Subgroups — C₂×C₃²×C₄×C₃⋊C₈ #13162356

Order 1728.

1404 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₃²×C₄×C₃⋊C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₈	—	3	1	no	maximal
5	C₂×C₃×C₄×C₃⋊C₈	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³×C₄	C₄	1	2	yes	—
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃³×C₄²	C₄	1	2	yes	—
9	C₃³×C₄²	C₂²	1	2	yes	—
10	C₂×C₃²×C₃⋊C₈	C₄	1	8	yes	direct product factor
11	C₂×C₃²×C₃⋊C₈	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	—	3	2	no	—
13	C₂×C₃²×C₄²	—	2	4	no	—
14	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
15	C₂×C₃²×C₄²	C₂×C₃	1	4	yes	—
16	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	8	yes	—
17	C₃²×C₄×C₈	—	3	4	no	—
18	C₃×C₄×C₃⋊C₈	C₂×C₃	1	16	yes	direct product factor
19	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃³×C₄	C₈	1	8	yes	complemented
21	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
23	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
24	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
25	C₃²×C₃⋊C₈	C₂×C₄	1	8	yes	direct product factor
26	C₂×C₃×C₄×C₈	—	3	4	no	—
27	C₂×C₄×C₃⋊C₈	C₃²	1	1	yes	characteristic, direct product factor
28	C₂²×C₃²×C₄	—	2	8	no	—
29	C₂²×C₃²×C₄	—	2	4	no	—
30	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
31	C₂²×C₃²×C₄	C₃×C₄	1	8	yes	—
32	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
34	C₂×C₃²×C₈	—	3	8	no	—
35	C₂×C₃²×C₈	—	3	4	no	—
36	C₃²×C₄²	—	2	8	no	—
37	C₃²×C₄²	—	2	8	no	—
38	C₃²×C₄²	Dic₃	1	2	yes	—
39	C₃²×C₄²	C₃×C₄	1	8	yes	—
40	C₃²×C₄²	C₂×D₃	1	2	yes	—
41	C₃²×C₄²	C₂²×C₃	1	8	yes	—
42	C₂×C₃×C₃⋊C₈	C₃×C₄	1	32	yes	direct product factor
43	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	16	yes	—
44	C₂²×C₃³	C₄²	1	2	yes	—
45	C₂²×C₃³	C₂×C₈	1	4	yes	—
46	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
47	C₃³×C₄	C₄²	1	2	yes	—
48	C₃³×C₄	C₂×C₈	1	8	yes	complemented
49	C₃³×C₄	C₂²×C₄	1	2	yes	—
50	C₂²×C₃×C₈	—	3	8	no	—
51	C₂×C₃×C₄²	—	2	4	no	—
52	C₂×C₃×C₄²	C₃×D₃	1	4	yes	—
53	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	C₂×C₃²	1	2	yes	—
55	C₃×C₄×C₈	—	3	16	no	—
56	C₄×C₃⋊C₈	C₂×C₃²	1	4	yes	direct product factor
57	C₂³×C₃²	—	2	4	no	—
58	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
59	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
60	C₂×C₃²×C₄	—	2	32	no	—
61	C₂×C₃²×C₄	—	2	16	no	—
62	C₂×C₃²×C₄	—	2	16	no	—
63	C₂×C₃²×C₄	—	2	4	no	—
64	C₂×C₃²×C₄	—	2	4	no	—
65	C₂×C₃²×C₄	C₃⋊C₈	1	8	yes	direct product factor
66	C₂×C₃²×C₄	C₃×C₈	1	32	yes	complemented
67	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
68	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
69	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×C₃×C₄	1	16	yes	—
71	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₃²×C₄	C₂×C₃×C₄	1	16	yes	—
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
75	C₃²×C₈	—	3	8	no	—
76	C₃×C₃⋊C₈	C₂×C₃×C₄	1	32	yes	direct product factor
77	C₂×C₄×C₈	—	3	1	no	Sylow
78	C₂×C₃³	C₄×C₈	1	4	yes	complemented
79	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₃³	C₂²×C₈	1	2	yes	—
81	C₂²×C₃×C₄	—	2	8	no	—
82	C₂²×C₃×C₄	—	2	4	no	—
83	C₂²×C₃×C₄	C₃×Dic₃	1	8	yes	—
84	C₂²×C₃×C₄	C₃²×C₄	1	2	yes	—
85	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
87	C₂×C₃×C₈	—	3	32	no	—
88	C₂×C₃×C₈	—	3	16	no	—
89	C₃×C₄²	—	2	8	no	—
90	C₃×C₄²	—	2	8	no	—
91	C₃×C₄²	C₃×Dic₃	1	8	yes	—
92	C₃×C₄²	C₃²×C₄	1	2	yes	—
93	C₃×C₄²	C₂×C₃×D₃	1	8	yes	—
94	C₃×C₄²	C₂²×C₃²	1	2	yes	—
95	C₂×C₃⋊C₈	C₃²×C₄	1	8	yes	direct product factor
96	C₂×C₃⋊C₈	C₂²×C₃²	1	4	yes	—
97	C₂²×C₃²	—	2	16	no	—
98	C₂²×C₃²	—	2	8	no	—
99	C₂²×C₃²	—	2	4	no	—
100	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
101	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
102	C₂²×C₃²	C₃×C₄²	1	8	yes	—
103	C₂²×C₃²	C₂×C₃×C₈	1	16	yes	—
104	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
105	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
106	C₃²×C₄	—	2	32	no	—
107	C₃²×C₄	—	2	8	no	—
108	C₃²×C₄	—	2	8	no	—
109	C₃²×C₄	C₂×C₃⋊C₈	1	8	yes	direct product factor
110	C₃²×C₄	C₄×Dic₃	1	2	yes	—
111	C₃²×C₄	C₃×C₄²	1	8	yes	—
112	C₃²×C₄	C₂×C₃×C₈	1	32	yes	complemented
113	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
114	C₃²×C₄	C₂²×C₃×C₄	1	8	yes	—
115	C₂²×C₈	—	3	2	no	—
116	C₂×C₄²	C₃²×D₃	1	1	yes	characteristic
117	C₄×C₈	—	3	4	no	—
118	C₃³	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₃	—	2	4	no	—
120	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
121	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	32	no	—
123	C₂×C₃×C₄	—	2	16	no	—
124	C₂×C₃×C₄	—	2	16	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	C₃×C₃⋊C₈	1	32	yes	direct product factor
128	C₂×C₃×C₄	C₃²×C₈	1	8	yes	complemented
129	C₂×C₃×C₄	C₃×C₄×D₃	1	16	yes	—
130	C₂×C₃×C₄	C₂×C₃×Dic₃	1	16	yes	—
131	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
132	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
133	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
135	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
136	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
137	C₃×C₈	—	3	32	no	—
138	C₃⋊C₈	C₂×C₃²×C₄	1	8	yes	direct product factor
139	C₂×C₃²	—	2	8	no	—
140	C₂×C₃²	—	2	4	no	—
141	C₂×C₃²	—	2	16	no	—
142	C₂×C₃²	C₄×C₃⋊C₈	1	4	yes	direct product factor
143	C₂×C₃²	C₃×C₄×C₈	1	16	yes	complemented
144	C₂×C₃²	C₂²×C₃⋊C₈	1	2	yes	—
145	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
146	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
147	C₂×C₃²	C₂²×C₃×C₈	1	8	yes	—
148	C₂²×C₄	C₃²×Dic₃	1	2	yes	—
149	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
150	C₂×C₈	—	3	8	no	—
151	C₂×C₈	—	3	4	no	—
152	C₄²	C₃²×Dic₃	1	2	yes	—
153	C₄²	C₂×C₃²×D₃	1	2	yes	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	16	no	—
156	C₂²×C₃	—	2	8	no	—
157	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	16	yes	—
158	C₂²×C₃	C₃×C₄×Dic₃	1	8	yes	—
159	C₂²×C₃	C₃²×C₄²	1	2	yes	—
160	C₂²×C₃	C₂×C₃²×C₈	1	4	yes	—
161	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
162	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
163	C₃×C₄	—	2	8	no	—
164	C₃×C₄	—	2	32	no	—
165	C₃×C₄	—	2	8	no	—
166	C₃×C₄	C₂×C₃×C₃⋊C₈	1	32	yes	direct product factor
167	C₃×C₄	C₃×C₄×Dic₃	1	8	yes	—
168	C₃×C₄	C₃²×C₄²	1	2	yes	—
169	C₃×C₄	C₂×C₃²×C₈	1	8	yes	complemented
170	C₃×C₄	C₂×C₃×C₄×D₃	1	8	yes	—
171	C₃×C₄	C₂²×C₃²×C₄	1	2	yes	—
172	C₃²	—	2	4	no	—
173	C₃²	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic, direct product factor
174	C₃²	C₂×C₃×C₄×C₈	1	4	yes	complemented
175	C₂³	C₂×C₃²×Dic₃	1	1	yes	characteristic
176	C₂×C₄	C₃²×C₃⋊C₈	1	8	yes	direct product factor
177	C₂×C₄	C₃²×C₄×D₃	1	4	yes	—
178	C₂×C₄	C₂×C₃²×Dic₃	1	4	yes	—
179	C₂×C₄	C₂×C₃²×Dic₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
181	C₈	—	3	8	no	—
182	C₂×C₃	—	2	8	no	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	—	2	16	no	—
185	C₂×C₃	C₃×C₄×C₃⋊C₈	1	16	yes	direct product factor
186	C₂×C₃	C₃²×C₄×C₈	1	4	yes	complemented
187	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	8	yes	—
188	C₂×C₃	C₂×C₃×C₄×Dic₃	1	4	yes	—
189	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
190	C₂×C₃	C₂²×C₃²×C₈	1	2	yes	—
191	C₂²	C₂×C₃²×C₃⋊C₈	1	4	yes	—
192	C₂²	C₃²×C₄×Dic₃	1	2	yes	—
193	C₂²	C₂²×C₃²×Dic₃	1	1	yes	characteristic
194	C₄	C₂×C₃²×C₃⋊C₈	1	8	yes	direct product factor
195	C₄	C₃²×C₄×Dic₃	1	2	yes	—
196	C₄	C₂×C₃²×C₄×D₃	1	2	yes	—
197	C₃	—	2	4	no	—
198	C₃	C₂×C₃×C₄×C₃⋊C₈	1	4	yes	direct product factor
199	C₃	C₂×C₃²×C₄×C₈	1	1	yes	characteristic, complemented
200	C₂	C₃²×C₄×C₃⋊C₈	1	4	yes	direct product factor
201	C₂	C₂²×C₃²×C₃⋊C₈	1	2	yes	—
202	C₂	C₂×C₃²×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₄²	D₃
commutator, hypocenter	C₃	C₂×C₃²×C₄×C₈
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₄²	C₂
socle, prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄×C₃⋊C₈	C₂	direct (N × H)
5. C₂×C₃×C₄×C₃⋊C₈	C₃	direct (N × H)
10. C₂×C₃²×C₃⋊C₈	C₄	direct (N × H)
18. C₃×C₄×C₃⋊C₈	C₂×C₃	direct (N × H)
20. C₂×C₃³×C₄	C₈	semidirect
25. C₃²×C₃⋊C₈	C₂×C₄	direct (N × H)
27. C₂×C₄×C₃⋊C₈	C₃²	direct (N × H)
42. C₂×C₃×C₃⋊C₈	C₃×C₄	direct (N × H)
48. C₃³×C₄	C₂×C₈	semidirect
56. C₄×C₃⋊C₈	C₂×C₃²	direct (N × H)
65. C₂×C₃²×C₄	C₃⋊C₈	direct (N × H)
66. C₂×C₃²×C₄	C₃×C₈	semidirect
76. C₃×C₃⋊C₈	C₂×C₃×C₄	direct (N × H)
78. C₂×C₃³	C₄×C₈	semidirect
95. C₂×C₃⋊C₈	C₃²×C₄	direct (N × H)
109. C₃²×C₄	C₂×C₃⋊C₈	direct (N × H)
112. C₃²×C₄	C₂×C₃×C₈	semidirect
118. C₃³	C₂×C₄×C₈	semidirect
127. C₂×C₃×C₄	C₃×C₃⋊C₈	direct (N × H)
128. C₂×C₃×C₄	C₃²×C₈	semidirect
138. C₃⋊C₈	C₂×C₃²×C₄	direct (N × H)
142. C₂×C₃²	C₄×C₃⋊C₈	direct (N × H)
143. C₂×C₃²	C₃×C₄×C₈	semidirect
166. C₃×C₄	C₂×C₃×C₃⋊C₈	direct (N × H)
169. C₃×C₄	C₂×C₃²×C₈	semidirect
173. C₃²	C₂×C₄×C₃⋊C₈	direct (N × H)
174. C₃²	C₂×C₃×C₄×C₈	semidirect
176. C₂×C₄	C₃²×C₃⋊C₈	direct (N × H)
185. C₂×C₃	C₃×C₄×C₃⋊C₈	direct (N × H)
186. C₂×C₃	C₃²×C₄×C₈	semidirect
194. C₄	C₂×C₃²×C₃⋊C₈	direct (N × H)
198. C₃	C₂×C₃×C₄×C₃⋊C₈	direct (N × H)
199. C₃	C₂×C₃²×C₄×C₈	semidirect
200. C₂	C₃²×C₄×C₃⋊C₈	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×C₃³×C₄²	2
C₂²×C₃²×C₃⋊C₈	2
C₃²×C₄×C₃⋊C₈	2
C₂×C₃²×C₄×C₈	3
C₂×C₃×C₄×C₃⋊C₈	3

Subgroups — C2×C32×C4×C3⋊C8 #13162356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄×C₃⋊C₈ #13162356