Finite Group Databasebeta

← #11918954

Subgroups — #11918954

Order 1152.

4272 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596758C212yesmaximal, complemented
2C32×#576C211yesmaximal, characteristic, complemented
3#10596307C212yesmaximal, complemented
4#10596246C216yesmaximal, complemented
5#10596244C216yesmaximal, complemented
6#10596242C213yesmaximal, complemented
7C3⋊D3×#135C213yesmaximal, complemented
8#10596228C213yesmaximal, complemented
9C2×#64121C213yesmaximal, complemented
10#10596224C211yesmaximal, characteristic, complemented
11C3⋊D3×#131C211yesmaximal, characteristic, complemented
12#87203—34nomaximal
13C32×H(5,𝔽2)C2212yescomplemented
14#64367—22no—
15#64367C2216yescomplemented
16#64367C2212yescomplemented
17#41×C3⋊D3—22no—
18#41×C3⋊D3C2211yescharacteristic
19#41×C3⋊D3C2216yescomplemented
20#64361—26no—
21#64361C2216yescomplemented
22C2×#3830—26no—
23C2×#3830C2216yescomplemented
24C32×AGL(1,ℤ/8)C2216yescomplemented
25C32×#135C2213yescomplemented
26C2×C32×D8C2213yescomplemented
27C32×#131C2211yescharacteristic, complemented
28#64160C2216yescomplemented
29#64159C2216yescomplemented
30#64157C2211yescharacteristic, complemented
31C2×#3754C2216yescomplemented
32#64141C2216yescomplemented
33#64128C2211yescharacteristic, complemented
34Q16×C3⋊D3C2213yescomplemented
35#64125C2216yescomplemented
36#64124C22112yescomplemented
37SD16×C3⋊D3C2216yescomplemented
38#64121C2213yescomplemented
39#64121C2216yescomplemented
40#64121C2216yescomplemented
41#64121C2216yescomplemented
42#64120C22112yescomplemented
43D8×C3⋊D3C2213yescomplemented
44#64118C2216yescomplemented
45#64116C2213yescomplemented
46M16×C3⋊D3C2213yescomplemented
47C2×#3746C2213yescomplemented
48#64113C2213yescomplemented
49#64110C2213yescomplemented
50C2×C8×C3⋊D3C2213yescomplemented
51#5966—38no—
52C3×#576D314yescomplemented
53#5836—38no—
54#5775—324no—
55#5773—324no—
56#5771—312no—
57D3×#135—312no—
58#5757—312no—
59C2×#912—312no—
60#5753—34no—
61D3×#131—34no—
62C32×#41—22no—
63C32×#41C2316yes—
64C32×#41C2311yescharacteristic
65C2×C32×D4—26no—
66C2×C32×D4C2316yes—
67C2×#638—26no—
68C2×#638—212no—
69C22×C3⋊Dic3—26no—
70#3832D411yescharacteristic
71#3832C2312yes—
72Q8×C3⋊D3—22no—
73Q8×C3⋊D3D411yescharacteristic
74Q8×C3⋊D3C2316yes—
75#3830—26no—
76#3830—212no—
77#3830—26no—
78#3830—26no—
79#3830—26no—
80#3830—26no—
81#3830D413yes—
82#3830C2316yes—
83#3830C23112yes—
84D4×C3⋊D3—26no—
85D4×C3⋊D3—26no—
86D4×C3⋊D3D413yes—
87D4×C3⋊D3C2316yes—
88#3828—26no—
89#3828—26no—
90#3828C2313yes—
91#3828C2316yes—
92C2×C4×C3⋊D3—26no—
93C2×C4×C3⋊D3C2313yes—
94C2×#634—26no—
95C2×#634C2316yes—
96C32×Q16C2311yescharacteristic
97C32×SD16C2316yes—
98C32×D8C2313yes—
99C32×D8C2316yes—
100C32×M16C2313yes—
101C2×C32×C8C2313yes—
102#3756C2316yes—
103#3755C2312yes—
104#3754C23112yes—
105#3754C2316yes—
106#3753C2316yes—
107#3748C2313yes—
108C2×C3⋊(C3⋊C8)C2313yes—
109#3746C2313yes—
110#3746C2316yes—
111#3745C2311yescharacteristic
112#3744C2316yes—
113#3743C2316yes—
114C8×C3⋊D3C2311yescharacteristic
115C8×C3⋊D3C2313yes—
116C8×C3⋊D3C2313yes—
117C8×C3⋊D3C2313yes—
118#3584—91noSylow
119C3×H(5,𝔽2)C2×D318yescomplemented
120#1010—38no—
121#1010—324no—
122#1010—68no—
123D3×#41—34no—
124D3×#41—324no—
125D3×#41—68no—
126#1004—324no—
127#1004—624no—
128C2×#237—324no—
129C2×#237—624no—
130C3×AGL(1,ℤ/8)C2×D3124yescomplemented
131C3×#135C2×D3112yescomplemented
132C2×C3×D8C2×D3112yescomplemented
133C3×#131C2×D314yescomplemented
134#951—324no—
135#950—324no—
136#948—34no—
137C2×#214—324no—
138#932—324no—
139#919—34no—
140D3×Q16—312no—
141#916—324no—
142#915—348no—
143D3×SD16—324no—
144#912—312no—
145#912—324no—
146#912—324no—
147#912—324no—
148#911—348no—
149D3×D8—312no—
150#909—324no—
151#907—312no—
152D3×M16—312no—
153C2×Dic12—312no—
154#904—312no—
155#901—312no—
156C2×D3×C8—312no—
157C23×C32—26no—
158C22×C3⋊D3—26no—
159C22×C3⋊D3C2×D413yes—
160C32×Q8C2×D411yescharacteristic
161C32×Q8C2412yes—
162C32×D4—26no—
163C32×D4—26no—
164C32×D4C2×D413yes—
165C32×D4C2416yes—
166C2×C32×C4—26no—
167C2×C32×C4C2413yes—
168#638—26no—
169#638—212no—
170#638—26no—
171#638—26no—
172#638C2×D416yes—
173C2×C3⋊Dic3—26no—
174C2×C3⋊Dic3—212no—
175C2×C3⋊Dic3—26no—
176C2×C3⋊Dic3C2×D413yes—
177C4⋊(C3⋊D3)—22no—
178C4⋊(C3⋊D3)C2×D413yes—
179C4⋊(C3⋊D3)C2412yes—
180C4×C3⋊D3—26no—
181C4×C3⋊D3—26no—
182C4×C3⋊D3—22no—
183C4×C3⋊D3C2×D413yes—
184C4×C3⋊D3C2×D413yes—
185C4×C3⋊D3C2411yescharacteristic
186C4×C3⋊D3C2413yes—
187#634—26no—
188#634—26no—
189#634C2×D413yes—
190#634C2416yes—
191C32×C8C2411yescharacteristic
192C32×C8C2413yes—
193C3⋊(C3⋊C8)C2411yescharacteristic
194C3⋊(C3⋊C8)C2413yes—
195#585—92no—
196#578—93no—
197#578—91no—
198#577—92no—
199#577—96no—
200#576—93no—
201#576C3⋊D311yescharacteristic, complemented
202#575—96no—
203C2×#135—93no—
204C2×#135—93no—
205C2×#131—91no—
206C3×#41—28no—
207C3×#41C22×D3124yes—
208C3×#41C22×D314yes—
209C2×C3×D4—224no—
210C2×C3×D4C22×D3124yes—
211C2×#67—648no—
212C2×#67—624no—
213C22×Dic3—624no—
214#239—34no—
215#239—38no—
216D3×Q8—34no—
217D3×Q8—324no—
218D3×Q8—68no—
219#237—312no—
220#237—348no—
221#237—324no—
222#237—624no—
223#237—624no—
224#237—624no—
225#237—624no—
226#237—648no—
227#237—624no—
228D3×D4—312no—
229D3×D4—324no—
230D3×D4—624no—
231D3×D4—624no—
232#235—312no—
233#235—324no—
234#235—624no—
235#235—624no—
236C2×C4×D3—312no—
237C2×C4×D3—624no—
238C2×Dic6—324no—
239C2×Dic6—624no—
240C3×Q16C22×D314yes—
241C3×SD16C22×D3124yes—
242C3×D8C22×D3112yes—
243C3×D8C22×D3124yes—
244C3×M16C22×D3112yes—
245C2×C3×C8C22×D3112yes—
246#216—324no—
247#215—38no—
248#214—348no—
249#214—324no—
250#213—324no—
251#208—312no—
252C2×C3⋊C8—312no—
253Dic12—312no—
254Dic12—324no—
255D24—34no—
256#204—324no—
257ΓL(1,𝔽25)—324no—
258D3×C8—34no—
259D3×C8—312no—
260D3×C8—312no—
261D3×C8—312no—
262C22×C32—26no—
263C22×C32—86no—
264C22×C32C22×D413yes—
265C2×C3⋊D3—22no—
266C2×C3⋊D3—43no—
267C2×C3⋊D3—86no—
268C2×C3⋊D3C22×D413yes—
269C2×C3⋊D3C22×D411yescharacteristic
270C32×C4—22no—
271C32×C4C22×D413yes—
272C32×C4C2511yescharacteristic
273C3⋊Dic3—26no—
274C3⋊Dic3C22×D413yes—
275C3⋊Dic3C22×D411yescharacteristic
2762−1+4—96no—
2772−1+4—92no—
2782−1+4—182no—
279H(5,𝔽2)—96no—
280H(5,𝔽2)—186no—
281H(5,𝔽2)C2×C3⋊D312yescomplemented
282C2×#41—91no—
283C2×#41—96no—
284C2×#41—96no—
285C2×#41—186no—
286C2×#41—182no—
287#137—96no—
288#137—912no—
289#137—96no—
290AGL(1,ℤ/8)—96no—
291AGL(1,ℤ/8)—912no—
292AGL(1,ℤ/8)C2×C3⋊D316yescomplemented
293#135—96no—
294#135—91no—
295#135—96no—
296#135—93no—
297#135—96no—
298#135—96no—
299#135—96no—
300#135—93no—
301#135C2×C3⋊D313yescomplemented
302C2×Q16—93no—
303C2×Q16—93no—
304C2×SD16—96no—
305C2×SD16—96no—
306C2×D8—93no—
307C2×D8C2×C3⋊D313yescomplemented
308#131—91no—
309#131—93no—
310#131—93no—
311#131C2×C3⋊D311yescharacteristic, complemented
312C2×M16—93no—
313C22×C8—93no—
314C23×C3—224no—
315C22×D3—312no—
316C22×D3—624no—
317C3×Q8D3×D414yes—
318C3×Q8C23×D318yes—
319C3×D4—224no—
320C3×D4—224no—
321C3×D4D3×D4112yes—
322C3×D4C23×D3124yes—
323C2×C3×C4—224no—
324C2×C3×C4C23×D3112yes—
325#67—324no—
326#67—624no—
327#67—624no—
328#67—648no—
329#67—624no—
330C2×Dic3—312no—
331C2×Dic3—648no—
332C2×Dic3—624no—
333C2×Dic3—624no—
334D12—312no—
335D12—38no—
336D12—68no—
337C4×D3—312no—
338C4×D3—312no—
339C4×D3—34no—
340C4×D3—312no—
341C4×D3—68no—
342C4×D3—624no—
343C4×D3—624no—
344Dic6—312no—
345Dic6—324no—
346Dic6—624no—
347Dic6—624no—
348C3×C8C23×D314yes—
349C3×C8C23×D3112yes—
350C3⋊C8—34no—
351C3⋊C8—312no—
352C2×C32—23no—
353C2×C32—46no—
354C2×C32C23×D411yescharacteristic
355C3⋊D3—21no—
356C3⋊D3—23no—
357C3⋊D3—42no—
358#41—22no—
359#41—91no—
360#41—93no—
361#41—93no—
362#41—92no—
363#41—96no—
364#41—912no—
365#41—96no—
366#41—186no—
367#41—186no—
368#41—186no—
369#41—186no—
370#41—186no—
371#41—1812no—
372#41—186no—
373#41—186no—
374#41C22×C3⋊D316yes—
375#41C22×C3⋊D311yescharacteristic
376C2×Q8—91no—
377C2×Q8—96no—
378C2×Q8—96no—
379C2×Q8—186no—
380C2×Q8—182no—
381C2×D4—26no—
382C2×D4—93no—
383C2×D4—96no—
384C2×D4—186no—
385C2×D4—186no—
386C2×D4—1812no—
387C2×D4—186no—
388C2×D4C22×C3⋊D316yes—
389C22×C4—93no—
390C22×C4—186no—
391C22×C4—186no—
392Q16—96no—
393Q16—93no—
394Q16—96no—
395Q16C22×C3⋊D311yescharacteristic
396SD16—92no—
397SD16—96no—
398SD16—912no—
399SD16—96no—
400SD16C22×C3⋊D316yes—
401D8—96no—
402D8—91no—
403D8C22×C3⋊D313yes—
404D8C22×C3⋊D316yes—
405M16—93no—
406M16—96no—
407M16C22×C3⋊D313yes—
408C2×C8—93no—
409C2×C8—91no—
410C2×C8—93no—
411C2×C8—93no—
412C2×C8—93no—
413C2×C8C22×C3⋊D313yes—
414C22×C3—224no—
415C22×C3—824no—
416C22×C3C2×D3×D4112yes—
417C2×D3—34no—
418C2×D3—312no—
419C2×D3—68no—
420C2×D3—1212no—
421C2×D3—2424no—
422C3×C4—28no—
423C3×C4C2×D3×D4112yes—
424C3×C4C24×D314yes—
425Dic3—312no—
426Dic3—34no—
427Dic3—624no—
428C32#358411yescharacteristic, complemented, Sylow
429C23—26no—
430C23—93no—
431C23—186no—
432Q8—93no—
433Q8—96no—
434Q8—186no—
435Q8—186no—
436Q8D4×C3⋊D311yescharacteristic
437Q8C23×C3⋊D312yes—
438D4—26no—
439D4—26no—
440D4—96no—
441D4—93no—
442D4—92no—
443D4—182no—
444D4—186no—
445D4—186no—
446D4—1812no—
447D4—186no—
448D4D4×C3⋊D313yes—
449D4C23×C3⋊D316yes—
450C2×C4—26no—
451C2×C4—93no—
452C2×C4—93no—
453C2×C4—93no—
454C2×C4—91no—
455C2×C4—93no—
456C2×C4—182no—
457C2×C4—186no—
458C2×C4—186no—
459C2×C4—1812no—
460C2×C4—186no—
461C2×C4—186no—
462C2×C4C23×C3⋊D313yes—
463C8—91no—
464C8—93no—
465C8C23×C3⋊D311yescharacteristic
466C8C23×C3⋊D313yes—
467C2×C3—212no—
468C2×C3—424no—
469C2×C3C22×D3×D414yes—
470D3—64no—
471D3—612no—
472D3—128no—
473C22—26no—
474C22—86no—
475C22—91no—
476C22—93no—
477C22—182no—
478C22—363no—
479C22—726no—
480C22C2×D4×C3⋊D313yes—
481C4—22no—
482C4—91no—
483C4—93no—
484C4—186no—
485C4C2×D4×C3⋊D313yes—
486C4C24×C3⋊D311yescharacteristic
487C3#8720314yescomplemented
488C2—23no—
489C2—46no—
490C2—181no—
491C2—183no—
492C2—362no—
493C2C22×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C22×D4×C3⋊D3
commutator, cosocleC32×C4C25
FrattiniC4C24×C3⋊D3
FittingC32×#576C2
socleC2×C32C23×D4
hypercenter#576C3⋊D3
hypocenterC32#3584
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ #41 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #10596758C2semidirect
2. C32×#576C2semidirect
3. #10596307C2semidirect
4. #10596246C2semidirect
5. #10596244C2semidirect
6. #10596242C2semidirect
7. C3⋊D3×#135C2semidirect
8. #10596228C2semidirect
9. C2×#64121C2semidirect
10. #10596224C2semidirect
11. C3⋊D3×#131C2semidirect
13. C32×H(5,𝔽2)C22semidirect
15. #64367C22semidirect
16. #64367C22semidirect
19. #41×C3⋊D3C22semidirect
21. #64361C22semidirect
23. C2×#3830C22semidirect
24. C32×AGL(1,ℤ/8)C22semidirect
25. C32×#135C22semidirect
26. C2×C32×D8C22semidirect
27. C32×#131C22semidirect
28. #64160C22semidirect
29. #64159C22semidirect
30. #64157C22semidirect
31. C2×#3754C22semidirect
32. #64141C22semidirect
33. #64128C22semidirect
34. Q16×C3⋊D3C22semidirect
35. #64125C22semidirect
36. #64124C22semidirect
37. SD16×C3⋊D3C22semidirect
38. #64121C22semidirect
39. #64121C22semidirect
40. #64121C22semidirect
41. #64121C22semidirect
42. #64120C22semidirect
43. D8×C3⋊D3C22semidirect
44. #64118C22semidirect
45. #64116C22semidirect
46. M16×C3⋊D3C22semidirect
47. C2×#3746C22semidirect
48. #64113C22semidirect
49. #64110C22semidirect
50. C2×C8×C3⋊D3C22semidirect
52. C3×#576D3semidirect
119. C3×H(5,𝔽2)C2×D3semidirect
130. C3×AGL(1,ℤ/8)C2×D3semidirect
131. C3×#135C2×D3semidirect
132. C2×C3×D8C2×D3semidirect
133. C3×#131C2×D3semidirect
201. #576C3⋊D3semidirect
281. H(5,𝔽2)C2×C3⋊D3semidirect
292. AGL(1,ℤ/8)C2×C3⋊D3semidirect
301. #135C2×C3⋊D3semidirect
307. C2×D8C2×C3⋊D3semidirect
311. #131C2×C3⋊D3semidirect
428. C32#3584semidirect
487. C3#87203semidirect

Maximal subgroups

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

subgroupindex
#105967582
C32×#5762
#105963072
#105962462
#105962442
#105962422
C3⋊D3×#1352
#105962282
C2×#641212
#105962242
C3⋊D3×#1312
#872033