Finite Group Databasebeta

← #10594706

Subgroups — #10594706

Order 576.

547 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#235C211yesmaximal, characteristic
2#64116C211yesmaximal, characteristic, complemented
3C3×#907C211yesmaximal, characteristic, complemented
4C3×D3×M16C211yesmaximal, characteristic, complemented
5C2×#3711C211yesmaximal, characteristic, complemented
6#63817C211yesmaximal, characteristic, complemented
7#63815C211yesmaximal, characteristic, complemented
8#63804C212yesmaximal, complemented
9#63803C212yesmaximal, complemented
10#63791C212yesmaximal, complemented
11D3×ΓL(1,𝔽25)C212yesmaximal, complemented
12#5750—31nomaximal
13#5745—31nomaximal
14#3828C2211yescharacteristic
15C3×#235C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17D3×#67C412yes—
18#3804C412yes—
19D3×D12C411yescharacteristic
20C4×D32C2212yes—
21#3799C411yescharacteristic
22#3798C2212yes—
23#3795C411yescharacteristic
24D3×Dic6C411yescharacteristic
25C32×M16C2211yescharacteristic, complemented
26C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
27#3743C2212yescomplemented
28C8×C3⋊D3C2212yescomplemented
29C3×#208C2211yescharacteristic, complemented
30C2×C3×C3⋊C8C2211yescharacteristic, complemented
31C3×ΓL(1,𝔽25)C2212yes—
32C3×ΓL(1,𝔽25)C2212yescomplemented
33C3×D3×C8C2212yescomplemented
34C3×D3×C8C2212yescomplemented
35#3712C2212yescomplemented
36#3711C2211yescharacteristic
37#3711C2211yescharacteristic
38#3711C2212yescomplemented
39#3711C2211yescharacteristic, complemented
40#3711C2211yescharacteristic, complemented
41#3710C2212yescomplemented
42D3×C3⋊C8C2212yescomplemented
43D3×#41—31no—
44C2×#235—31no—
45C3×#131—31no—
46C2×C3×M16—31no—
47#948—31no—
48C2×#208—31no—
49#907—31no—
50#907—31no—
51#907—31no—
52#907—31no—
53#907—32no—
54#907—61no—
55#907D311yescharacteristic, complemented
56D3×M16—32no—
57D3×M16—32no—
58D3×M16—32no—
59D3×M16D311yescharacteristic, complemented
60#901—31no—
61#901—32no—
62#901—32no—
63#901—32no—
64C2×ΓL(1,𝔽25)—31no—
65C2×ΓL(1,𝔽25)—32no—
66C22×C3×D3C2×C411yescharacteristic
67C2×D32C2×C412yes—
68C2×C32×C4C2311yescharacteristic
69#638C2×C412yes—
70C4⋊(C3⋊D3)C2×C411yescharacteristic
71C4×C3⋊D3C2312yes—
72#634C2×C411yescharacteristic
73C3×#67C2×C412yes—
74C2×C3×Dic3C2×C411yescharacteristic
75C3×D12C2×C411yescharacteristic
76C3×C4×D3C2×C411yescharacteristic
77C3×C4×D3C2×C411yescharacteristic
78C3×C4×D3C2312yes—
79C3×C4×D3C2311yescharacteristic
80C3×C4×D3C2311yescharacteristic
81C3×Dic6C2×C411yescharacteristic
82#627C2×C412yes—
83#626C2×C412yes—
84#626C2×C412yes—
85#625C2×C412yes—
86#624C2×C412yes—
87D3×Dic3C2×C412yes—
88D3×Dic3C2×C412yes—
89C32×C8C2312yes—
90C3⋊(C3⋊C8)C2311yescharacteristic
91C3⋊(C3⋊C8)C2311yescharacteristic
92C3×C3⋊C8C2312yes—
93C3×C3⋊C8C2311yescharacteristic
94C3×C3⋊C8C2311yescharacteristic
95#568—91noSylow
96C3×#41—31no—
97C22×C3×C4—31no—
98C2×#67—32no—
99#239—31no—
100D3×Q8—31no—
101#237—32no—
102#237—31no—
103D3×D4—32no—
104D3×D4—31no—
105#235—31no—
106#235—31no—
107#235—32no—
108#235—31no—
109#235—32no—
110#235—32no—
111#235—31no—
112#235—61no—
113#235C2×D311yescharacteristic
114C2×D12—31no—
115C2×C4×D3—32no—
116C2×C4×D3—32no—
117C2×C4×D3C2×D311yescharacteristic
118C2×Dic6—31no—
119C3×M16—21no—
120C3×M16—31no—
121C3×M16—32no—
122C3×M16—32no—
123C3×M16C2×D311yescharacteristic, complemented
124C3×M16C2×D311yescharacteristic, complemented
125C2×C3×C8—31no—
126C2×C3×C8—32no—
127C2×C3×C8—32no—
128#208—31no—
129#208—31no—
130#208—31no—
131#208—31no—
132#208—32no—
133#208C2×D311yescharacteristic, complemented
134C2×C3⋊C8—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—32no—
137C2×C3⋊C8—61no—
138C2×C3⋊C8C2×D311yescharacteristic, complemented
139ΓL(1,𝔽25)—31no—
140ΓL(1,𝔽25)—31no—
141ΓL(1,𝔽25)—31no—
142ΓL(1,𝔽25)—31no—
143ΓL(1,𝔽25)—32no—
144ΓL(1,𝔽25)—32no—
145ΓL(1,𝔽25)—32no—
146ΓL(1,𝔽25)—32no—
147ΓL(1,𝔽25)—32no—
148ΓL(1,𝔽25)—62no—
149ΓL(1,𝔽25)C2×D312yes—
150ΓL(1,𝔽25)C2×D312yescomplemented
151D3×C8—32no—
152D3×C8—32no—
153D3×C8—32no—
154D3×C8—32no—
155D3×C8—32no—
156D3×C8—62no—
157D3×C8C2×D312yescomplemented
158D3×C8C2×D312yescomplemented
159C22×C32C22×C411yescharacteristic
160C2×C3⋊D3C22×C412yes—
161C2×C3×D3—41no—
162C2×C3×D3C22×C411yescharacteristic
163C2×C3×D3C22×C412yes—
164C2×C3×D3C22×C411yescharacteristic
165D32—42no—
166C32×C4C22×C411yescharacteristic
167C32×C4C2411yescharacteristic
168C3⋊Dic3C22×C412yes—
169C3×Dic3C22×C411yescharacteristic
170C3×Dic3C22×C412yes—
171C3×Dic3C22×C411yescharacteristic
172C2×#41—91no—
173#131—31no—
174#131—92no—
175#131—91no—
176#131—92no—
177#131—91no—
178#131—91no—
179C2×M16—31no—
180C2×M16—92no—
181C2×M16—91no—
182C2×M16—92no—
183C23×C3—31no—
184C22×D3—32no—
185C22×D3—32no—
186C22×D3C4×D311yescharacteristic
187C3×Q8—31no—
188C3×D4—32no—
189C3×D4—31no—
190C2×C3×C4—21no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4C22×D311yescharacteristic
198C2×C3×C4C22×D311yescharacteristic
199#67—32no—
200#67—32no—
201#67—32no—
202#67—32no—
203#67—32no—
204#67—32no—
205#67—62no—
206#67C4×D312yes—
207C2×Dic3—32no—
208C2×Dic3—32no—
209C2×Dic3C4×D311yescharacteristic
210D12—32no—
211D12—32no—
212D12—31no—
213D12—31no—
214D12—61no—
215D12C4×D311yescharacteristic
216C4×D3—32no—
217C4×D3—32no—
218C4×D3—32no—
219C4×D3—32no—
220C4×D3—32no—
221C4×D3—32no—
222C4×D3—62no—
223C4×D3C4×D311yescharacteristic
224C4×D3C4×D311yescharacteristic
225C4×D3C22×D312yes—
226C4×D3C22×D311yescharacteristic
227C4×D3C22×D311yescharacteristic
228Dic6—31no—
229Dic6—31no—
230Dic6—32no—
231Dic6—32no—
232Dic6—61no—
233Dic6C4×D311yescharacteristic
234C3×C8—22no—
235C3×C8—32no—
236C3×C8—31no—
237C3×C8—31no—
238C3×C8C22×D312yes—
239C3×C8C22×D312yes—
240C3⋊C8—31no—
241C3⋊C8—31no—
242C3⋊C8—31no—
243C3⋊C8—31no—
244C3⋊C8—61no—
245C3⋊C8—61no—
246C3⋊C8C22×D311yescharacteristic
247C3⋊C8C22×D312yes—
248C3⋊C8C22×D311yescharacteristic
249C2×C32—21no—
250C2×C32C23×C411yescharacteristic
251C3⋊D3—22no—
252C3×D3—21no—
253C3×D3—22no—
254C3×D3—21no—
255#41—31no—
256#41—92no—
257#41—92no—
258#41—91no—
259#41—91no—
260#41—91no—
261C2×Q8—91no—
262C2×D4—91no—
263C2×D4—92no—
264C22×C4—31no—
265C22×C4—92no—
266M16—31no—
267M16—32no—
268M16—32no—
269M16—92no—
270M16—91no—
271M16—91no—
272M16—91no—
273M16—91no—
274M16—92no—
275M16—92no—
276M16D3211yescharacteristic, complemented
277C2×C8—31no—
278C2×C8—32no—
279C2×C8—32no—
280C2×C8—92no—
281C2×C8—92no—
282C2×C8—91no—
283C2×C8—92no—
284C22×C3—21no—
285C22×C3—31no—
286C22×C3—32no—
287C22×C3—31no—
288C22×C3—121no—
289C22×C3C2×C4×D311yescharacteristic
290C22×C3C2×C4×D311yescharacteristic
291C2×D3—32no—
292C2×D3—32no—
293C2×D3—41no—
294C2×D3—62no—
295C2×D3—122no—
296C2×D3—122no—
297C2×D3C2×C4×D311yescharacteristic
298C2×D3C2×C4×D312yes—
299C2×D3C2×C4×D311yescharacteristic
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—31no—
303C3×C4—32no—
304C3×C4—31no—
305C3×C4C2×C4×D311yescharacteristic
306C3×C4C2×C4×D311yescharacteristic
307C3×C4C23×D311yescharacteristic
308C3×C4C23×D311yescharacteristic
309Dic3—32no—
310Dic3—32no—
311Dic3—62no—
312Dic3C2×C4×D311yescharacteristic
313Dic3C2×C4×D312yes—
314Dic3C2×C4×D311yescharacteristic
315C32#56811yescharacteristic, complemented, Sylow
316C23—31no—
317C23—92no—
318Q8—31no—
319Q8—92no—
320Q8—91no—
321D4—32no—
322D4—31no—
323D4—92no—
324D4—92no—
325D4—92no—
326D4—92no—
327D4—91no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—92no—
335C2×C4—92no—
336C2×C4—92no—
337C2×C4—92no—
338C2×C4C2×D3211yescharacteristic
339C8—31no—
340C8—31no—
341C8—32no—
342C8—91no—
343C8—91no—
344C8C2×D3212yes—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—62no—
350C2×C3—61no—
351C2×C3—61no—
352C2×C3C22×C4×D311yescharacteristic
353C2×C3C22×C4×D311yescharacteristic
354D3—21no—
355D3—21no—
356D3—22no—
357D3—62no—
358D3—62no—
359D3—122no—
360C22—31no—
361C22—31no—
362C22—32no—
363C22—92no—
364C22—121no—
365C22—362no—
366C22C4×D3211yescharacteristic
367C4—31no—
368C4—31no—
369C4—32no—
370C4—92no—
371C4C4×D3211yescharacteristic
372C4C22×D3211yescharacteristic
373C3—21no—
374C3#574511yescharacteristic, complemented
375C3#575011yescharacteristic, complemented
376C2—21no—
377C2—61no—
378C2—61no—
379C2—62no—
380C2—182no—
381C2C2×C4×D3211yescharacteristic
subgroupG/H
center, FrattiniC4C22×D32
commutator, socleC2×C32C23×C4
FittingC32×M16C22
prime radicalD3×#235C2
cosocleC32×C4C24
hypercenterM16D32
hypocenterC32#568
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. #64116C2semidirect
3. C3×#907C2semidirect
4. C3×D3×M16C2semidirect
5. C2×#3711C2semidirect
6. #63817C2semidirect
7. #63815C2semidirect
8. #63804C2semidirect
9. #63803C2semidirect
10. #63791C2semidirect
11. D3×ΓL(1,𝔽25)C2semidirect
25. C32×M16C22semidirect
26. C2×C3⋊(C3⋊C8)C22semidirect
27. #3743C22semidirect
28. C8×C3⋊D3C22semidirect
29. C3×#208C22semidirect
30. C2×C3×C3⋊C8C22semidirect
32. C3×ΓL(1,𝔽25)C22semidirect
33. C3×D3×C8C22semidirect
34. C3×D3×C8C22semidirect
35. #3712C22semidirect
38. #3711C22semidirect
39. #3711C22semidirect
40. #3711C22semidirect
41. #3710C22semidirect
42. D3×C3⋊C8C22semidirect
55. #907D3semidirect
59. D3×M16D3semidirect
123. C3×M16C2×D3semidirect
124. C3×M16C2×D3semidirect
133. #208C2×D3semidirect
138. C2×C3⋊C8C2×D3semidirect
150. ΓL(1,𝔽25)C2×D3semidirect
157. D3×C8C2×D3semidirect
158. D3×C8C2×D3semidirect
276. M16D32semidirect
315. C32#568semidirect
374. C3#5745semidirect
375. C3#5750semidirect

Maximal subgroups

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

subgroupindex
D3×#2352
#641162
C3×#9072
C3×D3×M162
C2×#37112
#638172
#638152
#638042
#638032
#637912
D3×ΓL(1,𝔽25)2
#57503
#57453