Subgroups — C₂×#11722052 #13160402

Order 1728.

1148 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11722052	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63644	—	3	1	no	maximal
6	C₂×#63573	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
9	C₂²×C₃×C₃⋊Dic₃	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	1	no	—
11	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
12	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
14	C₂×C₃²×C₃⋊C₈	C₄	1	4	yes	complemented
15	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₃²×C₈	—	3	1	no	—
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
18	C₂×C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
19	C₂×C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
20	C₂×C₄⋊(C₃⋊Dic₃)	D₃	1	1	yes	characteristic
21	C₂×C₃×C₄⋊Dic₃	—	3	4	no	—
22	C₂²×C₃×C₃⋊C₈	D₃	1	4	yes	—
23	#63644	—	3	4	no	—
24	#63573	—	3	16	no	—
25	C₂³×C₃³	D₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	Q₈	1	2	yes	—
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
32	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
33	C₂×C₃×C₃⋊Dic₃	—	4	2	no	—
34	C₃²×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
35	C₂×#817	—	9	4	no	—
36	C₂×#808	—	9	1	no	—
37	C₂²×C₃²×C₄	—	2	4	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
40	C₂²×C₃⋊Dic₃	—	2	1	no	—
41	C₂²×C₃⋊Dic₃	—	6	8	no	—
42	C₂²×C₃⋊Dic₃	—	6	4	no	—
43	C₂²×C₃⋊Dic₃	—	6	1	no	—
44	C₂²×C₃×Dic₃	—	6	4	no	—
45	C₂×C₃²×C₈	—	3	4	no	—
46	C₂×C₃²×C₈	—	3	2	no	—
47	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
48	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
49	C₄⋊(C₃⋊Dic₃)	—	3	8	no	—
50	C₄⋊(C₃⋊Dic₃)	—	6	8	no	—
51	C₄⋊(C₃⋊Dic₃)	—	6	8	no	—
52	C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
53	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
54	C₄⋊(C₃⋊Dic₃)	C₂×D₃	1	2	yes	—
55	C₃×C₄⋊Dic₃	—	3	8	no	—
56	C₃×C₄⋊Dic₃	—	6	4	no	—
57	C₂×C₃×C₃⋊C₈	Dic₃	1	16	yes	complemented
58	C₂×C₃×C₃⋊C₈	C₂×D₃	1	8	yes	—
59	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
60	C₂²×C₃³	SD₁₆	1	2	yes	—
61	C₂²×C₃³	SD₁₆	1	2	yes	—
62	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
63	C₃³×C₄	C₄⋊C₄	1	2	yes	—
64	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
67	C₃×C₃⋊Dic₃	—	4	2	no	—
68	C₂²×C₃×C₈	—	3	4	no	—
69	C₂×C₃×C₄⋊C₄	—	9	1	no	—
70	C₂×C₄⋊Dic₃	—	3	4	no	—
71	C₂×C₄⋊Dic₃	—	9	4	no	—
72	C₂×C₄⋊Dic₃	—	9	1	no	—
73	C₂×C₄⋊Dic₃	—	18	4	no	—
74	C₂²×C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic
75	#817	—	9	16	no	—
76	#808	—	9	4	no	—
77	C₂³×C₃²	—	2	4	no	—
78	C₂³×C₃²	D₁₂	1	4	yes	—
79	C₂³×C₃²	#67	1	1	yes	characteristic
80	C₂×C₃²×C₄	—	2	8	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	Dic₆	1	8	yes	—
85	C₂×C₃²×C₄	Dic₆	1	2	yes	—
86	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
87	C₂×C₃²×C₄	D₁₂	1	4	yes	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
89	C₂×C₃²×C₄	#67	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
91	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₃⋊Dic₃	—	2	2	no	—
93	C₂×C₃⋊Dic₃	—	4	2	no	—
94	C₂×C₃⋊Dic₃	—	6	16	no	—
95	C₂×C₃⋊Dic₃	—	6	8	no	—
96	C₂×C₃⋊Dic₃	—	6	2	no	—
97	C₂×C₃⋊Dic₃	—	12	16	no	—
98	C₂×C₃⋊Dic₃	—	12	8	no	—
99	C₂×C₃⋊Dic₃	—	12	2	no	—
100	C₂×C₃×Dic₃	—	6	8	no	—
101	C₂×C₃×Dic₃	—	12	8	no	—
102	C₃²×C₈	—	3	4	no	—
103	C₃×C₃⋊C₈	C₂×Dic₃	1	16	yes	complemented
104	C₂×#106	—	27	1	no	Sylow
105	C₂×C₃³	#106	1	4	yes	complemented
106	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	—	18	1	no	—
111	C₂²×C₃×C₄	D₃²	1	4	yes	—
112	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
113	C₂²×Dic₃	—	6	4	no	—
114	C₂²×Dic₃	—	18	8	no	—
115	C₂²×Dic₃	—	18	4	no	—
116	C₂²×Dic₃	—	18	1	no	—
117	C₂×C₃×C₈	—	3	16	no	—
118	C₂×C₃×C₈	—	3	8	no	—
119	C₃×C₄⋊C₄	—	9	2	no	—
120	C₃×C₄⋊C₄	—	18	1	no	—
121	C₄⋊Dic₃	—	3	8	no	—
122	C₄⋊Dic₃	—	6	4	no	—
123	C₄⋊Dic₃	—	9	8	no	—
124	C₄⋊Dic₃	—	9	2	no	—
125	C₄⋊Dic₃	—	18	8	no	—
126	C₄⋊Dic₃	—	18	8	no	—
127	C₄⋊Dic₃	—	18	4	no	—
128	C₄⋊Dic₃	—	18	1	no	—
129	C₂×C₃⋊C₈	C₃⋊Dic₃	1	4	yes	complemented
130	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
131	C₂²×C₃²	—	2	8	no	—
132	C₂²×C₃²	—	2	8	no	—
133	C₂²×C₃²	—	2	8	no	—
134	C₂²×C₃²	—	2	4	no	—
135	C₂²×C₃²	#204	1	8	yes	—
136	C₂²×C₃²	#204	1	8	yes	—
137	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
138	C₂²×C₃²	C₄⋊Dic₃	1	8	yes	—
139	C₂²×C₃²	#214	1	2	yes	—
140	C₂²×C₃²	#215	1	2	yes	—
141	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
142	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
143	C₃²×C₄	—	2	8	no	—
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
147	C₃²×C₄	C₄⋊Dic₃	1	8	yes	—
148	C₃²×C₄	C₂×Dic₆	1	4	yes	—
149	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
150	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
151	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
152	C₃⋊Dic₃	—	4	2	no	—
153	C₃⋊Dic₃	—	12	16	no	—
154	C₃⋊Dic₃	—	12	8	no	—
155	C₃⋊Dic₃	—	12	2	no	—
156	C₃×Dic₃	—	12	8	no	—
157	C₂²×C₈	—	3	1	no	—
158	C₂×C₄⋊C₄	—	9	1	no	—
159	C₂×C₄⋊C₄	—	27	1	no	—
160	#106	—	27	4	no	—
161	C₃³	C₂×#106	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	#626	1	4	yes	—
164	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	18	2	no	—
170	C₂×C₃×C₄	—	36	2	no	—
171	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
172	C₂×C₃×C₄	#626	1	4	yes	—
173	C₂×C₃×C₄	#627	1	8	yes	—
174	C₂×C₃×C₄	#634	1	2	yes	—
175	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
177	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
178	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
179	C₂×Dic₃	—	6	8	no	—
180	C₂×Dic₃	—	12	8	no	—
181	C₂×Dic₃	—	18	16	no	—
182	C₂×Dic₃	—	18	8	no	—
183	C₂×Dic₃	—	18	2	no	—
184	C₂×Dic₃	—	36	16	no	—
185	C₂×Dic₃	—	36	8	no	—
186	C₂×Dic₃	—	36	2	no	—
187	C₃×C₈	—	3	16	no	—
188	C₃⋊C₈	C₂×C₃⋊Dic₃	1	4	yes	complemented
189	C₂×C₃²	—	2	4	no	—
190	C₂×C₃²	—	2	16	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	2	4	no	—
193	C₂×C₃²	#808	1	4	yes	complemented
194	C₂×C₃²	#817	1	16	yes	complemented
195	C₂×C₃²	C₂×#204	1	4	yes	—
196	C₂×C₃²	C₂×#204	1	4	yes	—
197	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
198	C₂×C₃²	C₂×C₄⋊Dic₃	1	4	yes	—
199	C₂×C₃²	C₂×#214	1	1	yes	characteristic
200	C₂×C₃²	C₂×#215	1	1	yes	characteristic
201	C₂²×C₄	—	18	1	no	—
202	C₂²×C₄	—	54	1	no	—
203	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
204	C₂×C₈	—	3	4	no	—
205	C₂×C₈	—	3	2	no	—
206	C₄⋊C₄	—	9	2	no	—
207	C₄⋊C₄	—	18	1	no	—
208	C₄⋊C₄	—	27	2	no	—
209	C₄⋊C₄	—	54	1	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	2	8	no	—
212	C₂²×C₃	—	2	8	no	—
213	C₂²×C₃	—	2	8	no	—
214	C₂²×C₃	#3716	1	8	yes	—
215	C₂²×C₃	#3717	1	8	yes	—
216	C₂²×C₃	Dic₃⋊Dic₃	1	8	yes	—
217	C₂²×C₃	#3744	1	2	yes	—
218	C₂²×C₃	#3744	1	2	yes	—
219	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
220	C₂²×C₃	C₂×#626	1	4	yes	—
221	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	8	no	—
225	C₃×C₄	—	36	2	no	—
226	C₃×C₄	Dic₃⋊Dic₃	1	8	yes	—
227	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
228	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
229	C₃×C₄	C₂×#627	1	4	yes	—
230	C₃×C₄	C₂×#634	1	1	yes	characteristic
231	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
232	Dic₃	—	12	8	no	—
233	Dic₃	—	36	16	no	—
234	Dic₃	—	36	8	no	—
235	Dic₃	—	36	2	no	—
236	C₃²	—	2	4	no	—
237	C₃²	C₂×#808	1	1	yes	characteristic, complemented
238	C₃²	C₂×#817	1	4	yes	complemented
239	C₂³	#6292	1	1	yes	characteristic
240	C₂×C₄	—	18	2	no	—
241	C₂×C₄	—	36	2	no	—
242	C₂×C₄	—	54	2	no	—
243	C₂×C₄	—	108	2	no	—
244	C₂×C₄	D₃×C₃⋊Dic₃	1	2	yes	—
245	C₂×C₄	#6292	1	1	yes	characteristic
246	C₂×C₄	#6293	1	2	yes	—
247	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
248	C₈	—	3	4	no	—
249	C₂×C₃	—	2	16	no	—
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	2	4	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	#63573	1	16	yes	complemented
254	C₂×C₃	#63644	1	4	yes	complemented
255	C₂×C₃	C₂×#3716	1	4	yes	—
256	C₂×C₃	C₂×#3717	1	4	yes	—
257	C₂×C₃	C₂×Dic₃⋊Dic₃	1	4	yes	—
258	C₂×C₃	C₂×#3744	1	1	yes	characteristic
259	C₂×C₃	C₂×#3744	1	1	yes	characteristic
260	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
261	C₂²	#88796	1	2	yes	—
262	C₂²	#88797	1	2	yes	—
263	C₂²	Dic₃⋊(C₃⋊Dic₃)	1	2	yes	—
264	C₂²	C₂×#6292	1	1	yes	characteristic
265	C₄	—	36	2	no	—
266	C₄	—	108	2	no	—
267	C₄	Dic₃⋊(C₃⋊Dic₃)	1	2	yes	—
268	C₄	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
269	C₄	C₂×#6293	1	1	yes	characteristic
270	C₃	—	2	4	no	—
271	C₃	C₂×#63573	1	4	yes	complemented
272	C₃	C₂×#63644	1	1	yes	characteristic, complemented
273	C₂	#11722052	1	4	yes	direct product factor
274	C₂	C₂×#88796	1	1	yes	characteristic
275	C₂	C₂×#88797	1	1	yes	characteristic
276	C₂	C₂×Dic₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6292
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle, prime radical	C₂³×C₃³	D₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×#106
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11722052	C₂	direct (N × H)
14. C₂×C₃²×C₃⋊C₈	C₄	semidirect
34. C₃²×C₃⋊C₈	C₂×C₄	semidirect
57. C₂×C₃×C₃⋊C₈	Dic₃	semidirect
103. C₃×C₃⋊C₈	C₂×Dic₃	semidirect
105. C₂×C₃³	#106	semidirect
129. C₂×C₃⋊C₈	C₃⋊Dic₃	semidirect
161. C₃³	C₂×#106	semidirect
188. C₃⋊C₈	C₂×C₃⋊Dic₃	semidirect
193. C₂×C₃²	#808	semidirect
194. C₂×C₃²	#817	semidirect
237. C₃²	C₂×#808	semidirect
238. C₃²	C₂×#817	semidirect
253. C₂×C₃	#63573	semidirect
254. C₂×C₃	#63644	semidirect
271. C₃	C₂×#63573	semidirect
272. C₃	C₂×#63644	semidirect
273. C₂	#11722052	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
C₂×C₃×C₄⋊(C₃⋊Dic₃)	2
C₂²×C₃²×C₃⋊C₈	2
#11722052	2
C₂×#63644	3
C₂×#63573	3

Subgroups — C2×#11722052 #13160402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722052 #13160402