Subgroups — C₂×#11722053 #13160401

Order 1728.

1084 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#11722053	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63635	—	3	1	no	maximal
6	C₂×#63574	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₃⋊Dic₃	—	2	1	no	—
9	C₂²×C₃²×Dic₃	—	2	1	no	—
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
11	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
12	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
14	C₃²×C₄⋊Dic₃	—	2	1	no	—
15	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₃²×C₄⋊C₄	—	3	1	no	—
17	C₂×C₄⋊(C₃⋊Dic₃)	D₃	1	1	yes	characteristic
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	—	3	4	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	—	6	4	no	—
21	C₂×C₃×C₄⋊Dic₃	—	3	4	no	—
22	C₂×C₃×C₄⋊Dic₃	D₃	1	4	yes	—
23	#63635	—	3	4	no	—
24	#63574	—	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×C₃²×Dic₃	—	2	2	no	—
33	C₂×C₃²×Dic₃	—	4	2	no	—
34	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	4	yes	complemented
35	C₂×#808	—	9	4	no	—
36	C₂×#808	—	9	1	no	—
37	C₂²×C₃²×C₄	—	2	4	no	—
38	C₂²×C₃²×C₄	—	6	1	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃⋊Dic₃	—	2	1	no	—
42	C₂²×C₃×Dic₃	—	2	4	no	—
43	C₂²×C₃×Dic₃	—	6	4	no	—
44	C₃²×C₄⋊C₄	—	3	2	no	—
45	C₃²×C₄⋊C₄	—	6	1	no	—
46	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
47	C₄⋊(C₃⋊Dic₃)	C₂×D₃	1	2	yes	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	16	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
52	C₂×C₃⋊(C₃⋊C₈)	—	6	16	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
54	C₃×C₄⋊Dic₃	—	2	4	no	—
55	C₃×C₄⋊Dic₃	—	3	8	no	—
56	C₃×C₄⋊Dic₃	—	6	4	no	—
57	C₃×C₄⋊Dic₃	C₂×D₃	1	8	yes	—
58	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
59	C₂²×C₃³	SD₁₆	1	2	yes	—
60	C₂²×C₃³	SD₁₆	1	2	yes	—
61	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
62	C₃³×C₄	C₄⋊C₄	1	2	yes	—
63	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₃×C₃⋊Dic₃	—	4	2	no	—
66	C₃²×Dic₃	—	4	2	no	—
67	C₂×C₃×C₄⋊C₄	—	3	4	no	—
68	C₂×C₃×C₄⋊C₄	—	9	1	no	—
69	C₂×C₄⋊Dic₃	—	3	4	no	—
70	C₂×C₄⋊Dic₃	C₃⋊D₃	1	1	yes	characteristic
71	C₂²×C₃⋊C₈	—	9	4	no	—
72	C₂²×C₃⋊C₈	—	9	1	no	—
73	C₂²×C₃⋊C₈	—	18	4	no	—
74	#808	—	9	16	no	—
75	#808	—	9	4	no	—
76	C₂³×C₃²	—	2	4	no	—
77	C₂³×C₃²	#67	1	4	yes	—
78	C₂³×C₃²	#67	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	2	8	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	8	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	6	2	no	—
84	C₂×C₃²×C₄	—	12	2	no	—
85	C₂×C₃²×C₄	Dic₆	1	8	yes	—
86	C₂×C₃²×C₄	Dic₆	1	2	yes	—
87	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
88	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
89	C₂×C₃²×C₄	#67	1	4	yes	—
90	C₂×C₃²×C₄	#67	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₃⋊Dic₃	—	2	2	no	—
94	C₂×C₃⋊Dic₃	—	4	2	no	—
95	C₂×C₃×Dic₃	—	2	8	no	—
96	C₂×C₃×Dic₃	—	4	8	no	—
97	C₂×C₃×Dic₃	—	6	8	no	—
98	C₂×C₃×Dic₃	—	12	8	no	—
99	C₃⋊(C₃⋊C₈)	—	3	4	no	—
100	C₃⋊(C₃⋊C₈)	—	3	16	no	—
101	C₃⋊(C₃⋊C₈)	—	6	16	no	—
102	C₂×#106	—	27	1	no	Sylow
103	C₂×C₃³	#106	1	4	yes	complemented
104	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	4	no	—
108	C₂²×C₃×C₄	—	6	4	no	—
109	C₂²×C₃×C₄	—	18	1	no	—
110	C₂²×C₃×C₄	D₃²	1	4	yes	—
111	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₂²×Dic₃	—	2	1	no	—
113	C₂²×Dic₃	—	6	4	no	—
114	C₃×C₄⋊C₄	—	3	8	no	—
115	C₃×C₄⋊C₄	—	6	4	no	—
116	C₃×C₄⋊C₄	—	9	2	no	—
117	C₃×C₄⋊C₄	—	18	1	no	—
118	C₄⋊Dic₃	—	2	1	no	—
119	C₄⋊Dic₃	—	3	8	no	—
120	C₄⋊Dic₃	—	6	4	no	—
121	C₄⋊Dic₃	C₂×C₃⋊D₃	1	2	yes	—
122	C₂×C₃⋊C₈	—	9	16	no	—
123	C₂×C₃⋊C₈	—	9	4	no	—
124	C₂×C₃⋊C₈	—	9	8	no	—
125	C₂×C₃⋊C₈	—	9	2	no	—
126	C₂×C₃⋊C₈	—	18	16	no	—
127	C₂×C₃⋊C₈	—	18	8	no	—
128	C₂²×C₃²	—	2	8	no	—
129	C₂²×C₃²	—	2	8	no	—
130	C₂²×C₃²	—	2	8	no	—
131	C₂²×C₃²	—	2	4	no	—
132	C₂²×C₃²	Dic₃⋊C₄	1	8	yes	—
133	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
134	C₂²×C₃²	#214	1	8	yes	—
135	C₂²×C₃²	#214	1	2	yes	—
136	C₂²×C₃²	#215	1	8	yes	—
137	C₂²×C₃²	#215	1	2	yes	—
138	C₂²×C₃²	C₂×#67	1	4	yes	—
139	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
140	C₃²×C₄	—	2	8	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	2	4	no	—
143	C₃²×C₄	—	12	2	no	—
144	C₃²×C₄	Dic₃⋊C₄	1	8	yes	—
145	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
146	C₃²×C₄	C₂×Dic₆	1	4	yes	—
147	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
148	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
149	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₃⋊Dic₃	—	4	2	no	—
151	C₃×Dic₃	—	4	8	no	—
152	C₃×Dic₃	—	12	8	no	—
153	C₂²×C₈	—	27	1	no	—
154	C₂×C₄⋊C₄	—	3	1	no	—
155	C₂×C₄⋊C₄	—	9	1	no	—
156	#106	—	27	4	no	—
157	C₃³	C₂×#106	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₃	—	2	4	no	—
159	C₂³×C₃	#625	1	4	yes	—
160	C₂³×C₃	#638	1	1	yes	characteristic
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	6	8	no	—
166	C₂×C₃×C₄	—	12	8	no	—
167	C₂×C₃×C₄	—	18	2	no	—
168	C₂×C₃×C₄	—	36	2	no	—
169	C₂×C₃×C₄	#624	1	8	yes	—
170	C₂×C₃×C₄	#625	1	4	yes	—
171	C₂×C₃×C₄	#627	1	8	yes	—
172	C₂×C₃×C₄	#634	1	2	yes	—
173	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
174	C₂×C₃×C₄	#638	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
176	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
177	C₂×Dic₃	—	2	2	no	—
178	C₂×Dic₃	—	4	2	no	—
179	C₂×Dic₃	—	6	8	no	—
180	C₂×Dic₃	—	12	8	no	—
181	C₃⋊C₈	—	9	16	no	—
182	C₃⋊C₈	—	9	4	no	—
183	C₃⋊C₈	—	18	16	no	—
184	C₂×C₃²	—	2	4	no	—
185	C₂×C₃²	—	2	4	no	—
186	C₂×C₃²	—	2	16	no	—
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	#808	1	16	yes	complemented
189	C₂×C₃²	#808	1	4	yes	complemented
190	C₂×C₃²	C₂×Dic₃⋊C₄	1	4	yes	—
191	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
192	C₂×C₃²	C₂×#214	1	4	yes	—
193	C₂×C₃²	C₂×#214	1	1	yes	characteristic
194	C₂×C₃²	C₂×#215	1	4	yes	—
195	C₂×C₃²	C₂×#215	1	1	yes	characteristic
196	C₂²×C₄	—	6	1	no	—
197	C₂²×C₄	—	18	1	no	—
198	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
199	C₂×C₈	—	27	4	no	—
200	C₂×C₈	—	27	2	no	—
201	C₄⋊C₄	—	3	2	no	—
202	C₄⋊C₄	—	6	1	no	—
203	C₄⋊C₄	—	9	2	no	—
204	C₄⋊C₄	—	18	1	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	8	no	—
207	C₂²×C₃	—	2	8	no	—
208	C₂²×C₃	—	2	8	no	—
209	C₂²×C₃	#3715	1	8	yes	—
210	C₂²×C₃	#3715	1	8	yes	—
211	C₂²×C₃	#3724	1	8	yes	—
212	C₂²×C₃	#3750	1	2	yes	—
213	C₂²×C₃	#3754	1	2	yes	—
214	C₂²×C₃	#3755	1	2	yes	—
215	C₂²×C₃	C₂×#625	1	4	yes	—
216	C₂²×C₃	C₂×#638	1	1	yes	characteristic
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	2	8	no	—
220	C₃×C₄	—	12	8	no	—
221	C₃×C₄	—	36	2	no	—
222	C₃×C₄	#3724	1	8	yes	—
223	C₃×C₄	#3750	1	2	yes	—
224	C₃×C₄	C₂×#624	1	4	yes	—
225	C₃×C₄	C₂×#627	1	4	yes	—
226	C₃×C₄	C₂×#634	1	1	yes	characteristic
227	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
228	Dic₃	—	4	2	no	—
229	Dic₃	—	12	8	no	—
230	C₃²	—	2	4	no	—
231	C₃²	C₂×#808	1	4	yes	complemented
232	C₃²	C₂×#808	1	1	yes	characteristic, complemented
233	C₂³	#6290	1	1	yes	characteristic
234	C₂×C₄	—	6	2	no	—
235	C₂×C₄	—	12	2	no	—
236	C₂×C₄	—	18	2	no	—
237	C₂×C₄	—	36	2	no	—
238	C₂×C₄	#6289	1	2	yes	—
239	C₂×C₄	#6290	1	1	yes	characteristic
240	C₂×C₄	#6293	1	2	yes	—
241	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
242	C₈	—	27	4	no	—
243	C₂×C₃	—	2	16	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	2	4	no	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	#63574	1	16	yes	complemented
248	C₂×C₃	#63635	1	4	yes	complemented
249	C₂×C₃	C₂×#3715	1	4	yes	—
250	C₂×C₃	C₂×#3715	1	4	yes	—
251	C₂×C₃	C₂×#3724	1	4	yes	—
252	C₂×C₃	C₂×#3750	1	1	yes	characteristic
253	C₂×C₃	C₂×#3754	1	1	yes	characteristic
254	C₂×C₃	C₂×#3755	1	1	yes	characteristic
255	C₂²	#88792	1	2	yes	—
256	C₂²	#88793	1	2	yes	—
257	C₂²	#88806	1	2	yes	—
258	C₂²	C₂×#6290	1	1	yes	characteristic
259	C₄	—	12	2	no	—
260	C₄	—	36	2	no	—
261	C₄	#88806	1	2	yes	—
262	C₄	C₂×#6289	1	1	yes	characteristic
263	C₄	C₂×#6293	1	1	yes	characteristic
264	C₃	—	2	4	no	—
265	C₃	C₂×#63574	1	4	yes	complemented
266	C₃	C₂×#63635	1	1	yes	characteristic, complemented
267	C₂	#11722053	1	4	yes	direct product factor
268	C₂	C₂×#88792	1	1	yes	characteristic
269	C₂	C₂×#88793	1	1	yes	characteristic
270	C₂	C₂×#88806	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6290
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. #11722053	C₂	direct (N × H)
10. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
34. C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	semidirect
103. C₂×C₃³	#106	semidirect
157. C₃³	C₂×#106	semidirect
188. C₂×C₃²	#808	semidirect
189. C₂×C₃²	#808	semidirect
231. C₃²	C₂×#808	semidirect
232. C₃²	C₂×#808	semidirect
247. C₂×C₃	#63574	semidirect
248. C₂×C₃	#63635	semidirect
265. C₃	C₂×#63574	semidirect
266. C₃	C₂×#63635	semidirect
267. C₂	#11722053	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⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₃×C₄⋊(C₃⋊Dic₃)	2
C₂×C₃²×C₄⋊Dic₃	2
#11722053	2
C₂×#63635	3
C₂×#63574	3

Subgroups — C2×#11722053 #13160401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722053 #13160401