Subgroups — C₂×C₃²×#818 #13162540

Order 1728.

1084 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₃²×#818	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×#107	—	3	1	no	maximal
5	C₂×C₃×#818	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×Dic₃	—	2	2	no	—
8	C₂×C₃³×C₈	C₄	1	4	yes	complemented
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₃²×C₄⋊Dic₃	—	2	2	no	—
11	C₃²×C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	—	2	4	no	—
13	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
14	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	—
15	C₂×C₃²×C₄⋊C₄	—	3	2	no	—
16	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	8	yes	—
17	C₃²×#107	—	3	4	no	—
18	C₃×#818	C₂×C₃	1	16	yes	direct product factor
19	C₂³×C₃³	D₄	1	1	yes	characteristic
20	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
21	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃³×C₄	Q₈	1	2	yes	—
23	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃²×Dic₃	—	2	4	no	—
25	C₂×C₃²×Dic₃	—	4	4	no	—
26	C₃³×C₈	C₂×C₄	1	4	yes	complemented
27	C₂×C₃×#107	—	3	4	no	—
28	C₂×#818	C₃²	1	1	yes	characteristic, direct product factor
29	C₂²×C₃²×C₄	—	2	4	no	—
30	C₂²×C₃²×C₄	—	6	2	no	—
31	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
33	C₂²×C₃×Dic₃	—	2	8	no	—
34	C₂×C₃²×C₈	—	2	16	no	—
35	C₂×C₃²×C₈	—	2	8	no	—
36	C₂×C₃²×C₈	Dic₃	1	4	yes	complemented
37	C₂×C₃²×C₈	C₃×C₄	1	16	yes	complemented
38	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
40	C₃²×C₄⋊C₄	—	3	4	no	—
41	C₃²×C₄⋊C₄	—	6	2	no	—
42	C₃×C₄⋊Dic₃	—	2	8	no	—
43	C₃×C₄⋊Dic₃	C₂²×C₃	1	16	yes	—
44	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
45	C₂²×C₃³	D₈	1	2	yes	—
46	C₂²×C₃³	Q₁₆	1	2	yes	—
47	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
48	C₃³×C₄	C₄⋊C₄	1	2	yes	—
49	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
51	C₃²×Dic₃	—	4	4	no	—
52	C₂²×C₃×C₈	—	2	4	no	—
53	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
54	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
55	C₂×C₃×C₄⋊C₄	—	3	8	no	—
56	C₂×C₄⋊Dic₃	C₂×C₃²	1	2	yes	—
57	C₃×#107	—	3	16	no	—
58	#818	C₂×C₃²	1	4	yes	direct product factor
59	C₂³×C₃²	—	2	4	no	—
60	C₂³×C₃²	D₁₂	1	1	yes	characteristic
61	C₂³×C₃²	C₃×D₄	1	4	yes	—
62	C₂×C₃²×C₄	—	2	8	no	—
63	C₂×C₃²×C₄	—	2	4	no	—
64	C₂×C₃²×C₄	—	2	8	no	—
65	C₂×C₃²×C₄	—	2	4	no	—
66	C₂×C₃²×C₄	—	6	4	no	—
67	C₂×C₃²×C₄	—	12	4	no	—
68	C₂×C₃²×C₄	Dic₆	1	2	yes	—
69	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
71	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
72	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
73	C₂×C₃²×C₄	C₃×Q₈	1	8	yes	—
74	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
76	C₂×C₃×Dic₃	—	2	16	no	—
77	C₂×C₃×Dic₃	—	4	16	no	—
78	C₃²×C₈	—	2	16	no	—
79	C₃²×C₈	C₂×Dic₃	1	4	yes	complemented
80	C₃²×C₈	C₂×C₃×C₄	1	16	yes	complemented
81	C₂×#107	—	3	1	no	Sylow
82	C₂×C₃³	#107	1	4	yes	complemented
83	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
85	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	4	no	—
87	C₂²×C₃×C₄	—	6	8	no	—
88	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
89	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
90	C₂²×Dic₃	—	2	2	no	—
91	C₂×C₃×C₈	—	2	16	no	—
92	C₂×C₃×C₈	—	2	8	no	—
93	C₂×C₃×C₈	C₃×Dic₃	1	16	yes	complemented
94	C₂×C₃×C₈	C₃²×C₄	1	4	yes	complemented
95	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
96	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
97	C₃×C₄⋊C₄	—	3	16	no	—
98	C₃×C₄⋊C₄	—	6	8	no	—
99	C₄⋊Dic₃	—	2	2	no	—
100	C₄⋊Dic₃	C₂²×C₃²	1	4	yes	—
101	C₂²×C₃²	—	2	8	no	—
102	C₂²×C₃²	—	2	8	no	—
103	C₂²×C₃²	—	2	8	no	—
104	C₂²×C₃²	—	2	4	no	—
105	C₂²×C₃²	D₂₄	1	2	yes	—
106	C₂²×C₃²	Dic₁₂	1	2	yes	—
107	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
108	C₂²×C₃²	C₃×C₄⋊C₄	1	8	yes	—
109	C₂²×C₃²	C₃×D₈	1	8	yes	—
110	C₂²×C₃²	C₃×Q₁₆	1	8	yes	—
111	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
112	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
113	C₃²×C₄	—	2	8	no	—
114	C₃²×C₄	—	2	4	no	—
115	C₃²×C₄	—	2	4	no	—
116	C₃²×C₄	—	12	4	no	—
117	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
118	C₃²×C₄	C₃×C₄⋊C₄	1	8	yes	—
119	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
120	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
121	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
122	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
123	C₃×Dic₃	—	4	16	no	—
124	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	3	2	no	—
126	#107	—	3	4	no	—
127	C₃³	C₂×#107	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₃	—	2	4	no	—
129	C₂³×C₃	C₃×D₁₂	1	4	yes	—
130	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	8	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	6	16	no	—
136	C₂×C₃×C₄	—	12	16	no	—
137	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
138	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
140	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
141	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
143	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
144	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
145	C₂×Dic₃	—	2	4	no	—
146	C₂×Dic₃	—	4	4	no	—
147	C₃×C₈	—	2	16	no	—
148	C₃×C₈	C₂×C₃×Dic₃	1	16	yes	complemented
149	C₃×C₈	C₂×C₃²×C₄	1	4	yes	complemented
150	C₂×C₃²	—	2	4	no	—
151	C₂×C₃²	—	2	4	no	—
152	C₂×C₃²	—	2	4	no	—
153	C₂×C₃²	—	2	16	no	—
154	C₂×C₃²	#818	1	4	yes	direct product factor
155	C₂×C₃²	C₃×#107	1	16	yes	complemented
156	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
157	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
158	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
159	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
160	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
161	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
162	C₂²×C₄	—	6	2	no	—
163	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
164	C₂×C₈	C₃²×Dic₃	1	4	yes	complemented
165	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
166	C₄⋊C₄	—	3	4	no	—
167	C₄⋊C₄	—	6	2	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	—	2	8	no	—
170	C₂²×C₃	—	2	8	no	—
171	C₂²×C₃	—	2	8	no	—
172	C₂²×C₃	C₃×D₂₄	1	8	yes	—
173	C₂²×C₃	C₃×Dic₁₂	1	8	yes	—
174	C₂²×C₃	C₃×C₄⋊Dic₃	1	8	yes	—
175	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
176	C₂²×C₃	C₃²×D₈	1	2	yes	—
177	C₂²×C₃	C₃²×Q₁₆	1	2	yes	—
178	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
179	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
180	C₃×C₄	—	2	4	no	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	8	no	—
183	C₃×C₄	—	12	16	no	—
184	C₃×C₄	C₃×C₄⋊Dic₃	1	8	yes	—
185	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
186	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
187	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
188	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
189	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
190	Dic₃	—	4	4	no	—
191	C₃²	—	2	4	no	—
192	C₃²	C₂×#818	1	1	yes	characteristic, direct product factor
193	C₃²	C₂×C₃×#107	1	4	yes	complemented
194	C₂³	C₃²×D₁₂	1	1	yes	characteristic
195	C₂×C₄	—	6	4	no	—
196	C₂×C₄	—	12	4	no	—
197	C₂×C₄	C₃²×Dic₆	1	2	yes	—
198	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
200	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
201	C₈	C₂×C₃²×Dic₃	1	4	yes	complemented
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	—	2	16	no	—
204	C₂×C₃	—	2	4	no	—
205	C₂×C₃	—	2	4	no	—
206	C₂×C₃	C₃×#818	1	16	yes	direct product factor
207	C₂×C₃	C₃²×#107	1	4	yes	complemented
208	C₂×C₃	C₂×C₃×D₂₄	1	4	yes	—
209	C₂×C₃	C₂×C₃×Dic₁₂	1	4	yes	—
210	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
211	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
213	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
214	C₂²	C₃²×D₂₄	1	2	yes	—
215	C₂²	C₃²×Dic₁₂	1	2	yes	—
216	C₂²	C₃²×C₄⋊Dic₃	1	2	yes	—
217	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
218	C₄	—	12	4	no	—
219	C₄	C₃²×C₄⋊Dic₃	1	2	yes	—
220	C₄	C₂×C₃²×Dic₆	1	1	yes	characteristic
221	C₄	C₂²×C₃²×Dic₃	1	1	yes	characteristic
222	C₃	—	2	4	no	—
223	C₃	C₂×C₃×#818	1	4	yes	direct product factor
224	C₃	C₂×C₃²×#107	1	1	yes	characteristic, complemented
225	C₂	C₃²×#818	1	4	yes	direct product factor
226	C₂	C₂×C₃²×D₂₄	1	1	yes	characteristic
227	C₂	C₂×C₃²×Dic₁₂	1	1	yes	characteristic
228	C₂	C₂×C₃²×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃²	D₁₂
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle, prime radical	C₂³×C₃³	D₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₂×C₃²×#107
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
3. C₃²×#818	C₂	direct (N × H)
5. C₂×C₃×#818	C₃	direct (N × H)
8. C₂×C₃³×C₈	C₄	semidirect
18. C₃×#818	C₂×C₃	direct (N × H)
26. C₃³×C₈	C₂×C₄	semidirect
28. C₂×#818	C₃²	direct (N × H)
36. C₂×C₃²×C₈	Dic₃	semidirect
37. C₂×C₃²×C₈	C₃×C₄	semidirect
58. #818	C₂×C₃²	direct (N × H)
79. C₃²×C₈	C₂×Dic₃	semidirect
80. C₃²×C₈	C₂×C₃×C₄	semidirect
82. C₂×C₃³	#107	semidirect
93. C₂×C₃×C₈	C₃×Dic₃	semidirect
94. C₂×C₃×C₈	C₃²×C₄	semidirect
127. C₃³	C₂×#107	semidirect
148. C₃×C₈	C₂×C₃×Dic₃	semidirect
149. C₃×C₈	C₂×C₃²×C₄	semidirect
154. C₂×C₃²	#818	direct (N × H)
155. C₂×C₃²	C₃×#107	semidirect
164. C₂×C₈	C₃²×Dic₃	semidirect
192. C₃²	C₂×#818	direct (N × H)
193. C₃²	C₂×C₃×#107	semidirect
201. C₈	C₂×C₃²×Dic₃	semidirect
206. C₂×C₃	C₃×#818	direct (N × H)
207. C₂×C₃	C₃²×#107	semidirect
223. C₃	C₂×C₃×#818	direct (N × H)
224. C₃	C₂×C₃²×#107	semidirect
225. C₂	C₃²×#818	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×C₃³×C₈	2
C₂×C₃²×C₄⋊Dic₃	2
C₃²×#818	2
C₂×C₃²×#107	3
C₂×C₃×#818	3

Subgroups — C2×C32×#818 #13162540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#818 #13162540