Finite Group Databasebeta

← #10595461

Subgroups — #10595461

Order 576.

619 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64366C211yesmaximal, characteristic, complemented
2#64307C211yesmaximal, characteristic, complemented
3C3×#950C211yesmaximal, characteristic, complemented
4C3×#949C211yesmaximal, characteristic, complemented
5#63950C212yesmaximal, complemented
6#63941C212yesmaximal, complemented
7#63937C212yesmaximal, complemented
8#63926C212yesmaximal, complemented
9#63825C211yesmaximal, characteristic, complemented
10C2×#3714C211yesmaximal, characteristic, complemented
11#63817C211yesmaximal, characteristic, complemented
12#5774—31nomaximal
13#5769—31nomaximal
14C32×#41C2211yescharacteristic, complemented
15#3832C2212yescomplemented
16D4×C3⋊D3—22no—
17D4×C3⋊D3C2212yescomplemented
18#3828—21no—
19#3828C2211yescharacteristic
20C2×C4⋊(C3⋊D3)—21no—
21C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
22C3×#235C2211yescharacteristic, complemented
23C2×C3×D12C2211yescharacteristic, complemented
24D3×#67—22no—
25#3802C2212yescomplemented
26D3×D12—21no—
27#3796C2212yescomplemented
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#3717C2212yescomplemented
38#3714C2212yescomplemented
39#3714C2212yescomplemented
40#3714C2212yescomplemented
41#3712C2212yescomplemented
42#3710C2212yescomplemented
43#1009—31no—
44#1009—31no—
45#1009—31no—
46#1009—61no—
47#1004—31no—
48C3×AGL(1,ℤ/8)—31no—
49C3×#135—31no—
50#950D311yescharacteristic, complemented
51#949—31no—
52#949D311yescharacteristic, complemented
53C2×#213—31no—
54#919—32no—
55#916—32no—
56#914—32no—
57#914—32no—
58#914—32no—
59#911—32no—
60D3×D8—32no—
61D3×D8—32no—
62#908—31no—
63#907—31no—
64C2×D24—31no—
65#901—31no—
66C22×C3⋊D3—21no—
67C22×C3⋊D3—22no—
68C22×C3×D3—21no—
69C2×D32—22no—
70C32×Q8C2311yescharacteristic
71C32×D4—21no—
72C32×D4C2311yescharacteristic
73C2×C32×C4—21no—
74C2×C32×C4C2311yescharacteristic
75#638—22no—
76#638D412yes—
77C4⋊(C3⋊D3)—22no—
78C4⋊(C3⋊D3)—21no—
79C4⋊(C3⋊D3)D411yescharacteristic
80C4⋊(C3⋊D3)C2312yes—
81C4×C3⋊D3—22no—
82C4×C3⋊D3C2312yes—
83#634D411yescharacteristic
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#576—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—
104#239—32no—
105#239—32no—
106#239—62no—
107#237—32no—
108#237—61no—
109D3×D4—32no—
110D3×D4—32no—
111D3×D4—32no—
112D3×D4—32no—
113D3×D4—62no—
114D3×D4—61no—
115D3×D4—62no—
116D3×D4—62no—
117D3×D4—62no—
118D3×D4—64no—
119#235—31no—
120#235—31no—
121#235—61no—
122#235—61no—
123#235—61no—
124#235—61no—
125#235—62no—
126#235C2×D311yescharacteristic, complemented
127C2×D12—31no—
128C2×D12—31no—
129C2×D12—61no—
130C2×D12—61no—
131C2×D12—61no—
132C2×D12—61no—
133C2×D12—62no—
134C2×D12C2×D311yescharacteristic, complemented
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#216C2×D311yescharacteristic, complemented
144#215—32no—
145#215C2×D311yescharacteristic, complemented
146#215C2×D312yescomplemented
147#214C2×D311yescharacteristic, complemented
148#213—32no—
149#213—32no—
150#213—32no—
151#213C2×D311yescharacteristic, complemented
152#213C2×D312yescomplemented
153#208C2×D311yescharacteristic, complemented
154C2×C3⋊C8C2×D311yescharacteristic, complemented
155D24—32no—
156D24—32no—
157D24—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⋊D3—22no—
166C2×C3⋊D3—82no—
167C2×C3⋊D3—81no—
168C2×C3⋊D3C2×D412yes—
169C2×C3×D3—22no—
170C2×C3×D3—21no—
171C2×C3×D3—81no—
172D32—82no—
173C32×C4—21no—
174C32×C4C2×D411yescharacteristic
175C32×C4C2411yescharacteristic
176C3⋊Dic3C2×D412yes—
177C3×Dic3—21no—
178H(5,𝔽2)—91no—
179H(5,𝔽2)—91no—
180AGL(1,ℤ/8)—31no—
181AGL(1,ℤ/8)—91no—
182AGL(1,ℤ/8)—92no—
183AGL(1,ℤ/8)—92no—
184#135—31no—
185#135—92no—
186C2×D8—91no—
187C2×D8—92no—
188#131—91no—
189C23×C3—61no—
190C22×D3—21no—
191C22×D3—62no—
192C22×D3—62no—
193C22×D3—62no—
194C22×D3—62no—
195C22×D3—64no—
196C22×D3—62no—
197C22×D3—61no—
198C22×D3—61no—
199C3×Q8—21no—
200C3×Q8—31no—
201C3×Q8C22×D311yescharacteristic
202C3×Q8C22×D311yescharacteristic
203C3×D4—22no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—21no—
207C3×D4—31no—
208C3×D4—32no—
209C3×D4—61no—
210C3×D4—61no—
211C3×D4C22×D311yescharacteristic
212C3×D4C22×D311yescharacteristic
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—61no—
218C2×C3×C4C22×D311yescharacteristic
219C2×C3×C4C22×D311yescharacteristic
220#67—21no—
221#67—32no—
222#67—32no—
223#67—62no—
224#67—62no—
225#67—62no—
226#67—62no—
227#67—62no—
228#67—62no—
229#67—64no—
230C2×Dic3—62no—
231D12—21no—
232D12—31no—
233D12—31no—
234D12—32no—
235D12—32no—
236D12—61no—
237D12—62no—
238D12—62no—
239D12—62no—
240D12—62no—
241D12—64no—
242D12—62no—
243D12—61no—
244D12—61no—
245D12C22×D312yes—
246D12C22×D311yescharacteristic
247C4×D3—21no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—62no—
251C4×D3—62no—
252C4×D3—62no—
253C4×D3—62no—
254C4×D3—64no—
255Dic6—31no—
256Dic6—31no—
257Dic6—61no—
258Dic6C22×D311yescharacteristic
259C3×C8—31no—
260C3×C8—32no—
261C3×C8—31no—
262C3⋊C8C22×D311yescharacteristic
263C3⋊C8C22×D312yes—
264C3⋊C8C22×D311yescharacteristic
265C2×C32—21no—
266C2×C32—41no—
267C2×C32C22×D411yescharacteristic
268C3⋊D3—22no—
269C3⋊D3—42no—
270C3×D3—41no—
271C3×D3—42no—
272#41—31no—
273#41—91no—
274#41—92no—
275#41—92no—
276#41—181no—
277#41—181no—
278#41D3211yescharacteristic, complemented
279C2×D4—31no—
280C2×D4—92no—
281C2×D4—92no—
282C2×D4—91no—
283C2×D4—181no—
284C2×D4—181no—
285C2×D4—182no—
286C2×D4—182no—
287Q16—31no—
288SD16—31no—
289SD16—32no—
290SD16—31no—
291SD16—92no—
292D8—31no—
293D8—32no—
294D8—92no—
295D8—92no—
296D8—92no—
297M16—31no—
298M16—92no—
299C2×C8—31no—
300C2×C8—92no—
301C22×C3—22no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—62no—
306C22×C3—61no—
307C22×C3—241no—
308C22×C3D3×D411yescharacteristic
309C22×C3D3×D411yescharacteristic
310C2×D3—21no—
311C2×D3—22no—
312C2×D3—32no—
313C2×D3—32no—
314C2×D3—62no—
315C2×D3—64no—
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—81no—
319C2×D3—242no—
320C2×D3—242no—
321C2×D3—242no—
322C2×D3—242no—
323C2×D3—241no—
324C2×D3—241no—
325C2×D3—242no—
326C2×D3—244no—
327C3×C4—22no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—61no—
333C3×C4D3×D411yescharacteristic
334C3×C4D3×D411yescharacteristic
335C3×C4C23×D311yescharacteristic
336C3×C4C23×D311yescharacteristic
337Dic3—21no—
338Dic3—32no—
339Dic3—32no—
340Dic3—62no—
341C32#57611yescharacteristic, complemented, Sylow
342C23—61no—
343C23—182no—
344C23—182no—
345C23—181no—
346Q8—31no—
347Q8—91no—
348Q8C2×D3211yescharacteristic
349D4—21no—
350D4—32no—
351D4—31no—
352D4—61no—
353D4—61no—
354D4—92no—
355D4—92no—
356D4—91no—
357D4—182no—
358D4—182no—
359D4—182no—
360D4—182no—
361D4—181no—
362D4C2×D3211yescharacteristic
363C2×C4—21no—
364C2×C4—61no—
365C2×C4—92no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4C2×D3211yescharacteristic
369C8—31no—
370C8—31no—
371C8—32no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—41no—
376C2×C3—42no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—122no—
380C2×C3—121no—
381C2×C3C2×D3×D411yescharacteristic
382C2×C3C2×D3×D411yescharacteristic
383D3—41no—
384D3—42no—
385D3—62no—
386D3—62no—
387D3—122no—
388D3—124no—
389D3—122no—
390D3—122no—
391C22—21no—
392C22—61no—
393C22—62no—
394C22—92no—
395C22—182no—
396C22—241no—
397C22—721no—
398C22—722no—
399C22—722no—
400C22#381111yescharacteristic
401C4—21no—
402C4—61no—
403C4—92no—
404C4#381111yescharacteristic
405C4C22×D3211yescharacteristic
406C3—21no—
407C3#576911yescharacteristic, complemented
408C3#577411yescharacteristic, complemented
409C2—21no—
410C2—41no—
411C2—122no—
412C2—121no—
413C2—182no—
414C2—362no—
415C2C2×#381111yescharacteristic
subgroupG/H
centerC2C2×#3811
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC32×#41C22
socleC2×C32C22×D4
hypercenter#41D32
hypocenterC32#576
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. #64366C2semidirect
2. #64307C2semidirect
3. C3×#950C2semidirect
4. C3×#949C2semidirect
5. #63950C2semidirect
6. #63941C2semidirect
7. #63937C2semidirect
8. #63926C2semidirect
9. #63825C2semidirect
10. C2×#3714C2semidirect
11. #63817C2semidirect
14. C32×#41C22semidirect
15. #3832C22semidirect
17. D4×C3⋊D3C22semidirect
21. C2×C4⋊(C3⋊D3)C22semidirect
22. C3×#235C22semidirect
23. C2×C3×D12C22semidirect
25. #3802C22semidirect
27. #3796C22semidirect
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. #3717C22semidirect
38. #3714C22semidirect
39. #3714C22semidirect
40. #3714C22semidirect
41. #3712C22semidirect
42. #3710C22semidirect
50. #950D3semidirect
52. #949D3semidirect
99. C3×#41C2×D3semidirect
100. C3×#41C2×D3semidirect
126. #235C2×D3semidirect
134. C2×D12C2×D3semidirect
143. #216C2×D3semidirect
145. #215C2×D3semidirect
146. #215C2×D3semidirect
147. #214C2×D3semidirect
151. #213C2×D3semidirect
152. #213C2×D3semidirect
153. #208C2×D3semidirect
154. C2×C3⋊C8C2×D3semidirect
278. #41D32semidirect
341. C32#576semidirect
407. C3#5769semidirect
408. C3#5774semidirect

Maximal subgroups

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

subgroupindex
#643662
#643072
C3×#9502
C3×#9492
#639502
#639412
#639372
#639262
#638252
C2×#37142
#638172
#57743
#57693