Subgroups — C₂²×C₃²×Dic₁₂ #13179119

Order 1728.

1852 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Dic₆	C₂	1	2	yes	maximal
2	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×Dic₁₂	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃²×Q₁₆	—	3	1	no	maximal
5	C₂²×C₃×Dic₁₂	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×C₃²×Dic₆	—	2	6	no	—
9	C₂×C₃²×Dic₆	C₂²	1	12	yes	—
10	C₂×C₃³×C₈	C₂²	1	6	yes	—
11	C₃²×Dic₁₂	C₂²	1	16	yes	direct product factor
12	C₂²×C₃²×Q₈	—	3	2	no	—
13	C₂²×C₃×Dic₆	C₂×C₃	1	8	yes	—
14	C₂×C₃²×Q₁₆	—	3	12	no	—
15	C₂²×C₃²×C₈	—	2	4	no	—
16	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
17	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	—
18	C₂×C₃×Dic₁₂	C₂×C₃	1	48	yes	direct product factor
19	C₂³×C₃³	D₄	1	1	yes	characteristic
20	C₂×C₃³×C₄	D₄	1	3	yes	—
21	C₂×C₃³×C₄	C₂³	1	3	yes	—
22	C₂×C₃²×Dic₃	—	2	12	no	—
23	C₃²×Dic₆	—	2	12	no	—
24	C₃²×Dic₆	C₂³	1	8	yes	—
25	C₃³×C₈	C₂³	1	4	yes	—
26	C₂²×C₃×Q₁₆	—	3	4	no	—
27	C₂²×Dic₁₂	C₃²	1	1	yes	characteristic, direct product factor
28	C₂×C₃²×Q₈	—	3	12	no	—
29	C₂×C₃²×Q₈	—	6	6	no	—
30	C₂²×C₃²×C₄	—	2	4	no	—
31	C₂²×C₃²×C₄	—	6	2	no	—
32	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
34	C₂²×C₃×Dic₃	—	2	8	no	—
35	C₂×C₃×Dic₆	—	2	24	no	—
36	C₂×C₃×Dic₆	C₂²×C₃	1	48	yes	—
37	C₃²×Q₁₆	—	3	16	no	—
38	C₂×C₃²×C₈	—	2	24	no	—
39	C₂×C₃²×C₈	C₂×D₃	1	6	yes	—
40	C₂×C₃²×C₈	C₂²×C₃	1	24	yes	—
41	C₃×Dic₁₂	C₂²×C₃	1	64	yes	direct product factor
42	C₂²×C₃³	Q₁₆	1	4	yes	complemented
43	C₂²×C₃³	C₂×D₄	1	3	yes	—
44	C₃³×C₄	C₂×D₄	1	3	yes	—
45	C₃³×C₄	C₂⁴	1	1	yes	characteristic
46	C₃²×Dic₃	—	2	8	no	—
47	C₂²×C₃×Q₈	—	3	8	no	—
48	C₂²×Dic₆	C₂×C₃²	1	2	yes	—
49	C₂×C₃×Q₁₆	—	3	48	no	—
50	C₂²×C₃×C₈	—	2	4	no	—
51	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
52	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
53	C₂×Dic₁₂	C₂×C₃²	1	12	yes	direct product factor
54	C₂³×C₃²	—	2	4	no	—
55	C₂³×C₃²	D₁₂	1	1	yes	characteristic
56	C₂³×C₃²	C₃×D₄	1	4	yes	—
57	C₃²×Q₈	—	3	8	no	—
58	C₃²×Q₈	—	6	12	no	—
59	C₂×C₃²×C₄	—	2	12	no	—
60	C₂×C₃²×C₄	—	2	12	no	—
61	C₂×C₃²×C₄	—	6	12	no	—
62	C₂×C₃²×C₄	D₁₂	1	3	yes	—
63	C₂×C₃²×C₄	C₃×D₄	1	12	yes	—
64	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
66	C₂×C₃×Dic₃	—	2	48	no	—
67	C₃×Dic₆	—	2	48	no	—
68	C₃×Dic₆	C₂³×C₃	1	32	yes	—
69	C₃²×C₈	—	2	16	no	—
70	C₃²×C₈	C₂²×D₃	1	4	yes	—
71	C₃²×C₈	C₂³×C₃	1	16	yes	—
72	C₂²×Q₁₆	—	3	1	no	Sylow
73	C₂×C₃³	C₂×Q₁₆	1	6	yes	complemented
74	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₃×Q₈	—	3	48	no	—
76	C₂×C₃×Q₈	—	6	24	no	—
77	C₂²×C₃×C₄	—	2	4	no	—
78	C₂²×C₃×C₄	—	6	8	no	—
79	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
80	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
81	C₂²×Dic₃	—	2	2	no	—
82	C₂×Dic₆	—	2	6	no	—
83	C₂×Dic₆	C₂²×C₃²	1	12	yes	—
84	C₃×Q₁₆	—	3	64	no	—
85	C₂×C₃×C₈	—	2	24	no	—
86	C₂×C₃×C₈	C₂×C₃×D₃	1	24	yes	—
87	C₂×C₃×C₈	C₂²×C₃²	1	6	yes	—
88	Dic₁₂	C₂²×C₃²	1	16	yes	direct product factor
89	C₂²×C₃²	—	2	16	no	—
90	C₂²×C₃²	—	2	12	no	—
91	C₂²×C₃²	Dic₁₂	1	4	yes	direct product factor
92	C₂²×C₃²	C₃×Q₁₆	1	16	yes	complemented
93	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
94	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
95	C₃²×C₄	—	2	4	no	—
96	C₃²×C₄	—	2	12	no	—
97	C₃²×C₄	—	6	8	no	—
98	C₃²×C₄	C₂×D₁₂	1	3	yes	—
99	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
100	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
101	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
102	C₃×Dic₃	—	2	32	no	—
103	C₂²×Q₈	—	3	2	no	—
104	C₂×Q₁₆	—	3	12	no	—
105	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
106	C₃³	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₃	—	2	4	no	—
108	C₂³×C₃	C₃×D₁₂	1	4	yes	—
109	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
110	C₃×Q₈	—	3	32	no	—
111	C₃×Q₈	—	6	48	no	—
112	C₂×C₃×C₄	—	2	12	no	—
113	C₂×C₃×C₄	—	2	12	no	—
114	C₂×C₃×C₄	—	6	48	no	—
115	C₂×C₃×C₄	C₃×D₁₂	1	12	yes	—
116	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
117	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
118	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
119	C₂×Dic₃	—	2	12	no	—
120	Dic₆	—	2	12	no	—
121	Dic₆	C₂³×C₃²	1	8	yes	—
122	C₃×C₈	—	2	16	no	—
123	C₃×C₈	C₂²×C₃×D₃	1	16	yes	—
124	C₃×C₈	C₂³×C₃²	1	4	yes	—
125	C₂×C₃²	—	2	24	no	—
126	C₂×C₃²	—	2	4	no	—
127	C₂×C₃²	C₂×Dic₁₂	1	6	yes	direct product factor
128	C₂×C₃²	C₂×C₃×Q₁₆	1	24	yes	complemented
129	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
130	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
131	C₂×Q₈	—	3	12	no	—
132	C₂×Q₈	—	6	6	no	—
133	C₂²×C₄	—	6	2	no	—
134	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
135	Q₁₆	—	3	16	no	—
136	C₂×C₈	C₂×C₃²×D₃	1	6	yes	—
137	C₂²×C₃	—	2	12	no	—
138	C₂²×C₃	—	2	16	no	—
139	C₂²×C₃	C₃×Dic₁₂	1	16	yes	direct product factor
140	C₂²×C₃	C₃²×Q₁₆	1	4	yes	complemented
141	C₂²×C₃	C₂×C₃×D₁₂	1	12	yes	—
142	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
143	C₃×C₄	—	2	4	no	—
144	C₃×C₄	—	2	12	no	—
145	C₃×C₄	—	6	32	no	—
146	C₃×C₄	C₂×C₃×D₁₂	1	12	yes	—
147	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
148	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
149	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
150	Dic₃	—	2	8	no	—
151	C₃²	—	2	4	no	—
152	C₃²	C₂²×Dic₁₂	1	1	yes	characteristic, direct product factor
153	C₃²	C₂²×C₃×Q₁₆	1	4	yes	complemented
154	C₂³	C₃²×D₁₂	1	1	yes	characteristic
155	Q₈	—	3	8	no	—
156	Q₈	—	6	12	no	—
157	C₂×C₄	—	6	12	no	—
158	C₂×C₄	C₃²×D₁₂	1	3	yes	—
159	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
160	C₈	C₂²×C₃²×D₃	1	4	yes	—
161	C₂×C₃	—	2	4	no	—
162	C₂×C₃	—	2	24	no	—
163	C₂×C₃	C₂×C₃×Dic₁₂	1	24	yes	direct product factor
164	C₂×C₃	C₂×C₃²×Q₁₆	1	6	yes	complemented
165	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
166	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
167	C₂²	C₃²×Dic₁₂	1	4	yes	direct product factor
168	C₂²	C₂×C₃²×D₁₂	1	3	yes	—
169	C₄	—	6	8	no	—
170	C₄	C₂×C₃²×D₁₂	1	3	yes	—
171	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
172	C₃	—	2	4	no	—
173	C₃	C₂²×C₃×Dic₁₂	1	4	yes	direct product factor
174	C₃	C₂²×C₃²×Q₁₆	1	1	yes	characteristic, complemented
175	C₂	C₂×C₃²×Dic₁₂	1	6	yes	direct product factor
176	C₂	C₂²×C₃²×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃²	D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle, prime radical	C₂³×C₃³	D₄
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₂²×C₃²×Q₁₆
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. C₂×C₃²×Dic₁₂	C₂	direct (N × H)
5. C₂²×C₃×Dic₁₂	C₃	direct (N × H)
11. C₃²×Dic₁₂	C₂²	direct (N × H)
18. C₂×C₃×Dic₁₂	C₂×C₃	direct (N × H)
27. C₂²×Dic₁₂	C₃²	direct (N × H)
41. C₃×Dic₁₂	C₂²×C₃	direct (N × H)
42. C₂²×C₃³	Q₁₆	semidirect
53. C₂×Dic₁₂	C₂×C₃²	direct (N × H)
73. C₂×C₃³	C₂×Q₁₆	semidirect
88. Dic₁₂	C₂²×C₃²	direct (N × H)
91. C₂²×C₃²	Dic₁₂	direct (N × H)
92. C₂²×C₃²	C₃×Q₁₆	semidirect
106. C₃³	C₂²×Q₁₆	semidirect
127. C₂×C₃²	C₂×Dic₁₂	direct (N × H)
128. C₂×C₃²	C₂×C₃×Q₁₆	semidirect
139. C₂²×C₃	C₃×Dic₁₂	direct (N × H)
140. C₂²×C₃	C₃²×Q₁₆	semidirect
152. C₃²	C₂²×Dic₁₂	direct (N × H)
153. C₃²	C₂²×C₃×Q₁₆	semidirect
163. C₂×C₃	C₂×C₃×Dic₁₂	direct (N × H)
164. C₂×C₃	C₂×C₃²×Q₁₆	semidirect
167. C₂²	C₃²×Dic₁₂	direct (N × H)
173. C₃	C₂²×C₃×Dic₁₂	direct (N × H)
174. C₃	C₂²×C₃²×Q₁₆	semidirect
175. C₂	C₂×C₃²×Dic₁₂	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×C₃²×Dic₆	2
C₂²×C₃³×C₈	2
C₂×C₃²×Dic₁₂	2
C₂²×C₃²×Q₁₆	3
C₂²×C₃×Dic₁₂	3

Subgroups — C22×C32×Dic12 #13179119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×Dic₁₂ #13179119