Finite Group Databasebeta

← #13163681

Subgroups — #13163681

Order 1728.

2268 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 407 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×#89076C211yesmaximal, characteristic, complemented
2C33×#121C211yesmaximal, characteristic, complemented
3C2×#88861C211yesmaximal, characteristic, complemented
4C2×#88860C211yesmaximal, characteristic, complemented
5#11722200C211yesmaximal, characteristic, complemented
6#11722180C211yesmaximal, characteristic, complemented
7#11722179C211yesmaximal, characteristic, complemented
8#10592736—313nomaximal
9C2×C33×D4—21no—
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×#6312—21no—
13#89076—22no—
14#89076—22no—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)—21no—
17C2×#6308C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C33×C22⋊C4—21no—
20#88861—22no—
21#88860—22no—
22C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C2×#3828—313no—
25C32×#121D3113yescomplemented
26C2×#3754—313no—
27C2×#3753—313no—
28#63657—313no—
29#63637—313no—
30#63636—313no—
31C23×C33—21no—
32C23×C33D411yescharacteristic
33C22×C3⋊(C3⋊D3)—21no—
34C33×D4—21no—
35C33×D4—21no—
36C33×D4—42no—
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42#6312—24no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yescomplemented
46C4×C3⋊(C3⋊D3)—24no—
47#6308—21no—
48#6308D412yescomplemented
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51#5037—913no—
52C2×C32×D4—213no—
53C2×C32×D4C2×D3113yes—
54C22×C32×C4C2×D3113yes—
55C2×#638—613no—
56#3828—626no—
57#3828—626no—
58C2×C4⋊(C3⋊D3)—313no—
59C2×C4×C3⋊D3—613no—
60C2×#634—313no—
61C32×C4⋊C4C2×D3113yes—
62C32×C22⋊C4—213no—
63#3754—626no—
64#3753—626no—
65C2×C3⋊(C3⋊C8)—313no—
66C2×C3⋊(C3⋊C8)—313no—
67C22×C33—21no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33—41no—
71C22×C33—41no—
72C22×C33—41no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—22no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C33×C4—21no—
78C33×C4—41no—
79C33×C4C2×D411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C2×#235—913no—
83C3×#121C3⋊D3113yescomplemented
84C2×#214—913no—
85C2×#213—913no—
86#830—913no—
87#810—913no—
88#809—913no—
89C23×C32—213no—
90C23×C32#67113yes—
91C22×C3⋊D3—613no—
92C32×D4—213no—
93C32×D4—213no—
94C32×D4—426no—
95C2×C32×C4—213no—
96C2×C32×C4—213no—
97C2×C32×C4—213no—
98C2×C32×C4#67113yes—
99C2×C32×C4C22×D3113yes—
100#638—652no—
101C2×C3⋊Dic3—613no—
102C4⋊(C3⋊D3)—326no—
103C4⋊(C3⋊D3)—613no—
104C4×C3⋊D3—652no—
105#634—326no—
106#634—613no—
107C3⋊(C3⋊C8)—613no—
108C3⋊(C3⋊C8)—613no—
109#449—271noSylow
110C2×C33—41no—
111C2×C33—81no—
112C2×C33#12011yescharacteristic
113C2×C33#13511yescharacteristic
114C2×C33AGL(1,ℤ/8)11yescharacteristic
115C3⋊(C3⋊D3)—42no—
116C2×C3×D4—213no—
117C2×C3×D4C2×C3⋊D3113yes—
118C22×C3×C4C2×C3⋊D3113yes—
119C2×#67—1813no—
120#235—1826no—
121#235—1826no—
122C2×D12—913no—
123C2×C4×D3—1813no—
124C2×Dic6—913no—
125C3×C4⋊C4C2×C3⋊D3113yes—
126C3×C22⋊C4—213no—
127#214—1826no—
128#213—1826no—
129C2×C3⋊C8—913no—
130C2×C3⋊C8—913no—
131C22×C32—213no—
132C22×C32—213no—
133C22×C32—213no—
134C22×C32—413no—
135C22×C32—413no—
136C22×C32—413no—
137C22×C32C2×#67113yes—
138C2×C3⋊D3—626no—
139C2×C3⋊D3—1213no—
140C2×C3⋊D3—1213no—
141C32×C4—213no—
142C32×C4—413no—
143C32×C4D3×D4113yes—
144C32×C4D3×D4113yes—
145C3⋊Dic3—626no—
146C2×#41—271no—
147C2×SD16—271no—
148C2×D8—271no—
149#121C3⋊(C3⋊D3)11yescharacteristic, complemented
150#103—271no—
151#102—271no—
152C22⋊C8—271no—
153C33#44911yescharacteristic, complemented, Sylow
154C23×C3—213no—
155C23×C3#638113yes—
156C22×D3—1813no—
157C3×D4—213no—
158C3×D4—213no—
159C3×D4—426no—
160C2×C3×C4—213no—
161C2×C3×C4—213no—
162C2×C3×C4—213no—
163C2×C3×C4#638113yes—
164C2×C3×C4C22×C3⋊D3113yes—
165#67—1852no—
166C2×Dic3—1813no—
167D12—926no—
168D12—1813no—
169C4×D3—1852no—
170Dic6—926no—
171Dic6—1813no—
172C3⋊C8—1813no—
173C3⋊C8—1813no—
174C2×C32—413no—
175C2×C32—813no—
176C2×C32#932113yes—
177C2×C32#937113yes—
178C2×C32#950113yes—
179C3⋊D3—1226no—
180#41—542no—
181#41—542no—
182C2×Q8—271no—
183C2×D4—21no—
184C2×D4—271no—
185C2×D4—541no—
186C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
187C22×C4—541no—
188C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
189SD16—542no—
190D8—542no—
191C2×C8—271no—
192C2×C8—271no—
193C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
194C22⋊C4—21no—
195C22×C3—213no—
196C22×C3—213no—
197C22×C3—213no—
198C22×C3—413no—
199C22×C3—413no—
200C22×C3—413no—
201C22×C3C2×#638113yes—
202C2×D3—1826no—
203C2×D3—3613no—
204C2×D3—3613no—
205C3×C4—213no—
206C3×C4—413no—
207C3×C4D4×C3⋊D3113yes—
208C3×C4D4×C3⋊D3113yes—
209Dic3—1826no—
210C32#5037113yescomplemented
211C23—21no—
212C23—541no—
213C23#631211yescharacteristic
214Q8—272no—
215Q8—541no—
216D4—21no—
217D4—21no—
218D4—42no—
219D4—272no—
220D4—544no—
221D4—541no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—544no—
226C2×C4—541no—
227C2×C4#631211yescharacteristic
228C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
229C8—541no—
230C8—541no—
231C2×C3—413no—
232C2×C3—813no—
233C2×C3#64141113yes—
234C2×C3#64146113yes—
235C2×C3#64159113yes—
236D3—3626no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—542no—
244C22—1081no—
245C22—1081no—
246C22C2×#631211yescharacteristic
247C4—21no—
248C4—41no—
249C4—542no—
250C4D4×C3⋊(C3⋊D3)11yescharacteristic
251C4D4×C3⋊(C3⋊D3)11yescharacteristic
252C3#10592736113yescomplemented
253C2—41no—
254C2—81no—
255C2—1082no—
256C2#1172397311yescharacteristic
257C2#1172397811yescharacteristic
258C2#1172399111yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#121C2
socleC22×C33C2×D4
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C33×#121C2semidirect
3. C2×#88861C2semidirect
4. C2×#88860C2semidirect
5. #11722200C2semidirect
6. #11722180C2semidirect
7. #11722179C2semidirect
25. C32×#121D3semidirect
45. C4⋊(C3⋊(C3⋊D3))D4semidirect
48. #6308D4semidirect
83. C3×#121C3⋊D3semidirect
149. #121C3⋊(C3⋊D3)semidirect
153. C33#449semidirect
210. C32#5037semidirect
252. C3#10592736semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C33×#1212
C2×#888612
C2×#888602
#117222002
#117221802
#117221792
#105927363