Finite Group Databasebeta

← #10595462

Subgroups — #10595462

Order 576.

589 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64367C211yesmaximal, characteristic, complemented
2#64307C211yesmaximal, characteristic, complemented
3C3×#950C211yesmaximal, characteristic, complemented
4C3×#949C211yesmaximal, characteristic, complemented
5#63951C212yesmaximal, complemented
6#63940C212yesmaximal, complemented
7#63936C212yesmaximal, complemented
8#63927C212yesmaximal, complemented
9#63831C211yesmaximal, characteristic, complemented
10C2×#3716C211yesmaximal, characteristic, complemented
11#63817C211yesmaximal, characteristic, complemented
12#5775—31nomaximal
13#5770—31nomaximal
14C32×#41C2211yescharacteristic, complemented
15Q8×C3⋊D3C2212yescomplemented
16#3830—22no—
17#3830C2212yescomplemented
18#3828—21no—
19#3828C2211yescharacteristic
20C2×#634—21no—
21C2×#634C2211yescharacteristic, complemented
22C3×#235C2211yescharacteristic
23C2×C3×D12C2211yescharacteristic
24D3×#67—22no—
25#3802C2212yes—
26D3×D12—21no—
27#3796C2212yes—
28#3795—21no—
29C3×#216C2211yescharacteristic, complemented
30C3×#215C2212yescomplemented
31C3×#215C2211yescharacteristic, complemented
32C3×#214C2211yescharacteristic, complemented
33C3×#213C2212yescomplemented
34C3×#213C2211yescharacteristic, complemented
35C3×#208C2211yescharacteristic, complemented
36C2×C3×C3⋊C8C2211yescharacteristic, complemented
37#3719C2212yescomplemented
38#3716C2212yescomplemented
39#3716C2212yescomplemented
40#3716C2212yescomplemented
41#3712C2212yescomplemented
42#3710C2212yescomplemented
43#1010—31no—
44#1010—31no—
45#1010—61no—
46#1009—31no—
47#1004—31no—
48C3×AGL(1,ℤ/8)—31no—
49C3×#135—31no—
50#951—31no—
51#950D311yescharacteristic, complemented
52#949D311yescharacteristic, complemented
53C2×#214—31no—
54#918—32no—
55#916—32no—
56#915—32no—
57D3×SD16—32no—
58D3×SD16—32no—
59#912—32no—
60#911—32no—
61#911—32no—
62#909—31no—
63#907—31no—
64C2×#204—31no—
65#901—31no—
66C22×C3×D3—21no—
67C2×D32—22no—
68C32×Q8C2311yescharacteristic
69C32×D4—21no—
70C32×D4C2311yescharacteristic
71C2×C32×C4—21no—
72C2×C32×C4C2311yescharacteristic
73#638—22no—
74#638D412yes—
75C2×C3⋊Dic3—21no—
76C2×C3⋊Dic3—22no—
77C4⋊(C3⋊D3)D411yescharacteristic
78C4×C3⋊D3—22no—
79C4×C3⋊D3C2312yes—
80#634—21no—
81#634—22no—
82#634D411yescharacteristic
83#634C2312yes—
84C3×#67—21no—
85C3×D12—21no—
86C3×D12C2312yes—
87C3×D12C2311yescharacteristic
88C3×C4×D3—21no—
89C3×Dic6C2311yescharacteristic
90#626—22no—
91#625—22no—
92D3×Dic3—22no—
93C3×C3⋊C8C2311yescharacteristic
94C3×C3⋊C8C2312yes—
95C3×C3⋊C8C2311yescharacteristic
96#577—91noSylow
97C3×#41—21no—
98C3×#41—31no—
99C3×#41C2×D311yescharacteristic, complemented
100C3×#41C2×D311yescharacteristic, complemented
101C2×C3×D4—31no—
102C2×#67—62no—
103#239—32no—
104D3×Q8—32no—
105D3×Q8—32no—
106D3×Q8—62no—
107#237—32no—
108#237—32no—
109#237—32no—
110#237—62no—
111#237—61no—
112#237—62no—
113#237—62no—
114#237—64no—
115D3×D4—32no—
116D3×D4—32no—
117D3×D4—61no—
118D3×D4—62no—
119#235—31no—
120#235—31no—
121#235—61no—
122#235—61no—
123#235—61no—
124#235—61no—
125#235—62no—
126#235C2×D311yescharacteristic
127C2×D12—61no—
128C2×D12C2×D311yescharacteristic
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6—61no—
132C2×Dic6—61no—
133C2×Dic6—61no—
134C2×Dic6—62no—
135C3×Q16—31no—
136C3×SD16—31no—
137C3×SD16—32no—
138C3×SD16—31no—
139C3×D8—31no—
140C3×D8—32no—
141C3×M16—31no—
142C2×C3×C8—31no—
143#216—32no—
144#216C2×D311yescharacteristic, complemented
145#215C2×D311yescharacteristic, complemented
146#215C2×D312yescomplemented
147#214—32no—
148#214—32no—
149#214—32no—
150#214C2×D311yescharacteristic, complemented
151#213C2×D311yescharacteristic, complemented
152#213C2×D312yescomplemented
153#208C2×D311yescharacteristic, complemented
154C2×C3⋊C8C2×D311yescharacteristic, complemented
155Dic12—32no—
156#204—32no—
157#204—32no—
158#204—32no—
159ΓL(1,𝔽25)—32no—
160ΓL(1,𝔽25)—32no—
161D3×C8—32no—
162D3×C8—32no—
163C22×C32—21no—
164C22×C32C2×D411yescharacteristic
165C2×C3⋊D3C2×D412yes—
166C2×C3×D3—22no—
167C2×C3×D3—21no—
168C2×C3×D3—81no—
169D32—82no—
170C32×C4—21no—
171C32×C4C2×D411yescharacteristic
172C32×C4C2411yescharacteristic
173C3⋊Dic3—22no—
174C3⋊Dic3C2×D412yes—
175C3×Dic3—21no—
1762−1+4—91no—
177H(5,𝔽2)—91no—
178#137—91no—
179#137—92no—
180AGL(1,ℤ/8)—31no—
181AGL(1,ℤ/8)—92no—
182#135—31no—
183#135—92no—
184C2×SD16—91no—
185C2×SD16—92no—
186#131—91no—
187C23×C3—61no—
188C22×D3—21no—
189C22×D3—62no—
190C22×D3—62no—
191C3×Q8—21no—
192C3×Q8—31no—
193C3×Q8C22×D311yescharacteristic
194C3×Q8C22×D311yescharacteristic
195C3×D4—22no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—31no—
200C3×D4—32no—
201C3×D4—61no—
202C3×D4—61no—
203C3×D4C22×D311yescharacteristic
204C3×D4C22×D311yescharacteristic
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—61no—
210C2×C3×C4C22×D311yescharacteristic
211C2×C3×C4C22×D311yescharacteristic
212#67—21no—
213#67—32no—
214#67—32no—
215#67—62no—
216#67—62no—
217#67—62no—
218#67—62no—
219#67—62no—
220#67—62no—
221#67—64no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—64no—
226C2×Dic3—62no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229D12—21no—
230D12—31no—
231D12—31no—
232D12—61no—
233D12—62no—
234D12C22×D312yes—
235D12C22×D311yescharacteristic
236C4×D3—21no—
237C4×D3—32no—
238C4×D3—32no—
239C4×D3—62no—
240C4×D3—62no—
241C4×D3—62no—
242C4×D3—62no—
243C4×D3—64no—
244Dic6—31no—
245Dic6—31no—
246Dic6—32no—
247Dic6—32no—
248Dic6—61no—
249Dic6—62no—
250Dic6—62no—
251Dic6—62no—
252Dic6—62no—
253Dic6—64no—
254Dic6—61no—
255Dic6—61no—
256Dic6C22×D311yescharacteristic
257C3×C8—31no—
258C3×C8—32no—
259C3×C8—31no—
260C3⋊C8C22×D311yescharacteristic
261C3⋊C8C22×D312yes—
262C3⋊C8C22×D311yescharacteristic
263C2×C32—21no—
264C2×C32—41no—
265C2×C32C22×D411yescharacteristic
266C3⋊D3—22no—
267C3×D3—41no—
268C3×D3—42no—
269#41—31no—
270#41—91no—
271#41—92no—
272#41—92no—
273#41—181no—
274#41—181no—
275#41—182no—
276#41D3211yescharacteristic, complemented
277C2×Q8—92no—
278C2×Q8—91no—
279C2×Q8—181no—
280C2×D4—31no—
281C2×D4—92no—
282C2×D4—181no—
283C2×D4—182no—
284Q16—31no—
285Q16—92no—
286SD16—31no—
287SD16—32no—
288SD16—31no—
289SD16—92no—
290SD16—92no—
291SD16—92no—
292D8—31no—
293D8—32no—
294M16—31no—
295M16—92no—
296C2×C8—31no—
297C2×C8—92no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—62no—
303C22×C3—61no—
304C22×C3—241no—
305C22×C3D3×D411yescharacteristic
306C22×C3D3×D411yescharacteristic
307C2×D3—21no—
308C2×D3—22no—
309C2×D3—32no—
310C2×D3—32no—
311C2×D3—62no—
312C2×D3—81no—
313C2×D3—242no—
314C2×D3—242no—
315C3×C4—22no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—61no—
321C3×C4D3×D411yescharacteristic
322C3×C4D3×D411yescharacteristic
323C3×C4C23×D311yescharacteristic
324C3×C4C23×D311yescharacteristic
325Dic3—21no—
326Dic3—32no—
327Dic3—32no—
328Dic3—62no—
329Dic3—64no—
330Dic3—62no—
331Dic3—62no—
332C32#57711yescharacteristic, complemented, Sylow
333C23—61no—
334C23—182no—
335Q8—31no—
336Q8—91no—
337Q8—92no—
338Q8—182no—
339Q8—181no—
340Q8C2×D3211yescharacteristic
341D4—21no—
342D4—32no—
343D4—31no—
344D4—61no—
345D4—61no—
346D4—92no—
347D4—91no—
348D4—182no—
349D4—182no—
350D4—182no—
351D4C2×D3211yescharacteristic
352C2×C4—21no—
353C2×C4—61no—
354C2×C4—92no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—181no—
359C2×C4C2×D3211yescharacteristic
360C8—31no—
361C8—31no—
362C8—32no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—41no—
367C2×C3—42no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—122no—
371C2×C3—121no—
372C2×C3C2×D3×D411yescharacteristic
373C2×C3C2×D3×D411yescharacteristic
374D3—41no—
375D3—42no—
376D3—62no—
377D3—62no—
378D3—122no—
379C22—21no—
380C22—61no—
381C22—62no—
382C22—92no—
383C22—241no—
384C22—722no—
385C22#381111yescharacteristic
386C4—21no—
387C4—61no—
388C4—92no—
389C4—182no—
390C4#381111yescharacteristic
391C4C22×D3211yescharacteristic
392C3—21no—
393C3#577011yescharacteristic, complemented
394C3#577511yescharacteristic, complemented
395C2—21no—
396C2—41no—
397C2—122no—
398C2—121no—
399C2—182no—
400C2C2×#381111yescharacteristic
subgroupG/H
centerC2C2×#3811
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC32×#41C22
socleC2×C32C22×D4
hypercenter#41D32
hypocenterC32#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #64367C2semidirect
2. #64307C2semidirect
3. C3×#950C2semidirect
4. C3×#949C2semidirect
5. #63951C2semidirect
6. #63940C2semidirect
7. #63936C2semidirect
8. #63927C2semidirect
9. #63831C2semidirect
10. C2×#3716C2semidirect
11. #63817C2semidirect
14. C32×#41C22semidirect
15. Q8×C3⋊D3C22semidirect
17. #3830C22semidirect
21. C2×#634C22semidirect
29. C3×#216C22semidirect
30. C3×#215C22semidirect
31. C3×#215C22semidirect
32. C3×#214C22semidirect
33. C3×#213C22semidirect
34. C3×#213C22semidirect
35. C3×#208C22semidirect
36. C2×C3×C3⋊C8C22semidirect
37. #3719C22semidirect
38. #3716C22semidirect
39. #3716C22semidirect
40. #3716C22semidirect
41. #3712C22semidirect
42. #3710C22semidirect
51. #950D3semidirect
52. #949D3semidirect
99. C3×#41C2×D3semidirect
100. C3×#41C2×D3semidirect
144. #216C2×D3semidirect
145. #215C2×D3semidirect
146. #215C2×D3semidirect
150. #214C2×D3semidirect
151. #213C2×D3semidirect
152. #213C2×D3semidirect
153. #208C2×D3semidirect
154. C2×C3⋊C8C2×D3semidirect
276. #41D32semidirect
332. C32#577semidirect
393. C3#5770semidirect
394. C3#5775semidirect

Maximal subgroups

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

subgroupindex
#643672
#643072
C3×#9502
C3×#9492
#639512
#639402
#639362
#639272
#638312
C2×#37162
#638172
#57753
#57703