Subgroups — C₂×C₄×C₃⋊(C₉⋊C₈) #13144224

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
3	C₄×C₃⋊(C₉⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	maximal
5	C₂×C₄×C₉⋊C₈	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₉	C₄	1	2	yes	—
9	C₃×C₄²×C₉	C₂²	1	2	yes	—
10	C₂×C₃⋊(C₉⋊C₈)	C₄	1	8	yes	direct product factor
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	—
12	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
13	C₂²×C₃⋊(C₃⋊C₈)	—	3	2	no	—
14	C₄×C₃⋊(C₃⋊C₈)	—	3	4	no	—
15	C₂×C₄²×C₉	D₃	1	3	yes	—
16	C₂²×C₉⋊C₈	—	3	6	no	—
17	C₄×C₉⋊C₈	—	3	12	no	—
18	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	C₈	1	8	yes	complemented
20	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₃⋊(C₉⋊C₈)	C₂×C₄	1	8	yes	direct product factor
25	C₂×C₄×C₃⋊C₈	—	9	3	no	—
26	C₂×C₄×C₃⋊C₈	—	9	1	no	—
27	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₃²×C₄²	Dic₃	1	2	yes	—
30	C₃²×C₄²	C₂×D₃	1	2	yes	—
31	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
33	C₂²×C₄×C₉	Dic₃	1	6	yes	—
34	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
35	C₄²×C₉	Dic₃	1	6	yes	—
36	C₄²×C₉	C₂×D₃	1	6	yes	—
37	C₂×C₉⋊C₈	—	3	24	no	—
38	C₂×C₉⋊C₈	—	3	12	no	—
39	C₂²×C₃×C₉	C₄²	1	2	yes	—
40	C₂²×C₃×C₉	C₂×C₈	1	4	yes	—
41	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₄×C₉	C₄²	1	2	yes	—
43	C₃×C₄×C₉	C₂×C₈	1	8	yes	complemented
44	C₃×C₄×C₉	C₂²×C₄	1	2	yes	—
45	C₂×C₃×C₄²	D₉	1	3	yes	—
46	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
47	C₂²×C₃⋊C₈	—	9	6	no	—
48	C₂²×C₃⋊C₈	—	9	2	no	—
49	C₄×C₃⋊C₈	—	9	12	no	—
50	C₄×C₃⋊C₈	—	9	4	no	—
51	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₃⋊C₈	1	8	yes	—
53	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
54	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
55	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₂³×C₉	C₂×Dic₃	1	3	yes	—
58	C₃⋊(C₃⋊C₈)	—	3	8	no	—
59	C₂×C₄×C₉	C₃⋊C₈	1	24	yes	complemented
60	C₂×C₄×C₉	C₄×D₃	1	12	yes	—
61	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
62	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
63	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
64	C₉⋊C₈	—	3	24	no	—
65	C₂×C₄×C₈	—	27	1	no	Sylow
66	C₂×C₃×C₉	C₄×C₈	1	4	yes	complemented
67	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₃×C₉	C₂²×C₈	1	2	yes	—
69	C₂²×C₃×C₄	Dic₉	1	6	yes	—
70	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
71	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
72	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
73	C₃×C₄²	Dic₉	1	6	yes	—
74	C₃×C₄²	C₂×D₉	1	6	yes	—
75	C₃×C₄²	C₃⋊Dic₃	1	2	yes	—
76	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
77	C₂×C₃⋊C₈	—	9	24	no	—
78	C₂×C₃⋊C₈	—	9	12	no	—
79	C₂×C₃⋊C₈	—	9	8	no	—
80	C₂×C₃⋊C₈	—	9	4	no	—
81	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
82	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
83	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
84	C₃²×C₄	C₂×C₃⋊C₈	1	8	yes	—
85	C₃²×C₄	C₄×Dic₃	1	2	yes	—
86	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
87	C₂²×C₉	C₂×C₃⋊C₈	1	12	yes	—
88	C₂²×C₉	C₄×Dic₃	1	6	yes	—
89	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
90	C₄×C₉	C₂×C₃⋊C₈	1	24	yes	complemented
91	C₄×C₉	C₄×Dic₃	1	6	yes	—
92	C₄×C₉	C₂×C₄×D₃	1	6	yes	—
93	C₂²×C₈	—	27	2	no	—
94	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
95	C₄×C₈	—	27	4	no	—
96	C₃×C₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₃	C₂×Dic₉	1	3	yes	—
98	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
99	C₂×C₃×C₄	C₉⋊C₈	1	24	yes	complemented
100	C₂×C₃×C₄	C₄×D₉	1	12	yes	—
101	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
102	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
103	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	8	yes	—
104	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
105	C₂×C₃×C₄	C₄×C₃⋊D₃	1	4	yes	—
106	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
107	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
109	C₃⋊C₈	—	9	24	no	—
110	C₃⋊C₈	—	9	8	no	—
111	C₂×C₃²	C₄×C₃⋊C₈	1	4	yes	—
112	C₂×C₃²	C₂²×C₃⋊C₈	1	2	yes	—
113	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
114	C₂×C₉	C₄×C₃⋊C₈	1	12	yes	complemented
115	C₂×C₉	C₂²×C₃⋊C₈	1	6	yes	—
116	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
117	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
118	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
119	C₂×C₈	—	27	8	no	—
120	C₂×C₈	—	27	4	no	—
121	C₄²	C₃⋊Dic₉	1	2	yes	—
122	C₄²	C₂×C₃⋊D₉	1	2	yes	—
123	C₂²×C₃	C₂×C₉⋊C₈	1	12	yes	—
124	C₂²×C₃	C₄×Dic₉	1	6	yes	—
125	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
126	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
127	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
128	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
129	C₃×C₄	C₂×C₉⋊C₈	1	24	yes	complemented
130	C₃×C₄	C₄×Dic₉	1	6	yes	—
131	C₃×C₄	C₂×C₄×D₉	1	6	yes	—
132	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	8	yes	—
133	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
134	C₃×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
135	C₃²	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
136	C₉	C₂×C₄×C₃⋊C₈	1	3	yes	complemented
137	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
138	C₂×C₄	C₃⋊(C₉⋊C₈)	1	8	yes	direct product factor
139	C₂×C₄	C₄×C₃⋊D₉	1	4	yes	—
140	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
141	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
142	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
143	C₈	—	27	8	no	—
144	C₂×C₃	C₄×C₉⋊C₈	1	12	yes	complemented
145	C₂×C₃	C₂²×C₉⋊C₈	1	6	yes	—
146	C₂×C₃	C₂×C₄×Dic₉	1	3	yes	—
147	C₂×C₃	C₄×C₃⋊(C₃⋊C₈)	1	4	yes	—
148	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	2	yes	—
149	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
150	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	4	yes	—
151	C₂²	C₄×C₃⋊Dic₉	1	2	yes	—
152	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
153	C₄	C₂×C₃⋊(C₉⋊C₈)	1	8	yes	direct product factor
154	C₄	C₄×C₃⋊Dic₉	1	2	yes	—
155	C₄	C₂×C₄×C₃⋊D₉	1	2	yes	—
156	C₃	C₂×C₄×C₉⋊C₈	1	3	yes	complemented
157	C₃	C₂×C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
158	C₂	C₄×C₃⋊(C₉⋊C₈)	1	4	yes	direct product factor
159	C₂	C₂²×C₃⋊(C₉⋊C₈)	1	2	yes	—
160	C₂	C₂×C₄×C₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄²	C₃⋊D₉
commutator, hypocenter	C₃×C₉	C₂×C₄×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×C₃⋊(C₉⋊C₈)	C₂	direct (N × H)
10. C₂×C₃⋊(C₉⋊C₈)	C₄	direct (N × H)
19. C₂×C₃×C₄×C₉	C₈	semidirect
24. C₃⋊(C₉⋊C₈)	C₂×C₄	direct (N × H)
43. C₃×C₄×C₉	C₂×C₈	semidirect
59. C₂×C₄×C₉	C₃⋊C₈	semidirect
66. C₂×C₃×C₉	C₄×C₈	semidirect
90. C₄×C₉	C₂×C₃⋊C₈	semidirect
96. C₃×C₉	C₂×C₄×C₈	semidirect
99. C₂×C₃×C₄	C₉⋊C₈	semidirect
114. C₂×C₉	C₄×C₃⋊C₈	semidirect
129. C₃×C₄	C₂×C₉⋊C₈	semidirect
136. C₉	C₂×C₄×C₃⋊C₈	semidirect
138. C₂×C₄	C₃⋊(C₉⋊C₈)	direct (N × H)
144. C₂×C₃	C₄×C₉⋊C₈	semidirect
153. C₄	C₂×C₃⋊(C₉⋊C₈)	direct (N × H)
156. C₃	C₂×C₄×C₉⋊C₈	semidirect
158. C₂	C₄×C₃⋊(C₉⋊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×C₃×C₄²×C₉	2
C₂²×C₃⋊(C₉⋊C₈)	2
C₄×C₃⋊(C₉⋊C₈)	2
C₂×C₄×C₃⋊(C₃⋊C₈)	3
C₂×C₄×C₉⋊C₈	3

Subgroups — C2×C4×C3⋊(C9⋊C8) #13144224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃⋊(C₉⋊C₈) #13144224