Finite Group Databasebeta

← #13163465

Subgroups — #13163465

Order 1728.

2128 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 2128 conjugacy classes of subgroups — too many to draw. Only its 351 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#11724684C211yesmaximal, characteristic
2#11723992C211yesmaximal, characteristic, complemented
3#11723950C211yesmaximal, characteristic, complemented
4#11723916C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6#11722193C211yesmaximal, characteristic, complemented
7#11722175C211yesmaximal, characteristic, complemented
8#10592524—313nomaximal
9C33×#41C2211yescharacteristic
10#89080—21no—
11D4×C3⋊(C3⋊D3)—22no—
12D4×C3⋊(C3⋊D3)—21no—
13#89076—21no—
14#89076C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))—21no—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×#6308C2211yescharacteristic, complemented
18C33×M16C2211yescharacteristic, complemented
19C33×C42C2211yescharacteristic, complemented
20#88863—21no—
21#88861—21no—
22#88859—22no—
23#88855C2211yescharacteristic, complemented
24#88853—21no—
25#88851—21no—
26#64366—313no—
27#64160—313no—
28#64118—313no—
29#64084—313no—
30C32×#104D3113yescomplemented
31#63650—313no—
32#63632—313no—
33C22×C3⋊(C3⋊D3)—22no—
34C22×C3⋊(C3⋊D3)D412yes—
35C33×Q8D411yescharacteristic
36C33×D4—21no—
37C33×D4D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41#6312—22no—
42#6312—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊(C3⋊D3))—22no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
48C4×C3⋊(C3⋊D3)—22no—
49C4×C3⋊(C3⋊D3)—21no—
50#6308—21no—
51#6308—21no—
52#6308D411yescharacteristic
53C33×C8—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55#4825—913no—
56C32×#41C2×D3113yes—
57#3832—613no—
58D4×C3⋊D3—626no—
59D4×C3⋊D3—613no—
60#3828—313no—
61#3828—613no—
62C2×C4⋊(C3⋊D3)—313no—
63C2×C4⋊(C3⋊D3)—613no—
64C2×#634—313no—
65C32×M16C2×D3113yescomplemented
66C32×C42C2×D3113yescomplemented
67#3756—613no—
68#3754—613no—
69#3752—626no—
70#3748—313no—
71#3746—613no—
72#3744—613no—
73C22×C33—21no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—22no—
76C2×C3⋊(C3⋊D3)—21no—
77C2×C3⋊(C3⋊D3)—42no—
78C2×C3⋊(C3⋊D3)—82no—
79C33×C4—21no—
80C33×C4—41no—
81C33×C4C2×D411yescharacteristic
82C33×C4C2×D411yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3⋊(C3⋊Dic3)—41no—
85#1009—913no—
86#951—913no—
87#909—913no—
88#875—913no—
89C3×#104C3⋊D3113yescomplemented
90#823—913no—
91#805—913no—
92C22×C3⋊D3—326no—
93C22×C3⋊D3—626no—
94C32×Q8D12113yes—
95C32×D4—213no—
96C32×D4D12113yes—
97C2×C32×C4—213no—
98C2×C32×C4—213no—
99C2×C32×C4C22×D3113yes—
100#638—626no—
101#638—613no—
102C2×C3⋊Dic3—613no—
103C4⋊(C3⋊D3)—313no—
104C4⋊(C3⋊D3)—626no—
105C4⋊(C3⋊D3)—613no—
106C4⋊(C3⋊D3)—613no—
107C4×C3⋊D3—626no—
108C4×C3⋊D3—613no—
109#634—313no—
110#634—613no—
111#634—613no—
112C32×C8—213no—
113C3⋊(C3⋊C8)—613no—
114#455—271noSylow
115C2×C33—21no—
116C2×C33—41no—
117C2×C33#12011yescharacteristic
118C3⋊(C3⋊D3)—42no—
119C3⋊(C3⋊D3)—41no—
120C3×#41C2×C3⋊D3113yes—
121#239—1813no—
122D3×D4—1826no—
123D3×D4—1813no—
124#235—913no—
125#235—1813no—
126C2×D12—913no—
127C2×D12—1813no—
128C2×Dic6—913no—
129C3×M16C2×C3⋊D3113yescomplemented
130C3×C42C2×C3⋊D3113yescomplemented
131#216—1813no—
132#214—1813no—
133(C2×D3)⋊C4—1826no—
134#208—913no—
135Dic12—1813no—
136#204—1813no—
137C22×C32—213no—
138C22×C32D3×D4113yes—
139C2×C3⋊D3—613no—
140C2×C3⋊D3—626no—
141C2×C3⋊D3—1226no—
142C2×C3⋊D3—2426no—
143C32×C4—213no—
144C32×C4—413no—
145C32×C4C2×D12113yes—
146C32×C4D3×D4113yes—
147C3⋊Dic3—613no—
148C3⋊Dic3—1213no—
149H(5,𝔽2)—271no—
150#137—271no—
151#137—271no—
152#124—271no—
153#104—271no—
154#104C3⋊(C3⋊D3)11yescharacteristic, complemented
155#100—271no—
156C33#45511yescharacteristic, complemented, Sylow
157C22×D3—926no—
158C22×D3—1826no—
159C3×Q8C4⋊(C3⋊D3)113yes—
160C3×D4—213no—
161C3×D4C4⋊(C3⋊D3)113yes—
162C2×C3×C4—213no—
163C2×C3×C4—213no—
164C2×C3×C4C22×C3⋊D3113yes—
165#67—1826no—
166#67—1813no—
167C2×Dic3—1813no—
168D12—913no—
169D12—1826no—
170D12—1813no—
171D12—1813no—
172C4×D3—1826no—
173C4×D3—1813no—
174Dic6—913no—
175Dic6—1813no—
176Dic6—1813no—
177C3×C8—213no—
178C3⋊C8—1813no—
179C2×C32—213no—
180C2×C32—413no—
181C2×C32#882113yes—
182C3⋊D3—1226no—
183C3⋊D3—1213no—
184#41—271no—
185#41—541no—
186#41—541no—
187#41C2×C3⋊(C3⋊D3)11yescharacteristic
188C2×Q8—271no—
189C2×D4—271no—
190C2×D4—541no—
191C2×D4—542no—
192C2×D4—541no—
193Q16—541no—
194Q16—541no—
195SD16—541no—
196SD16—541no—
197M16—271no—
198M16C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
199C22⋊C4—542no—
200C42C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
201C22×C3—213no—
202C22×C3D4×C3⋊D3113yes—
203C2×D3—1813no—
204C2×D3—1826no—
205C2×D3—3626no—
206C2×D3—7226no—
207C3×C4—213no—
208C3×C4—413no—
209C3×C4C2×C4⋊(C3⋊D3)113yes—
210C3×C4D4×C3⋊D3113yes—
211Dic3—1813no—
212Dic3—3613no—
213C32#4825113yescomplemented
214C23—272no—
215C23—542no—
216Q8—271no—
217Q8—541no—
218Q8—541no—
219Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic
220D4—21no—
221D4—271no—
222D4—542no—
223D4—542no—
224D4—541no—
225D4—541no—
226D4—541no—
227D4C4⋊(C3⋊(C3⋊D3))11yescharacteristic
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—542no—
231C2×C4—541no—
232C2×C4—541no—
233C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
234C8—21no—
235C8—541no—
236C2×C3—213no—
237C2×C3—413no—
238C2×C3#64091113yes—
239D3—3626no—
240D3—3613no—
241C22—21no—
242C22—541no—
243C22—542no—
244C22—1082no—
245C22—2162no—
246C22D4×C3⋊(C3⋊D3)11yescharacteristic
247C4—21no—
248C4—41no—
249C4—541no—
250C4—1081no—
251C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
252C4D4×C3⋊(C3⋊D3)11yescharacteristic
253C3#10592524113yescomplemented
254C2—21no—
255C2—41no—
256C2—1082no—
257C2—1081no—
258C2#1172392311yescharacteristic
subgroupG/H
centerC2#11723923
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#104C2
socleC2×C33#120
prime radical#11724684C2
hypercenter#104C3⋊(C3⋊D3)
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #11723992C2semidirect
3. #11723950C2semidirect
4. #11723916C2semidirect
5. C33×#104C2semidirect
6. #11722193C2semidirect
7. #11722175C2semidirect
17. C2×#6308C22semidirect
18. C33×M16C22semidirect
19. C33×C42C22semidirect
23. #88855C22semidirect
30. C32×#104D3semidirect
65. C32×M16C2×D3semidirect
66. C32×C42C2×D3semidirect
89. C3×#104C3⋊D3semidirect
129. C3×M16C2×C3⋊D3semidirect
130. C3×C42C2×C3⋊D3semidirect
154. #104C3⋊(C3⋊D3)semidirect
156. C33#455semidirect
198. M16C2×C3⋊(C3⋊D3)semidirect
200. C42C2×C3⋊(C3⋊D3)semidirect
213. C32#4825semidirect
253. C3#10592524semidirect

Maximal subgroups

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

subgroupindex
#117246842
#117239922
#117239502
#117239162
C33×#1042
#117221932
#117221752
#105925243