	—	2	2	no	—
65	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₃³×C₄	C₄⋊C₄	1	2	yes	—
68	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
70	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
71	C₃×C₃⋊Dic₃	—	4	2	no	—
72	C₂²×C₃×C₈	—	3	4	no	—
73	C₃×#117	—	9	1	no	—
74	#926	—	3	4	no	—
75	#926	—	9	4	no	—
76	#926	—	9	1	no	—
77	#926	—	18	4	no	—
78	C₂²×C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic
79	#818	—	9	8	no	—
80	#817	—	9	8	no	—
81	#808	—	9	2	no	—
82	#807	—	9	2	no	—
83	C₂³×C₃²	—	2	4	no	—
84	C₂³×C₃²	D₁₂	1	4	yes	—
85	C₂³×C₃²	#67	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	2	8	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	8	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	Dic₆	1	8	yes	—
91	C₂×C₃²×C₄	Dic₆	1	2	yes	—
92	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
93	C₂×C₃²×C₄	D₁₂	1	4	yes	—
94	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
95	C₂×C₃²×C₄	#67	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
97	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	—	2	2	no	—
99	C₂×C₃⋊Dic₃	—	6	16	no	—
100	C₂×C₃⋊Dic₃	—	6	8	no	—
101	C₂×C₃⋊Dic₃	—	6	2	no	—
102	C₂×C₃×Dic₃	—	6	8	no	—
103	C₃²×C₈	—	3	2	no	—
104	C₃²×C₈	—	3	2	no	—
105	C₃×C₃⋊C₈	C₂×Dic₃	1	8	yes	—
106	C₃×C₃⋊C₈	C₂×Dic₃	1	8	yes	—
107	#427	—	27	1	no	Sylow
108	C₂×C₃³	—	2	2	no	—
109	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₃³	#135	1	2	yes	—
111	C₂²×C₃×C₄	—	2	4	no	—
112	C₂²×C₃×C₄	D₃²	1	4	yes	—
113	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
114	C₂×C₃×C₈	—	3	16	no	—
115	C₂×C₃×C₈	—	3	4	no	—
116	C₂×C₃×C₈	—	3	4	no	—
117	C₃×C₄⋊C₄	—	9	2	no	—
118	C₃×C₂²⋊C₄	—	18	1	no	—
119	C₃×C₄²	—	18	1	no	—
120	#217	—	6	4	no	—
121	#217	—	18	8	no	—
122	#217	—	18	4	no	—
123	#217	—	18	1	no	—
124	C₄⋊Dic₃	—	3	8	no	—
125	C₄⋊Dic₃	—	9	8	no	—
126	C₄⋊Dic₃	—	9	2	no	—
127	C₄⋊Dic₃	—	18	8	no	—
128	C₄×Dic₃	—	6	4	no	—
129	C₄×Dic₃	—	18	8	no	—
130	C₄×Dic₃	—	18	4	no	—
131	C₄×Dic₃	—	18	1	no	—
132	C₂×C₃⋊C₈	C₃⋊Dic₃	1	4	yes	complemented
133	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
134	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
135	C₂²×C₃²	—	2	2	no	—
136	C₂²×C₃²	—	2	8	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	2	8	no	—
139	C₂²×C₃²	—	4	8	no	—
140	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
141	C₂²×C₃²	C₄⋊Dic₃	1	8	yes	—
142	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
143	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
144	C₃²×C₄	—	2	8	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
148	C₃²×C₄	C₄⋊Dic₃	1	8	yes	—
149	C₃²×C₄	C₂×Dic₆	1	4	yes	—
150	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
151	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
152	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
153	C₃⋊Dic₃	—	4	2	no	—
154	C₃⋊Dic₃	—	12	16	no	—
155	C₃⋊Dic₃	—	12	8	no	—
156	C₃⋊Dic₃	—	12	2	no	—
157	C₃×Dic₃	—	12	8	no	—
158	C₂²×C₈	—	3	1	no	—
159	#117	—	9	1	no	—
160	#117	—	27	1	no	—
161	#107	—	27	2	no	—
162	#106	—	27	2	no	—
163	C₃³	#427	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₃	—	2	4	no	—
165	C₂³×C₃	#626	1	4	yes	—
166	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	8	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	18	2	no	—
172	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
173	C₂×C₃×C₄	#626	1	4	yes	—
174	C₂×C₃×C₄	#627	1	8	yes	—
175	C₂×C₃×C₄	#634	1	2	yes	—
176	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
178	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
179	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
180	C₂×Dic₃	—	6	8	no	—
181	C₂×Dic₃	—	18	16	no	—
182	C₂×Dic₃	—	18	8	no	—
183	C₂×Dic₃	—	18	2	no	—
184	C₃×C₈	—	3	8	no	—
185	C₃×C₈	—	3	8	no	—
186	C₃⋊C₈	C₂×C₃⋊Dic₃	1	2	yes	—
187	C₃⋊C₈	C₂×C₃⋊Dic₃	1	2	yes	—
188	C₂×C₃²	—	2	2	no	—
189	C₂×C₃²	—	2	8	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	—	2	8	no	—
192	C₂×C₃²	—	4	8	no	—
193	C₂×C₃²	#904	1	8	yes	—
194	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
195	C₂×C₃²	C₂×C₄⋊Dic₃	1	4	yes	—
196	C₂×C₃²	#950	1	2	yes	—
197	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
198	C₂×C₈	—	3	4	no	—
199	C₂×C₈	—	3	1	no	—
200	C₂×C₈	—	3	1	no	—
201	C₄⋊C₄	—	9	2	no	—
202	C₄⋊C₄	—	27	2	no	—
203	C₂²⋊C₄	—	18	1	no	—
204	C₂²⋊C₄	—	54	1	no	—
205	C₄²	—	18	1	no	—
206	C₄²	—	54	1	no	—
207	C₂²×C₃	—	2	8	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	4	8	no	—
212	C₂²×C₃	Dic₃⋊Dic₃	1	8	yes	—
213	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
214	C₂²×C₃	C₂×#626	1	4	yes	—
215	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	8	no	—
219	C₃×C₄	—	36	2	no	—
220	C₃×C₄	Dic₃⋊Dic₃	1	8	yes	—
221	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
222	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
223	C₃×C₄	C₂×#627	1	4	yes	—
224	C₃×C₄	C₂×#634	1	1	yes	characteristic
225	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
226	Dic₃	—	12	8	no	—
227	Dic₃	—	36	16	no	—
228	Dic₃	—	36	8	no	—
229	Dic₃	—	36	2	no	—
230	C₃²	—	2	4	no	—
231	C₃²	#4998	1	1	yes	characteristic, complemented
232	C₃²	#5106	1	4	yes	complemented
233	C₂³	#6292	1	1	yes	characteristic
234	C₂×C₄	—	18	2	no	—
235	C₂×C₄	—	54	2	no	—
236	C₂×C₄	D₃×C₃⋊Dic₃	1	2	yes	—
237	C₂×C₄	#6292	1	1	yes	characteristic
238	C₂×C₄	#6293	1	2	yes	—
239	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
240	C₈	—	3	2	no	—
241	C₈	—	3	2	no	—
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	2	8	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	2	8	no	—
246	C₂×C₃	—	4	8	no	—
247	C₂×C₃	#63829	1	8	yes	—
248	C₂×C₃	C₂×Dic₃⋊Dic₃	1	4	yes	—
249	C₂×C₃	#64113	1	2	yes	—
250	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
251	C₂²	—	2	2	no	—
252	C₂²	Dic₃⋊(C₃⋊Dic₃)	1	2	yes	—
253	C₂²	C₂×#6292	1	1	yes	characteristic
254	C₄	—	36	2	no	—
255	C₄	—	108	2	no	—
256	C₄	Dic₃⋊(C₃⋊Dic₃)	1	2	yes	—
257	C₄	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
258	C₄	C₂×#6293	1	1	yes	characteristic
259	C₃	—	2	4	no	—
260	C₃	#10591122	1	4	yes	complemented
261	C₃	#10592805	1	1	yes	characteristic, complemented
262	C₂	—	2	2	no	—
263	C₂	#11723369	1	2	yes	—
264	C₂	C₂×Dic₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6292
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂³×C₃³	D₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	#427
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11722055	C₂	semidirect
5. #11722052	C₂	semidirect
15. C₂×C₃²×C₃⋊C₈	C₄	semidirect
61. C₂×C₃×C₃⋊C₈	Dic₃	semidirect
132. C₂×C₃⋊C₈	C₃⋊Dic₃	semidirect
163. C₃³	#427	semidirect
231. C₃²	#4998	semidirect
232. C₃²	#5106	semidirect
260. C₃	#10591122	semidirect
261. C₃	#10592805	semidirect

Maximal subgroups

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

subgroup	index
#11723967	2
C₃×#64135	2
C₂²×C₃²×C₃⋊C₈	2
#11722055	2
#11722052	2
#10592805	3
#10591122	3