Finite Group Databasebeta

← #13179876

Subgroups — #13179876

Order 1728.

3612 conjugacy classes of subgroups; only the 835 normal subgroups are shown.

normal subgroups only

This group has 3612 conjugacy classes of subgroups — too many to draw. Only its 835 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#11724685C211yesmaximal, characteristic, complemented
2#11724684C211yesmaximal, characteristic, complemented
3C33×#131C211yesmaximal, characteristic, complemented
4#11723950C213yesmaximal, complemented
5#11723949C213yesmaximal, complemented
6#11723945C213yesmaximal, complemented
7C2×#88851C213yesmaximal, complemented
8#10596222—313nomaximal
9C33×#41C2211yescharacteristic
10#89080—21no—
11Q8×C3⋊(C3⋊D3)—21no—
12#89078—23no—
13D4×C3⋊(C3⋊D3)—23no—
14#89076C2213yes—
15#89076C2213yescomplemented
16C2×C4⋊(C3⋊(C3⋊D3))C2213yes—
17C2×#6308C2213yescomplemented
18C33×M16C2213yescomplemented
19C2×C33×C8C2213yescomplemented
20#88853C2213yescomplemented
21#88852C2213yescomplemented
22#88851C2213yescomplemented
23#88851C2216yescomplemented
24#88851C2211yescharacteristic, complemented
25#64367—313no—
26#64366—313no—
27C32×#131D3113yescomplemented
28#64118—339no—
29#64117—339no—
30#64113—339no—
31C2×#3744—339no—
32C22×C3⋊(C3⋊D3)—23no—
33C33×Q8D411yescharacteristic
34C33×D4D413yes—
35C2×C33×C4C2313yes—
36#6312—23no—
37#6312—23no—
38C2×C3⋊(C3⋊Dic3)—23no—
39C4⋊(C3⋊(C3⋊D3))—23no—
40C4⋊(C3⋊(C3⋊D3))C2313yes—
41C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
42C4×C3⋊(C3⋊D3)—23no—
43C4×C3⋊(C3⋊D3)—23no—
44#6308—23no—
45#6308C2313yes—
46#6308C2311yescharacteristic
47C33×C8C2311yescharacteristic
48C33×C8C2313yes—
49#5751—913no—
50C32×#41C2×D3113yes—
51#3832—613no—
52Q8×C3⋊D3—613no—
53#3830—639no—
54D4×C3⋊D3—639no—
55#3828—339no—
56#3828—339no—
57C2×C4⋊(C3⋊D3)—339no—
58C2×#634—339no—
59C32×M16C2×D3139yescomplemented
60C2×C32×C8C2×D3139yescomplemented
61#3746—339no—
62#3745—339no—
63#3744—339no—
64#3744—378no—
65#3744—313no—
66C22×C33C2×D413yes—
67C2×C3⋊(C3⋊D3)—21no—
68C2×C3⋊(C3⋊D3)—23no—
69C2×C3⋊(C3⋊D3)—83no—
70C33×C4C2×D413yes—
71C33×C4C2411yescharacteristic
72C3⋊(C3⋊Dic3)—23no—
73C3⋊(C3⋊Dic3)—21no—
74#1010—913no—
75#1009—913no—
76C3×#131C3⋊D3113yescomplemented
77#909—939no—
78#908—939no—
79#904—939no—
80C2×#204—939no—
81C22×C3⋊D3—639no—
82C32×Q8D12113yes—
83C32×D4D12139yes—
84C2×C32×C4C22×D3139yes—
85#638—639no—
86#638—639no—
87C2×C3⋊Dic3—639no—
88C4⋊(C3⋊D3)—339no—
89C4⋊(C3⋊D3)—313no—
90C4⋊(C3⋊D3)—639no—
91C4×C3⋊D3—639no—
92C4×C3⋊D3—639no—
93#634—339no—
94#634—313no—
95#634—639no—
96C32×C8C22×D3113yes—
97C32×C8C22×D3139yes—
98#577—271noSylow
99C2×C33—23no—
100C2×C33C22×D411yescharacteristic
101C3⋊(C3⋊D3)—41no—
102C3⋊(C3⋊D3)—43no—
103C3×#41C2×C3⋊D3113yes—
104#239—1813no—
105D3×Q8—1813no—
106#237—1839no—
107D3×D4—1839no—
108#235—939no—
109#235—939no—
110C2×D12—939no—
111C2×Dic6—939no—
112C3×M16C2×C3⋊D3139yescomplemented
113C2×C3×C8C2×C3⋊D3139yescomplemented
114Dic12—939no—
115D24—939no—
116#204—939no—
117#204—978no—
118#204—913no—
119C22×C32C2×D12139yes—
120C2×C3⋊D3—613no—
121C2×C3⋊D3—639no—
122C2×C3⋊D3—2439no—
123C32×C4C2×D12139yes—
124C32×C4C23×D3113yes—
125C3⋊Dic3—639no—
126C3⋊Dic3—613no—
1272−1+4—271no—
128H(5,𝔽2)—271no—
129#137—273no—
130AGL(1,ℤ/8)—273no—
131#135—273no—
132C2×SD16—273no—
133#131C3⋊(C3⋊D3)11yescharacteristic, complemented
134C33#57711yescharacteristic, complemented, Sylow
135C22×D3—1839no—
136C3×Q8C4⋊(C3⋊D3)113yes—
137C3×D4C4⋊(C3⋊D3)139yes—
138C2×C3×C4C22×C3⋊D3139yes—
139#67—1839no—
140#67—1839no—
141C2×Dic3—1839no—
142D12—939no—
143D12—913no—
144D12—1839no—
145C4×D3—1839no—
146C4×D3—1839no—
147Dic6—939no—
148Dic6—913no—
149Dic6—1839no—
150C3×C8C22×C3⋊D3113yes—
151C3×C8C22×C3⋊D3139yes—
152C2×C32—239no—
153C2×C32C22×D12113yes—
154C3⋊D3—1213no—
155C3⋊D3—1239no—
156#41—273no—
157#41—273no—
158#41—541no—
159#41—543no—
160#41C2×C3⋊(C3⋊D3)11yescharacteristic
161C2×Q8—273no—
162C2×Q8—541no—
163C2×D4—273no—
164C2×D4—543no—
165Q16—273no—
166SD16—273no—
167SD16—276no—
168SD16—271no—
169D8—273no—
170M16C2×C3⋊(C3⋊D3)13yescomplemented
171C2×C8C2×C3⋊(C3⋊D3)13yescomplemented
172C22×C3C2×C4⋊(C3⋊D3)139yes—
173C2×D3—1813no—
174C2×D3—1839no—
175C2×D3—7239no—
176C3×C4C2×C4⋊(C3⋊D3)139yes—
177C3×C4C23×C3⋊D3113yes—
178Dic3—1839no—
179Dic3—1813no—
180C32#5751113yescomplemented
181C23—543no—
182Q8—273no—
183Q8—271no—
184Q8—543no—
185Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic
186D4—271no—
187D4—273no—
188D4—543no—
189D4—543no—
190D4—543no—
191D4C4⋊(C3⋊(C3⋊D3))13yes—
192C2×C4—543no—
193C2×C4—543no—
194C2×C4—543no—
195C2×C4C22×C3⋊(C3⋊D3)13yes—
196C8C22×C3⋊(C3⋊D3)11yescharacteristic
197C8C22×C3⋊(C3⋊D3)13yes—
198C2×C3—239no—
199C2×C3C22×C4⋊(C3⋊D3)113yes—
200D3—3639no—
201D3—3613no—
202C22—541no—
203C22—543no—
204C22—2163no—
205C22C2×C4⋊(C3⋊(C3⋊D3))13yes—
206C4—541no—
207C4—543no—
208C4C2×C4⋊(C3⋊(C3⋊D3))13yes—
209C4C23×C3⋊(C3⋊D3)11yescharacteristic
210C3#10596222113yescomplemented
211C2—23no—
212C2—1083no—
213C2—1081no—
214C2C22×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
subgroupG/H
centerC2C22×C4⋊(C3⋊(C3⋊D3))
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC33×#131C2
socleC2×C33C22×D4
hypercenter#131C3⋊(C3⋊D3)
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #41 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #11724685C2semidirect
2. #11724684C2semidirect
3. C33×#131C2semidirect
4. #11723950C2semidirect
5. #11723949C2semidirect
6. #11723945C2semidirect
7. C2×#88851C2semidirect
15. #89076C22semidirect
17. C2×#6308C22semidirect
18. C33×M16C22semidirect
19. C2×C33×C8C22semidirect
20. #88853C22semidirect
21. #88852C22semidirect
22. #88851C22semidirect
23. #88851C22semidirect
24. #88851C22semidirect
27. C32×#131D3semidirect
59. C32×M16C2×D3semidirect
60. C2×C32×C8C2×D3semidirect
76. C3×#131C3⋊D3semidirect
112. C3×M16C2×C3⋊D3semidirect
113. C2×C3×C8C2×C3⋊D3semidirect
133. #131C3⋊(C3⋊D3)semidirect
134. C33#577semidirect
170. M16C2×C3⋊(C3⋊D3)semidirect
171. C2×C8C2×C3⋊(C3⋊D3)semidirect
180. C32#5751semidirect
210. C3#10596222semidirect

Maximal subgroups

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

subgroupindex
#117246852
#117246842
C33×#1312
#117239502
#117239492
#117239452
C2×#888512
#105962223