Subgroups — C₂²×C₃²×D₂₄ #13179117

Order 1728.

2332 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×D₂₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃²×D₈	—	3	1	no	maximal
5	C₂²×C₃×D₂₄	C₃	1	4	yes	maximal, direct product factor
6	C₂³×C₃²×D₃	—	2	2	no	—
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃²×D₁₂	—	2	6	no	—
9	C₂×C₃²×D₁₂	C₂²	1	12	yes	complemented
10	C₂×C₃³×C₈	C₂²	1	6	yes	complemented
11	C₃²×D₂₄	C₂²	1	16	yes	direct product factor
12	C₂²×C₃²×D₄	—	3	2	no	—
13	C₂²×C₃×D₁₂	C₂×C₃	1	8	yes	complemented
14	C₂×C₃²×D₈	—	3	12	no	—
15	C₂²×C₃²×C₈	—	2	4	no	—
16	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic, complemented
17	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	complemented
18	C₂×C₃×D₂₄	C₂×C₃	1	48	yes	direct product factor
19	C₂³×C₃³	D₄	1	1	yes	characteristic
20	C₂²×C₃²×D₃	—	2	12	no	—
21	C₂²×C₃²×D₃	—	4	8	no	—
22	C₂×C₃³×C₄	D₄	1	3	yes	—
23	C₂×C₃³×C₄	C₂³	1	3	yes	—
24	C₃²×D₁₂	—	2	12	no	—
25	C₃²×D₁₂	C₂³	1	8	yes	complemented
26	C₃³×C₈	C₂³	1	4	yes	complemented
27	C₂²×C₃×D₈	—	3	4	no	—
28	C₂²×D₂₄	C₃²	1	1	yes	characteristic, direct product factor
29	C₂⁴×C₃²	—	6	2	no	—
30	C₂³×C₃×D₃	—	2	8	no	—
31	C₂×C₃²×D₄	—	3	12	no	—
32	C₂×C₃²×D₄	—	6	6	no	—
33	C₂²×C₃²×C₄	—	2	4	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
36	C₂×C₃×D₁₂	—	2	24	no	—
37	C₂×C₃×D₁₂	C₂²×C₃	1	48	yes	complemented
38	C₃²×D₈	—	3	16	no	—
39	C₂×C₃²×C₈	—	2	24	no	—
40	C₂×C₃²×C₈	C₂×D₃	1	6	yes	complemented
41	C₂×C₃²×C₈	C₂²×C₃	1	24	yes	complemented
42	C₃×D₂₄	C₂²×C₃	1	64	yes	direct product factor
43	C₂²×C₃³	D₈	1	4	yes	complemented
44	C₂²×C₃³	C₂×D₄	1	3	yes	—
45	C₂×C₃²×D₃	—	2	8	no	—
46	C₂×C₃²×D₃	—	4	24	no	—
47	C₃³×C₄	C₂×D₄	1	3	yes	—
48	C₃³×C₄	C₂⁴	1	1	yes	characteristic
49	C₂²×C₃×D₄	—	3	8	no	—
50	C₂²×D₁₂	C₂×C₃²	1	2	yes	complemented
51	C₂×C₃×D₈	—	3	48	no	—
52	C₂²×C₃×C₈	—	2	4	no	—
53	C₂²×C₃×C₈	C₃×D₃	1	4	yes	complemented
54	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
55	C₂×D₂₄	C₂×C₃²	1	12	yes	direct product factor
56	C₂³×C₃²	—	2	4	no	—
57	C₂³×C₃²	—	6	12	no	—
58	C₂³×C₃²	—	12	8	no	—
59	C₂³×C₃²	D₁₂	1	1	yes	characteristic
60	C₂³×C₃²	C₃×D₄	1	4	yes	—
61	C₂²×C₃×D₃	—	2	48	no	—
62	C₂²×C₃×D₃	—	4	32	no	—
63	C₃²×D₄	—	3	8	no	—
64	C₃²×D₄	—	6	12	no	—
65	C₂×C₃²×C₄	—	2	12	no	—
66	C₂×C₃²×C₄	—	2	12	no	—
67	C₂×C₃²×C₄	D₁₂	1	3	yes	—
68	C₂×C₃²×C₄	C₃×D₄	1	12	yes	—
69	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
70	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
71	C₃×D₁₂	—	2	48	no	—
72	C₃×D₁₂	C₂³×C₃	1	32	yes	complemented
73	C₃²×C₈	—	2	16	no	—
74	C₃²×C₈	C₂²×D₃	1	4	yes	complemented
75	C₃²×C₈	C₂³×C₃	1	16	yes	complemented
76	C₂²×D₈	—	3	1	no	Sylow
77	C₂×C₃³	C₂×D₈	1	6	yes	complemented
78	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
79	C₃²×D₃	—	4	8	no	—
80	C₂⁴×C₃	—	6	8	no	—
81	C₂³×D₃	—	2	2	no	—
82	C₂×C₃×D₄	—	3	48	no	—
83	C₂×C₃×D₄	—	6	24	no	—
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
87	C₂×D₁₂	—	2	6	no	—
88	C₂×D₁₂	C₂²×C₃²	1	12	yes	complemented
89	C₃×D₈	—	3	64	no	—
90	C₂×C₃×C₈	—	2	24	no	—
91	C₂×C₃×C₈	C₂×C₃×D₃	1	24	yes	complemented
92	C₂×C₃×C₈	C₂²×C₃²	1	6	yes	complemented
93	D₂₄	C₂²×C₃²	1	16	yes	direct product factor
94	C₂²×C₃²	—	2	16	no	—
95	C₂²×C₃²	—	2	12	no	—
96	C₂²×C₃²	—	6	8	no	—
97	C₂²×C₃²	—	12	24	no	—
98	C₂²×C₃²	D₂₄	1	4	yes	direct product factor
99	C₂²×C₃²	C₃×D₈	1	16	yes	complemented
100	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
101	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
102	C₂×C₃×D₃	—	2	32	no	—
103	C₂×C₃×D₃	—	4	96	no	—
104	C₃²×C₄	—	2	4	no	—
105	C₃²×C₄	—	2	12	no	—
106	C₃²×C₄	C₂×D₁₂	1	3	yes	—
107	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
108	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
109	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
110	C₂²×D₄	—	3	2	no	—
111	C₂×D₈	—	3	12	no	—
112	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic, complemented
113	C₃³	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₃	—	2	4	no	—
115	C₂³×C₃	—	6	48	no	—
116	C₂³×C₃	—	12	32	no	—
117	C₂³×C₃	C₃×D₁₂	1	4	yes	—
118	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
119	C₂²×D₃	—	2	12	no	—
120	C₂²×D₃	—	4	8	no	—
121	C₃×D₄	—	3	32	no	—
122	C₃×D₄	—	6	48	no	—
123	C₂×C₃×C₄	—	2	12	no	—
124	C₂×C₃×C₄	—	2	12	no	—
125	C₂×C₃×C₄	C₃×D₁₂	1	12	yes	—
126	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
127	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
128	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
129	D₁₂	—	2	12	no	—
130	D₁₂	C₂³×C₃²	1	8	yes	complemented
131	C₃×C₈	—	2	16	no	—
132	C₃×C₈	C₂²×C₃×D₃	1	16	yes	complemented
133	C₃×C₈	C₂³×C₃²	1	4	yes	complemented
134	C₂×C₃²	—	2	24	no	—
135	C₂×C₃²	—	2	4	no	—
136	C₂×C₃²	—	12	8	no	—
137	C₂×C₃²	C₂×D₂₄	1	6	yes	direct product factor
138	C₂×C₃²	C₂×C₃×D₈	1	24	yes	complemented
139	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
140	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
141	C₃×D₃	—	4	32	no	—
142	C₂⁴	—	6	2	no	—
143	C₂×D₄	—	3	12	no	—
144	C₂×D₄	—	6	6	no	—
145	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
146	D₈	—	3	16	no	—
147	C₂×C₈	C₂×C₃²×D₃	1	6	yes	complemented
148	C₂²×C₃	—	2	12	no	—
149	C₂²×C₃	—	2	16	no	—
150	C₂²×C₃	—	6	32	no	—
151	C₂²×C₃	—	12	96	no	—
152	C₂²×C₃	C₃×D₂₄	1	16	yes	direct product factor
153	C₂²×C₃	C₃²×D₈	1	4	yes	complemented
154	C₂²×C₃	C₂×C₃×D₁₂	1	12	yes	—
155	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
156	C₂×D₃	—	2	8	no	—
157	C₂×D₃	—	4	24	no	—
158	C₃×C₄	—	2	4	no	—
159	C₃×C₄	—	2	12	no	—
160	C₃×C₄	C₂×C₃×D₁₂	1	12	yes	—
161	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
162	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
163	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
164	C₃²	—	2	4	no	—
165	C₃²	C₂²×D₂₄	1	1	yes	characteristic, direct product factor
166	C₃²	C₂²×C₃×D₈	1	4	yes	complemented
167	C₂³	—	6	12	no	—
168	C₂³	—	12	8	no	—
169	C₂³	C₃²×D₁₂	1	1	yes	characteristic
170	D₄	—	3	8	no	—
171	D₄	—	6	12	no	—
172	C₂×C₄	C₃²×D₁₂	1	3	yes	—
173	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
174	C₈	C₂²×C₃²×D₃	1	4	yes	complemented
175	C₂×C₃	—	2	4	no	—
176	C₂×C₃	—	2	24	no	—
177	C₂×C₃	—	12	32	no	—
178	C₂×C₃	C₂×C₃×D₂₄	1	24	yes	direct product factor
179	C₂×C₃	C₂×C₃²×D₈	1	6	yes	complemented
180	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
181	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
182	D₃	—	4	8	no	—
183	C₂²	—	6	8	no	—
184	C₂²	—	12	24	no	—
185	C₂²	C₃²×D₂₄	1	4	yes	direct product factor
186	C₂²	C₂×C₃²×D₁₂	1	3	yes	—
187	C₄	C₂×C₃²×D₁₂	1	3	yes	—
188	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
189	C₃	—	2	4	no	—
190	C₃	C₂²×C₃×D₂₄	1	4	yes	direct product factor
191	C₃	C₂²×C₃²×D₈	1	1	yes	characteristic, complemented
192	C₂	—	12	8	no	—
193	C₂	C₂×C₃²×D₂₄	1	6	yes	direct product factor
194	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	C₂³×C₃³	D₄
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₂²×C₃²×D₈
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
1. C₂²×C₃²×D₁₂	C₂	semidirect
2. C₂²×C₃³×C₈	C₂	semidirect
3. C₂×C₃²×D₂₄	C₂	direct (N × H)
5. C₂²×C₃×D₂₄	C₃	direct (N × H)
9. C₂×C₃²×D₁₂	C₂²	semidirect
10. C₂×C₃³×C₈	C₂²	semidirect
11. C₃²×D₂₄	C₂²	direct (N × H)
13. C₂²×C₃×D₁₂	C₂×C₃	semidirect
16. C₂²×C₃²×C₈	D₃	semidirect
17. C₂²×C₃²×C₈	C₂×C₃	semidirect
18. C₂×C₃×D₂₄	C₂×C₃	direct (N × H)
25. C₃²×D₁₂	C₂³	semidirect
26. C₃³×C₈	C₂³	semidirect
28. C₂²×D₂₄	C₃²	direct (N × H)
37. C₂×C₃×D₁₂	C₂²×C₃	semidirect
40. C₂×C₃²×C₈	C₂×D₃	semidirect
41. C₂×C₃²×C₈	C₂²×C₃	semidirect
42. C₃×D₂₄	C₂²×C₃	direct (N × H)
43. C₂²×C₃³	D₈	semidirect
50. C₂²×D₁₂	C₂×C₃²	semidirect
53. C₂²×C₃×C₈	C₃×D₃	semidirect
54. C₂²×C₃×C₈	C₂×C₃²	semidirect
55. C₂×D₂₄	C₂×C₃²	direct (N × H)
72. C₃×D₁₂	C₂³×C₃	semidirect
74. C₃²×C₈	C₂²×D₃	semidirect
75. C₃²×C₈	C₂³×C₃	semidirect
77. C₂×C₃³	C₂×D₈	semidirect
88. C₂×D₁₂	C₂²×C₃²	semidirect
91. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
92. C₂×C₃×C₈	C₂²×C₃²	semidirect
93. D₂₄	C₂²×C₃²	direct (N × H)
98. C₂²×C₃²	D₂₄	direct (N × H)
99. C₂²×C₃²	C₃×D₈	semidirect
112. C₂²×C₈	C₃²×D₃	semidirect
113. C₃³	C₂²×D₈	semidirect
130. D₁₂	C₂³×C₃²	semidirect
132. C₃×C₈	C₂²×C₃×D₃	semidirect
133. C₃×C₈	C₂³×C₃²	semidirect
137. C₂×C₃²	C₂×D₂₄	direct (N × H)
138. C₂×C₃²	C₂×C₃×D₈	semidirect
147. C₂×C₈	C₂×C₃²×D₃	semidirect
152. C₂²×C₃	C₃×D₂₄	direct (N × H)
153. C₂²×C₃	C₃²×D₈	semidirect
165. C₃²	C₂²×D₂₄	direct (N × H)
166. C₃²	C₂²×C₃×D₈	semidirect
174. C₈	C₂²×C₃²×D₃	semidirect
178. C₂×C₃	C₂×C₃×D₂₄	direct (N × H)
179. C₂×C₃	C₂×C₃²×D₈	semidirect
185. C₂²	C₃²×D₂₄	direct (N × H)
190. C₃	C₂²×C₃×D₂₄	direct (N × H)
191. C₃	C₂²×C₃²×D₈	semidirect
193. C₂	C₂×C₃²×D₂₄	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×D₁₂	2
C₂²×C₃³×C₈	2
C₂×C₃²×D₂₄	2
C₂²×C₃²×D₈	3
C₂²×C₃×D₂₄	3

Subgroups — C22×C32×D24 #13179117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×D₂₄ #13179117