Finite Group Databasebeta

← #107×C3⋊(C3⋊D3) #13163519

Subgroups — #107×C3⋊(C3⋊D3) #13163519

Order 1728.

1820 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 1820 conjugacy classes of subgroups — too many to draw. Only its 427 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C4⋊C4×C3⋊(C3⋊D3)C212yesmaximal
3C33×#107C211yesmaximal, characteristic, complemented
4#11722188C211yesmaximal, characteristic, complemented
5#11722177C212yesmaximal, complemented
6C3⋊D3×#107—313nomaximal
7C2×C4×C3⋊(C3⋊D3)—22no—
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×C33×C8C2211yescharacteristic
10C33×C4⋊C4C2212yes—
11C4⋊(C3⋊(C3⋊Dic3))C2212yes—
12#88857—22no—
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C8×C3⋊(C3⋊D3)C414yescomplemented
15C2×C8×C3⋊D3—313no—
16C4⋊C4×C3⋊D3—326no—
17C32×#107D3113yescomplemented
18#63645—313no—
19#63634—326no—
20C22×C3⋊(C3⋊D3)D411yescharacteristic
21C2×C33×C4—22no—
22C2×C33×C4C2311yescharacteristic
23C2×C3⋊(C3⋊Dic3)—22no—
24C2×C3⋊(C3⋊Dic3)D411yescharacteristic
25C4×C3⋊(C3⋊D3)—44no—
26C4×C3⋊(C3⋊D3)C2×C412yes—
27C4×C3⋊(C3⋊D3)Q812yes—
28C33×C8C2×C412yescomplemented
29C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
30D3×#107—913no—
31C2×C4×C3⋊D3—313no—
32C2×C4×C3⋊D3—626no—
33C2×C32×C8C2×D3113yes—
34C32×C4⋊C4C2×D3126yes—
35C4⋊(C3⋊Dic3)—326no—
36#3750—626no—
37C2×C3⋊(C3⋊C8)—313no—
38C8×C3⋊D3—352no—
39C22×C33C2×D411yescharacteristic
40C2×C3⋊(C3⋊D3)C4⋊C412yes—
41C2×C3⋊(C3⋊D3)D812yes—
42C2×C3⋊(C3⋊D3)Q1612yes—
43C33×C4—42no—
44C33×C4C22×C411yescharacteristic
45C33×C4C2×Q811yescharacteristic
46C3⋊(C3⋊Dic3)—42no—
47C3⋊(C3⋊Dic3)C4⋊C412yes—
48C2×D3×C8—913no—
49D3×C4⋊C4—926no—
50C3×#107C3⋊D3113yescomplemented
51#818—913no—
52#807—926no—
53C22×C3⋊D3—313no—
54C2×C32×C4—226no—
55C2×C32×C4C22×D3113yes—
56C2×C3⋊Dic3—313no—
57C2×C3⋊Dic3—626no—
58C4×C3⋊D3—326no—
59C4×C3⋊D3—326no—
60C4×C3⋊D3—1252no—
61C32×C8C4×D3126yescomplemented
62C3⋊(C3⋊C8)—326no—
63C2×#107—271noSylow
64C2×C33C2×C4⋊C411yescharacteristic
65C2×C33C2×D811yescharacteristic
66C2×C33C2×Q1611yescharacteristic
67C3⋊(C3⋊D3)#10714yesdirect product factor
68C2×C4×D3—913no—
69C2×C4×D3—1826no—
70C2×C3×C8C2×C3⋊D3113yes—
71C3×C4⋊C4C2×C3⋊D3126yes—
72C4⋊Dic3—926no—
73Dic3⋊C4—1826no—
74C2×C3⋊C8—913no—
75D3×C8—952no—
76C22×C32D3×D4113yes—
77C2×C3⋊D3—326no—
78C2×C3⋊D3—326no—
79C2×C3⋊D3—326no—
80C32×C4—426no—
81C32×C4C2×C4×D3113yes—
82C32×C4D3×Q8113yes—
83C3⋊Dic3—326no—
84C3⋊Dic3—1226no—
85C22×C8—271no—
86C2×C4⋊C4—272no—
87#107—271no—
88#107—272no—
89#107C3⋊(C3⋊D3)11yescharacteristic, direct product factor
90C33C2×#10711yescharacteristic, complemented, Sylow
91C22×D3—913no—
92C2×C3×C4—226no—
93C2×C3×C4C22×C3⋊D3113yes—
94C2×Dic3—913no—
95C2×Dic3—1826no—
96C4×D3—926no—
97C4×D3—926no—
98C4×D3—3652no—
99C3×C8C4×C3⋊D3126yescomplemented
100C3⋊C8—926no—
101C2×C32D3×C4⋊C4113yes—
102C2×C32D3×D8113yes—
103C2×C32D3×Q16113yes—
104C3⋊D3—352no—
105C22×C4—271no—
106C22×C4—542no—
107C2×C8—271no—
108C2×C8—274no—
109C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
110C4⋊C4—272no—
111C4⋊C4—542no—
112C4⋊C4C2×C3⋊(C3⋊D3)12yes—
113C22×C3D4×C3⋊D3113yes—
114C2×D3—926no—
115C2×D3—926no—
116C2×D3—926no—
117C3×C4—426no—
118C3×C4C2×C4×C3⋊D3113yes—
119C3×C4Q8×C3⋊D3113yes—
120Dic3—926no—
121Dic3—3626no—
122C32D3×#107113yescomplemented
123C23—271no—
124C2×C4—22no—
125C2×C4—271no—
126C2×C4—272no—
127C2×C4—272no—
128C2×C4—542no—
129C2×C4—1084no—
130C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
131C8—272no—
132C8C4×C3⋊(C3⋊D3)12yescomplemented
133C2×C3C4⋊C4×C3⋊D3113yes—
134C2×C3D8×C3⋊D3113yes—
135C2×C3Q16×C3⋊D3113yes—
136D3—952no—
137C22—272no—
138C22—272no—
139C22—272no—
140C22D4×C3⋊(C3⋊D3)11yescharacteristic
141C4—42no—
142C4—272no—
143C4—1082no—
144C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
145C4Q8×C3⋊(C3⋊D3)11yescharacteristic
146C3C3⋊D3×#107113yescomplemented
147C2—274no—
148C2C4⋊C4×C3⋊(C3⋊D3)11yescharacteristic
149C2D8×C3⋊(C3⋊D3)11yescharacteristic
150C2Q16×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#107C2
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenter#107C3⋊(C3⋊D3)
hypocenterC33C2×#107
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. C33×#107C2semidirect
4. #11722188C2semidirect
5. #11722177C2semidirect
14. C8×C3⋊(C3⋊D3)C4semidirect
17. C32×#107D3semidirect
28. C33×C8C2×C4semidirect
29. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
50. C3×#107C3⋊D3semidirect
61. C32×C8C4×D3semidirect
67. C3⋊(C3⋊D3)#107direct (N × H)
89. #107C3⋊(C3⋊D3)direct (N × H)
90. C33C2×#107semidirect
99. C3×C8C4×C3⋊D3semidirect
122. C32D3×#107semidirect
132. C8C4×C3⋊(C3⋊D3)semidirect
146. C3C3⋊D3×#107semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
C4⋊C4×C3⋊(C3⋊D3)2
C33×#1072
#117221882
#117221772
C3⋊D3×#1073