Subgroups — #13163859

Order 1728.

4004 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88864	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88864	C₂	1	2	yes	maximal, complemented
5	#11722201	C₂	1	2	yes	maximal, complemented
6	#10592919	—	3	13	no	maximal
7	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
8	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
9	C₂×C₃³×D₄	—	2	2	no	—
10	C₂×C₃³×D₄	—	2	2	no	—
11	C₂×C₃³×D₄	C₄	1	4	yes	complemented
12	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
14	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
15	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
16	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
17	#88864	—	2	2	no	—
18	#88864	—	2	4	no	—
19	C₂²×C₃²×D₄	D₃	1	13	yes	—
20	C₂³×C₃⋊Dic₃	—	3	13	no	—
21	C₂×#3757	—	3	26	no	—
22	C₂×#3757	—	3	13	no	—
23	#63658	—	3	26	no	—
24	C₂³×C₃³	—	2	2	no	—
25	C₂³×C₃³	—	2	2	no	—
26	C₂³×C₃³	—	2	2	no	—
27	C₂³×C₃³	—	2	1	no	—
28	C₂³×C₃³	—	2	1	no	—
29	C₂³×C₃³	—	2	2	no	—
30	C₂³×C₃³	C₂×C₄	1	2	yes	—
31	C₂³×C₃³	C₂×C₄	1	2	yes	complemented
32	C₂³×C₃³	D₄	1	2	yes	—
33	C₂³×C₃³	D₄	1	2	yes	—
34	C₂³×C₃³	C₂³	1	1	yes	characteristic
35	C₃³×D₄	—	2	8	no	—
36	C₂×C₃³×C₄	—	2	1	no	—
37	C₂×C₃³×C₄	—	2	1	no	—
38	C₂×C₃³×C₄	C₂×C₄	1	2	yes	complemented
39	C₂×C₃⋊(C₃⋊Dic₃)	—	2	8	no	—
40	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
41	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
42	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
43	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
45	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	4	yes	complemented
46	#5220	—	9	13	no	—
47	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
48	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
49	C₂×C₃²×D₄	—	2	26	no	—
50	C₂×C₃²×D₄	—	2	26	no	—
51	C₂×C₃²×D₄	Dic₃	1	52	yes	complemented
52	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
53	C₂²×C₃⋊Dic₃	—	3	26	no	—
54	C₂²×C₃⋊Dic₃	—	3	26	no	—
55	C₂²×C₃⋊Dic₃	—	6	52	no	—
56	C₂²×C₃⋊Dic₃	—	6	26	no	—
57	#3757	—	6	52	no	—
58	#3757	—	6	26	no	—
59	C₂²×C₃³	—	2	4	no	—
60	C₂²×C₃³	—	2	2	no	—
61	C₂²×C₃³	—	2	4	no	—
62	C₂²×C₃³	—	2	2	no	—
63	C₂²×C₃³	—	2	2	no	—
64	C₂²×C₃³	—	2	4	no	—
65	C₂²×C₃³	—	4	1	no	—
66	C₂²×C₃³	—	4	2	no	—
67	C₂²×C₃³	—	4	1	no	—
68	C₂²×C₃³	C₂²⋊C₄	1	4	yes	complemented
69	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃³	C₂×D₄	1	2	yes	—
73	C₂²×C₃³	#41	1	2	yes	—
74	C₃³×C₄	—	2	2	no	—
75	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
76	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
77	C₂²×C₃×D₄	C₃⋊D₃	1	13	yes	—
78	C₂³×Dic₃	—	9	13	no	—
79	C₂×#217	—	9	26	no	—
80	C₂×#217	—	9	13	no	—
81	C₃⋊#95	—	9	26	no	—
82	C₂³×C₃²	—	2	26	no	—
83	C₂³×C₃²	—	2	26	no	—
84	C₂³×C₃²	—	2	26	no	—
85	C₂³×C₃²	—	2	26	no	—
86	C₂³×C₃²	—	2	13	no	—
87	C₂³×C₃²	—	2	13	no	—
88	C₂³×C₃²	C₂×Dic₃	1	26	yes	—
89	C₂³×C₃²	C₂×Dic₃	1	26	yes	complemented
90	C₂³×C₃²	#67	1	26	yes	—
91	C₂³×C₃²	#67	1	26	yes	—
92	C₂³×C₃²	C₂²×D₃	1	13	yes	—
93	C₃²×D₄	—	2	104	no	—
94	C₂×C₃²×C₄	—	2	13	no	—
95	C₂×C₃²×C₄	—	2	13	no	—
96	C₂×C₃²×C₄	C₂×Dic₃	1	26	yes	complemented
97	C₂×C₃⋊Dic₃	—	3	52	no	—
98	C₂×C₃⋊Dic₃	—	6	104	no	—
99	C₂×C₃⋊Dic₃	—	6	26	no	—
100	C₂×C₃⋊Dic₃	—	6	26	no	—
101	C₂×C₃⋊Dic₃	—	6	26	no	—
102	C₂×C₃⋊Dic₃	—	6	26	no	—
103	C₂×C₃⋊Dic₃	—	6	26	no	—
104	#386	—	27	1	no	Sylow
105	C₂×C₃³	—	2	4	no	—
106	C₂×C₃³	—	4	2	no	—
107	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₃³	C₄×D₄	1	2	yes	—
109	C₂×C₃³	#120	1	1	yes	characteristic
110	C₂×C₃³	#121	1	2	yes	—
111	C₂×C₃³	#123	1	1	yes	characteristic
112	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
113	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
114	C₂×C₃×D₄	—	2	26	no	—
115	C₂×C₃×D₄	—	2	26	no	—
116	C₂×C₃×D₄	C₃⋊Dic₃	1	52	yes	complemented
117	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
118	C₂²×Dic₃	—	9	26	no	—
119	C₂²×Dic₃	—	9	26	no	—
120	C₂²×Dic₃	—	18	52	no	—
121	C₂²×Dic₃	—	18	26	no	—
122	#217	—	18	52	no	—
123	#217	—	18	26	no	—
124	C₂²×C₃²	—	2	52	no	—
125	C₂²×C₃²	—	2	26	no	—
126	C₂²×C₃²	—	2	26	no	—
127	C₂²×C₃²	—	2	52	no	—
128	C₂²×C₃²	—	2	52	no	—
129	C₂²×C₃²	—	2	26	no	—
130	C₂²×C₃²	—	4	13	no	—
131	C₂²×C₃²	—	4	13	no	—
132	C₂²×C₃²	—	4	26	no	—
133	C₂²×C₃²	#217	1	52	yes	complemented
134	C₂²×C₃²	D₃×D₄	1	26	yes	—
135	C₂²×C₃²	#237	1	26	yes	—
136	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
137	C₂²×C₃²	C₂×#67	1	13	yes	—
138	C₂²×C₃²	C₂×#67	1	13	yes	—
139	C₃²×C₄	—	2	26	no	—
140	C₃⋊Dic₃	—	6	52	no	—
141	C₃⋊Dic₃	—	12	26	no	—
142	C₂²×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
143	C₂³×C₄	—	27	1	no	—
144	C₂×C₂²⋊C₄	—	27	2	no	—
145	C₂×C₂²⋊C₄	—	27	1	no	—
146	#95	—	27	2	no	—
147	C₃³	#386	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₃	—	2	26	no	—
149	C₂³×C₃	—	2	26	no	—
150	C₂³×C₃	—	2	26	no	—
151	C₂³×C₃	—	2	26	no	—
152	C₂³×C₃	—	2	13	no	—
153	C₂³×C₃	—	2	13	no	—
154	C₂³×C₃	C₂×C₃⋊Dic₃	1	26	yes	—
155	C₂³×C₃	C₂×C₃⋊Dic₃	1	26	yes	complemented
156	C₂³×C₃	#638	1	26	yes	—
157	C₂³×C₃	#638	1	26	yes	—
158	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
159	C₃×D₄	—	2	104	no	—
160	C₂×C₃×C₄	—	2	13	no	—
161	C₂×C₃×C₄	—	2	13	no	—
162	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	26	yes	complemented
163	C₂×Dic₃	—	9	52	no	—
164	C₂×Dic₃	—	18	104	no	—
165	C₂×Dic₃	—	18	26	no	—
166	C₂×Dic₃	—	18	26	no	—
167	C₂×Dic₃	—	18	26	no	—
168	C₂×Dic₃	—	18	26	no	—
169	C₂×Dic₃	—	18	26	no	—
170	C₂×C₃²	—	2	52	no	—
171	C₂×C₃²	—	4	26	no	—
172	C₂×C₃²	D₄×Dic₃	1	26	yes	—
173	C₂×C₃²	#935	1	13	yes	—
174	C₂×C₃²	#937	1	13	yes	—
175	C₂×C₃²	#939	1	26	yes	—
176	C₂×C₃²	C₂×#217	1	13	yes	—
177	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
178	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	C₃⋊(C₃⋊Dic₃)	1	4	yes	complemented
182	C₂²×C₄	—	27	2	no	—
183	C₂²×C₄	—	27	2	no	—
184	C₂²×C₄	—	54	4	no	—
185	C₂²×C₄	—	54	2	no	—
186	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
187	C₂²⋊C₄	—	54	4	no	—
188	C₂²⋊C₄	—	54	2	no	—
189	C₂²×C₃	—	2	52	no	—
190	C₂²×C₃	—	2	26	no	—
191	C₂²×C₃	—	2	26	no	—
192	C₂²×C₃	—	2	52	no	—
193	C₂²×C₃	—	2	52	no	—
194	C₂²×C₃	—	2	26	no	—
195	C₂²×C₃	—	4	13	no	—
196	C₂²×C₃	—	4	13	no	—
197	C₂²×C₃	—	4	26	no	—
198	C₂²×C₃	#3757	1	52	yes	complemented
199	C₂²×C₃	D₄×C₃⋊D₃	1	26	yes	—
200	C₂²×C₃	#3830	1	26	yes	—
201	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
202	C₂²×C₃	C₂×#638	1	13	yes	—
203	C₂²×C₃	C₂×#638	1	13	yes	—
204	C₃×C₄	—	2	26	no	—
205	Dic₃	—	18	52	no	—
206	Dic₃	—	36	26	no	—
207	C₃²	#5220	1	13	yes	complemented
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
215	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
216	C₂³	#6312	1	2	yes	—
217	C₂³	#6312	1	2	yes	—
218	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
219	D₄	—	2	8	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	27	4	no	—
223	C₂×C₄	—	54	8	no	—
224	C₂×C₄	—	54	2	no	—
225	C₂×C₄	—	54	2	no	—
226	C₂×C₄	—	54	2	no	—
227	C₂×C₄	—	54	2	no	—
228	C₂×C₄	—	54	2	no	—
229	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	complemented
230	C₂×C₃	—	2	52	no	—
231	C₂×C₃	—	4	26	no	—
232	C₂×C₃	D₄×C₃⋊Dic₃	1	26	yes	—
233	C₂×C₃	#64144	1	13	yes	—
234	C₂×C₃	#64146	1	13	yes	—
235	C₂×C₃	#64148	1	26	yes	—
236	C₂×C₃	C₂×#3757	1	13	yes	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	2	no	—
246	C₂²	#88864	1	4	yes	complemented
247	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
248	C₂²	#89078	1	2	yes	—
249	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
250	C₂²	C₂×#6312	1	1	yes	characteristic
251	C₂²	C₂×#6312	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	54	4	no	—
254	C₄	—	108	2	no	—
255	C₃	#10592919	1	13	yes	complemented
256	C₂	—	2	4	no	—
257	C₂	—	4	2	no	—
258	C₂	D₄×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
259	C₂	#11723976	1	1	yes	characteristic
260	C₂	#11723978	1	1	yes	characteristic
261	C₂	#11723980	1	2	yes	—
262	C₂	C₂×#88864	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊(C₃⋊D₃)
commutator	C₂²×C₃³	C₂²×C₄
Fitting, prime radical	C₂²×C₃³×D₄	C₂
socle, cosocle	C₂³×C₃³	C₂³
hypercenter	C₂²×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#386
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
3. C₂×#88864	C₂	semidirect
4. C₂×#88864	C₂	semidirect
5. #11722201	C₂	semidirect
11. C₂×C₃³×D₄	C₄	semidirect
31. C₂³×C₃³	C₂×C₄	semidirect
38. C₂×C₃³×C₄	C₂×C₄	semidirect
45. C₂×C₃⋊(C₃⋊Dic₃)	D₄	semidirect
51. C₂×C₃²×D₄	Dic₃	semidirect
68. C₂²×C₃³	C₂²⋊C₄	semidirect
89. C₂³×C₃²	C₂×Dic₃	semidirect
96. C₂×C₃²×C₄	C₂×Dic₃	semidirect
116. C₂×C₃×D₄	C₃⋊Dic₃	semidirect
133. C₂²×C₃²	#217	semidirect
147. C₃³	#386	semidirect
155. C₂³×C₃	C₂×C₃⋊Dic₃	semidirect
162. C₂×C₃×C₄	C₂×C₃⋊Dic₃	semidirect
181. C₂×D₄	C₃⋊(C₃⋊Dic₃)	semidirect
198. C₂²×C₃	#3757	semidirect
207. C₃²	#5220	semidirect
215. C₂³	C₂×C₃⋊(C₃⋊Dic₃)	semidirect
229. C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	semidirect
246. C₂²	#88864	semidirect
255. C₃	#10592919	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃³×D₄	2
C₂³×C₃⋊(C₃⋊Dic₃)	2
C₂×#88864	2
#11722201	2
#10592919	3