Subgroups — C₂×#11722018 #13160277

Order 1728.

1404 conjugacy classes of subgroups.

The subgroup lattice has 244 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	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11722018	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₈×C₃⋊Dic₃	—	3	1	no	maximal
6	C₂×#63553	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
9	C₄×C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
10	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₃²×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃²×C₈	—	3	1	no	—
16	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
17	C₂×C₄×C₃⋊Dic₃	—	3	4	no	—
18	C₂×C₄×C₃⋊Dic₃	—	6	4	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₃⋊C₈	—	3	4	no	—
21	C₂²×C₃×C₃⋊C₈	D₃	1	4	yes	—
22	C₈×C₃⋊Dic₃	—	3	4	no	—
23	#63553	—	3	16	no	—
24	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊(C₃⋊Dic₃)	C₈	1	8	yes	complemented
30	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
31	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	complemented
32	C₃²×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₈×Dic₃	—	9	4	no	—
34	C₂×C₈×Dic₃	—	9	1	no	—
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	3	2	no	—
39	C₂²×C₃⋊Dic₃	—	3	8	no	—
40	C₂²×C₃⋊Dic₃	—	6	8	no	—
41	C₂×C₃²×C₈	—	3	4	no	—
42	C₂×C₃²×C₈	—	3	2	no	—
43	C₄×C₃⋊Dic₃	—	3	2	no	—
44	C₄×C₃⋊Dic₃	—	3	8	no	—
45	C₄×C₃⋊Dic₃	—	3	2	no	—
46	C₄×C₃⋊Dic₃	—	3	8	no	—
47	C₄×C₃⋊Dic₃	—	6	8	no	—
48	C₄×C₃⋊Dic₃	—	6	8	no	—
49	C₂×C₃⋊(C₃⋊C₈)	Dic₃	1	4	yes	complemented
50	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₃⋊C₈	—	3	16	no	—
52	C₂×C₃×C₃⋊C₈	—	3	8	no	—
53	C₂×C₃×C₃⋊C₈	Dic₃	1	16	yes	complemented
54	C₂×C₃×C₃⋊C₈	C₂×D₃	1	8	yes	—
55	C₂²×C₃³	C₄²	1	2	yes	—
56	C₂²×C₃³	C₂×C₈	1	4	yes	—
57	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
58	C₃³×C₄	C₄²	1	2	yes	—
59	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₃⋊(C₃⋊Dic₃)	C₂×C₈	1	8	yes	complemented
62	C₂²×C₃×C₈	—	3	4	no	—
63	C₂²×C₃×C₈	—	9	1	no	—
64	C₂×C₄×Dic₃	—	9	4	no	—
65	C₂×C₄×Dic₃	—	9	1	no	—
66	C₂×C₄×Dic₃	—	18	4	no	—
67	C₂²×C₃⋊C₈	—	3	4	no	—
68	C₂²×C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic
69	C₈×Dic₃	—	9	16	no	—
70	C₈×Dic₃	—	9	4	no	—
71	C₂³×C₃²	—	2	4	no	—
72	C₂³×C₃²	C₄×D₃	1	4	yes	—
73	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	—	2	8	no	—
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
79	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
80	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
81	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
83	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₃⋊Dic₃	—	3	8	no	—
87	C₂×C₃⋊Dic₃	—	3	4	no	—
88	C₂×C₃⋊Dic₃	—	3	32	no	—
89	C₂×C₃⋊Dic₃	—	3	16	no	—
90	C₂×C₃⋊Dic₃	—	6	16	no	—
91	C₂×C₃⋊Dic₃	—	6	32	no	—
92	C₃²×C₈	—	3	4	no	—
93	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	4	yes	complemented
94	C₃×C₃⋊C₈	—	3	16	no	—
95	C₃×C₃⋊C₈	C₂×Dic₃	1	16	yes	complemented
96	C₂×C₄×C₈	—	27	1	no	Sylow
97	C₂×C₃³	C₄×C₈	1	4	yes	complemented
98	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₃³	C₂²×C₈	1	2	yes	—
100	C₂²×C₃×C₄	—	2	4	no	—
101	C₂²×C₃×C₄	D₃²	1	4	yes	—
102	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
103	C₂²×Dic₃	—	9	8	no	—
104	C₂²×Dic₃	—	9	2	no	—
105	C₂²×Dic₃	—	18	8	no	—
106	C₂×C₃×C₈	—	3	16	no	—
107	C₂×C₃×C₈	—	3	8	no	—
108	C₂×C₃×C₈	—	9	4	no	—
109	C₂×C₃×C₈	—	9	2	no	—
110	C₄×Dic₃	—	9	8	no	—
111	C₄×Dic₃	—	9	2	no	—
112	C₄×Dic₃	—	9	8	no	—
113	C₄×Dic₃	—	9	2	no	—
114	C₄×Dic₃	—	18	8	no	—
115	C₄×Dic₃	—	18	8	no	—
116	C₂×C₃⋊C₈	—	3	16	no	—
117	C₂×C₃⋊C₈	—	3	8	no	—
118	C₂×C₃⋊C₈	C₃⋊Dic₃	1	4	yes	complemented
119	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
120	C₂²×C₃²	—	2	16	no	—
121	C₂²×C₃²	—	2	8	no	—
122	C₂²×C₃²	—	2	4	no	—
123	C₂²×C₃²	D₃×C₈	1	16	yes	—
124	C₂²×C₃²	D₃×C₈	1	4	yes	—
125	C₂²×C₃²	C₄×Dic₃	1	8	yes	—
126	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
127	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
128	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
129	C₃²×C₄	—	2	8	no	—
130	C₃²×C₄	—	2	4	no	—
131	C₃²×C₄	—	2	4	no	—
132	C₃²×C₄	C₄×Dic₃	1	8	yes	—
133	C₃²×C₄	C₄×Dic₃	1	2	yes	—
134	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
135	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
137	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₃⋊Dic₃	—	3	8	no	—
139	C₃⋊Dic₃	—	3	32	no	—
140	C₃⋊Dic₃	—	6	32	no	—
141	C₂²×C₈	—	3	1	no	—
142	C₂²×C₈	—	9	1	no	—
143	C₂×C₄²	—	27	1	no	—
144	C₄×C₈	—	27	4	no	—
145	C₃³	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₃	—	2	4	no	—
147	C₂³×C₃	#624	1	4	yes	—
148	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	8	no	—
150	C₂×C₃×C₄	—	2	8	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
154	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
155	C₂×C₃×C₄	#624	1	4	yes	—
156	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
157	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
159	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
160	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
161	C₂×Dic₃	—	9	32	no	—
162	C₂×Dic₃	—	9	8	no	—
163	C₂×Dic₃	—	9	16	no	—
164	C₂×Dic₃	—	9	4	no	—
165	C₂×Dic₃	—	18	16	no	—
166	C₂×Dic₃	—	18	32	no	—
167	C₃×C₈	—	3	16	no	—
168	C₃×C₈	—	9	4	no	—
169	C₃⋊C₈	—	3	16	no	—
170	C₃⋊C₈	C₂×C₃⋊Dic₃	1	4	yes	complemented
171	C₂×C₃²	—	2	16	no	—
172	C₂×C₃²	—	2	8	no	—
173	C₂×C₃²	—	2	4	no	—
174	C₂×C₃²	C₈×Dic₃	1	16	yes	complemented
175	C₂×C₃²	C₈×Dic₃	1	4	yes	complemented
176	C₂×C₃²	C₂×D₃×C₈	1	8	yes	—
177	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
178	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
179	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
180	C₂²×C₄	—	27	2	no	—
181	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
182	C₂×C₈	—	3	4	no	—
183	C₂×C₈	—	3	2	no	—
184	C₂×C₈	—	9	4	no	—
185	C₂×C₈	—	9	2	no	—
186	C₄²	—	27	2	no	—
187	C₄²	—	27	2	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	2	16	no	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	#3710	1	16	yes	—
192	C₂²×C₃	Dic₃²	1	8	yes	—
193	C₂²×C₃	C₈×C₃⋊D₃	1	4	yes	—
194	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
195	C₂²×C₃	C₂×#624	1	4	yes	—
196	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	8	no	—
200	C₃×C₄	Dic₃²	1	8	yes	—
201	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
202	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
203	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
204	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
205	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
206	Dic₃	—	9	32	no	—
207	Dic₃	—	9	8	no	—
208	Dic₃	—	18	32	no	—
209	C₃²	—	2	4	no	—
210	C₃²	C₂×C₈×Dic₃	1	4	yes	complemented
211	C₃²	C₂×C₈×Dic₃	1	1	yes	characteristic, complemented
212	C₂³	#6289	1	1	yes	characteristic
213	C₂×C₄	—	27	4	no	—
214	C₂×C₄	—	27	8	no	—
215	C₂×C₄	D₃×C₃⋊Dic₃	1	2	yes	—
216	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
217	C₂×C₄	#6289	1	1	yes	characteristic
218	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
219	C₈	—	3	4	no	—
220	C₈	—	9	4	no	—
221	C₂×C₃	—	2	16	no	—
222	C₂×C₃	—	2	4	no	—
223	C₂×C₃	—	2	8	no	—
224	C₂×C₃	#63553	1	16	yes	complemented
225	C₂×C₃	C₈×C₃⋊Dic₃	1	4	yes	complemented
226	C₂×C₃	C₂×#3710	1	8	yes	—
227	C₂×C₃	C₂×Dic₃²	1	4	yes	—
228	C₂×C₃	C₂×C₈×C₃⋊D₃	1	2	yes	—
229	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
230	C₂²	#88785	1	4	yes	—
231	C₂²	Dic₃×C₃⋊Dic₃	1	2	yes	—
232	C₂²	C₂×#6289	1	1	yes	characteristic
233	C₄	—	27	8	no	—
234	C₄	Dic₃×C₃⋊Dic₃	1	2	yes	—
235	C₄	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
236	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
237	C₃	—	2	4	no	—
238	C₃	C₂×#63553	1	4	yes	complemented
239	C₃	C₂×C₈×C₃⋊Dic₃	1	1	yes	characteristic, complemented
240	C₂	#11722018	1	4	yes	direct product factor
241	C₂	C₂×#88785	1	2	yes	—
242	C₂	C₂×Dic₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×C₃⋊D₃
commutator, hypocenter	C₃³	C₂×C₄×C₈
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle, prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11722018	C₂	direct (N × H)
11. C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	semidirect
13. C₂×C₃²×C₃⋊C₈	C₄	semidirect
29. C₂×C₃⋊(C₃⋊Dic₃)	C₈	semidirect
31. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
32. C₃²×C₃⋊C₈	C₂×C₄	semidirect
49. C₂×C₃⋊(C₃⋊C₈)	Dic₃	semidirect
53. C₂×C₃×C₃⋊C₈	Dic₃	semidirect
61. C₃⋊(C₃⋊Dic₃)	C₂×C₈	semidirect
93. C₃⋊(C₃⋊C₈)	C₂×Dic₃	semidirect
95. C₃×C₃⋊C₈	C₂×Dic₃	semidirect
97. C₂×C₃³	C₄×C₈	semidirect
118. C₂×C₃⋊C₈	C₃⋊Dic₃	semidirect
145. C₃³	C₂×C₄×C₈	semidirect
170. C₃⋊C₈	C₂×C₃⋊Dic₃	semidirect
174. C₂×C₃²	C₈×Dic₃	semidirect
175. C₂×C₃²	C₈×Dic₃	semidirect
210. C₃²	C₂×C₈×Dic₃	semidirect
211. C₃²	C₂×C₈×Dic₃	semidirect
224. C₂×C₃	#63553	semidirect
225. C₂×C₃	C₈×C₃⋊Dic₃	semidirect
238. C₃	C₂×#63553	semidirect
239. C₃	C₂×C₈×C₃⋊Dic₃	semidirect
240. C₂	#11722018	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
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₂²×C₃²×C₃⋊C₈	2
#11722018	2
C₂×C₈×C₃⋊Dic₃	3
C₂×#63553	3

Subgroups — C2×#11722018 #13160277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11722018 #13160277