Subgroups — C₂³×C₉×C₃⋊C₈ #13153525

Order 1728.

1250 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	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₃⋊C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₉	—	3	1	no	maximal
5	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄×C₉	C₄	1	7	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	7	yes	—
8	C₂×C₉×C₃⋊C₈	C₂²	1	28	yes	direct product factor
9	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
10	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	14	yes	—
11	C₂³×C₄×C₉	—	2	1	no	—
12	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
13	C₂²×C₈×C₉	—	3	14	no	—
14	C₂³×C₃×C₉	C₈	1	8	yes	complemented
15	C₂³×C₃×C₉	C₂×C₄	1	7	yes	—
16	C₂×C₃×C₄×C₉	C₂×C₄	1	21	yes	—
17	C₂×C₃×C₄×C₉	C₂³	1	7	yes	—
18	C₉×C₃⋊C₈	C₂³	1	8	yes	direct product factor
19	C₂³×C₃×C₈	—	3	1	no	—
20	C₂³×C₃⋊C₈	C₉	1	1	yes	characteristic, direct product factor
21	C₂⁴×C₃²	C₃×C₄	1	1	yes	characteristic
22	C₂²×C₃²×C₄	C₃×C₄	1	7	yes	—
23	C₂²×C₃²×C₄	C₂²×C₃	1	7	yes	—
24	C₂⁴×C₉	—	2	1	no	—
25	C₂⁴×C₉	Dic₃	1	1	yes	characteristic
26	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	28	yes	—
27	C₂²×C₄×C₉	—	2	7	no	—
28	C₂²×C₄×C₉	—	2	7	no	—
29	C₂²×C₄×C₉	Dic₃	1	7	yes	—
30	C₂²×C₄×C₉	C₂×D₃	1	7	yes	—
31	C₂×C₈×C₉	—	3	28	no	—
32	C₂²×C₃×C₉	C₂×C₈	1	28	yes	complemented
33	C₂²×C₃×C₉	C₂²×C₄	1	7	yes	—
34	C₃×C₄×C₉	C₂²×C₄	1	7	yes	—
35	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
36	C₂³×C₃×C₄	—	2	1	no	—
37	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
38	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
39	C₂²×C₃×C₈	—	3	14	no	—
40	C₂²×C₃⋊C₈	C₂×C₉	1	14	yes	direct product factor
41	C₂³×C₃²	C₃×C₈	1	8	yes	—
42	C₂³×C₃²	C₂×C₃×C₄	1	7	yes	—
43	C₂×C₃²×C₄	C₂×C₃×C₄	1	21	yes	—
44	C₂×C₃²×C₄	C₂³×C₃	1	7	yes	—
45	C₂³×C₉	—	2	8	no	—
46	C₂³×C₉	—	2	7	no	—
47	C₂³×C₉	C₃⋊C₈	1	8	yes	direct product factor
48	C₂³×C₉	C₂×Dic₃	1	7	yes	—
49	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
50	C₂×C₄×C₉	—	2	21	no	—
51	C₂×C₄×C₉	—	2	7	no	—
52	C₂×C₄×C₉	C₂×Dic₃	1	21	yes	—
53	C₂×C₄×C₉	C₂²×D₃	1	7	yes	—
54	C₈×C₉	—	3	8	no	—
55	C₂³×C₈	—	3	1	no	Sylow
56	C₂×C₃×C₉	C₂²×C₈	1	14	yes	complemented
57	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
58	C₂⁴×C₃	—	2	1	no	—
59	C₂⁴×C₃	C₄×C₉	1	1	yes	characteristic
60	C₂⁴×C₃	C₃×Dic₃	1	1	yes	characteristic
61	C₂²×C₃×C₄	—	2	7	no	—
62	C₂²×C₃×C₄	—	2	7	no	—
63	C₂²×C₃×C₄	C₄×C₉	1	7	yes	—
64	C₂²×C₃×C₄	C₂²×C₉	1	7	yes	—
65	C₂²×C₃×C₄	C₃×Dic₃	1	7	yes	—
66	C₂²×C₃×C₄	C₂×C₃×D₃	1	7	yes	—
67	C₂×C₃×C₈	—	3	28	no	—
68	C₂×C₃⋊C₈	C₂²×C₉	1	28	yes	direct product factor
69	C₂²×C₃²	C₂×C₃×C₈	1	28	yes	—
70	C₂²×C₃²	C₂²×C₃×C₄	1	7	yes	—
71	C₃²×C₄	C₂²×C₃×C₄	1	7	yes	—
72	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
73	C₂²×C₉	—	2	28	no	—
74	C₂²×C₉	—	2	7	no	—
75	C₂²×C₉	C₂×C₃⋊C₈	1	28	yes	direct product factor
76	C₂²×C₉	C₂²×Dic₃	1	7	yes	—
77	C₄×C₉	—	2	1	no	—
78	C₄×C₉	—	2	7	no	—
79	C₄×C₉	C₂²×Dic₃	1	7	yes	—
80	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
81	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
82	C₂²×C₈	—	3	14	no	—
83	C₃×C₉	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
84	C₂³×C₃	—	2	8	no	—
85	C₂³×C₃	—	2	7	no	—
86	C₂³×C₃	C₈×C₉	1	8	yes	complemented
87	C₂³×C₃	C₂×C₄×C₉	1	7	yes	—
88	C₂³×C₃	C₃×C₃⋊C₈	1	8	yes	—
89	C₂³×C₃	C₂×C₃×Dic₃	1	7	yes	—
90	C₂×C₃×C₄	—	2	21	no	—
91	C₂×C₃×C₄	—	2	7	no	—
92	C₂×C₃×C₄	C₂×C₄×C₉	1	21	yes	—
93	C₂×C₃×C₄	C₂³×C₉	1	7	yes	—
94	C₂×C₃×C₄	C₂×C₃×Dic₃	1	21	yes	—
95	C₂×C₃×C₄	C₂²×C₃×D₃	1	7	yes	—
96	C₃×C₈	—	3	8	no	—
97	C₃⋊C₈	C₂³×C₉	1	8	yes	direct product factor
98	C₂×C₃²	C₂²×C₃×C₈	1	14	yes	—
99	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
100	C₂×C₉	—	2	14	no	—
101	C₂×C₉	—	2	1	no	—
102	C₂×C₉	C₂²×C₃⋊C₈	1	14	yes	direct product factor
103	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
104	C₂⁴	C₉×Dic₃	1	1	yes	characteristic
105	C₂²×C₄	C₉×Dic₃	1	7	yes	—
106	C₂²×C₄	C₂×D₃×C₉	1	7	yes	—
107	C₂×C₈	—	3	28	no	—
108	C₂²×C₃	—	2	7	no	—
109	C₂²×C₃	—	2	28	no	—
110	C₂²×C₃	C₂×C₈×C₉	1	28	yes	complemented
111	C₂²×C₃	C₂²×C₄×C₉	1	7	yes	—
112	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	28	yes	—
113	C₂²×C₃	C₂²×C₃×Dic₃	1	7	yes	—
114	C₃×C₄	—	2	1	no	—
115	C₃×C₄	—	2	7	no	—
116	C₃×C₄	C₂²×C₄×C₉	1	7	yes	—
117	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
118	C₃×C₄	C₂²×C₃×Dic₃	1	7	yes	—
119	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
120	C₃²	C₂³×C₃×C₈	1	1	yes	characteristic
121	C₉	—	2	1	no	—
122	C₉	C₂³×C₃⋊C₈	1	1	yes	characteristic, direct product factor
123	C₂³	C₉×C₃⋊C₈	1	8	yes	direct product factor
124	C₂³	C₂×C₉×Dic₃	1	7	yes	—
125	C₂×C₄	C₂×C₉×Dic₃	1	21	yes	—
126	C₂×C₄	C₂²×D₃×C₉	1	7	yes	—
127	C₈	—	3	8	no	—
128	C₂×C₃	—	2	14	no	—
129	C₂×C₃	—	2	1	no	—
130	C₂×C₃	C₂²×C₈×C₉	1	14	yes	complemented
131	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
132	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	14	yes	—
133	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
134	C₂²	C₂×C₉×C₃⋊C₈	1	28	yes	direct product factor
135	C₂²	C₂²×C₉×Dic₃	1	7	yes	—
136	C₄	C₂²×C₉×Dic₃	1	7	yes	—
137	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
138	C₃	—	2	1	no	—
139	C₃	C₂³×C₈×C₉	1	1	yes	characteristic, complemented
140	C₃	C₂³×C₃×C₃⋊C₈	1	1	yes	characteristic
141	C₂	C₂²×C₉×C₃⋊C₈	1	14	yes	direct product factor
142	C₂	C₂³×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄×C₉	D₃
commutator, hypocenter	C₃	C₂³×C₈×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle, prime radical	C₂⁴×C₃²	C₃×C₄
cosocle	C₃²×C₄	C₂⁴×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×C₃⋊C₈	C₂	direct (N × H)
8. C₂×C₉×C₃⋊C₈	C₂²	direct (N × H)
14. C₂³×C₃×C₉	C₈	semidirect
18. C₉×C₃⋊C₈	C₂³	direct (N × H)
20. C₂³×C₃⋊C₈	C₉	direct (N × H)
32. C₂²×C₃×C₉	C₂×C₈	semidirect
40. C₂²×C₃⋊C₈	C₂×C₉	direct (N × H)
47. C₂³×C₉	C₃⋊C₈	direct (N × H)
56. C₂×C₃×C₉	C₂²×C₈	semidirect
68. C₂×C₃⋊C₈	C₂²×C₉	direct (N × H)
75. C₂²×C₉	C₂×C₃⋊C₈	direct (N × H)
83. C₃×C₉	C₂³×C₈	semidirect
86. C₂³×C₃	C₈×C₉	semidirect
97. C₃⋊C₈	C₂³×C₉	direct (N × H)
102. C₂×C₉	C₂²×C₃⋊C₈	direct (N × H)
110. C₂²×C₃	C₂×C₈×C₉	semidirect
122. C₉	C₂³×C₃⋊C₈	direct (N × H)
123. C₂³	C₉×C₃⋊C₈	direct (N × H)
130. C₂×C₃	C₂²×C₈×C₉	semidirect
134. C₂²	C₂×C₉×C₃⋊C₈	direct (N × H)
139. C₃	C₂³×C₈×C₉	semidirect
141. C₂	C₂²×C₉×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
C₂³×C₃×C₄×C₉	2
C₂²×C₉×C₃⋊C₈	2
C₂³×C₃×C₃⋊C₈	3
C₂³×C₈×C₉	3

Subgroups — C23×C9×C3⋊C8 #13153525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₉×C₃⋊C₈ #13153525