Finite Group Databasebeta

← #10594713

Subgroups — #10594713

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
1#64307C211yesmaximal, characteristic, complemented
2#64298C211yesmaximal, characteristic, complemented
3#64118C211yesmaximal, characteristic, complemented
4C3×#908C211yesmaximal, characteristic, complemented
5C3×#907C211yesmaximal, characteristic, complemented
6#63829C211yesmaximal, characteristic, complemented
7C2×#3716C211yesmaximal, characteristic, complemented
8#63807C212yesmaximal, complemented
9#63800C212yesmaximal, complemented
10#63795C212yesmaximal, complemented
11D3×#204C212yesmaximal, complemented
12#5775—31nomaximal
13#5751—31nomaximal
14#3828C2211yescharacteristic
15C2×#634C2211yescharacteristic, complemented
16C3×#235C2211yescharacteristic
17C3×#235C2211yescharacteristic, complemented
18C2×C3×D12C2211yescharacteristic
19D3×#67—22no—
20#3805—22no—
21#3802—21no—
22D3×D12C2212yes—
23#3796—21no—
24#3795—21no—
25#3795C2212yes—
26#3795C2212yescomplemented
27D3×Dic6—21no—
28D3×Dic6C2212yescomplemented
29C32×M16C2211yescharacteristic, complemented
30#3746C2212yescomplemented
31#3744C2212yescomplemented
32C2×C3×C3⋊C8C2211yescharacteristic, complemented
33C3×D24C2212yescomplemented
34C3×#204C2212yescomplemented
35C3×ΓL(1,𝔽25)C2212yescomplemented
36C3×D3×C8C2212yescomplemented
37#3719C2211yescharacteristic, complemented
38#3717C2211yescharacteristic, complemented
39#3716C2212yescomplemented
40#3716C2212yescomplemented
41#3716C2211yescharacteristic, complemented
42#3714C2211yescharacteristic, complemented
43#1010—31no—
44#1010—31no—
45#1009—31no—
46#1004—31no—
47C3×AGL(1,ℤ/8)—31no—
48C3×#131—31no—
49#950—31no—
50C2×#214—31no—
51#915—32no—
52D3×SD16—32no—
53#912—32no—
54#911—32no—
55#909—31no—
56#909—31no—
57#909—32no—
58#909—61no—
59#908—32no—
60#908D311yescharacteristic, complemented
61#907D311yescharacteristic, complemented
62#904—31no—
63#904—32no—
64C2×#204—31no—
65C2×#204—32no—
66C22×C3×D3—21no—
67C2×D32—22no—
68C2×C32×C4C2311yescharacteristic
69#638—21no—
70C2×C3⋊Dic3—21no—
71C4⋊(C3⋊D3)C2311yescharacteristic
72C4×C3⋊D3—21no—
73#634—21no—
74#634C2312yes—
75#634C2311yescharacteristic
76C3×#67—21no—
77C3×#67D412yes—
78C3×D12—21no—
79C3×D12D411yescharacteristic
80C3×D12C2311yescharacteristic
81C3×D12C2312yes—
82C3×C4×D3—21no—
83C3×C4×D3C2312yes—
84C3×Dic6D411yescharacteristic
85C3×Dic6C2311yescharacteristic
86#627—22no—
87#626—22no—
88#625—22no—
89#625—22no—
90D3×Dic3—22no—
91D3×Dic3—22no—
92D3×Dic3—22no—
93C32×C8C2312yes—
94C3×C3⋊C8C2311yescharacteristic
95C3×C3⋊C8C2311yescharacteristic
96#577—91noSylow
97C3×#41—31no—
98C3×#41—31no—
99C2×C3×D4—31no—
100C2×#67—62no—
101#239—61no—
102D3×Q8—32no—
103D3×Q8—61no—
104#237—32no—
105#237—32no—
106#237—61no—
107#237—62no—
108#237—62no—
109#237—61no—
110D3×D4—32no—
111D3×D4—61no—
112D3×D4—62no—
113D3×D4—61no—
114#235—31no—
115#235—31no—
116#235—32no—
117#235—32no—
118#235—61no—
119#235—61no—
120#235C2×D311yescharacteristic
121#235C2×D311yescharacteristic, complemented
122C2×D12—32no—
123C2×D12C2×D311yescharacteristic
124C2×Dic6—31no—
125C2×Dic6—31no—
126C2×Dic6—32no—
127C2×Dic6—61no—
128C2×Dic6—61no—
129C3×SD16—32no—
130C3×D8—32no—
131C3×M16—21no—
132C3×M16—32no—
133C3×M16C2×D311yescharacteristic, complemented
134C3×M16C2×D311yescharacteristic, complemented
135C2×C3×C8—31no—
136C2×C3×C8—32no—
137#216—31no—
138#215—31no—
139#214—32no—
140#214—31no—
141#214—32no—
142#213—31no—
143C2×C3⋊C8C2×D311yescharacteristic, complemented
144Dic12—31no—
145Dic12—32no—
146Dic12—32no—
147Dic12—62no—
148D24—31no—
149D24C2×D312yescomplemented
150#204—32no—
151#204—32no—
152#204—31no—
153#204—31no—
154#204—32no—
155#204—32no—
156#204—62no—
157#204C2×D312yescomplemented
158ΓL(1,𝔽25)C2×D312yescomplemented
159D3×C8C2×D312yescomplemented
160C22×C32C2×D411yescharacteristic
161C2×C3⋊D3—21no—
162C2×C3×D3—21no—
163C2×C3×D3—22no—
164C2×C3×D3—81no—
165C2×C3×D3C2×D412yes—
166D32—82no—
167C32×C4C2×D411yescharacteristic
168C32×C4C2411yescharacteristic
169C3⋊Dic3—21no—
170C3⋊Dic3—22no—
171C3×Dic3—21no—
172C3×Dic3C2×D412yes—
1732−1+4—91no—
174H(5,𝔽2)—91no—
175#137—92no—
176#137—91no—
177AGL(1,ℤ/8)—31no—
178AGL(1,ℤ/8)—92no—
179#135—92no—
180#135—91no—
181C2×SD16—92no—
182C2×SD16—91no—
183#131—31no—
184C23×C3—61no—
185C22×D3—21no—
186C22×D3—62no—
187C22×D3—62no—
188C3×Q8—31no—
189C3×Q8—31no—
190C3×D4—32no—
191C3×D4—31no—
192C3×D4—31no—
193C3×D4—32no—
194C3×D4—61no—
195C3×D4—61no—
196C2×C3×C4—21no—
197C2×C3×C4—32no—
198C2×C3×C4—61no—
199C2×C3×C4C22×D311yescharacteristic
200C2×C3×C4C22×D311yescharacteristic
201#67—21no—
202#67—62no—
203#67—62no—
204#67—61no—
205#67—61no—
206#67—62no—
207#67—62no—
208#67—62no—
209#67—121no—
210#67D1212yes—
211C2×Dic3—62no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—121no—
217D12—21no—
218D12—31no—
219D12—31no—
220D12—61no—
221D12—62no—
222D12D1211yescharacteristic
223D12C22×D312yes—
224D12C22×D311yescharacteristic
225C4×D3—21no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—62no—
229C4×D3—61no—
230C4×D3—61no—
231C4×D3—121no—
232C4×D3C22×D312yes—
233Dic6—32no—
234Dic6—31no—
235Dic6—32no—
236Dic6—31no—
237Dic6—62no—
238Dic6—61no—
239Dic6—62no—
240Dic6—61no—
241Dic6—61no—
242Dic6—62no—
243Dic6—121no—
244Dic6D1211yescharacteristic
245Dic6C22×D311yescharacteristic
246C3×C8—22no—
247C3×C8—31no—
248C3×C8—31no—
249C3×C8C22×D312yes—
250C3×C8C22×D312yes—
251C3⋊C8C22×D311yescharacteristic
252C3⋊C8C22×D311yescharacteristic
253C2×C32—21no—
254C2×C32C22×D411yescharacteristic
255C3⋊D3—41no—
256C3×D3—22no—
257C3×D3—42no—
258C3×D3—41no—
259#41—31no—
260#41—31no—
261#41—92no—
262#41—92no—
263#41—91no—
264#41—181no—
265#41—182no—
266#41—181no—
267C2×Q8—92no—
268C2×Q8—91no—
269C2×Q8—181no—
270C2×D4—31no—
271C2×D4—92no—
272C2×D4—181no—
273C2×D4—182no—
274Q16—91no—
275Q16—92no—
276SD16—32no—
277SD16—92no—
278SD16—92no—
279SD16—91no—
280SD16—91no—
281SD16—92no—
282D8—32no—
283D8—91no—
284M16—32no—
285M16D3211yescharacteristic, complemented
286C2×C8—31no—
287C2×C8—32no—
288C22×C3—21no—
289C22×C3—32no—
290C22×C3—61no—
291C22×C3—62no—
292C22×C3—241no—
293C22×C3C2×D1211yescharacteristic
294C22×C3D3×D411yescharacteristic
295C2×D3—21no—
296C2×D3—22no—
297C2×D3—61no—
298C2×D3—61no—
299C2×D3—81no—
300C2×D3—121no—
301C2×D3—242no—
302C2×D3—242no—
303C2×D3C2×D1212yes—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—32no—
307C3×C4—61no—
308C3×C4C2×D1211yescharacteristic
309C3×C4D3×D411yescharacteristic
310C3×C4C23×D311yescharacteristic
311C3×C4C23×D311yescharacteristic
312Dic3—21no—
313Dic3—62no—
314Dic3—61no—
315Dic3—62no—
316Dic3—61no—
317Dic3—122no—
318Dic3—121no—
319Dic3C2×D1212yes—
320C32#57711yescharacteristic, complemented, Sylow
321C23—61no—
322C23—182no—
323Q8—31no—
324Q8—31no—
325Q8—92no—
326Q8—91no—
327Q8—182no—
328Q8—181no—
329D4—32no—
330D4—31no—
331D4—32no—
332D4—31no—
333D4—61no—
334D4—61no—
335D4—91no—
336D4—182no—
337D4—182no—
338D4—182no—
339D4—181no—
340C2×C4—32no—
341C2×C4—61no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4—182no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4C2×D3211yescharacteristic
348C8—31no—
349C8—31no—
350C8C2×D3212yes—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—41no—
355C2×C3—62no—
356C2×C3—122no—
357C2×C3—121no—
358C2×C3C22×D1211yescharacteristic
359C2×C3C2×D3×D411yescharacteristic
360D3—22no—
361D3—42no—
362D3—41no—
363D3—121no—
364D3—121no—
365D3—241no—
366C22—32no—
367C22—61no—
368C22—62no—
369C22—181no—
370C22—241no—
371C22—722no—
372C22D3×D1211yescharacteristic
373C4—32no—
374C4—61no—
375C4—181no—
376C4—182no—
377C4D3×D1211yescharacteristic
378C4C22×D3211yescharacteristic
379C3—21no—
380C3#575111yescharacteristic, complemented
381C3#577511yescharacteristic, complemented
382C2—21no—
383C2—62no—
384C2—122no—
385C2—121no—
386C2—361no—
387C2C2×D3×D1211yescharacteristic
subgroupG/H
centerC2C2×D3×D12
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC32×M16C22
socleC2×C32C22×D4
hypercenterM16D32
hypocenterC32#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #64307C2semidirect
2. #64298C2semidirect
3. #64118C2semidirect
4. C3×#908C2semidirect
5. C3×#907C2semidirect
6. #63829C2semidirect
7. C2×#3716C2semidirect
8. #63807C2semidirect
9. #63800C2semidirect
10. #63795C2semidirect
11. D3×#204C2semidirect
15. C2×#634C22semidirect
17. C3×#235C22semidirect
26. #3795C22semidirect
28. D3×Dic6C22semidirect
29. C32×M16C22semidirect
30. #3746C22semidirect
31. #3744C22semidirect
32. C2×C3×C3⋊C8C22semidirect
33. C3×D24C22semidirect
34. C3×#204C22semidirect
35. C3×ΓL(1,𝔽25)C22semidirect
36. C3×D3×C8C22semidirect
37. #3719C22semidirect
38. #3717C22semidirect
39. #3716C22semidirect
40. #3716C22semidirect
41. #3716C22semidirect
42. #3714C22semidirect
60. #908D3semidirect
61. #907D3semidirect
121. #235C2×D3semidirect
133. C3×M16C2×D3semidirect
134. C3×M16C2×D3semidirect
143. C2×C3⋊C8C2×D3semidirect
149. D24C2×D3semidirect
157. #204C2×D3semidirect
158. ΓL(1,𝔽25)C2×D3semidirect
159. D3×C8C2×D3semidirect
285. M16D32semidirect
320. C32#577semidirect
380. C3#5751semidirect
381. C3#5775semidirect

Maximal subgroups

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

subgroupindex
#643072
#642982
#641182
C3×#9082
C3×#9072
#638292
C2×#37162
#638072
#638002
#637952
D3×#2042
#57753
#57513