Subgroups — C₂×C₃²×#817 #13162539

Order 1728.

1084 conjugacy classes of subgroups.

The subgroup lattice has 214 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×#817	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×#106	—	3	1	no	maximal
5	C₂×C₃×#817	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×#106	—	3	4	no	—
18	C₃×#817	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×#106	—	3	4	no	—
28	C₂×#817	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	SD₁₆	1	4	yes	—
46	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
47	C₃³×C₄	C₄⋊C₄	1	2	yes	—
48	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
50	C₃²×Dic₃	—	4	4	no	—
51	C₂²×C₃×C₈	—	2	4	no	—
52	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
53	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	3	8	no	—
55	C₂×C₄⋊Dic₃	C₂×C₃²	1	2	yes	—
56	C₃×#106	—	3	16	no	—
57	#817	C₂×C₃²	1	4	yes	direct product factor
58	C₂³×C₃²	—	2	4	no	—
59	C₂³×C₃²	D₁₂	1	1	yes	characteristic
60	C₂³×C₃²	C₃×D₄	1	4	yes	—
61	C₂×C₃²×C₄	—	2	8	no	—
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	—	2	8	no	—
64	C₂×C₃²×C₄	—	2	4	no	—
65	C₂×C₃²×C₄	—	6	4	no	—
66	C₂×C₃²×C₄	—	12	4	no	—
67	C₂×C₃²×C₄	Dic₆	1	2	yes	—
68	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
70	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
71	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
72	C₂×C₃²×C₄	C₃×Q₈	1	8	yes	—
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
75	C₂×C₃×Dic₃	—	2	16	no	—
76	C₂×C₃×Dic₃	—	4	16	no	—
77	C₃²×C₈	—	2	16	no	—
78	C₃²×C₈	C₂×Dic₃	1	4	yes	complemented
79	C₃²×C₈	C₂×C₃×C₄	1	16	yes	complemented
80	C₂×#106	—	3	1	no	Sylow
81	C₂×C₃³	#106	1	4	yes	complemented
82	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃³	C₂×SD₁₆	1	2	yes	—
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	—	6	8	no	—
86	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
87	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
88	C₂²×Dic₃	—	2	2	no	—
89	C₂×C₃×C₈	—	2	16	no	—
90	C₂×C₃×C₈	—	2	8	no	—
91	C₂×C₃×C₈	C₃×Dic₃	1	16	yes	complemented
92	C₂×C₃×C₈	C₃²×C₄	1	4	yes	complemented
93	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
94	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
95	C₃×C₄⋊C₄	—	3	16	no	—
96	C₃×C₄⋊C₄	—	6	8	no	—
97	C₄⋊Dic₃	—	2	2	no	—
98	C₄⋊Dic₃	C₂²×C₃²	1	4	yes	—
99	C₂²×C₃²	—	2	16	no	—
100	C₂²×C₃²	—	2	8	no	—
101	C₂²×C₃²	—	2	4	no	—
102	C₂²×C₃²	#204	1	4	yes	—
103	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
104	C₂²×C₃²	C₃×C₄⋊C₄	1	8	yes	—
105	C₂²×C₃²	C₃×SD₁₆	1	16	yes	—
106	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
107	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
108	C₃²×C₄	—	2	8	no	—
109	C₃²×C₄	—	2	4	no	—
110	C₃²×C₄	—	2	4	no	—
111	C₃²×C₄	—	12	4	no	—
112	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
113	C₃²×C₄	C₃×C₄⋊C₄	1	8	yes	—
114	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
115	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
116	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
117	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
118	C₃×Dic₃	—	4	16	no	—
119	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	3	2	no	—
121	#106	—	3	4	no	—
122	C₃³	C₂×#106	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	C₃×D₁₂	1	4	yes	—
125	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	6	16	no	—
131	C₂×C₃×C₄	—	12	16	no	—
132	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
133	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
134	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
135	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
136	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
138	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
139	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
140	C₂×Dic₃	—	2	4	no	—
141	C₂×Dic₃	—	4	4	no	—
142	C₃×C₈	—	2	16	no	—
143	C₃×C₈	C₂×C₃×Dic₃	1	16	yes	complemented
144	C₃×C₈	C₂×C₃²×C₄	1	4	yes	complemented
145	C₂×C₃²	—	2	8	no	—
146	C₂×C₃²	—	2	4	no	—
147	C₂×C₃²	—	2	16	no	—
148	C₂×C₃²	#817	1	4	yes	direct product factor
149	C₂×C₃²	C₃×#106	1	16	yes	complemented
150	C₂×C₃²	C₂×#204	1	2	yes	—
151	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
152	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
153	C₂×C₃²	C₂×C₃×SD₁₆	1	8	yes	—
154	C₂²×C₄	—	6	2	no	—
155	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
156	C₂×C₈	C₃²×Dic₃	1	4	yes	complemented
157	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
158	C₄⋊C₄	—	3	4	no	—
159	C₄⋊C₄	—	6	2	no	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	16	no	—
162	C₂²×C₃	—	2	8	no	—
163	C₂²×C₃	C₃×#204	1	16	yes	—
164	C₂²×C₃	C₃×C₄⋊Dic₃	1	8	yes	—
165	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
166	C₂²×C₃	C₃²×SD₁₆	1	4	yes	—
167	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
168	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
169	C₃×C₄	—	2	4	no	—
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	—	2	8	no	—
172	C₃×C₄	—	12	16	no	—
173	C₃×C₄	C₃×C₄⋊Dic₃	1	8	yes	—
174	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
175	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
176	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
177	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
178	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
179	Dic₃	—	4	4	no	—
180	C₃²	—	2	4	no	—
181	C₃²	C₂×#817	1	1	yes	characteristic, direct product factor
182	C₃²	C₂×C₃×#106	1	4	yes	complemented
183	C₂³	C₃²×D₁₂	1	1	yes	characteristic
184	C₂×C₄	—	6	4	no	—
185	C₂×C₄	—	12	4	no	—
186	C₂×C₄	C₃²×Dic₆	1	2	yes	—
187	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
188	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
189	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
190	C₈	C₂×C₃²×Dic₃	1	4	yes	complemented
191	C₂×C₃	—	2	4	no	—
192	C₂×C₃	—	2	16	no	—
193	C₂×C₃	—	2	8	no	—
194	C₂×C₃	C₃×#817	1	16	yes	direct product factor
195	C₂×C₃	C₃²×#106	1	4	yes	complemented
196	C₂×C₃	C₂×C₃×#204	1	8	yes	—
197	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
198	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₃	C₂×C₃²×SD₁₆	1	2	yes	—
200	C₂²	C₃²×#204	1	4	yes	—
201	C₂²	C₃²×C₄⋊Dic₃	1	2	yes	—
202	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
203	C₄	—	12	4	no	—
204	C₄	C₃²×C₄⋊Dic₃	1	2	yes	—
205	C₄	C₂×C₃²×Dic₆	1	1	yes	characteristic
206	C₄	C₂²×C₃²×Dic₃	1	1	yes	characteristic
207	C₃	—	2	4	no	—
208	C₃	C₂×C₃×#817	1	4	yes	direct product factor
209	C₃	C₂×C₃²×#106	1	1	yes	characteristic, complemented
210	C₂	C₃²×#817	1	4	yes	direct product factor
211	C₂	C₂×C₃²×#204	1	2	yes	—
212	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×#106
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×#817	C₂	direct (N × H)
5. C₂×C₃×#817	C₃	direct (N × H)
8. C₂×C₃³×C₈	C₄	semidirect
18. C₃×#817	C₂×C₃	direct (N × H)
26. C₃³×C₈	C₂×C₄	semidirect
28. C₂×#817	C₃²	direct (N × H)
36. C₂×C₃²×C₈	Dic₃	semidirect
37. C₂×C₃²×C₈	C₃×C₄	semidirect
57. #817	C₂×C₃²	direct (N × H)
78. C₃²×C₈	C₂×Dic₃	semidirect
79. C₃²×C₈	C₂×C₃×C₄	semidirect
81. C₂×C₃³	#106	semidirect
91. C₂×C₃×C₈	C₃×Dic₃	semidirect
92. C₂×C₃×C₈	C₃²×C₄	semidirect
122. C₃³	C₂×#106	semidirect
143. C₃×C₈	C₂×C₃×Dic₃	semidirect
144. C₃×C₈	C₂×C₃²×C₄	semidirect
148. C₂×C₃²	#817	direct (N × H)
149. C₂×C₃²	C₃×#106	semidirect
156. C₂×C₈	C₃²×Dic₃	semidirect
181. C₃²	C₂×#817	direct (N × H)
182. C₃²	C₂×C₃×#106	semidirect
190. C₈	C₂×C₃²×Dic₃	semidirect
194. C₂×C₃	C₃×#817	direct (N × H)
195. C₂×C₃	C₃²×#106	semidirect
208. C₃	C₂×C₃×#817	direct (N × H)
209. C₃	C₂×C₃²×#106	semidirect
210. C₂	C₃²×#817	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×#817	2
C₂×C₃²×#106	3
C₂×C₃×#817	3

Subgroups — C2×C32×#817 #13162539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#817 #13162539