Finite Group Databasebeta

← #13180980

Subgroups — #13180980

Order 1728.

924 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3830C211yesmaximal, characteristic
2#11724515C211yesmaximal, characteristic, complemented
3C3×#64287C211yesmaximal, characteristic, complemented
4C2×#88925C212yesmaximal, complemented
5#11724267C212yesmaximal, complemented
6#11724266C212yesmaximal, complemented
7D3×#3792C212yesmaximal, complemented
8#11724255C211yesmaximal, characteristic, complemented
9#11724254C211yesmaximal, characteristic, complemented
10#11724251C211yesmaximal, characteristic, complemented
11#11724250C211yesmaximal, characteristic, complemented
12#10596640—31nomaximal
13#89078C411yescharacteristic
14C3×#3830C2211yescharacteristic
15C32×D3×D4C411yescharacteristic
16D3×#638C412yes—
17#89029C412yes—
18#89028C2212yes—
19C2×D3×C3⋊Dic3C2212yes—
20C4×D3×C3⋊D3C2211yescharacteristic
21#89015C411yescharacteristic
22#89014C2211yescharacteristic
23D3×#634C411yescharacteristic
24#88993C2212yescomplemented
25C2×C3⋊#622C2212yescomplemented
26#88989C2211yescharacteristic, complemented
27#88988C2211yescharacteristic, complemented
28C3×#3792C2212yescomplemented
29C2×C3×#622C2212yescomplemented
30C3×#3788C2211yescharacteristic, complemented
31C3×#3787C2211yescharacteristic, complemented
32#88926C2211yescharacteristic, complemented
33#88926C2211yescharacteristic, complemented
34#88925C2211yescharacteristic
35#88925C2211yescharacteristic
36#88925C2212yes—
37#88925C2212yescomplemented
38#88925C2212yescomplemented
39#88924C2212yescomplemented
40D3×#622C2212yescomplemented
41D3×#622C2212yescomplemented
42C2×#3830—31no—
43D3×#237—62no—
44C2×#3792—32no—
45C2×#3792—32no—
46#64287—31no—
47#64287—32no—
48#64287—32no—
49#64287—31no—
50#64287—31no—
51#64287D311yescharacteristic, complemented
52#64283—31no—
53#64283—31no—
54C22×C32×D3C2×C412yes—
55C2×D3×C3⋊D3C2×C411yescharacteristic
56C33×D4C2×C411yescharacteristic
57#6312C2×C412yes—
58C2×C3⋊(C3⋊Dic3)C2×C412yes—
59C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
60#6308C2×C411yescharacteristic
61C3×#638C2×C412yes—
62C2×C3×C3⋊Dic3C2312yes—
63C3×C4×C3⋊D3C2311yescharacteristic
64C3×#634C2×C411yescharacteristic
65C32×#67C2×C412yes—
66C32×D12C2×C411yescharacteristic
67C32×C4×D3C2×C411yescharacteristic
68#6293C2×C412yes—
69#6291C2×C412yes—
70#6290C2×C412yes—
71#6289C2311yescharacteristic
72Dic3×C3⋊D3C2×C411yescharacteristic
73D3×C3⋊Dic3C2×C412yes—
74D3×C3⋊Dic3C2×C412yes—
75D3×C3⋊Dic3C2312yes—
76D3×C3⋊Dic3C2311yescharacteristic
77C3⋊#622C2311yescharacteristic
78C3⋊#622C2312yes—
79C3⋊#622C2311yescharacteristic
80C3×#622C2311yescharacteristic
81C3×#622C2312yes—
82C3×#622C2311yescharacteristic
83#5750—91nomaximal
84C2×C32×D4—31no—
85C2×#638—32no—
86C22×C3⋊Dic3—32no—
87#3830—31no—
88#3830—32no—
89#3830—32no—
90#3830—31no—
91#3830—31no—
92#3830—62no—
93#3830—122no—
94#3830C2×D311yescharacteristic
95C2×C4×C3⋊D3—31no—
96C2×#634—31no—
97C3×#237—62no—
98C3×D3×D4—22no—
99D3×#67—64no—
100#3805—64no—
101#3804—64no—
102C2×D3×Dic3—64no—
103C4×D32—62no—
104#3796—62no—
105#3795—62no—
106D3×Dic6—62no—
107#3792—32no—
108#3792—32no—
109#3792—31no—
110#3792—31no—
111#3792—32no—
112#3792—32no—
113#3792C2×D312yescomplemented
114C2×#622—32no—
115C2×#622—32no—
116C2×#622—32no—
117C2×#622C2×D312yescomplemented
118#3788—31no—
119#3788—31no—
120#3788—31no—
121#3788C2×D311yescharacteristic, complemented
122#3787—31no—
123#3787—32no—
124#3787C2×D311yescharacteristic, complemented
125C22×C33C22×C412yes—
126C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
127C2×C3×C3⋊D3C22×C411yescharacteristic
128C2×C32×D3—42no—
129C2×C32×D3C22×C411yescharacteristic
130C2×C32×D3C22×C412yes—
131D3×C3⋊D3—41no—
132C33×C4C22×C411yescharacteristic
133C3⋊(C3⋊Dic3)C22×C411yescharacteristic
134C3⋊(C3⋊Dic3)C22×C412yes—
135C3×C3⋊Dic3C22×C412yes—
136C3×C3⋊Dic3C2411yescharacteristic
137C32×Dic3C22×C411yescharacteristic
138D3×#41—91no—
139C2×#237—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—
150C23×C32—32no—
151C22×C3⋊D3—31no—
152C22×C3×D3—24no—
153C2×D32—62no—
154C32×D4—22no—
155C32×D4—32no—
156C32×D4—31no—
157C32×D4—42no—
158C32×D4C4×D311yescharacteristic
159C2×C32×C4—31no—
160#638—32no—
161#638—32no—
162#638—32no—
163#638—64no—
164#638—124no—
165#638C4×D312yes—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3—124no—
173C2×C3⋊Dic3C22×D312yes—
174C4×C3⋊D3—31no—
175C4×C3⋊D3—31no—
176C4×C3⋊D3—31no—
177C4×C3⋊D3—62no—
178C4×C3⋊D3—122no—
179C4×C3⋊D3C22×D311yescharacteristic
180#634—31no—
181#634—32no—
182#634—62no—
183#634—122no—
184#634C4×D311yescharacteristic
185C3×#67—24no—
186C3×#67—64no—
187C2×C3×Dic3—64no—
188C3×D12—22no—
189C3×C4×D3—22no—
190C3×C4×D3—62no—
191C3×Dic6—62no—
192#627—64no—
193#626—64no—
194#625—64no—
195#624—62no—
196D3×Dic3—64no—
197D3×Dic3—64no—
198D3×Dic3—64no—
199D3×Dic3—62no—
200D3×Dic3—62no—
201#622—31no—
202#622—32no—
203#622—31no—
204#622C22×D312yes—
205#622C22×D311yescharacteristic
206#622C22×D311yescharacteristic
207#568—271noSylow
208C2×C33—22no—
209C2×C33C23×C411yescharacteristic
210C3⋊(C3⋊D3)—21no—
211C3×C3⋊D3—21no—
212C32×D3—21no—
213C32×D3—22no—
214C3×#41—91no—
215C2×C3×D4—62no—
216C2×#67—184no—
217C22×Dic3—184no—
218#239—91no—
219D3×Q8—91no—
220#237—62no—
221#237—91no—
222#237—92no—
223#237—182no—
224#237—182no—
225#237—184no—
226#237—184no—
227#237—182no—
228#237—362no—
229D3×D4—92no—
230D3×D4#15711yescharacteristic
231#235—92no—
232#235—91no—
233C2×C4×D3—92no—
234C2×C4×D3—91no—
235C2×C4×D3—182no—
236C2×Dic6—182no—
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)—92no—
248ΓL(1,𝔽25)—92no—
249ΓL(1,𝔽25)—92no—
250ΓL(1,𝔽25)—91no—
251ΓL(1,𝔽25)—91no—
252D3×C8—92no—
253D3×C8—92no—
254D3×C8—92no—
255C22×C32—24no—
256C22×C32—31no—
257C22×C32—32no—
258C22×C32—44no—
259C22×C32—122no—
260C22×C32C2×C4×D312yes—
261C2×C3⋊D3—31no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—122no—
264C2×C3⋊D3—121no—
265C2×C3⋊D3C2×C4×D311yescharacteristic
266C2×C3×D3—22no—
267C2×C3×D3—24no—
268C2×C3×D3—48no—
269C2×C3×D3—62no—
270D32—124no—
271C32×C4—22no—
272C32×C4—31no—
273C32×C4—42no—
274C32×C4C2×C4×D311yescharacteristic
275C3⋊Dic3—32no—
276C3⋊Dic3—31no—
277C3⋊Dic3—62no—
278C3⋊Dic3—64no—
279C3⋊Dic3—124no—
280C3⋊Dic3—122no—
281C3⋊Dic3C2×C4×D312yes—
282C3⋊Dic3C23×D311yescharacteristic
283C3×Dic3—22no—
284C3×Dic3—64no—
285C3×Dic3—62no—
286C2×#41—271no—
287#131—91no—
288#131—271no—
289#131—272no—
290#131—272no—
291#131—271no—
292#131—271no—
293C2×M16—271no—
294C2×M16—272no—
295C2×M16—271no—
296C2×M16—272no—
297C33#56811yescharacteristic, complemented, Sylow
298C23×C3—64no—
299C22×D3—91no—
300C22×D3—182no—
301C22×D3C2×#15712yes—
302C3×Q8—91no—
303C3×D4—22no—
304C3×D4—42no—
305C3×D4—62no—
306C3×D4—64no—
307C3×D4—92no—
308C3×D4C2×#15711yescharacteristic
309C2×C3×C4—62no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312#67—64no—
313#67—92no—
314#67—92no—
315#67—184no—
316#67—184no—
317#67—184no—
318#67—364no—
319#67C2×#15712yes—
320C2×Dic3—64no—
321C2×Dic3—92no—
322C2×Dic3—91no—
323C2×Dic3—184no—
324C2×Dic3—182no—
325C2×Dic3—184no—
326C2×Dic3—184no—
327C2×Dic3—184no—
328C2×Dic3—364no—
329D12—92no—
330D12C2×#15711yescharacteristic
331C4×D3—62no—
332C4×D3—92no—
333C4×D3—91no—
334C4×D3—92no—
335C4×D3—92no—
336C4×D3—91no—
337C4×D3—91no—
338C4×D3—182no—
339C4×D3—182no—
340C4×D3—182no—
341C4×D3—362no—
342C4×D3C2×#15711yescharacteristic
343Dic6—62no—
344Dic6—91no—
345Dic6—92no—
346Dic6—184no—
347Dic6—182no—
348Dic6—362no—
349C3×C8—91no—
350C3×C8—92no—
351C3×C8—91no—
352C3⋊C8—91no—
353C3⋊C8—92no—
354C3⋊C8—91no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—42no—
358C2×C32—44no—
359C2×C32—61no—
360C2×C32—62no—
361C2×C32—84no—
362C2×C32C22×C4×D311yescharacteristic
363C3⋊D3—21no—
364C3⋊D3—61no—
365C3⋊D3—122no—
366C3⋊D3—242no—
367C3×D3—24no—
368C3×D3—44no—
369C3×D3—122no—
370#41—91no—
371#41—272no—
372#41—271no—
373#41—272no—
374#41—271no—
375#41—271no—
376C2×Q8—271no—
377C2×D4—31no—
378C2×D4—272no—
379C22×C4—271no—
380C22×C4—272no—
381M16—91no—
382M16—92no—
383M16—271no—
384M16—271no—
385M16—272no—
386M16—271no—
387M16—271no—
388M16—272no—
389M16—272no—
390M16—271no—
391M16—272no—
392C2×C8—91no—
393C2×C8—92no—
394C2×C8—272no—
395C2×C8—272no—
396C2×C8—272no—
397C2×C8—271no—
398C2×C8—272no—
399C22×C3—24no—
400C22×C3—44no—
401C22×C3—62no—
402C22×C3—64no—
403C22×C3—91no—
404C22×C3—128no—
405C22×C3C22×#15712yes—
406C2×D3—42no—
407C2×D3—62no—
408C2×D3—91no—
409C2×D3—182no—
410C2×D3—362no—
411C2×D3—361no—
412C2×D3—364no—
413C2×D3C22×#15711yescharacteristic
414C2×D3C22×#15712yes—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—62no—
418C3×C4—91no—
419C3×C4—92no—
420C3×C4C22×#15711yescharacteristic
421Dic3—64no—
422Dic3—62no—
423Dic3—92no—
424Dic3—91no—
425Dic3—184no—
426Dic3—182no—
427Dic3—364no—
428Dic3—362no—
429Dic3C22×#15711yescharacteristic
430C32—22no—
431C32—42no—
432C32#575011yescharacteristic, complemented
433C23—32no—
434C23—271no—
435Q8—91no—
436Q8—272no—
437Q8—271no—
438D4—32no—
439D4—31no—
440D4—92no—
441D4—272no—
442D4—272no—
443D4—272no—
444D4D3×#15711yescharacteristic
445C2×C4—31no—
446C2×C4—92no—
447C2×C4—91no—
448C2×C4—271no—
449C2×C4—271no—
450C2×C4—272no—
451C2×C4—271no—
452C2×C4—272no—
453C2×C4—272no—
454C2×C4—272no—
455C2×C4—271no—
456C8—91no—
457C8—91no—
458C8—92no—
459C8—272no—
460C8—271no—
461C8—271no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—44no—
465C2×C3—42no—
466C2×C3—64no—
467C2×C3—84no—
468C2×C3—124no—
469C2×C3—181no—
470C2×C3C23×#15711yescharacteristic
471D3—21no—
472D3—22no—
473D3—122no—
474D3—181no—
475D3—362no—
476D3—722no—
477C22—31no—
478C22—32no—
479C22—91no—
480C22—122no—
481C22—271no—
482C22—1081no—
483C22C2×D3×#15712yes—
484C4—31no—
485C4—91no—
486C4—92no—
487C4—271no—
488C4—272no—
489C4C2×D3×#15711yescharacteristic
490C3—22no—
491C3—42no—
492C3#1059664011yescharacteristic, complemented
493C2—22no—
494C2—61no—
495C2—62no—
496C2—181no—
497C2—541no—
498C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC33×D4C2×C4
prime radicalD3×#3830C2
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×#88925C2semidirect
5. #11724267C2semidirect
6. #11724266C2semidirect
7. D3×#3792C2semidirect
8. #11724255C2semidirect
9. #11724254C2semidirect
10. #11724251C2semidirect
11. #11724250C2semidirect
24. #88993C22semidirect
25. C2×C3⋊#622C22semidirect
26. #88989C22semidirect
27. #88988C22semidirect
28. C3×#3792C22semidirect
29. C2×C3×#622C22semidirect
30. C3×#3788C22semidirect
31. C3×#3787C22semidirect
32. #88926C22semidirect
33. #88926C22semidirect
37. #88925C22semidirect
38. #88925C22semidirect
39. #88924C22semidirect
40. D3×#622C22semidirect
41. D3×#622C22semidirect
51. #64287D3semidirect
113. #3792C2×D3semidirect
117. C2×#622C2×D3semidirect
121. #3788C2×D3semidirect
124. #3787C2×D3semidirect
297. C33#568semidirect
432. C32#5750semidirect
492. C3#10596640semidirect

Maximal subgroups

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

subgroupindex
D3×#38302
#117245152
C3×#642872
C2×#889252
#117242672
#117242662
D3×#37922
#117242552
#117242542
#117242512
#117242502
#105966403
#57509