Finite Group Databasebeta

← #10596177

Subgroups — #10596177

Order 576.

702 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#126C211yesmaximal, characteristic, complemented
2#64105C211yesmaximal, characteristic, complemented
3#64103C211yesmaximal, characteristic, complemented
4#64102C211yesmaximal, characteristic, complemented
5#64101C211yesmaximal, characteristic, complemented
6#64098C211yesmaximal, characteristic, complemented
7#64096C211yesmaximal, characteristic, complemented
8#64094C211yesmaximal, characteristic, complemented
9#64093C211yesmaximal, characteristic
10#64092C211yesmaximal, characteristic, complemented
11#64090C211yesmaximal, characteristic, complemented
12#64090C211yesmaximal, characteristic, complemented
13#64086C211yesmaximal, characteristic, complemented
14#64085C211yesmaximal, characteristic, complemented
15C42×C3⋊D3C211yesmaximal, characteristic, complemented
16#5706—34nomaximal
17C2×#638C2211yescharacteristic
18C2×#638C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic
23C2×C4×C3⋊D3C2211yescharacteristic
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×#634C2211yescharacteristic
26C32×C4⋊C4C2211yescharacteristic
27C32×C4⋊C4C2211yescharacteristic
28C32×C4⋊C4C2211yescharacteristic
29C32×C22⋊C4C2211yescharacteristic
30C32×C22⋊C4C2211yescharacteristic
31C32×C22⋊C4C2211yescharacteristic
32C32×C42C2211yescharacteristic
33#3757C2211yescharacteristic
34#3752C2211yescharacteristic
35#3752C2211yescharacteristic
36#3752C2211yescharacteristic
37#3752C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40C4⋊(C3⋊Dic3)C2211yescharacteristic
41#3750C2211yescharacteristic
42#3750C2211yescharacteristic
43#3750C2211yescharacteristic
44#3750C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47C4×C3⋊Dic3C2211yescharacteristic
48C4×C3⋊Dic3C2211yescharacteristic
49C4×C3⋊Dic3C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#126D314yescomplemented
53#896—34no—
54#894—34no—
55#893—34no—
56#892—34no—
57#889—34no—
58#887—34no—
59#885—34no—
60#884—34no—
61#883—34no—
62#67⋊C4—34no—
63#67⋊C4—34no—
64#877—34no—
65#876—34no—
66C42×D3—34no—
67C23×C32C2311yescharacteristic
68C22×C3⋊D3C2311yescharacteristic
69C22×C3⋊D3C2311yescharacteristic
70C2×C32×C4C2311yescharacteristic
71C2×C32×C4C2311yescharacteristic
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76#638—22no—
77#638—22no—
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3C2311yescharacteristic
80C2×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C4⋊(C3⋊D3)—22no—
86C4×C3⋊D3—22no—
87C4×C3⋊D3—22no—
88C4×C3⋊D3—22no—
89C4×C3⋊D3—22no—
90#634—22no—
91#529—91noSylow
92C2×#67—34no—
93C2×#67—34no—
94C22×Dic3—34no—
95C2×D12—34no—
96C2×C4×D3—34no—
97C2×C4×D3—34no—
98C2×C4×D3—34no—
99C2×C4×D3—34no—
100C2×Dic6—34no—
101C3×C4⋊C4C2×D314yes—
102C3×C4⋊C4C2×D314yes—
103C3×C4⋊C4C2×D314yes—
104C3×C22⋊C4C2×D314yes—
105C3×C22⋊C4C2×D314yes—
106C3×C22⋊C4C2×D314yes—
107C3×C42C2×D314yes—
108#217—34no—
109(C2×D3)⋊C4—34no—
110(C2×D3)⋊C4—34no—
111(C2×D3)⋊C4—34no—
112(C2×D3)⋊C4—34no—
113(C2×D3)⋊C4—34no—
114(C2×D3)⋊C4—34no—
115C4⋊Dic3—34no—
116Dic3⋊C4—34no—
117Dic3⋊C4—34no—
118Dic3⋊C4—34no—
119Dic3⋊C4—34no—
120Dic3⋊C4—34no—
121Dic3⋊C4—34no—
122C4×Dic3—34no—
123C4×Dic3—34no—
124C4×Dic3—34no—
125C4×Dic3—34no—
126C4×Dic3—34no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32C2411yescharacteristic
131C2×C3⋊D3—21no—
132C2×C3⋊D3—21no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3#4112yes—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—21no—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3⋊Dic3—21no—
146C3⋊Dic3—21no—
147C3⋊Dic3#4112yes—
148C3⋊Dic3#4112yes—
149#126—91no—
150#126C3⋊D311yescharacteristic, complemented
151#125—91no—
152#124—91no—
153#123—91no—
154#123—91no—
155C22⋊Q8—91no—
156#121—91no—
157C4×Q8—91no—
158C4×D4—91no—
159C4×D4—91no—
160C4×D4—91no—
161#117—91no—
162#117—91no—
163C2×C42—91no—
164C23×C3C22×D314yes—
165C22×D3—34no—
166C22×D3—34no—
167C2×C3×C4C22×D314yes—
168C2×C3×C4C22×D314yes—
169C2×C3×C4C22×D314yes—
170C2×C3×C4C22×D314yes—
171C2×C3×C4C22×D314yes—
172C2×C3×C4C22×D314yes—
173#67—68no—
174#67—68no—
175C2×Dic3—34no—
176C2×Dic3—34no—
177C2×Dic3—34no—
178C2×Dic3—34no—
179C2×Dic3—34no—
180C2×Dic3—34no—
181C2×Dic3—68no—
182D12—68no—
183C4×D3—68no—
184C4×D3—68no—
185C4×D3—68no—
186C4×D3—68no—
187Dic6—68no—
188C2×C32—41no—
189C2×C32C2×#4111yescharacteristic
190C2×C32C2×#4111yescharacteristic
191C2×C32C2×#4111yescharacteristic
192C3⋊D3—22no—
193C3⋊D3—41no—
194C2×Q8—91no—
195C2×D4—91no—
196C2×D4—91no—
197C2×D4—91no—
198C22×C4—91no—
199C22×C4—91no—
200C22×C4—91no—
201C22×C4—91no—
202C22×C4—91no—
203C4⋊C4—91no—
204C4⋊C4—91no—
205C4⋊C4—91no—
206C4⋊C4—91no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C4⋊C4—91no—
210C4⋊C4C2×C3⋊D311yescharacteristic
211C4⋊C4C2×C3⋊D311yescharacteristic
212C4⋊C4C2×C3⋊D311yescharacteristic
213C22⋊C4—91no—
214C22⋊C4—91no—
215C22⋊C4—91no—
216C22⋊C4—91no—
217C22⋊C4—91no—
218C22⋊C4—91no—
219C22⋊C4—91no—
220C22⋊C4C2×C3⋊D311yescharacteristic
221C22⋊C4C2×C3⋊D311yescharacteristic
222C22⋊C4C2×C3⋊D311yescharacteristic
223C42—91no—
224C42—91no—
225C42—91no—
226C42—91no—
227C42—91no—
228C42C2×C3⋊D311yescharacteristic
229C22×C3—24no—
230C22×C3—24no—
231C22×C3—24no—
232C22×C3C23×D314yes—
233C2×D3—38no—
234C2×D3—64no—
235C2×D3—64no—
236C2×D3—64no—
237C2×D3—64no—
238C2×D3—64no—
239C3×C4—24no—
240C3×C4—24no—
241C3×C4—24no—
242C3×C4—24no—
243C3×C4—24no—
244C3×C4—24no—
245Dic3—38no—
246Dic3—38no—
247Dic3—64no—
248Dic3—64no—
249Dic3—64no—
250Dic3—64no—
251C32#52911yescharacteristic, complemented, Sylow
252C23—91no—
253C23—91no—
254C23C22×C3⋊D311yescharacteristic
255Q8—182no—
256D4—182no—
257D4—182no—
258D4—182no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—182no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4C22×C3⋊D311yescharacteristic
271C2×C4C22×C3⋊D311yescharacteristic
272C2×C4C22×C3⋊D311yescharacteristic
273C2×C4C22×C3⋊D311yescharacteristic
274C2×C4C22×C3⋊D311yescharacteristic
275C2×C4C22×C3⋊D311yescharacteristic
276C2×C3—44no—
277C2×C3D3×#4114yes—
278C2×C3D3×#4114yes—
279C2×C3D3×#4114yes—
280D3—68no—
281D3—124no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—92no—
286C22—181no—
287C22—181no—
288C22—181no—
289C22—181no—
290C22—181no—
291C22C23×C3⋊D311yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—92no—
299C4—92no—
300C4—181no—
301C4—181no—
302C4—181no—
303C4—181no—
304C3#570614yescomplemented
305C2—41no—
306C2—182no—
307C2—361no—
308C2#41×C3⋊D311yescharacteristic
309C2#41×C3⋊D311yescharacteristic
310C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
prime radical#64093C2
hypercenter#126C3⋊D3
hypocenterC32#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C32×#126C2semidirect
2. #64105C2semidirect
3. #64103C2semidirect
4. #64102C2semidirect
5. #64101C2semidirect
6. #64098C2semidirect
7. #64096C2semidirect
8. #64094C2semidirect
10. #64092C2semidirect
11. #64090C2semidirect
12. #64090C2semidirect
13. #64086C2semidirect
14. #64085C2semidirect
15. C42×C3⋊D3C2semidirect
52. C3×#126D3semidirect
150. #126C3⋊D3semidirect
251. C32#529semidirect
304. C3#5706semidirect

Maximal subgroups

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

subgroupindex
C32×#1262
#641052
#641032
#641022
#641012
#640982
#640962
#640942
#640932
#640922
#640902
#640862
#640852
C42×C3⋊D32
#57063