Finite Group Databasebeta

← #13180990

Subgroups — #13180990

Order 1728.

924 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 170 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
1D3×#3832C211yesmaximal, characteristic
2#11724517C211yesmaximal, characteristic, complemented
3C3×#64289C211yesmaximal, characteristic, complemented
4#11724255C213yesmaximal, complemented
5#11724254C213yesmaximal, complemented
6#11724253C213yesmaximal, complemented
7#11724252C213yesmaximal, complemented
8#10596644—31nomaximal
9#89080C411yescharacteristic
10C3×#3832C2211yescharacteristic
11C32×D3×Q8C411yescharacteristic
12D3×C4⋊(C3⋊D3)C413yes—
13C4×D3×C3⋊D3C2213yes—
14#89018C2213yes—
15#89017C413yes—
16#88989C2213yescomplemented
17#88988C2213yescomplemented
18C3×#3788C2213yescomplemented
19C3×#3787C2213yescomplemented
20#88926C2213yescomplemented
21#88926C2213yescomplemented
22#88925C2211yescharacteristic
23#88925C2213yes—
24#88924C2216yescomplemented
25C2×#3832—31no—
26D3×#239—62no—
27#64289—31no—
28#64289—33no—
29#64289—33no—
30#64289D311yescharacteristic, complemented
31#64283—33no—
32#64283—33no—
33C2×D3×C3⋊D3C2×C413yes—
34C33×Q8C2×C411yescharacteristic
35C4⋊(C3⋊(C3⋊D3))C2×C413yes—
36C4×C3⋊(C3⋊D3)C2×C413yes—
37C3×C4⋊(C3⋊D3)C2×C413yes—
38C3×C4×C3⋊D3C2313yes—
39C32×C4×D3C2×C413yes—
40C32×Dic6C2×C413yes—
41#6292C2×C416yes—
42#6289C2313yes—
43Dic3×C3⋊D3C2×C413yes—
44D3×C3⋊Dic3C2311yescharacteristic
45C3⋊#622C2313yes—
46C3⋊#622C2311yescharacteristic
47C3×#622C2313yes—
48C3×#622C2311yescharacteristic
49#5750—91nomaximal
50C2×C32×Q8—31no—
51#3832—31no—
52#3832—33no—
53#3832—33no—
54#3832—62no—
55#3832—122no—
56#3832C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)—33no—
58C2×C4×C3⋊D3—33no—
59C3×#239—62no—
60C3×D3×Q8—22no—
61D3×D12—66no—
62C4×D32—66no—
63#3799—66no—
64#3796—66no—
65#3792—33no—
66#3792—31no—
67#3788—33no—
68#3788—33no—
69#3788—33no—
70#3788C2×D313yescomplemented
71#3787—33no—
72#3787—36no—
73#3787C2×D313yescomplemented
74C2×C3⋊(C3⋊D3)C22×C413yes—
75C2×C3×C3⋊D3C22×C413yes—
76C2×C32×D3C22×C411yescharacteristic
77D3×C3⋊D3—43no—
78C33×C4C22×C413yes—
79C3⋊(C3⋊Dic3)C22×C411yescharacteristic
80C3×C3⋊Dic3C2411yescharacteristic
81C32×Dic3C22×C413yes—
82D3×#41—91no—
83C2×#239—182no—
84C3×#131—91no—
85#948—91no—
86#907—93no—
87D3×M16—93no—
88#901—93no—
89C2×ΓL(1,𝔽25)—93no—
90C22×C3⋊D3—33no—
91C2×D32—66no—
92C32×Q8—22no—
93C32×Q8—33no—
94C32×Q8—42no—
95C32×Q8C4×D311yescharacteristic
96C2×C32×C4—33no—
97C2×C3⋊Dic3—31no—
98C4⋊(C3⋊D3)—36no—
99C4⋊(C3⋊D3)—33no—
100C4⋊(C3⋊D3)—66no—
101C4⋊(C3⋊D3)—126no—
102C4⋊(C3⋊D3)C4×D313yes—
103C4×C3⋊D3—33no—
104C4×C3⋊D3—33no—
105C4×C3⋊D3—33no—
106C4×C3⋊D3—66no—
107C4×C3⋊D3—126no—
108C4×C3⋊D3C22×D313yes—
109C3×D12—66no—
110C3×C4×D3—26no—
111C3×C4×D3—66no—
112C3×Dic6—26no—
113#626—612no—
114#624—66no—
115D3×Dic3—62no—
116D3×Dic3—66no—
117#622—33no—
118#622—31no—
119#622C22×D313yes—
120#622C22×D311yescharacteristic
121#568—271noSylow
122C2×C33C23×C411yescharacteristic
123C3⋊(C3⋊D3)—23no—
124C3×C3⋊D3—23no—
125C32×D3—21no—
126C3×#41—91no—
127C2×C3×Q8—62no—
128#239—62no—
129#239—91no—
130#239—186no—
131#239—186no—
132#239—182no—
133#239—362no—
134D3×Q8#15711yescharacteristic
135#237—93no—
136D3×D4—93no—
137#235—93no—
138C2×D12—186no—
139C2×C4×D3—93no—
140C2×C4×D3—186no—
141C3×M16—93no—
142C2×C3×C8—93no—
143#208—93no—
144C2×C3⋊C8—93no—
145ΓL(1,𝔽25)—91no—
146ΓL(1,𝔽25)—93no—
147ΓL(1,𝔽25)—93no—
148ΓL(1,𝔽25)—93no—
149D3×C8—96no—
150C22×C32—31no—
151C2×C3⋊D3—33no—
152C2×C3⋊D3—66no—
153C2×C3⋊D3—126no—
154C2×C3⋊D3—123no—
155C2×C3⋊D3C2×C4×D313yes—
156C2×C3×D3—22no—
157C2×C3×D3—66no—
158D32—1212no—
159C32×C4—26no—
160C32×C4—33no—
161C32×C4—46no—
162C32×C4C2×C4×D313yes—
163C3⋊Dic3—31no—
164C3⋊Dic3—62no—
165C3⋊Dic3—122no—
166C3⋊Dic3C23×D311yescharacteristic
167C3×Dic3—26no—
168C3×Dic3—62no—
169C2×#41—271no—
170#131—91no—
171#131—273no—
172#131—273no—
173#131—271no—
174C2×M16—273no—
175C2×M16—273no—
176C33#56811yescharacteristic, complemented, Sylow
177C22×D3—93no—
178C22×D3—186no—
179C3×Q8—22no—
180C3×Q8—42no—
181C3×Q8—66no—
182C3×Q8C2×#15711yescharacteristic
183C3×D4—93no—
184C2×C3×C4—66no—
185C2×C3×C4—93no—
186#67—96no—
187C2×Dic3—93no—
188C2×Dic3—182no—
189D12—66no—
190D12—93no—
191D12—186no—
192D12—1812no—
193D12—366no—
194C4×D3—66no—
195C4×D3—91no—
196C4×D3—93no—
197C4×D3—93no—
198C4×D3—186no—
199C4×D3—186no—
200C4×D3—186no—
201C4×D3—366no—
202C4×D3C2×#15713yes—
203Dic6C2×#15713yes—
204C3×C8—93no—
205C3×C8—91no—
206C3⋊C8—93no—
207C3⋊C8—91no—
208C2×C32—22no—
209C2×C32—42no—
210C2×C32—61no—
211C2×C32C22×C4×D311yescharacteristic
212C3⋊D3—23no—
213C3⋊D3—63no—
214C3⋊D3—126no—
215C3⋊D3—246no—
216C3×D3—24no—
217C3×D3—126no—
218#41—91no—
219#41—273no—
220#41—273no—
221#41—271no—
222C2×Q8—31no—
223C2×D4—273no—
224C22×C4—273no—
225M16—93no—
226M16—271no—
227M16—273no—
228M16—273no—
229M16—273no—
230M16—273no—
231C2×C8—93no—
232C2×C8—276no—
233C2×C8—273no—
234C22×C3—62no—
235C22×C3—93no—
236C2×D3—66no—
237C2×D3—93no—
238C2×D3—186no—
239C2×D3—366no—
240C2×D3—363no—
241C2×D3—3612no—
242C2×D3C22×#15711yescharacteristic
243C3×C4—26no—
244C3×C4—46no—
245C3×C4—66no—
246C3×C4—91no—
247C3×C4C22×#15713yes—
248Dic3—62no—
249Dic3—91no—
250Dic3—182no—
251Dic3—362no—
252Dic3C22×#15713yes—
253C32—22no—
254C32—42no—
255C32#575011yescharacteristic, complemented
256C23—273no—
257Q8—33no—
258Q8D3×#15711yescharacteristic
259D4—93no—
260D4—276no—
261D4—273no—
262C2×C4—33no—
263C2×C4—93no—
264C2×C4—273no—
265C2×C4—273no—
266C2×C4—271no—
267C2×C4—273no—
268C8—91no—
269C8—93no—
270C8—273no—
271C8—271no—
272C2×C3—22no—
273C2×C3—42no—
274C2×C3—64no—
275C2×C3—183no—
276C2×C3C23×#15711yescharacteristic
277D3—21no—
278D3—126no—
279D3—183no—
280D3—366no—
281D3—726no—
282C22—31no—
283C22—93no—
284C22—273no—
285C22—1083no—
286C4—33no—
287C4—91no—
288C4—271no—
289C4C2×D3×#15713yes—
290C3—22no—
291C3—42no—
292C3#1059664411yescharacteristic, complemented
293C2—61no—
294C2—183no—
295C2—543no—
296C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC33×Q8C2×C4
prime radicalD3×#3832C2
cosocleC3×C3⋊Dic3C24
hypercenterQ8D3×#157
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
2. #11724517C2semidirect
3. C3×#64289C2semidirect
4. #11724255C2semidirect
5. #11724254C2semidirect
6. #11724253C2semidirect
7. #11724252C2semidirect
16. #88989C22semidirect
17. #88988C22semidirect
18. C3×#3788C22semidirect
19. C3×#3787C22semidirect
20. #88926C22semidirect
21. #88926C22semidirect
24. #88924C22semidirect
30. #64289D3semidirect
70. #3788C2×D3semidirect
73. #3787C2×D3semidirect
176. C33#568semidirect
255. C32#5750semidirect
292. C3#10596644semidirect

Maximal subgroups

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

subgroupindex
D3×#38322
#117245172
C3×#642892
#117242552
#117242542
#117242532
#117242522
#105966443
#57509