Finite Group Databasebeta

← #13170499

Subgroups — #13170499

Order 1728.

1246 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11723350C212yesmaximal, complemented
5C3×#63814C212yesmaximal, complemented
6#11722822C214yesmaximal, complemented
7D3×#3711C214yesmaximal, complemented
8#10595441—32nomaximal
9#10594670—31nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C32×#235C2212yes—
12C3×#3811C412yes—
13C3×#3805C412yes—
14C3×#3802C411yescharacteristic
15C3×C4×D32C2212yes—
16C3×#3798C2212yes—
17C3×#3797C411yescharacteristic
18C3×#3796C412yes—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C3×#3748C2212yescomplemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88809C2212yes—
23#88808C2212yescomplemented
24#88787C2212yes—
25#88787C2212yescomplemented
26#88786C2214yescomplemented
27D3×C3⋊(C3⋊C8)C2214yescomplemented
28C3×#3711C2214yescomplemented
29C3×D3×C3⋊C8C2214yescomplemented
30C3×D3×#41—32no—
31#64306D311yescharacteristic
32#64157—32no—
33C2×#3743—31no—
34C3×#948—32no—
35C3×#901—32no—
36C2×#3711—32no—
37C2×#3711—34no—
38C2×#3711—61no—
39#63818—31no—
40#63815—32no—
41#63815—32no—
42#63815—34no—
43#63815—62no—
44#63814—34no—
45#63814D312yescomplemented
46D3×#208—32no—
47D3×#208—34no—
48#63804—34no—
49D3×ΓL(1,𝔽25)—34no—
50C22×C3×C3⋊D3C2×C411yescharacteristic
51C2×C3×D32C2×C412yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3C2×C411yescharacteristic
54C3×C4×C3⋊D3C2×C411yescharacteristic
55C3×C4×C3⋊D3C2×C411yescharacteristic
56C3×C4×C3⋊D3C2311yescharacteristic
57C3×C4×C3⋊D3C2311yescharacteristic
58C32×#67C2×C414yes—
59C32×D12C2×C412yes—
60C32×C4×D3C2314yes—
61C32×Dic6C2×C412yes—
62C3×#627C2×C412yes—
63C3×#626C2×C414yes—
64C3×#625C2×C412yes—
65C3×#624C2×C412yes—
66C3×D3×Dic3C2×C414yes—
67C3⋊(C3⋊(C3⋊C8))C2312yes—
68C3×C3⋊(C3⋊C8)C2314yes—
69C32×C3⋊C8C2312yes—
70#5819—91no—
71#5750—92no—
72C32×#41—32no—
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C3×#239—32no—
75C3×D3×Q8—32no—
76C3×#237—34no—
77C3×#237—32no—
78C3×D3×D4—34no—
79C3×D3×D4—32no—
80C3×#235—22no—
81C3×#235—34no—
82C3×#235C2×D312yes—
83C3×#235C2×D312yes—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—34no—
86C2×C3×C4×D3—61no—
87#3811Dic312yes—
88#3805Dic312yes—
89#3802Dic311yescharacteristic
90C4×D32C2×D312yes—
91#3798C2×D312yes—
92#3797Dic311yescharacteristic
93#3796Dic312yes—
94C2×C32×C8—31no—
95#3748—34no—
96#3748C2×D312yescomplemented
97C2×C3⋊(C3⋊C8)—31no—
98C2×C3⋊(C3⋊C8)—32no—
99C2×C3⋊(C3⋊C8)—34no—
100C2×C3⋊(C3⋊C8)—62no—
101C2×C3⋊(C3⋊C8)—61no—
102C2×C3⋊(C3⋊C8)—121no—
103#3743—32no—
104#3743—32no—
105C3×#208—32no—
106C3×#208—32no—
107C3×#208—34no—
108C3×#208—62no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—34no—
111C2×C3×C3⋊C8C2×D312yescomplemented
112C3×ΓL(1,𝔽25)—34no—
113C3×D3×C8—34no—
114#3712—32no—
115#3711—34no—
116#3711—34no—
117#3711—34no—
118#3711—32no—
119#3711—32no—
120#3711—32no—
121#3711—32no—
122#3711—62no—
123#3711—62no—
124#3711—64no—
125#3711C2×D314yescomplemented
126#3710—32no—
127D3×C3⋊C8—34no—
128D3×C3⋊C8—34no—
129D3×C3⋊C8—34no—
130D3×C3⋊C8—64no—
131D3×C3⋊C8C2×D314yescomplemented
132C22×C33C22×C411yescharacteristic
133C2×C3×C3⋊D3—41no—
134C2×C3×C3⋊D3C22×C411yescharacteristic
135C2×C3×C3⋊D3C22×C411yescharacteristic
136C2×C32×D3C22×C414yes—
137C3×D32—42no—
138C33×C4C22×C411yescharacteristic
139C33×C4C2411yescharacteristic
140C3×C3⋊Dic3C22×C411yescharacteristic
141C3×C3⋊Dic3C22×C411yescharacteristic
142C32×Dic3C22×C414yes—
143C2×C3×#41—91no—
144D3×#41—32no—
145C3×#131—92no—
146#948—32no—
147#948—94no—
148#948—92no—
149#948—92no—
150#948—182no—
151C2×#208—94no—
152C2×#208—91no—
153C2×#208—91no—
154#907—94no—
155#907—92no—
156D3×M16—94no—
157D3×M16—92no—
158#901—32no—
159#901—94no—
160C2×ΓL(1,𝔽25)—94no—
161C2×ΓL(1,𝔽25)—92no—
162C2×ΓL(1,𝔽25)—181no—
163C22×C3⋊D3C2×Dic311yescharacteristic
164C22×C3×D3—32no—
165C22×C3×D3—34no—
166C22×C3×D3—61no—
167C2×D32C2×Dic312yes—
168C32×Q8—32no—
169C32×D4—34no—
170C32×D4—32no—
171C2×C32×C4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—34no—
174C2×C32×C4—41no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3C2×Dic311yescharacteristic
178C4×C3⋊D3C2×Dic311yescharacteristic
179C4×C3⋊D3C2×Dic311yescharacteristic
180C4×C3⋊D3C22×D311yescharacteristic
181C4×C3⋊D3C22×D311yescharacteristic
182C3×#67—24no—
183C3×#67—34no—
184C3×#67—34no—
185C3×#67C4×D314yes—
186C3×#67C2×Dic314yes—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—61no—
190C3×D12—22no—
191C3×D12—34no—
192C3×D12C4×D312yes—
193C3×D12C2×Dic312yes—
194C3×C4×D3—24no—
195C3×C4×D3—34no—
196C3×C4×D3—32no—
197C3×C4×D3—32no—
198C3×C4×D3—32no—
199C3×C4×D3—32no—
200C3×C4×D3—34no—
201C3×C4×D3—61no—
202C3×C4×D3—61no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3C22×D314yes—
206C3×C4×D3C22×D314yes—
207C3×Dic6—22no—
208C3×Dic6—34no—
209C3×Dic6C4×D312yes—
210C3×Dic6C2×Dic312yes—
211#627C2×Dic312yes—
212#626C2×Dic314yes—
213#625C2×Dic312yes—
214#624C2×Dic312yes—
215D3×Dic3C2×Dic314yes—
216C32×C8—32no—
217C3⋊(C3⋊C8)—32no—
218C3⋊(C3⋊C8)—32no—
219C3⋊(C3⋊C8)—32no—
220C3⋊(C3⋊C8)—62no—
221C3⋊(C3⋊C8)—62no—
222C3⋊(C3⋊C8)—62no—
223C3⋊(C3⋊C8)—122no—
224C3⋊(C3⋊C8)C22×D314yes—
225C3×C3⋊C8—22no—
226C3×C3⋊C8—34no—
227C3×C3⋊C8—34no—
228C3×C3⋊C8—64no—
229C3×C3⋊C8C22×D312yes—
230C3×C3⋊C8C22×D312yes—
231#568—271noSylow
232C2×C33—21no—
233C2×C33C23×C411yescharacteristic
234C3×C3⋊D3—21no—
235C3×C3⋊D3—21no—
236C32×D3—24no—
237C3×#41—32no—
238C3×#41—32no—
239C3×#41—62no—
240C3×#41—92no—
241C3×#41—92no—
242C3×#41—92no—
243C2×C3×Q8—91no—
244C2×C3×D4—92no—
245C2×C3×D4—91no—
246C22×C3×C4—92no—
247C22×C3×C4—91no—
248#239—32no—
249D3×Q8—32no—
250#237—34no—
251#237—32no—
252D3×D4—34no—
253D3×D4—32no—
254#235—34no—
255#235D3212yes—
256C2×C4×D3—32no—
257C2×C4×D3—34no—
258C2×C4×D3—61no—
259C3×M16—94no—
260C3×M16—92no—
261C2×C3×C8—32no—
262C2×C3×C8—61no—
263C2×C3×C8—94no—
264#208—32no—
265#208—32no—
266#208—34no—
267#208—62no—
268#208—92no—
269#208—94no—
270#208—94no—
271#208—92no—
272#208—92no—
273#208—184no—
274C2×C3⋊C8—34no—
275C2×C3⋊C8—92no—
276C2×C3⋊C8—94no—
277C2×C3⋊C8—94no—
278C2×C3⋊C8—92no—
279C2×C3⋊C8—91no—
280C2×C3⋊C8—184no—
281C2×C3⋊C8—181no—
282C2×C3⋊C8—182no—
283C2×C3⋊C8—361no—
284C2×C3⋊C8D3211yescharacteristic, complemented
285ΓL(1,𝔽25)—34no—
286ΓL(1,𝔽25)—94no—
287ΓL(1,𝔽25)—92no—
288ΓL(1,𝔽25)—92no—
289ΓL(1,𝔽25)—92no—
290ΓL(1,𝔽25)—92no—
291ΓL(1,𝔽25)—94no—
292ΓL(1,𝔽25)—182no—
293ΓL(1,𝔽25)—182no—
294D3×C8—34no—
295D3×C8—94no—
296D3×C8—94no—
297C22×C32—21no—
298C22×C32—22no—
299C22×C32—34no—
300C22×C32—41no—
301C22×C32C2×C4×D312yes—
302C22×C32C22×Dic311yescharacteristic
303C2×C3⋊D3—41no—
304C2×C3⋊D3C22×Dic311yescharacteristic
305C2×C3⋊D3C22×Dic311yescharacteristic
306C2×C3×D3—24no—
307C2×C3×D3—32no—
308C2×C3×D3—32no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—124no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314C2×C3×D3C2×C4×D314yes—
315C2×C3×D3C22×Dic314yes—
316D32—42no—
317C32×C4—21no—
318C32×C4—22no—
319C32×C4—21no—
320C32×C4—22no—
321C32×C4—34no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4C2×C4×D312yes—
325C32×C4C22×Dic311yescharacteristic
326C32×C4C23×D312yes—
327C32×C4C23×D311yescharacteristic
328C3⋊Dic3C22×Dic311yescharacteristic
329C3⋊Dic3C22×Dic311yescharacteristic
330C3×Dic3—24no—
331C3×Dic3—32no—
332C3×Dic3—32no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3C2×C4×D314yes—
336C3×Dic3C22×Dic314yes—
337C2×#41—91no—
338#131—92no—
339#131—274no—
340#131—272no—
341C2×M16—271no—
342C2×M16—274no—
343C2×M16—271no—
344C33#56811yescharacteristic, complemented, Sylow
345C23×C3—92no—
346C23×C3—91no—
347C22×D3—32no—
348C22×D3—34no—
349C22×D3—61no—
350C3×Q8—32no—
351C3×Q8—32no—
352C3×Q8—62no—
353C3×Q8—92no—
354C3×D4—34no—
355C3×D4—34no—
356C3×D4—32no—
357C3×D4—32no—
358C3×D4—62no—
359C3×D4—64no—
360C3×D4—94no—
361C3×D4—92no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—34no—
365C2×C3×C4—34no—
366C2×C3×C4—41no—
367C2×C3×C4—64no—
368C2×C3×C4—92no—
369C2×C3×C4—94no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3212yes—
377#67—34no—
378#67—34no—
379#67D3×Dic314yes—
380C2×Dic3—32no—
381C2×Dic3—34no—
382C2×Dic3—61no—
383D12—34no—
384D12D3×Dic312yes—
385C4×D3—34no—
386C4×D3—32no—
387C4×D3—32no—
388C4×D3—32no—
389C4×D3—32no—
390C4×D3—34no—
391C4×D3—61no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—61no—
395C4×D3C2×D3214yes—
396Dic6—34no—
397Dic6D3×Dic312yes—
398C3×C8—32no—
399C3×C8—32no—
400C3×C8—62no—
401C3×C8—94no—
402C3⋊C8—34no—
403C3⋊C8—34no—
404C3⋊C8—64no—
405C3⋊C8—92no—
406C3⋊C8—92no—
407C3⋊C8—92no—
408C3⋊C8—182no—
409C3⋊C8—182no—
410C3⋊C8—182no—
411C3⋊C8—362no—
412C3⋊C8C2×D3212yes—
413C2×C32—21no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—21no—
417C2×C32—22no—
418C2×C32—41no—
419C2×C32—42no—
420C2×C32—42no—
421C2×C32—64no—
422C2×C32C22×C4×D312yes—
423C2×C32C23×Dic311yescharacteristic
424C3⋊D3—21no—
425C3⋊D3—21no—
426C3×D3—24no—
427C3×D3—24no—
428C3×D3—44no—
429C3×D3—62no—
430C3×D3—62no—
431C3×D3—121no—
432C3×D3—121no—
433#41—32no—
434#41—92no—
435#41—92no—
436#41—92no—
437C2×Q8—91no—
438C2×D4—92no—
439C2×D4—91no—
440C22×C4—92no—
441C22×C4—91no—
442M16—94no—
443M16—92no—
444M16—272no—
445M16—272no—
446M16—272no—
447M16—274no—
448C2×C8—31no—
449C2×C8—94no—
450C2×C8—272no—
451C2×C8—274no—
452C2×C8—271no—
453C22×C3—21no—
454C22×C3—22no—
455C22×C3—34no—
456C22×C3—34no—
457C22×C3—41no—
458C22×C3—64no—
459C22×C3—91no—
460C22×C3—91no—
461C22×C3—361no—
462C22×C3—362no—
463C22×C3C4×D3211yescharacteristic
464C22×C3C2×D3×Dic312yes—
465C2×D3—32no—
466C2×D3—32no—
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—124no—
470C2×D3—122no—
471C2×D3—122no—
472C2×D3C2×D3×Dic314yes—
473C3×C4—22no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—34no—
478C3×C4—34no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—64no—
482C3×C4—91no—
483C3×C4—91no—
484C3×C4C4×D3211yescharacteristic
485C3×C4C2×D3×Dic312yes—
486C3×C4C22×D3211yescharacteristic
487C3×C4C22×D3212yes—
488Dic3—32no—
489Dic3—32no—
490Dic3—61no—
491Dic3—61no—
492Dic3C2×D3×Dic314yes—
493C32—21no—
494C32—22no—
495C32—41no—
496C32#575012yescomplemented
497C32#581911yescharacteristic, complemented
498C23—92no—
499C23—91no—
500Q8—32no—
501Q8—92no—
502D4—34no—
503D4—32no—
504D4—94no—
505D4—92no—
506C2×C4—34no—
507C2×C4—92no—
508C2×C4—94no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—91no—
514C2×C4D3311yescharacteristic
515C8—32no—
516C8—94no—
517C8—272no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—42no—
524C2×C3—42no—
525C2×C3—41no—
526C2×C3—64no—
527C2×C3—64no—
528C2×C3—124no—
529C2×C3—181no—
530C2×C3—181no—
531C2×C3C2×C4×D3211yescharacteristic
532C2×C3C22×D3×Dic312yes—
533D3—24no—
534D3—62no—
535D3—62no—
536D3—121no—
537D3—121no—
538C22—34no—
539C22—91no—
540C22—91no—
541C22—361no—
542C22—362no—
543C22D32×Dic311yescharacteristic
544C4—34no—
545C4—91no—
546C4—91no—
547C4D32×Dic311yescharacteristic
548C4C2×D3311yescharacteristic
549C3—21no—
550C3—22no—
551C3—41no—
552C3#1059467011yescharacteristic, complemented
553C3#1059544112yescomplemented
554C2—21no—
555C2—64no—
556C2—181no—
557C2—181no—
558C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, FrattiniC4C2×D33
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C23
prime radicalC3×#64306C2
cosocleC33×C4C24
hypercenterC2×C4D33
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C2×#88787C2semidirect
4. #11723350C2semidirect
5. C3×#63814C2semidirect
6. #11722822C2semidirect
7. D3×#3711C2semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C3×#3748C22semidirect
21. C2×C32×C3⋊C8C22semidirect
23. #88808C22semidirect
25. #88787C22semidirect
26. #88786C22semidirect
27. D3×C3⋊(C3⋊C8)C22semidirect
28. C3×#3711C22semidirect
29. C3×D3×C3⋊C8C22semidirect
45. #63814D3semidirect
96. #3748C2×D3semidirect
111. C2×C3×C3⋊C8C2×D3semidirect
125. #3711C2×D3semidirect
131. D3×C3⋊C8C2×D3semidirect
284. C2×C3⋊C8D32semidirect
344. C33#568semidirect
496. C32#5750semidirect
497. C32#5819semidirect
552. C3#10594670semidirect
553. C3#10595441semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117236562
C2×#887872
#117233502
C3×#638142
#117228222
D3×#37112
#105954413
#105946703