Finite Group Databasebeta

← #13155248

Subgroups — #13155248

Order 1728.

500 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 70 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#11723647C211yesmaximal, characteristic, complemented
2#11723341C211yesmaximal, characteristic, complemented
3#11722006C212yesmaximal, complemented
4C3×#63549C212yesmaximal, complemented
5C3×#63544C211yesmaximal, characteristic, complemented
6#10591040—31nomaximal
7#10590233—32nomaximal
8C4⋊#1113—21no—
9#89018—21no—
10#88852C2211yescharacteristic, complemented
11C3×#3745C2212yes—
12C3×C8×C3⋊D3C2211yescharacteristic
13C32×Dic12C2211yescharacteristic, complemented
14#88810—21no—
15#88795—21no—
16C32×C3⋊C16C2212yescomplemented
17#64128—31no—
18#63808—32no—
19#63808—61no—
20#63797—31no—
21D3×D24—32no—
22#63626—32no—
23C3×#828—32no—
24C3×#800—32no—
25C3×#798—32no—
26#63549D312yescomplemented
27#63546—32no—
28#63546—32no—
29#63546—62no—
30#63544D311yescharacteristic, complemented
31C2×#1113—41no—
32C4⋊(C3⋊(C3⋊D3))—21no—
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4×C3⋊D3D411yescharacteristic
35C32×Dic6—21no—
36#6295—41no—
37#6292—41no—
38#6289—41no—
39C33×C8C2311yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41#4914—92no—
42#4908—91no—
43#3832—61no—
44#3802—61no—
45D3×D12—62no—
46#3799—62no—
47#3799—62no—
48C32×Q16—31no—
49#3755—61no—
50#3745—31no—
51#3745—32no—
52#3745—61no—
53#3745—62no—
54#3745—121no—
55#3745C2×D312yes—
56C8×C3⋊D3C2×D311yescharacteristic
57C3×Dic12—21no—
58C3×Dic12C2×D312yescomplemented
59C3×D24—32no—
60C3×D24—32no—
61C3×D24—62no—
62C3×D3×C8—32no—
63C3×D3×C8—61no—
64#3717—62no—
65#3717—62no—
66#3714—62no—
67#3713—61no—
68C32×C16—32no—
69C3×C3⋊C16—22no—
70C3×C3⋊C16C2×D312yescomplemented
71C3×C3⋊C16C2×D312yescomplemented
72C2×C3⋊(C3⋊D3)—41no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3D811yescharacteristic
75#1113—82no—
76C33×C4C2×D411yescharacteristic
77C3×C3⋊Dic3D811yescharacteristic
78C32×Dic3—41no—
79#919—92no—
80#919—181no—
81D3×D8—92no—
82#904—91no—
83C2×D24—91no—
84C3×SD32—92no—
85C3×M32—91no—
86#828—32no—
87#826—92no—
88#800—32no—
89D48—92no—
90D48—92no—
91D48—182no—
92#798—32no—
93C2×D32—122no—
94C2×D32—121no—
95C32×Q8—61no—
96C4⋊(C3⋊D3)—22no—
97C4⋊(C3⋊D3)—61no—
98C4⋊(C3⋊D3)—62no—
99C4⋊(C3⋊D3)—62no—
100C4⋊(C3⋊D3)—122no—
101C4⋊(C3⋊D3)—122no—
102C4⋊(C3⋊D3)—121no—
103C4×C3⋊D3—121no—
104C4×C3⋊D3D1211yescharacteristic
105C3×D12—62no—
106C3×D12—62no—
107C3×D12—122no—
108C3×C4×D3—32no—
109C3×C4×D3—61no—
110C3×Dic6—22no—
111C3×Dic6—22no—
112#626—122no—
113#626—122no—
114#626—122no—
115#625—121no—
116#624—122no—
117#624—122no—
118C32×C8—22no—
119C32×C8—21no—
120C32×C8—41no—
121C32×C8C22×D311yescharacteristic
122C32×C8C22×D312yes—
123C3⋊(C3⋊C8)D1211yescharacteristic
124C3×C3⋊C8—32no—
125C3×C3⋊C8—61no—
126#509—271noSylow
127C2×C33C2×D811yescharacteristic
128C3⋊(C3⋊D3)—81no—
129C3×C3⋊D3—21no—
130C3×C3⋊D3—82no—
131#239—182no—
132#239—182no—
133D3×D4—182no—
134#235—181no—
135C2×D12—181no—
136C3×Q16—32no—
137C3×Q16—61no—
138C3×D8—92no—
139C2×C3×C8—91no—
140#215—182no—
141#215—182no—
142#213—182no—
143Dic12D3211yescharacteristic, complemented
144D24—32no—
145D24—32no—
146D24—62no—
147D24—91no—
148D24—92no—
149D24—181no—
150D24—182no—
151D24—181no—
152D24—361no—
153#204—181no—
154D3×C8—32no—
155D3×C8—61no—
156C3×C16—32no—
157C3×C16—32no—
158C3×C16—62no—
159C3⋊C16D3212yescomplemented
160C2×C3⋊D3—42no—
161C2×C3⋊D3—121no—
162C2×C3⋊D3—122no—
163C2×C3⋊D3—122no—
164C2×C3⋊D3—242no—
165C2×C3⋊D3—242no—
166C2×C3⋊D3D2411yescharacteristic
167C2×C3×D3—32no—
168C2×C3×D3—61no—
169C2×C3×D3—122no—
170C2×C3×D3—122no—
171C2×C3×D3—242no—
172D32—244no—
173D32—242no—
174C32×C4—22no—
175C32×C4—21no—
176C32×C4—41no—
177C32×C4—121no—
178C32×C4C2×D1211yescharacteristic
179C32×C4D3×D412yes—
180C3⋊Dic3D2411yescharacteristic
181C3×Dic3—32no—
182C3×Dic3—42no—
183C3×Dic3—42no—
184C3×Dic3—61no—
185#135—271no—
186C2×D8—271no—
187SD32—92no—
188D16—272no—
189M32—91no—
190C33#50911yescharacteristic, complemented, Sylow
191C22×D3—362no—
192C22×D3—361no—
193C3×Q8—62no—
194C3×Q8—62no—
195C3×D4—182no—
196C2×C3×C4—91no—
197#67—362no—
198#67—361no—
199D12—62no—
200D12—62no—
201D12—122no—
202D12—182no—
203D12—182no—
204D12—181no—
205D12—362no—
206D12—362no—
207D12—361no—
208D12—362no—
209D12—362no—
210C4×D3—32no—
211C4×D3—61no—
212C4×D3—362no—
213C4×D3—361no—
214C4×D3—362no—
215Dic6—21no—
216C3×C8—22no—
217C3×C8—21no—
218C3×C8—41no—
219C3×C8—91no—
220C3×C8C2×D3212yes—
221C3×C8C2×D3211yescharacteristic
222C3⋊C8—32no—
223C3⋊C8—61no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32—41no—
227C2×C32C2×D2411yescharacteristic
228C2×C32D3×D812yes—
229C3⋊D3—21no—
230C3⋊D3—82no—
231C3⋊D3—241no—
232C3⋊D3—242no—
233C3⋊D3—242no—
234C3⋊D3—482no—
235C3⋊D3—482no—
236C3×D3—62no—
237C3×D3—121no—
238C3×D3—242no—
239C3×D3—242no—
240C3×D3—482no—
241#41—541no—
242C2×D4—541no—
243Q16—31no—
244SD16—541no—
245D8—92no—
246D8—271no—
247D8—541no—
248C2×C8—91no—
249C16—32no—
250C22×C3—91no—
251C22×C3—362no—
252C2×D3—32no—
253C2×D3—61no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—242no—
257C2×D3—362no—
258C2×D3—362no—
259C2×D3—361no—
260C2×D3—722no—
261C2×D3—722no—
262C2×D3—724no—
263C2×D3—722no—
264C3×C4—21no—
265C3×C4—22no—
266C3×C4—41no—
267C3×C4—91no—
268C3×C4—122no—
269C3×C4—122no—
270C3×C4D3×D1212yes—
271C3×C4#381111yescharacteristic
272Dic3—32no—
273Dic3—41no—
274Dic3—61no—
275C32—22no—
276C32—21no—
277C32—41no—
278C32#490811yescharacteristic, complemented
279C32#491412yescomplemented
280C23—1081no—
281Q8—61no—
282D4—182no—
283D4—541no—
284D4—1081no—
285D4—1081no—
286C2×C4—91no—
287C2×C4—1081no—
288C8—91no—
289C8D3311yescharacteristic
290C2×C3—21no—
291C2×C3—22no—
292C2×C3—41no—
293C2×C3—181no—
294C2×C3—722no—
295C2×C3D3×D2412yes—
296C2×C3#6392611yescharacteristic
297D3—62no—
298D3—121no—
299D3—242no—
300D3—242no—
301D3—482no—
302D3—722no—
303D3—722no—
304D3—721no—
305D3—1442no—
306D3—1442no—
307C22—91no—
308C22—362no—
309C22—1081no—
310C22—2162no—
311C4—91no—
312C4—121no—
313C4#8895511yescharacteristic
314C3—21no—
315C3—22no—
316C3—41no—
317C3#1059023312yescomplemented
318C3#1059104011yescharacteristic, complemented
319C2—181no—
320C2—722no—
321C2—2161no—
322C2#1172283311yescharacteristic
subgroupG/H
centerC2#11722833
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11723647C2semidirect
2. #11723341C2semidirect
3. #11722006C2semidirect
4. C3×#63549C2semidirect
5. C3×#63544C2semidirect
10. #88852C22semidirect
13. C32×Dic12C22semidirect
16. C32×C3⋊C16C22semidirect
26. #63549D3semidirect
30. #63544D3semidirect
58. C3×Dic12C2×D3semidirect
70. C3×C3⋊C16C2×D3semidirect
71. C3×C3⋊C16C2×D3semidirect
143. Dic12D32semidirect
159. C3⋊C16D32semidirect
190. C33#509semidirect
278. C32#4908semidirect
279. C32#4914semidirect
317. C3#10590233semidirect
318. C3#10591040semidirect

Maximal subgroups

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

subgroupindex
#117236472
#117233412
#117220062
C3×#635492
C3×#635442
#105910403
#105902333