Finite Group Databasebeta

← #13163880

Subgroups — #13163880

Order 1728.

2268 conjugacy classes of subgroups; only the 515 normal subgroups are shown.

normal subgroups only

This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 515 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×C33×#41C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4#11723971C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722202C211yesmaximal, characteristic, complemented
7#11722200C211yesmaximal, characteristic, complemented
8#10592937—313nomaximal
9C33×#41—22no—
10C33×#41—22no—
11C2×C33×Q8C2211yescharacteristic
12C2×C33×D4—21no—
13C2×C33×D4C2211yescharacteristic
14C22×C33×C4—21no—
15C22×C33×C4C2211yescharacteristic
16C2×#6312—21no—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18#88862—22no—
19#88860—22no—
20#88859—21no—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C2×C32×#41D3113yescomplemented
25C2×#3755—313no—
26C2×#3753—313no—
27#64139—313no—
28#63662—313no—
29#63659—313no—
30#63657—313no—
31C23×C33—21no—
32C23×C33D411yescharacteristic
33C22×C3⋊(C3⋊D3)—21no—
34C33×Q8—21no—
35C33×Q8D412yescomplemented
36C33×D4—24no—
37C33×D4—21no—
38C33×D4D412yescomplemented
39C2×C33×C4—24no—
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4D411yescharacteristic
44C2×C33×C4C2311yescharacteristic
45#6312—42no—
46C2×C3⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51#5238—913no—
52C32×#41—226no—
53C32×#41—226no—
54C2×C32×Q8C2×D3113yes—
55C2×C32×D4—213no—
56C2×C32×D4C2×D3113yes—
57C22×C32×C4—213no—
58C22×C32×C4C2×D3113yes—
59C2×#638—613no—
60C2×C4⋊(C3⋊D3)—313no—
61#3755—626no—
62#3753—626no—
63#3752—613no—
64C4⋊(C3⋊Dic3)—313no—
65C2×C3⋊(C3⋊C8)—313no—
66C2×C3⋊(C3⋊C8)—313no—
67C22×C33—21no—
68C22×C33—22no—
69C22×C33—21no—
70C22×C33—21no—
71C22×C33—41no—
72C22×C33—41no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C33×C4—21no—
78C33×C4—22no—
79C33×C4C2×D411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C2×C3×#41C3⋊D3113yescomplemented
83C2×#215—913no—
84C2×#213—913no—
85#930—913no—
86#835—913no—
87#832—913no—
88#830—913no—
89C23×C32—213no—
90C23×C32#67113yes—
91C22×C3⋊D3—613no—
92C32×Q8—213no—
93C32×Q8#67126yescomplemented
94C32×D4—252no—
95C32×D4—213no—
96C32×D4#67126yescomplemented
97C2×C32×C4—252no—
98C2×C32×C4—213no—
99C2×C32×C4—213no—
100C2×C32×C4—213no—
101C2×C32×C4#67113yes—
102C2×C32×C4C22×D3113yes—
103#638—1226no—
104C2×C3⋊Dic3—613no—
105C4⋊(C3⋊D3)—613no—
106C4⋊(C3⋊D3)—613no—
107C3⋊(C3⋊C8)—613no—
108C3⋊(C3⋊C8)—613no—
109#449—271noSylow
110C2×C33—41no—
111C2×C33—42no—
112C2×C33#12011yescharacteristic
113C2×C33#13511yescharacteristic
114C2×C33AGL(1,ℤ/8)11yescharacteristic
115C3⋊(C3⋊D3)—81no—
116C3×#41—226no—
117C3×#41—226no—
118C2×C3×Q8C2×C3⋊D3113yes—
119C2×C3×D4—213no—
120C2×C3×D4C2×C3⋊D3113yes—
121C22×C3×C4—213no—
122C22×C3×C4C2×C3⋊D3113yes—
123C2×#67—1813no—
124C2×D12—913no—
125#215—1826no—
126#213—1826no—
127(C2×D3)⋊C4—1813no—
128C4⋊Dic3—913no—
129C2×C3⋊C8—913no—
130C2×C3⋊C8—913no—
131C22×C32—213no—
132C22×C32—213no—
133C22×C32—213no—
134C22×C32—226no—
135C22×C32—413no—
136C22×C32—413no—
137C22×C32C2×#67113yes—
138C2×C3⋊D3—1213no—
139C2×C3⋊D3—1213no—
140C2×C3⋊D3—1213no—
141C32×C4—213no—
142C32×C4—226no—
143C32×C4C2×#67113yes—
144C32×C4C2×#67113yes—
145C3⋊Dic3—1213no—
146C2×#41C3⋊(C3⋊D3)11yescharacteristic, complemented
147C2×SD16—271no—
148C2×D8—271no—
149#121—271no—
150#103—271no—
151#102—271no—
152C22⋊C8—271no—
153C33#44911yescharacteristic, complemented, Sylow
154C23×C3—213no—
155C23×C3#638113yes—
156C22×D3—1813no—
157C3×Q8—213no—
158C3×Q8#638126yescomplemented
159C3×D4—252no—
160C3×D4—213no—
161C3×D4#638126yescomplemented
162C2×C3×C4—252no—
163C2×C3×C4—213no—
164C2×C3×C4—213no—
165C2×C3×C4—213no—
166C2×C3×C4#638113yes—
167C2×C3×C4C22×C3⋊D3113yes—
168#67—3626no—
169C2×Dic3—1813no—
170D12—1813no—
171D12—1813no—
172C3⋊C8—1813no—
173C3⋊C8—1813no—
174C2×C32—413no—
175C2×C32—426no—
176C2×C32#949113yes—
177C2×C32#950113yes—
178C2×C32#953113yes—
179C3⋊D3—2413no—
180#41—22no—
181#41—22no—
182C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
183C2×D4—21no—
184C2×D4—271no—
185C2×D4—541no—
186C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
187C22×C4—21no—
188C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
189SD16—542no—
190D8—542no—
191C2×C8—271no—
192C2×C8—271no—
193C4⋊C4—271no—
194C22⋊C4—541no—
195C22×C3—213no—
196C22×C3—213no—
197C22×C3—213no—
198C22×C3—226no—
199C22×C3—413no—
200C22×C3—413no—
201C22×C3C2×#638113yes—
202C2×D3—3613no—
203C2×D3—3613no—
204C2×D3—3613no—
205C3×C4—213no—
206C3×C4—226no—
207C3×C4C2×#638113yes—
208C3×C4C2×#638113yes—
209Dic3—3613no—
210C32#5238113yescomplemented
211C23—21no—
212C23—541no—
213C23#631211yescharacteristic
214Q8—21no—
215Q8#631212yescomplemented
216D4—24no—
217D4—21no—
218D4—541no—
219D4—541no—
220D4—1082no—
221D4#631212yescomplemented
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—541no—
227C2×C4#631211yescharacteristic
228C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
229C8—541no—
230C8—541no—
231C2×C3—413no—
232C2×C3—426no—
233C2×C3#64158113yes—
234C2×C3#64159113yes—
235C2×C3#64162113yes—
236D3—7213no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—41no—
242C22—41no—
243C22—1081no—
244C22—1081no—
245C22—1081no—
246C22C2×#631211yescharacteristic
247C4—21no—
248C4—22no—
249C4—1081no—
250C4C2×#631211yescharacteristic
251C4C2×#631211yescharacteristic
252C3#10592937113yescomplemented
253C2—41no—
254C2—42no—
255C2—2161no—
256C2#1172399011yescharacteristic
257C2#1172399111yescharacteristic
258C2#1172399411yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC2×C33×#41C2
socleC22×C33C2×D4
hypercenterC2×#41C3⋊(C3⋊D3)
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C2×#88862C2semidirect
3. C2×#88860C2semidirect
4. #11723971C2semidirect
5. #11722205C2semidirect
6. #11722202C2semidirect
7. #11722200C2semidirect
24. C2×C32×#41D3semidirect
35. C33×Q8D4semidirect
38. C33×D4D4semidirect
82. C2×C3×#41C3⋊D3semidirect
93. C32×Q8#67semidirect
96. C32×D4#67semidirect
146. C2×#41C3⋊(C3⋊D3)semidirect
153. C33#449semidirect
158. C3×Q8#638semidirect
161. C3×D4#638semidirect
210. C32#5238semidirect
215. Q8#6312semidirect
221. D4#6312semidirect
252. C3#10592937semidirect

Maximal subgroups

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

subgroupindex
C2×C33×#412
C2×#888622
C2×#888602
#117239712
#117222052
#117222022
#117222002
#105929373