Finite Group Databasebeta

← #10595466

Subgroups — #10595466

Order 576.

546 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1009C211yesmaximal, characteristic, complemented
2#64298C211yesmaximal, characteristic, complemented
3#64160C211yesmaximal, characteristic, complemented
4C3×#950C211yesmaximal, characteristic, complemented
5#63947C212yesmaximal, complemented
6#63943C212yesmaximal, complemented
7#63932C212yesmaximal, complemented
8D3×#214C212yesmaximal, complemented
9C2×#3716C211yesmaximal, characteristic, complemented
10#63823C211yesmaximal, characteristic, complemented
11#63814C211yesmaximal, characteristic, complemented
12#5836—31nomaximal
13#5770—31nomaximal
14C32×#41C2211yescharacteristic
15C2×#634C2211yescharacteristic, complemented
16C3×#239C2212yes—
17C3×D3×D4—22no—
18C3×D3×D4C2212yes—
19C3×#235—21no—
20C3×#235C2211yescharacteristic
21C3×#235C2211yescharacteristic, complemented
22C2×C3×D12—21no—
23C2×C3×D12C2211yescharacteristic
24#3805—22no—
25#3795—21no—
26#3795C2212yescomplemented
27D3×Dic6—21no—
28D3×Dic6C2212yescomplemented
29#3756C2212yescomplemented
30#3754C2212yescomplemented
31#3748C2211yescharacteristic, complemented
32C3×#216C2211yescharacteristic, complemented
33C3×#215C2211yescharacteristic, complemented
34C3×#214C2211yescharacteristic, complemented
35C3×#213C2211yescharacteristic, complemented
36C2×C3×C3⋊C8C2211yescharacteristic, complemented
37#3716C2212yescomplemented
38#3716C2212yescomplemented
39#3715C2212yescomplemented
40#3713C2212yescomplemented
41#3711C2212yescomplemented
42D3×C3⋊C8C2212yescomplemented
43C3×H(5,𝔽2)—31no—
44#1010—31no—
45#1010—31no—
46#1009D311yescharacteristic, complemented
47C3×#135—31no—
48#951—31no—
49#951—31no—
50#951—32no—
51#951—61no—
52#950—32no—
53#950D311yescharacteristic, complemented
54#949—31no—
55#948—31no—
56C2×#214—32no—
57C2×#214—31no—
58#932—31no—
59#932—32no—
60#918—32no—
61#916—32no—
62D3×SD16—32no—
63#911—32no—
64C2×#204—31no—
65#901—31no—
66C22×C3×D3—21no—
67C22×C3×D3—22no—
68C32×Q8C2311yescharacteristic
69C32×D4—21no—
70C32×D4C2311yescharacteristic
71C2×C32×C4—21no—
72C2×C32×C4C2311yescharacteristic
73C2×C3⋊Dic3—21no—
74#634—21no—
75#634C2312yes—
76C3×#67—21no—
77C3×#67—22no—
78C3×#67D412yes—
79C3×D12—22no—
80C3×D12—21no—
81C3×D12D411yescharacteristic
82C3×D12C2311yescharacteristic
83C3×D12C2312yes—
84C3×C4×D3—21no—
85C3×C4×D3—22no—
86C3×C4×D3C2312yes—
87C3×Dic6D411yescharacteristic
88C3×Dic6C2311yescharacteristic
89#627—22no—
90#625—22no—
91D3×Dic3—22no—
92D3×Dic3—22no—
93C3⋊(C3⋊C8)C2312yes—
94C3×C3⋊C8C2311yescharacteristic
95C3×C3⋊C8C2311yescharacteristic
96#577—91noSylow
97C3×#41—21no—
98C3×#41—31no—
99C3×#41—32no—
100C3×#41—31no—
101C3×#41—61no—
102C3×#41C2×D311yescharacteristic
103C3×#41C2×D311yescharacteristic
104C2×C3×D4—32no—
105C2×C3×D4—31no—
106C2×C3×D4—61no—
107C2×C3×D4—62no—
108#239C2×D312yes—
109D3×Q8—32no—
110D3×Q8—61no—
111#237—32no—
112#237—61no—
113#237—62no—
114#237—62no—
115D3×D4—22no—
116D3×D4C2×D312yes—
117#235—21no—
118#235—32no—
119#235—61no—
120#235C2×D311yescharacteristic
121#235C2×D311yescharacteristic, complemented
122C2×D12—21no—
123C2×D12C2×D311yescharacteristic
124C2×Dic6—31no—
125C2×Dic6—31no—
126C2×Dic6—32no—
127C2×Dic6—61no—
128C2×Dic6—61no—
129C3×Q16—31no—
130C3×SD16—31no—
131C3×SD16—31no—
132C3×D8—31no—
133C2×C3×C8—31no—
134#216—32no—
135#216—32no—
136#216—62no—
137#216C2×D311yescharacteristic, complemented
138#215—32no—
139#215C2×D311yescharacteristic, complemented
140#214—32no—
141#214—32no—
142#214—32no—
143#214—32no—
144#214—32no—
145#214—62no—
146#214C2×D311yescharacteristic, complemented
147#213—32no—
148#213—32no—
149#213C2×D311yescharacteristic, complemented
150#208—31no—
151#208—31no—
152#208—32no—
153#208—61no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8C2×D311yescharacteristic, complemented
156#204—32no—
157#204—32no—
158ΓL(1,𝔽25)—32no—
159D3×C8—32no—
160C22×C32—21no—
161C22×C32C2×D411yescharacteristic
162C2×C3×D3—21no—
163C2×C3×D3—22no—
164C2×C3×D3—82no—
165C2×C3×D3—81no—
166C2×C3×D3C2×D412yes—
167C32×C4—21no—
168C32×C4C2×D411yescharacteristic
169C32×C4C2411yescharacteristic
170C3⋊Dic3—22no—
171C3×Dic3—21no—
172C3×Dic3C2×D412yes—
1732−1+4—91no—
174H(5,𝔽2)—31no—
175#137—92no—
176#137—91no—
177AGL(1,ℤ/8)—92no—
178AGL(1,ℤ/8)—91no—
179#135—31no—
180#135—92no—
181C2×SD16—91no—
182C2×SD16—92no—
183#131—91no—
184C23×C3—62no—
185C23×C3—61no—
186C22×D3—21no—
187C22×D3—22no—
188C3×Q8—21no—
189C3×Q8—31no—
190C3×Q8—31no—
191C3×Q8C22×D311yescharacteristic
192C3×Q8C22×D311yescharacteristic
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—32no—
197C3×D4—31no—
198C3×D4—31no—
199C3×D4—32no—
200C3×D4—41no—
201C3×D4—62no—
202C3×D4—62no—
203C3×D4—61no—
204C3×D4—61no—
205C3×D4C22×D311yescharacteristic
206C3×D4C22×D311yescharacteristic
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—32no—
211C2×C3×C4—41no—
212C2×C3×C4—62no—
213C2×C3×C4—61no—
214C2×C3×C4C22×D311yescharacteristic
215C2×C3×C4C22×D311yescharacteristic
216#67—21no—
217#67—22no—
218#67—62no—
219#67—62no—
220#67#6712yes—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—62no—
225C2×Dic3—121no—
226D12—22no—
227D12—21no—
228D12#6711yescharacteristic
229D12C22×D312yes—
230D12C22×D311yescharacteristic
231C4×D3—21no—
232C4×D3—22no—
233C4×D3—62no—
234C4×D3—62no—
235C4×D3C22×D312yes—
236Dic6—32no—
237Dic6—32no—
238Dic6—62no—
239Dic6—62no—
240Dic6—61no—
241Dic6—61no—
242Dic6—62no—
243Dic6—121no—
244Dic6#6711yescharacteristic
245Dic6C22×D311yescharacteristic
246C3×C8—31no—
247C3×C8—31no—
248C3⋊C8—32no—
249C3⋊C8—32no—
250C3⋊C8—62no—
251C3⋊C8C22×D311yescharacteristic
252C3⋊C8C22×D311yescharacteristic
253C2×C32—21no—
254C2×C32—41no—
255C2×C32C22×D411yescharacteristic
256C3×D3—22no—
257C3×D3—41no—
258C3×D3—42no—
259#41—31no—
260#41—32no—
261#41—31no—
262#41—61no—
263#41—92no—
264#41—181no—
265#41—182no—
266#41D3211yescharacteristic
267C2×Q8—92no—
268C2×Q8—91no—
269C2×Q8—181no—
270C2×D4—32no—
271C2×D4—31no—
272C2×D4—61no—
273C2×D4—62no—
274Q16—31no—
275Q16—92no—
276SD16—31no—
277SD16—31no—
278SD16—92no—
279SD16—92no—
280SD16—92no—
281SD16—92no—
282D8—31no—
283D8—92no—
284M16—92no—
285M16—91no—
286C2×C8—31no—
287C2×C8—92no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—32no—
292C22×C3—41no—
293C22×C3—62no—
294C22×C3—61no—
295C22×C3—241no—
296C22×C3—242no—
297C22×C3D3×D411yescharacteristic
298C22×C3C2×#6711yescharacteristic
299C2×D3—21no—
300C2×D3—22no—
301C2×D3—82no—
302C2×D3—81no—
303C2×D3C2×#6712yes—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—32no—
309C3×C4—41no—
310C3×C4—61no—
311C3×C4D3×D411yescharacteristic
312C3×C4C2×#6711yescharacteristic
313C3×C4C23×D311yescharacteristic
314C3×C4C23×D311yescharacteristic
315Dic3—21no—
316Dic3—62no—
317Dic3—62no—
318Dic3—122no—
319Dic3C2×#6712yes—
320C32#57711yescharacteristic, complemented, Sylow
321C23—62no—
322C23—61no—
323Q8—31no—
324Q8—31no—
325Q8—92no—
326Q8—182no—
327Q8—181no—
328Q8C2×D3211yescharacteristic
329D4—21no—
330D4—32no—
331D4—31no—
332D4—32no—
333D4—31no—
334D4—62no—
335D4—62no—
336D4—61no—
337D4—61no—
338D4—182no—
339D4C2×D3211yescharacteristic
340C2×C4—21no—
341C2×C4—32no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—182no—
345C2×C4—182no—
346C2×C4—181no—
347C2×C4C2×D3211yescharacteristic
348C8—31no—
349C8—31no—
350C8—92no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—62no—
358C2×C3—81no—
359C2×C3—122no—
360C2×C3—121no—
361C2×C3C2×D3×D411yescharacteristic
362C2×C3C22×#6711yescharacteristic
363D3—22no—
364D3—41no—
365D3—42no—
366C22—21no—
367C22—32no—
368C22—61no—
369C22—62no—
370C22—241no—
371C22—242no—
372C22D3×#6711yescharacteristic
373C4—21no—
374C4—32no—
375C4—61no—
376C4—182no—
377C4D3×#6711yescharacteristic
378C4C22×D3211yescharacteristic
379C3—21no—
380C3#577011yescharacteristic, complemented
381C3#583611yescharacteristic, complemented
382C2—21no—
383C2—41no—
384C2—62no—
385C2—122no—
386C2—121no—
387C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC2C2×D3×#67
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. C3×#1009C2semidirect
2. #64298C2semidirect
3. #64160C2semidirect
4. C3×#950C2semidirect
5. #63947C2semidirect
6. #63943C2semidirect
7. #63932C2semidirect
8. D3×#214C2semidirect
9. C2×#3716C2semidirect
10. #63823C2semidirect
11. #63814C2semidirect
15. C2×#634C22semidirect
21. C3×#235C22semidirect
26. #3795C22semidirect
28. D3×Dic6C22semidirect
29. #3756C22semidirect
30. #3754C22semidirect
31. #3748C22semidirect
32. C3×#216C22semidirect
33. C3×#215C22semidirect
34. C3×#214C22semidirect
35. C3×#213C22semidirect
36. C2×C3×C3⋊C8C22semidirect
37. #3716C22semidirect
38. #3716C22semidirect
39. #3715C22semidirect
40. #3713C22semidirect
41. #3711C22semidirect
42. D3×C3⋊C8C22semidirect
46. #1009D3semidirect
53. #950D3semidirect
121. #235C2×D3semidirect
137. #216C2×D3semidirect
139. #215C2×D3semidirect
146. #214C2×D3semidirect
149. #213C2×D3semidirect
155. C2×C3⋊C8C2×D3semidirect
320. C32#577semidirect
380. C3#5770semidirect
381. C3#5836semidirect

Maximal subgroups

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

subgroupindex
C3×#10092
#642982
#641602
C3×#9502
#639472
#639432
#639322
D3×#2142
C2×#37162
#638232
#638142
#58363
#57703