Subgroups — C₃²×C₄×C₅₁₂ #72196152

Order 18432.

384 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₂₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₅₁₂	C₂	1	2	yes	maximal
3	C₃×C₄×C₅₁₂	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₄×C₁₂₈	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₂₅₆	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₂₅₆	C₂²	1	1	yes	characteristic
7	C₃²×C₅₁₂	C₄	1	4	yes	direct product factor
8	C₃×C₄×C₂₅₆	C₂×C₃	1	4	yes	—
9	C₂×C₃×C₅₁₂	C₂×C₃	1	8	yes	—
10	C₃²×C₄×C₆₄	C₈	1	1	yes	characteristic
11	C₂×C₃²×C₁₂₈	C₈	1	1	yes	characteristic
12	C₂×C₃²×C₁₂₈	C₂×C₄	1	1	yes	characteristic
13	C₃²×C₂₅₆	C₈	1	2	yes	—
14	C₃²×C₂₅₆	C₂×C₄	1	2	yes	—
15	C₄×C₅₁₂	C₃²	1	1	yes	characteristic, direct product factor, Sylow
16	C₃×C₄×C₁₂₈	C₃×C₄	1	4	yes	—
17	C₂×C₃×C₂₅₆	C₃×C₄	1	4	yes	—
18	C₂×C₃×C₂₅₆	C₂²×C₃	1	4	yes	—
19	C₃×C₅₁₂	C₃×C₄	1	16	yes	direct product factor
20	C₄×C₂₅₆	C₂×C₃²	1	1	yes	characteristic
21	C₂×C₅₁₂	C₂×C₃²	1	2	yes	—
22	C₂×C₃²×C₆₄	C₁₆	1	1	yes	characteristic
23	C₂×C₃²×C₆₄	C₂×C₈	1	1	yes	characteristic
24	C₃²×C₄×C₃₂	C₁₆	1	1	yes	characteristic
25	C₃²×C₁₂₈	C₁₆	1	2	yes	—
26	C₃²×C₁₂₈	C₄²	1	1	yes	characteristic
27	C₃²×C₁₂₈	C₂×C₈	1	1	yes	characteristic
28	C₂×C₃×C₁₂₈	C₃×C₈	1	4	yes	—
29	C₂×C₃×C₁₂₈	C₂×C₃×C₄	1	4	yes	—
30	C₃×C₄×C₆₄	C₃×C₈	1	4	yes	—
31	C₃×C₂₅₆	C₃×C₈	1	8	yes	—
32	C₃×C₂₅₆	C₂×C₃×C₄	1	8	yes	—
33	C₂×C₃²×C₃₂	C₃₂	1	1	yes	characteristic
34	C₂×C₃²×C₃₂	C₂×C₁₆	1	1	yes	characteristic
35	C₃²×C₄×C₁₆	C₃₂	1	1	yes	characteristic
36	C₃²×C₆₄	C₃₂	1	2	yes	—
37	C₃²×C₆₄	C₄×C₈	1	1	yes	characteristic
38	C₃²×C₆₄	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₂₅₆	C₃²×C₄	1	1	yes	characteristic
40	C₂×C₂₅₆	C₂²×C₃²	1	1	yes	characteristic
41	C₄×C₁₂₈	C₃²×C₄	1	1	yes	characteristic
42	C₅₁₂	C₃²×C₄	1	4	yes	direct product factor
43	C₂×C₃×C₆₄	C₃×C₁₆	1	4	yes	—
44	C₂×C₃×C₆₄	C₂×C₃×C₈	1	4	yes	—
45	C₃×C₄×C₃₂	C₃×C₁₆	1	4	yes	—
46	C₃×C₁₂₈	C₃×C₁₆	1	8	yes	—
47	C₃×C₁₂₈	C₃×C₄²	1	4	yes	—
48	C₃×C₁₂₈	C₂×C₃×C₈	1	4	yes	—
49	C₂×C₃²×C₁₆	C₆₄	1	1	yes	characteristic
50	C₂×C₃²×C₁₆	C₂×C₃₂	1	1	yes	characteristic
51	C₃²×C₄×C₈	C₆₄	1	1	yes	characteristic
52	C₃²×C₃₂	C₆₄	1	2	yes	—
53	C₃²×C₃₂	C₄×C₁₆	1	1	yes	characteristic
54	C₃²×C₃₂	C₂×C₃₂	1	1	yes	characteristic
55	C₂×C₁₂₈	C₃²×C₈	1	1	yes	characteristic
56	C₂×C₁₂₈	C₂×C₃²×C₄	1	1	yes	characteristic
57	C₄×C₆₄	C₃²×C₈	1	1	yes	characteristic
58	C₂₅₆	C₃²×C₈	1	2	yes	—
59	C₂₅₆	C₂×C₃²×C₄	1	2	yes	—
60	C₂×C₃×C₃₂	C₃×C₃₂	1	4	yes	—
61	C₂×C₃×C₃₂	C₂×C₃×C₁₆	1	4	yes	—
62	C₃×C₄×C₁₆	C₃×C₃₂	1	4	yes	—
63	C₃×C₆₄	C₃×C₃₂	1	8	yes	—
64	C₃×C₆₄	C₃×C₄×C₈	1	4	yes	—
65	C₃×C₆₄	C₂×C₃×C₁₆	1	4	yes	—
66	C₂×C₃²×C₈	C₁₂₈	1	1	yes	characteristic
67	C₂×C₃²×C₈	C₂×C₆₄	1	1	yes	characteristic
68	C₃²×C₄²	C₁₂₈	1	1	yes	characteristic
69	C₃²×C₁₆	C₁₂₈	1	2	yes	—
70	C₃²×C₁₆	C₄×C₃₂	1	1	yes	characteristic
71	C₃²×C₁₆	C₂×C₆₄	1	1	yes	characteristic
72	C₂×C₆₄	C₃²×C₁₆	1	1	yes	characteristic
73	C₂×C₆₄	C₂×C₃²×C₈	1	1	yes	characteristic
74	C₄×C₃₂	C₃²×C₁₆	1	1	yes	characteristic
75	C₁₂₈	C₃²×C₁₆	1	2	yes	—
76	C₁₂₈	C₃²×C₄²	1	1	yes	characteristic
77	C₁₂₈	C₂×C₃²×C₈	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₃×C₆₄	1	4	yes	—
79	C₂×C₃×C₁₆	C₂×C₃×C₃₂	1	4	yes	—
80	C₃×C₄×C₈	C₃×C₆₄	1	4	yes	—
81	C₃×C₃₂	C₃×C₆₄	1	8	yes	—
82	C₃×C₃₂	C₃×C₄×C₁₆	1	4	yes	—
83	C₃×C₃₂	C₂×C₃×C₃₂	1	4	yes	—
84	C₂×C₃²×C₄	C₂₅₆	1	2	yes	—
85	C₂×C₃²×C₄	C₂×C₁₂₈	1	1	yes	characteristic
86	C₃²×C₈	C₂₅₆	1	2	yes	—
87	C₃²×C₈	C₄×C₆₄	1	1	yes	characteristic
88	C₃²×C₈	C₂×C₁₂₈	1	1	yes	characteristic
89	C₂×C₃₂	C₃²×C₃₂	1	1	yes	characteristic
90	C₂×C₃₂	C₂×C₃²×C₁₆	1	1	yes	characteristic
91	C₄×C₁₆	C₃²×C₃₂	1	1	yes	characteristic
92	C₆₄	C₃²×C₃₂	1	2	yes	—
93	C₆₄	C₃²×C₄×C₈	1	1	yes	characteristic
94	C₆₄	C₂×C₃²×C₁₆	1	1	yes	characteristic
95	C₂×C₃×C₈	C₃×C₁₂₈	1	4	yes	—
96	C₂×C₃×C₈	C₂×C₃×C₆₄	1	4	yes	—
97	C₃×C₄²	C₃×C₁₂₈	1	4	yes	—
98	C₃×C₁₆	C₃×C₁₂₈	1	8	yes	—
99	C₃×C₁₆	C₃×C₄×C₃₂	1	4	yes	—
100	C₃×C₁₆	C₂×C₃×C₆₄	1	4	yes	—
101	C₂²×C₃²	C₂×C₂₅₆	1	1	yes	characteristic
102	C₃²×C₄	C₅₁₂	1	4	yes	direct product factor
103	C₃²×C₄	C₄×C₁₂₈	1	1	yes	characteristic
104	C₃²×C₄	C₂×C₂₅₆	1	1	yes	characteristic
105	C₂×C₁₆	C₃²×C₆₄	1	1	yes	characteristic
106	C₂×C₁₆	C₂×C₃²×C₃₂	1	1	yes	characteristic
107	C₄×C₈	C₃²×C₆₄	1	1	yes	characteristic
108	C₃₂	C₃²×C₆₄	1	2	yes	—
109	C₃₂	C₃²×C₄×C₁₆	1	1	yes	characteristic
110	C₃₂	C₂×C₃²×C₃₂	1	1	yes	characteristic
111	C₂×C₃×C₄	C₃×C₂₅₆	1	8	yes	—
112	C₂×C₃×C₄	C₂×C₃×C₁₂₈	1	4	yes	—
113	C₃×C₈	C₃×C₂₅₆	1	8	yes	—
114	C₃×C₈	C₃×C₄×C₆₄	1	4	yes	—
115	C₃×C₈	C₂×C₃×C₁₂₈	1	4	yes	—
116	C₂×C₃²	C₂×C₅₁₂	1	2	yes	—
117	C₂×C₃²	C₄×C₂₅₆	1	1	yes	characteristic
118	C₂×C₈	C₃²×C₁₂₈	1	1	yes	characteristic
119	C₂×C₈	C₂×C₃²×C₆₄	1	1	yes	characteristic
120	C₄²	C₃²×C₁₂₈	1	1	yes	characteristic
121	C₁₆	C₃²×C₁₂₈	1	2	yes	—
122	C₁₆	C₃²×C₄×C₃₂	1	1	yes	characteristic
123	C₁₆	C₂×C₃²×C₆₄	1	1	yes	characteristic
124	C₂²×C₃	C₂×C₃×C₂₅₆	1	4	yes	—
125	C₃×C₄	C₃×C₅₁₂	1	16	yes	direct product factor
126	C₃×C₄	C₂×C₃×C₂₅₆	1	4	yes	—
127	C₃×C₄	C₃×C₄×C₁₂₈	1	4	yes	—
128	C₃²	C₄×C₅₁₂	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×C₄	C₃²×C₂₅₆	1	2	yes	—
130	C₂×C₄	C₂×C₃²×C₁₂₈	1	1	yes	characteristic
131	C₈	C₃²×C₂₅₆	1	2	yes	—
132	C₈	C₂×C₃²×C₁₂₈	1	1	yes	characteristic
133	C₈	C₃²×C₄×C₆₄	1	1	yes	characteristic
134	C₂×C₃	C₂×C₃×C₅₁₂	1	8	yes	—
135	C₂×C₃	C₃×C₄×C₂₅₆	1	4	yes	—
136	C₂²	C₂×C₃²×C₂₅₆	1	1	yes	characteristic
137	C₄	C₃²×C₅₁₂	1	4	yes	direct product factor
138	C₄	C₂×C₃²×C₂₅₆	1	1	yes	characteristic
139	C₄	C₃²×C₄×C₁₂₈	1	1	yes	characteristic
140	C₃	C₃×C₄×C₅₁₂	1	4	yes	direct product factor
141	C₂	C₂×C₃²×C₅₁₂	1	2	yes	—
142	C₂	C₃²×C₄×C₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₂₅₆	C₂²×C₃²
socle, prime radical	C₂²×C₃²	C₂×C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₅₁₂	C₃	direct (N × H)
7. C₃²×C₅₁₂	C₄	direct (N × H)
15. C₄×C₅₁₂	C₃²	direct (N × H)
19. C₃×C₅₁₂	C₃×C₄	direct (N × H)
42. C₅₁₂	C₃²×C₄	direct (N × H)
102. C₃²×C₄	C₅₁₂	direct (N × H)
125. C₃×C₄	C₃×C₅₁₂	direct (N × H)
128. C₃²	C₄×C₅₁₂	direct (N × H)
137. C₄	C₃²×C₅₁₂	direct (N × H)
140. C₃	C₃×C₄×C₅₁₂	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₄×C₂₅₆	2
C₂×C₃²×C₅₁₂	2
C₃×C₄×C₅₁₂	3

Subgroups — C32×C4×C512 #72196152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₅₁₂ #72196152