Finite Group Databasebeta

← #13170519

Subgroups — #13170519

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
1#11724618C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic, complemented
3#11723354C212yesmaximal, complemented
4C3×#63818C211yesmaximal, characteristic, complemented
5C3×#63817C212yesmaximal, complemented
6#11722826C214yesmaximal, complemented
7#11722820C214yesmaximal, complemented
8#10594706—32nomaximal
9#10594670—31nomaximal
10C3×#3828C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12#89049C412yes—
13#89046C412yes—
14C4⋊#1113C411yescharacteristic
15C4×#1113C2212yes—
16#89042C2212yes—
17#89041C412yes—
18#89040C411yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C32×#208C2212yescomplemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88787C2212yes—
23#88787C2214yescomplemented
24#88787C2212yescomplemented
25C3⋊D3×C3⋊C8C2214yescomplemented
26C3×#3712C2212yes—
27C3×#3712C2214yescomplemented
28C3×#3710C2214yescomplemented
29C3×#3710C2212yescomplemented
30#64306—31no—
31D3×#235—32no—
32#64116—32no—
33C2×#3743—31no—
34C3×#907—32no—
35C3×D3×M16—32no—
36C3×#901—32no—
37C2×#3711—32no—
38C2×#3711—61no—
39#63818D311yescharacteristic, complemented
40#63817D312yescomplemented
41#63815—32no—
42#63815—32no—
43#63815—62no—
44#63804—34no—
45#63804—34no—
46#63803—34no—
47#63791—34no—
48D3×ΓL(1,𝔽25)—34no—
49D3×ΓL(1,𝔽25)—34no—
50C22×C3×C3⋊D3C2×C411yescharacteristic
51C2×#1113C2×C412yes—
52C2×C33×C4C2311yescharacteristic
53C3×#638C2×C414yes—
54C2×C3×C3⋊Dic3C2×C411yescharacteristic
55C3×C4⋊(C3⋊D3)C2×C412yes—
56C3×C4×C3⋊D3C2×C411yescharacteristic
57C3×C4×C3⋊D3C2×C411yescharacteristic
58C3×C4×C3⋊D3C2311yescharacteristic
59C3×C4×C3⋊D3C2314yes—
60C3×C4×C3⋊D3C2311yescharacteristic
61C3×#634C2×C412yes—
62#6296C2×C412yes—
63#6295C2×C414yes—
64#6295C2×C412yes—
65#6294C2×C412yes—
66#6294C2×C414yes—
67C3⋊(C3⋊(C3⋊C8))C2312yes—
68C32×C3⋊C8C2314yes—
69C32×C3⋊C8C2312yes—
70#5750—92no—
71#5745—91no—
72#3828C2×D312yes—
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C3×#235—32no—
75C3×#235—32no—
76C3×#235—62no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—61no—
79#3811—32no—
80D3×#67—34no—
81#3805—32no—
82#3804—34no—
83#3802—31no—
84D3×D12—32no—
85C4×D32—32no—
86C4×D32—34no—
87#3799—32no—
88#3798—34no—
89#3798—32no—
90#3797—31no—
91#3796—32no—
92#3795—32no—
93D3×Dic6—32no—
94C32×M16—32no—
95C2×C32×C8—31no—
96C2×C3⋊(C3⋊C8)—31no—
97C2×C3⋊(C3⋊C8)—32no—
98C2×C3⋊(C3⋊C8)—62no—
99C2×C3⋊(C3⋊C8)—61no—
100C2×C3⋊(C3⋊C8)—121no—
101#3743—32no—
102#3743—34no—
103#3743—32no—
104C8×C3⋊D3—34no—
105C3×#208—22no—
106C3×#208C2×D312yescomplemented
107C3×#208C2×D312yescomplemented
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8C2×D312yescomplemented
110C3×ΓL(1,𝔽25)—34no—
111C3×ΓL(1,𝔽25)—34no—
112C3×ΓL(1,𝔽25)—34no—
113C3×D3×C8—34no—
114C3×D3×C8—34no—
115C3×D3×C8—34no—
116#3712C2×D312yes—
117#3712C2×D314yescomplemented
118#3711—32no—
119#3711—34no—
120#3711—32no—
121#3711—34no—
122#3711—32no—
123#3711—32no—
124#3711—64no—
125#3711—62no—
126#3711—62no—
127#3710C2×D314yescomplemented
128#3710C2×D312yescomplemented
129D3×C3⋊C8—34no—
130D3×C3⋊C8—34no—
131D3×C3⋊C8—64no—
132C22×C33C22×C411yescharacteristic
133C2×C3×C3⋊D3—41no—
134C2×C3×C3⋊D3C22×C411yescharacteristic
135C2×C3×C3⋊D3C22×C414yes—
136C2×C3×C3⋊D3C22×C411yescharacteristic
137#1113—42no—
138C33×C4C22×C411yescharacteristic
139C33×C4C2411yescharacteristic
140C3×C3⋊Dic3C22×C411yescharacteristic
141C3×C3⋊Dic3C22×C414yes—
142C3×C3⋊Dic3C22×C411yescharacteristic
143D3×#41—92no—
144C2×#235—91no—
145C3×#131—92no—
146C2×C3×M16—91no—
147#948—92no—
148C2×#208—91no—
149#907—32no—
150#907—94no—
151#907—92no—
152#907—92no—
153#907—182no—
154D3×M16—32no—
155D3×M16—94no—
156D3×M16—94no—
157#901—32no—
158#901—94no—
159#901—94no—
160C2×ΓL(1,𝔽25)—94no—
161C2×ΓL(1,𝔽25)—92no—
162C2×ΓL(1,𝔽25)—181no—
163C22×C3⋊D3C4×D311yescharacteristic
164C22×C3×D3—32no—
165C22×C3×D3—61no—
166C2×D32—34no—
167C2×D32—32no—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—41no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173#638C4×D314yes—
174C2×C3⋊Dic3C4×D311yescharacteristic
175C4⋊(C3⋊D3)C4×D312yes—
176C4×C3⋊D3C4×D311yescharacteristic
177C4×C3⋊D3C4×D311yescharacteristic
178C4×C3⋊D3C22×D311yescharacteristic
179C4×C3⋊D3C22×D314yes—
180C4×C3⋊D3C22×D311yescharacteristic
181#634C4×D312yes—
182C3×#67—34no—
183C3×#67—34no—
184C3×#67—64no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—61no—
187C3×D12—32no—
188C3×D12—32no—
189C3×D12—62no—
190C3×C4×D3—32no—
191C3×C4×D3—32no—
192C3×C4×D3—34no—
193C3×C4×D3—34no—
194C3×C4×D3—32no—
195C3×C4×D3—32no—
196C3×C4×D3—64no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—61no—
200C3×C4×D3—61no—
201C3×Dic6—32no—
202C3×Dic6—32no—
203C3×Dic6—62no—
204#627—34no—
205#627—32no—
206#626—34no—
207#626—34no—
208#626—34no—
209#625—34no—
210#625—32no—
211#624—32no—
212#624—34no—
213D3×Dic3—34no—
214D3×Dic3—34no—
215D3×Dic3—34no—
216C32×C8—34no—
217C32×C8—32no—
218C3⋊(C3⋊C8)—32no—
219C3⋊(C3⋊C8)—32no—
220C3⋊(C3⋊C8)—32no—
221C3⋊(C3⋊C8)—62no—
222C3⋊(C3⋊C8)—62no—
223C3⋊(C3⋊C8)—62no—
224C3⋊(C3⋊C8)—122no—
225C3×C3⋊C8—24no—
226C3×C3⋊C8—22no—
227C3×C3⋊C8C22×D314yes—
228C3×C3⋊C8C22×D314yes—
229C3×C3⋊C8C22×D312yes—
230C3×C3⋊C8C22×D312yes—
231#568—271noSylow
232C2×C33—21no—
233C2×C33C23×C411yescharacteristic
234C3×C3⋊D3—21no—
235C3×C3⋊D3—24no—
236C3×C3⋊D3—21no—
237C3×#41—92no—
238C22×C3×C4—91no—
239C2×#67—92no—
240#239—92no—
241D3×Q8—92no—
242#237—92no—
243#237—94no—
244D3×D4—92no—
245D3×D4—94no—
246#235—32no—
247#235—32no—
248#235—62no—
249#235—92no—
250#235—92no—
251#235—94no—
252#235—92no—
253C2×D12—91no—
254C2×C4×D3—32no—
255C2×C4×D3—61no—
256C2×C4×D3—94no—
257C2×C4×D3—92no—
258C2×Dic6—91no—
259C3×M16—32no—
260C3×M16—32no—
261C3×M16—62no—
262C3×M16—92no—
263C3×M16—94no—
264C2×C3×C8—32no—
265C2×C3×C8—61no—
266C2×C3×C8—94no—
267C2×C3×C8—92no—
268#208—92no—
269#208—94no—
270#208—92no—
271#208D3212yescomplemented
272C2×C3⋊C8—92no—
273C2×C3⋊C8—91no—
274C2×C3⋊C8—94no—
275C2×C3⋊C8—181no—
276C2×C3⋊C8—182no—
277C2×C3⋊C8—361no—
278C2×C3⋊C8D3211yescharacteristic, complemented
279ΓL(1,𝔽25)—34no—
280ΓL(1,𝔽25)—34no—
281ΓL(1,𝔽25)—34no—
282ΓL(1,𝔽25)—92no—
283ΓL(1,𝔽25)—92no—
284ΓL(1,𝔽25)—92no—
285ΓL(1,𝔽25)—92no—
286ΓL(1,𝔽25)—94no—
287ΓL(1,𝔽25)—94no—
288ΓL(1,𝔽25)—182no—
289ΓL(1,𝔽25)—182no—
290ΓL(1,𝔽25)—184no—
291D3×C8—34no—
292D3×C8—34no—
293D3×C8—34no—
294D3×C8—94no—
295D3×C8—94no—
296D3×C8—184no—
297C22×C32—21no—
298C22×C32—22no—
299C22×C32—41no—
300C22×C32C2×C4×D312yes—
301C22×C32C2×C4×D311yescharacteristic
302C2×C3⋊D3—41no—
303C2×C3⋊D3C2×C4×D311yescharacteristic
304C2×C3⋊D3C2×C4×D314yes—
305C2×C3⋊D3C2×C4×D311yescharacteristic
306C2×C3×D3—32no—
307C2×C3×D3—34no—
308C2×C3×D3—34no—
309C2×C3×D3—32no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—64no—
313C2×C3×D3—122no—
314C2×C3×D3—122no—
315D32—122no—
316D32—124no—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4C2×C4×D311yescharacteristic
324C32×C4C2×C4×D312yes—
325C32×C4C23×D311yescharacteristic
326C32×C4C23×D312yes—
327C3⋊Dic3C2×C4×D311yescharacteristic
328C3⋊Dic3C2×C4×D314yes—
329C3⋊Dic3C2×C4×D311yescharacteristic
330C3×Dic3—32no—
331C3×Dic3—34no—
332C3×Dic3—34no—
333C3×Dic3—32no—
334C3×Dic3—64no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C2×#41—271no—
338#131—92no—
339#131—274no—
340#131—272no—
341C2×M16—91no—
342C2×M16—271no—
343C2×M16—274no—
344C33#56811yescharacteristic, complemented, Sylow
345C23×C3—91no—
346C22×D3—32no—
347C22×D3—61no—
348C22×D3—92no—
349C22×D3—94no—
350C3×Q8—92no—
351C3×D4—94no—
352C3×D4—92no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—41no—
356C2×C3×C4—91no—
357C2×C3×C4—94no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3212yes—
364#67—34no—
365#67—34no—
366#67—64no—
367#67—94no—
368#67—94no—
369#67—94no—
370C2×Dic3—32no—
371C2×Dic3—61no—
372C2×Dic3—92no—
373C2×Dic3—94no—
374D12—32no—
375D12—32no—
376D12—62no—
377D12—92no—
378D12—94no—
379C4×D3—32no—
380C4×D3—34no—
381C4×D3—34no—
382C4×D3—32no—
383C4×D3—32no—
384C4×D3—32no—
385C4×D3—61no—
386C4×D3—61no—
387C4×D3—64no—
388C4×D3—61no—
389C4×D3—61no—
390C4×D3—94no—
391C4×D3—94no—
392C4×D3—94no—
393Dic6—32no—
394Dic6—32no—
395Dic6—62no—
396Dic6—92no—
397Dic6—94no—
398C3×C8—32no—
399C3×C8—34no—
400C3×C8—34no—
401C3×C8—32no—
402C3×C8—64no—
403C3×C8—62no—
404C3⋊C8—92no—
405C3⋊C8—92no—
406C3⋊C8—92no—
407C3⋊C8—182no—
408C3⋊C8—182no—
409C3⋊C8—182no—
410C3⋊C8—362no—
411C3⋊C8C2×D3214yes—
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×C32C22×C4×D311yescharacteristic
422C2×C32C22×C4×D312yes—
423C3⋊D3—21no—
424C3⋊D3—21no—
425C3⋊D3—24no—
426C3×D3—64no—
427C3×D3—62no—
428C3×D3—64no—
429C3×D3—62no—
430C3×D3—124no—
431C3×D3—121no—
432C3×D3—121no—
433#41—92no—
434#41—272no—
435#41—272no—
436#41—272no—
437C2×Q8—271no—
438C2×D4—271no—
439C2×D4—272no—
440C22×C4—91no—
441C22×C4—272no—
442M16—32no—
443M16—92no—
444M16—94no—
445M16—272no—
446M16—272no—
447M16—274no—
448C2×C8—31no—
449C2×C8—92no—
450C2×C8—94no—
451C2×C8—274no—
452C2×C8—271no—
453C22×C3—21no—
454C22×C3—22no—
455C22×C3—41no—
456C22×C3—91no—
457C22×C3—94no—
458C22×C3—91no—
459C22×C3—361no—
460C22×C3C4×D3212yes—
461C22×C3C4×D3211yescharacteristic
462C2×D3—32no—
463C2×D3—32no—
464C2×D3—34no—
465C2×D3—34no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—64no—
469C2×D3—122no—
470C2×D3—122no—
471C2×D3—362no—
472C2×D3—364no—
473C3×C4—22no—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—21no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—91no—
480C3×C4—94no—
481C3×C4—91no—
482C3×C4C4×D3211yescharacteristic
483C3×C4C4×D3212yes—
484C3×C4C22×D3211yescharacteristic
485C3×C4C22×D3212yes—
486Dic3—34no—
487Dic3—34no—
488Dic3—32no—
489Dic3—32no—
490Dic3—64no—
491Dic3—61no—
492Dic3—61no—
493C32—21no—
494C32—22no—
495C32—41no—
496C32#574511yescharacteristic, complemented
497C32#575012yescomplemented
498C23—91no—
499C23—272no—
500Q8—92no—
501Q8—272no—
502D4—94no—
503D4—92no—
504D4—272no—
505D4—274no—
506C2×C4—91no—
507C2×C4—91no—
508C2×C4—94no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—272no—
513C2×C4—274no—
514C2×C4D3311yescharacteristic
515C8—32no—
516C8—34no—
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—181no—
527C2×C3—181no—
528C2×C3—184no—
529C2×C3C2×C4×D3211yescharacteristic
530C2×C3C2×C4×D3212yes—
531D3—64no—
532D3—62no—
533D3—62no—
534D3—64no—
535D3—121no—
536D3—121no—
537D3—124no—
538C22—91no—
539C22—91no—
540C22—94no—
541C22—361no—
542C22—1082no—
543C22#8894911yescharacteristic
544C4—91no—
545C4—91no—
546C4—94no—
547C4#8894911yescharacteristic
548C4C2×D3311yescharacteristic
549C3—21no—
550C3—22no—
551C3—41no—
552C3#1059467011yescharacteristic, complemented
553C3#1059470612yescomplemented
554C2—21no—
555C2—181no—
556C2—181no—
557C2—184no—
558C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC4C2×D33
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C23
prime radical#11724618C2
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. C2×#88787C2semidirect
3. #11723354C2semidirect
4. C3×#63818C2semidirect
5. C3×#63817C2semidirect
6. #11722826C2semidirect
7. #11722820C2semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C32×#208C22semidirect
21. C2×C32×C3⋊C8C22semidirect
23. #88787C22semidirect
24. #88787C22semidirect
25. C3⋊D3×C3⋊C8C22semidirect
27. C3×#3712C22semidirect
28. C3×#3710C22semidirect
29. C3×#3710C22semidirect
39. #63818D3semidirect
40. #63817D3semidirect
106. C3×#208C2×D3semidirect
107. C3×#208C2×D3semidirect
109. C2×C3×C3⋊C8C2×D3semidirect
117. #3712C2×D3semidirect
127. #3710C2×D3semidirect
128. #3710C2×D3semidirect
271. #208D32semidirect
278. C2×C3⋊C8D32semidirect
344. C33#568semidirect
496. C32#5745semidirect
497. C32#5750semidirect
552. C3#10594670semidirect
553. C3#10594706semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×#887872
#117233542
C3×#638182
C3×#638172
#117228262
#117228202
#105947063
#105946703