Finite Group Databasebeta

← #13163493

Subgroups — #13163493

Order 1728.

1344 conjugacy classes of subgroups; only the 353 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 353 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
1#11723929C211yesmaximal, characteristic
2#11723928C211yesmaximal, characteristic
3C33×#106C211yesmaximal, characteristic
4#11722186C211yesmaximal, characteristic
5#11722184C211yesmaximal, characteristic
6#11722180C211yesmaximal, characteristic
7#11722178C211yesmaximal, characteristic
8#10592553—313nomaximal
9C2×#6308—21no—
10C2×#6308C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16#88857—21no—
17C4×C3⋊(C3⋊Dic3)—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#64097—313no—
21#64096—313no—
22C32×#106D3113yes—
23#63643—313no—
24#63641—313no—
25#63637—313no—
26#63635—313no—
27C2×C33×C4—21no—
28C2×C33×C4—21no—
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)—21no—
31C2×C3⋊(C3⋊Dic3)—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34#6308—21no—
35#6308—42no—
36#6308Q812yescomplemented
37C33×C8—21no—
38C3⋊(C3⋊(C3⋊C8))—21no—
39#4854—913no—
40C2×#634—313no—
41C2×#634—613no—
42C2×C32×C8C2×D3113yes—
43C32×C4⋊C4C2×D3113yes—
44C32×C4⋊C4C2×D3113yes—
45C4⋊(C3⋊Dic3)—313no—
46#3750—613no—
47#3750—613no—
48C4×C3⋊Dic3—313no—
49C4×C3⋊Dic3—613no—
50C2×C3⋊(C3⋊C8)—313no—
51C22×C33C2×D411yescharacteristic
52C33×C4—41no—
53C33×C4—41no—
54C33×C4C2×Q811yescharacteristic
55C33×C4#4111yescharacteristic
56C3⋊(C3⋊Dic3)—22no—
57C3⋊(C3⋊Dic3)—21no—
58C3⋊(C3⋊Dic3)—41no—
59C3⋊(C3⋊Dic3)SD1612yes—
60#888—913no—
61#887—913no—
62C3×#106C3⋊D3113yes—
63#816—913no—
64Dic3⋊C8—913no—
65#810—913no—
66#808—913no—
67C2×C32×C4—213no—
68C2×C32×C4—213no—
69C2×C32×C4C22×D3113yes—
70C2×C3⋊Dic3—313no—
71C2×C3⋊Dic3—313no—
72C2×C3⋊Dic3—613no—
73C2×C3⋊Dic3—613no—
74#634—326no—
75#634—613no—
76#634—1226no—
77C32×C8—213no—
78C3⋊(C3⋊C8)—613no—
79#475—271noSylow
80C2×C33C22⋊Q811yescharacteristic
81C2×C33C2×SD1611yescharacteristic
82C2×C33#13711yescharacteristic
83C2×Dic6—913no—
84C2×Dic6—1813no—
85C2×C3×C8C2×C3⋊D3113yes—
86C3×C4⋊C4C2×C3⋊D3113yes—
87C3×C4⋊C4C2×C3⋊D3113yes—
88C4⋊Dic3—913no—
89Dic3⋊C4—1813no—
90Dic3⋊C4—1813no—
91C4×Dic3—913no—
92C4×Dic3—1813no—
93C2×C3⋊C8—913no—
94C22×C32D3×D4113yes—
95C32×C4—413no—
96C32×C4—413no—
97C32×C4#235113yes—
98C32×C4D3×Q8113yes—
99C3⋊Dic3—326no—
100C3⋊Dic3—626no—
101C3⋊Dic3—613no—
102C3⋊Dic3—1213no—
103C4⋊Q8—271no—
104C4×Q8—271no—
105#106—271no—
106#106C3⋊(C3⋊D3)11yescharacteristic
107C4⋊C8—271no—
108#103—271no—
109#103—271no—
110C33#47511yescharacteristic, complemented, Sylow
111C2×C3×C4—213no—
112C2×C3×C4—213no—
113C2×C3×C4C22×C3⋊D3113yes—
114C2×Dic3—913no—
115C2×Dic3—913no—
116C2×Dic3—1813no—
117C2×Dic3—1813no—
118Dic6—926no—
119Dic6—1813no—
120Dic6—3626no—
121C3×C8—213no—
122C3⋊C8—1813no—
123C2×C32#896113yes—
124C2×C32D3×SD16113yes—
125C2×C32#915113yes—
126C2×Q8—271no—
127C2×Q8—541no—
128C2×C8—271no—
129C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
130C4⋊C4—271no—
131C4⋊C4—541no—
132C4⋊C4—541no—
133C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
134C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
135C42—271no—
136C42—541no—
137C22×C3D4×C3⋊D3113yes—
138C3×C4—413no—
139C3×C4—413no—
140C3×C4#3828113yes—
141C3×C4Q8×C3⋊D3113yes—
142Dic3—926no—
143Dic3—1826no—
144Dic3—1813no—
145Dic3—3613no—
146C32#4854113yescomplemented
147Q8—272no—
148Q8—541no—
149Q8—1082no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—271no—
153C2×C4—271no—
154C2×C4—541no—
155C2×C4—541no—
156C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
157C8—21no—
158C8—541no—
159C2×C3#64105113yes—
160C2×C3SD16×C3⋊D3113yes—
161C2×C3#64124113yes—
162C22D4×C3⋊(C3⋊D3)11yescharacteristic
163C4—41no—
164C4—41no—
165C4—272no—
166C4—542no—
167C4—541no—
168C4—1081no—
169C4#8907611yescharacteristic
170C4Q8×C3⋊(C3⋊D3)11yescharacteristic
171C3#10592553113yescomplemented
172C2#1172393711yescharacteristic
173C2SD16×C3⋊(C3⋊D3)11yescharacteristic
174C2#1172395611yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#106C2
socle, prime radicalC22×C33C2×D4
hypercenter#106C3⋊(C3⋊D3)
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
36. #6308Q8semidirect
110. C33#475semidirect
146. C32#4854semidirect
171. C3#10592553semidirect

Maximal subgroups

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

subgroupindex
#117239292
#117239282
C33×#1062
#117221862
#117221842
#117221802
#117221782
#105925533