Finite Group Databasebeta

← #13180981

Subgroups — #13180981

Order 1728.

952 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724586C211yesmaximal, characteristic
2#11724515C211yesmaximal, characteristic, complemented
3C3×#64287C211yesmaximal, characteristic, complemented
4C2×#88926C212yesmaximal, complemented
5#11724269C212yesmaximal, complemented
6#11724267C212yesmaximal, complemented
7#11724266C212yesmaximal, complemented
8#11724254C211yesmaximal, characteristic, complemented
9#11724253C211yesmaximal, characteristic, complemented
10#11724252C211yesmaximal, characteristic, complemented
11D3×#3788C211yesmaximal, characteristic, complemented
12#10596648—31nomaximal
13D4×C3⋊(C3⋊D3)C411yescharacteristic
14C3×#3830C2211yescharacteristic
15C32×#237C411yescharacteristic
16#89039C412yes—
17C2×#6289C2212yes—
18#89031C412yes—
19#89028C2212yes—
20C4×D3×C3⋊D3C2211yescharacteristic
21#89022C411yescharacteristic
22#89019C411yescharacteristic
23#89018C2211yescharacteristic
24#88993C2212yescomplemented
25C2×C3⋊#622C2212yescomplemented
26#88989C2211yescharacteristic, complemented
27#88988C2211yescharacteristic, complemented
28C3×#3792C2212yescomplemented
29C2×C3×#622C2212yescomplemented
30C3×#3788C2211yescharacteristic, complemented
31C3×#3787C2211yescharacteristic, complemented
32#88926C2212yes—
33#88926C2211yescharacteristic
34#88926C2211yescharacteristic
35#88926C2212yescomplemented
36#88926C2212yescomplemented
37#88925C2211yescharacteristic, complemented
38#88925C2211yescharacteristic, complemented
39#88924C2212yescomplemented
40#88924C2212yescomplemented
41D3×#622C2212yescomplemented
42#41×C3⋊D3—31no—
43#64312—62no—
44#64289—31no—
45#64289—31no—
46#64287—31no—
47#64287—32no—
48#64287—32no—
49#64287D311yescharacteristic, complemented
50#64283—32no—
51#64283—31no—
52C2×#3788—32no—
53C2×#3788—31no—
54C22×C3⋊(C3⋊D3)C2×C412yes—
55C2×D3×C3⋊D3C2×C411yescharacteristic
56C33×D4C2×C411yescharacteristic
57#6312C2×C412yes—
58C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
59C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
60C3×#638C2×C412yes—
61C2×C3×C3⋊Dic3C2312yes—
62C3×C4×C3⋊D3C2311yescharacteristic
63C3×#634C2×C411yescharacteristic
64C32×#67C2×C412yes—
65C2×C32×Dic3C2×C412yes—
66C32×C4×D3C2×C411yescharacteristic
67C32×Dic6C2×C411yescharacteristic
68#6293C2×C412yes—
69#6292C2×C412yes—
70#6291C2×C412yes—
71#6289C2×C412yes—
72#6289C2×C412yes—
73#6289C2312yes—
74#6289C2311yescharacteristic
75Dic3×C3⋊D3C2×C411yescharacteristic
76D3×C3⋊Dic3C2311yescharacteristic
77C3⋊#622C2312yes—
78C3⋊#622C2311yescharacteristic
79C3⋊#622C2311yescharacteristic
80C3×#622C2312yes—
81C3×#622C2311yescharacteristic
82C3×#622C2311yescharacteristic
83#5750—91nomaximal
84C32×#41—31no—
85#3832—31no—
86Q8×C3⋊D3—31no—
87#3830—32no—
88#3830C2×D311yescharacteristic
89D4×C3⋊D3—31no—
90D4×C3⋊D3—32no—
91D4×C3⋊D3—62no—
92D4×C3⋊D3—122no—
93#3828—32no—
94#3828—31no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—31no—
97C3×#237—22no—
98C3×#237—62no—
99#3811—64no—
100C2×#624—64no—
101#3804—64no—
102#3804—64no—
103C4×D32—62no—
104#3799—62no—
105#3799—62no—
106#3797—62no—
107#3792—32no—
108#3792—31no—
109#3792—31no—
110#3792C2×D312yescomplemented
111C2×#622—32no—
112C2×#622—32no—
113C2×#622C2×D312yescomplemented
114#3788—31no—
115#3788—32no—
116#3788—31no—
117#3788—32no—
118#3788—32no—
119#3788—31no—
120#3788C2×D311yescharacteristic, complemented
121#3787—31no—
122#3787—32no—
123#3787—32no—
124#3787C2×D311yescharacteristic, complemented
125C22×C33C22×C412yes—
126C2×C3⋊(C3⋊D3)—42no—
127C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
128C2×C3⋊(C3⋊D3)C22×C412yes—
129C2×C3×C3⋊D3C22×C411yescharacteristic
130C2×C32×D3C22×C411yescharacteristic
131D3×C3⋊D3—41no—
132C33×C4C22×C411yescharacteristic
133C3⋊(C3⋊Dic3)C22×C411yescharacteristic
134C3×C3⋊Dic3C22×C412yes—
135C3×C3⋊Dic3C2411yescharacteristic
136C32×Dic3C22×C411yescharacteristic
137C32×Dic3C22×C412yes—
138D3×#41—91no—
139D3×#41—182no—
140C3×#131—91no—
141#948—91no—
142#907—91no—
143#907—92no—
144D3×M16—91no—
145D3×M16—92no—
146#901—91no—
147#901—92no—
148C2×ΓL(1,𝔽25)—91no—
149C2×ΓL(1,𝔽25)—92no—
150C22×C3⋊D3—32no—
151C22×C3⋊D3—31no—
152C22×C3⋊D3—64no—
153C22×C3⋊D3—124no—
154C2×D32—62no—
155C32×Q8—31no—
156C32×D4—22no—
157C32×D4—32no—
158C32×D4—42no—
159C32×D4C4×D311yescharacteristic
160C2×C32×C4—32no—
161C2×C32×C4—31no—
162#638—32no—
163#638—32no—
164#638—64no—
165#638—124no—
166#638C4×D312yes—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3C22×D312yes—
169C4⋊(C3⋊D3)—32no—
170C4⋊(C3⋊D3)—31no—
171C4⋊(C3⋊D3)—62no—
172C4⋊(C3⋊D3)—122no—
173C4×C3⋊D3—31no—
174C4×C3⋊D3—31no—
175C4×C3⋊D3—32no—
176C4×C3⋊D3—32no—
177C4×C3⋊D3—32no—
178C4×C3⋊D3—31no—
179C4×C3⋊D3—62no—
180C4×C3⋊D3—122no—
181C4×C3⋊D3C22×D311yescharacteristic
182#634—32no—
183#634C4×D311yescharacteristic
184C3×#67—24no—
185C3×#67—64no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—64no—
188C3×C4×D3—22no—
189C3×C4×D3—62no—
190C3×Dic6—22no—
191C3×Dic6—62no—
192#627—64no—
193#626—64no—
194#626—64no—
195#624—64no—
196#624—64no—
197#624—64no—
198#624—62no—
199D3×Dic3—62no—
200D3×Dic3—62no—
201#622—31no—
202#622—32no—
203#622—31no—
204#622C22×D311yescharacteristic
205#622C22×D312yes—
206#622C22×D311yescharacteristic
207#568—271noSylow
208C2×C33—22no—
209C2×C33C23×C411yescharacteristic
210C3⋊(C3⋊D3)—21no—
211C3⋊(C3⋊D3)—22no—
212C3×C3⋊D3—21no—
213C32×D3—21no—
214C3×#41—62no—
215C3×#41—91no—
216#239—91no—
217#239—182no—
218D3×Q8—91no—
219D3×Q8—182no—
220#237—62no—
221#237—92no—
222#237—184no—
223#237#15711yescharacteristic
224D3×D4—91no—
225D3×D4—92no—
226D3×D4—184no—
227D3×D4—182no—
228D3×D4—362no—
229#235—92no—
230#235—91no—
231#235—182no—
232#235—184no—
233C2×C4×D3—92no—
234C2×C4×D3—91no—
235C2×C4×D3—182no—
236C2×C4×D3—184no—
237C3×M16—91no—
238C3×M16—92no—
239C2×C3×C8—91no—
240C2×C3×C8—92no—
241#208—91no—
242#208—92no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—92no—
245ΓL(1,𝔽25)—91no—
246ΓL(1,𝔽25)—91no—
247ΓL(1,𝔽25)—91no—
248ΓL(1,𝔽25)—92no—
249ΓL(1,𝔽25)—91no—
250ΓL(1,𝔽25)—92no—
251ΓL(1,𝔽25)—92no—
252D3×C8—92no—
253D3×C8—92no—
254D3×C8—92no—
255C22×C32—24no—
256C22×C32—31no—
257C22×C32—44no—
258C22×C32C2×C4×D312yes—
259C2×C3⋊D3—31no—
260C2×C3⋊D3—32no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—64no—
263C2×C3⋊D3—122no—
264C2×C3⋊D3—124no—
265C2×C3⋊D3—121no—
266C2×C3⋊D3—122no—
267C2×C3⋊D3—128no—
268C2×C3⋊D3—248no—
269C2×C3⋊D3C2×C4×D311yescharacteristic
270C2×C3×D3—22no—
271C2×C3×D3—62no—
272D32—124no—
273C32×C4—22no—
274C32×C4—32no—
275C32×C4—31no—
276C32×C4—42no—
277C32×C4C2×C4×D311yescharacteristic
278C3⋊Dic3—31no—
279C3⋊Dic3—62no—
280C3⋊Dic3—122no—
281C3⋊Dic3C2×C4×D312yes—
282C3⋊Dic3C23×D311yescharacteristic
283C3×Dic3—22no—
284C3×Dic3—24no—
285C3×Dic3—64no—
286C3×Dic3—62no—
287C2×#41—271no—
288#131—91no—
289#131—272no—
290#131—271no—
291#131—272no—
292#131—271no—
293#131—271no—
294C2×M16—272no—
295C2×M16—271no—
296C2×M16—271no—
297C2×M16—272no—
298C33#56811yescharacteristic, complemented, Sylow
299C22×D3—92no—
300C22×D3—91no—
301C22×D3—182no—
302C22×D3—184no—
303C22×D3—364no—
304C3×Q8—62no—
305C3×Q8—91no—
306C3×D4—22no—
307C3×D4—42no—
308C3×D4—64no—
309C3×D4—92no—
310C3×D4C2×#15711yescharacteristic
311C2×C3×C4—62no—
312C2×C3×C4—64no—
313C2×C3×C4—92no—
314C2×C3×C4—91no—
315#67—64no—
316#67—92no—
317#67—92no—
318#67—184no—
319#67—184no—
320#67—364no—
321#67C2×#15712yes—
322C2×Dic3—64no—
323C2×Dic3—91no—
324C2×Dic3—182no—
325C2×Dic3C2×#15712yes—
326D12—92no—
327D12—91no—
328D12—182no—
329D12—184no—
330D12—362no—
331C4×D3—62no—
332C4×D3—91no—
333C4×D3—91no—
334C4×D3—92no—
335C4×D3—91no—
336C4×D3—92no—
337C4×D3—92no—
338C4×D3—182no—
339C4×D3—184no—
340C4×D3—184no—
341C4×D3—184no—
342C4×D3—182no—
343C4×D3—182no—
344C4×D3—362no—
345C4×D3C2×#15711yescharacteristic
346Dic6—62no—
347Dic6—92no—
348Dic6—184no—
349Dic6C2×#15711yescharacteristic
350C3×C8—92no—
351C3×C8—91no—
352C3×C8—91no—
353C3⋊C8—92no—
354C3⋊C8—91no—
355C3⋊C8—91no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—42no—
359C2×C32—44no—
360C2×C32—61no—
361C2×C32—84no—
362C2×C32C22×C4×D311yescharacteristic
363C3⋊D3—21no—
364C3⋊D3—61no—
365C3⋊D3—64no—
366C3⋊D3—62no—
367C3⋊D3—124no—
368C3⋊D3—124no—
369C3⋊D3—244no—
370C3×D3—42no—
371C3×D3—122no—
372#41—31no—
373#41—91no—
374#41—272no—
375#41—271no—
376#41—272no—
377#41—271no—
378C2×Q8—271no—
379C2×D4—272no—
380C2×D4—271no—
381C22×C4—272no—
382C22×C4—271no—
383M16—91no—
384M16—92no—
385M16—271no—
386M16—272no—
387M16—271no—
388M16—272no—
389M16—272no—
390M16—271no—
391M16—271no—
392M16—271no—
393M16—272no—
394C2×C8—91no—
395C2×C8—92no—
396C2×C8—272no—
397C2×C8—272no—
398C2×C8—272no—
399C2×C8—271no—
400C2×C8—272no—
401C22×C3—24no—
402C22×C3—44no—
403C22×C3—62no—
404C22×C3—91no—
405C22×C3C22×#15712yes—
406C2×D3—62no—
407C2×D3—91no—
408C2×D3—92no—
409C2×D3—182no—
410C2×D3—184no—
411C2×D3—362no—
412C2×D3—364no—
413C2×D3—361no—
414C2×D3—364no—
415C2×D3—362no—
416C2×D3—368no—
417C2×D3—728no—
418C2×D3C22×#15711yescharacteristic
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—64no—
422C3×C4—62no—
423C3×C4—91no—
424C3×C4—92no—
425C3×C4C22×#15711yescharacteristic
426Dic3—64no—
427Dic3—62no—
428Dic3—91no—
429Dic3—182no—
430Dic3—362no—
431Dic3C22×#15712yes—
432Dic3C22×#15711yescharacteristic
433C32—22no—
434C32—42no—
435C32#575011yescharacteristic, complemented
436C23—271no—
437C23—272no—
438Q8—31no—
439Q8—91no—
440Q8—272no—
441D4—32no—
442D4—92no—
443D4—272no—
444D4—272no—
445D4—272no—
446D4—271no—
447D4D3×#15711yescharacteristic
448C2×C4—32no—
449C2×C4—31no—
450C2×C4—92no—
451C2×C4—91no—
452C2×C4—272no—
453C2×C4—271no—
454C2×C4—272no—
455C2×C4—272no—
456C2×C4—271no—
457C2×C4—271no—
458C2×C4—271no—
459C8—91no—
460C8—91no—
461C8—92no—
462C8—271no—
463C8—272no—
464C8—271no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—44no—
468C2×C3—42no—
469C2×C3—84no—
470C2×C3—122no—
471C2×C3—181no—
472C2×C3C23×#15711yescharacteristic
473D3—21no—
474D3—122no—
475D3—184no—
476D3—181no—
477D3—182no—
478D3—364no—
479D3—364no—
480D3—724no—
481C22—31no—
482C22—91no—
483C22—271no—
484C22—272no—
485C22—1082no—
486C22—1081no—
487C22C2×D3×#15712yes—
488C4—31no—
489C4—32no—
490C4—91no—
491C4—92no—
492C4—271no—
493C4C2×D3×#15711yescharacteristic
494C3—22no—
495C3—42no—
496C3#1059664811yescharacteristic, complemented
497C2—22no—
498C2—61no—
499C2—181no—
500C2—541no—
501C2—542no—
502C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC33×D4C2×C4
prime radical#11724586C2
cosocleC3×C3⋊Dic3C24
hypercenterD4D3×#157
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #11724515C2semidirect
3. C3×#64287C2semidirect
4. C2×#88926C2semidirect
5. #11724269C2semidirect
6. #11724267C2semidirect
7. #11724266C2semidirect
8. #11724254C2semidirect
9. #11724253C2semidirect
10. #11724252C2semidirect
11. D3×#3788C2semidirect
24. #88993C22semidirect
25. C2×C3⋊#622C22semidirect
26. #88989C22semidirect
27. #88988C22semidirect
28. C3×#3792C22semidirect
29. C2×C3×#622C22semidirect
30. C3×#3788C22semidirect
31. C3×#3787C22semidirect
35. #88926C22semidirect
36. #88926C22semidirect
37. #88925C22semidirect
38. #88925C22semidirect
39. #88924C22semidirect
40. #88924C22semidirect
41. D3×#622C22semidirect
49. #64287D3semidirect
110. #3792C2×D3semidirect
113. C2×#622C2×D3semidirect
120. #3788C2×D3semidirect
124. #3787C2×D3semidirect
298. C33#568semidirect
435. C32#5750semidirect
496. C3#10596648semidirect

Maximal subgroups

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

subgroupindex
#117245862
#117245152
C3×#642872
C2×#889262
#117242692
#117242672
#117242662
#117242542
#117242532
#117242522
D3×#37882
#105966483
#57509