Finite Group Databasebeta

← #13149849

Subgroups — #13149849

Order 1728.

842 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722475C211yesmaximal, characteristic
2M16×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#907C211yesmaximal, characteristic, complemented
4C3×#63469C211yesmaximal, characteristic, complemented
5#11720877C211yesmaximal, characteristic, complemented
6C2×#88414C211yesmaximal, characteristic, complemented
7#11720858C211yesmaximal, characteristic, complemented
8#11720809C212yesmaximal, complemented
9#11720806C212yesmaximal, complemented
10D3×#3663C212yesmaximal, complemented
11D9×ΓL(1,𝔽25)C212yesmaximal, complemented
12#10594708—31nomaximal
13#10589852—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C9×#235C2211yescharacteristic
16C3×#3697C2211yescharacteristic
17#88668C412yes—
18#88663C412yes—
19#88646C411yescharacteristic
20C4×D3×D9C2212yes—
21#88640C2212yes—
22#88639C411yescharacteristic
23(C3⋊D9)⋊Q8C411yescharacteristic
24#88634C411yescharacteristic
25C3×C9×M16C2211yescharacteristic, complemented
26#88532C2211yescharacteristic, complemented
27#88523C2212yes—
28C8×C3⋊D9C2212yescomplemented
29C2×C9×C3⋊C8C2211yescharacteristic, complemented
30C2×C3×C9⋊C8C2211yescharacteristic, complemented
31C9×ΓL(1,𝔽25)C2212yescomplemented
32D3×C8×C9C2212yescomplemented
33C3×#3663C2212yescomplemented
34C3×C8×D9C2212yescomplemented
35#88420C2212yescomplemented
36D3×C9⋊C8C2212yescomplemented
37#88414C2211yescharacteristic
38#88414C2211yescharacteristic
39#88414C2211yescharacteristic, complemented
40#88414C2211yescharacteristic, complemented
41#88413C2212yescomplemented
42D9×C3⋊C8C2212yescomplemented
43#64306—31no—
44M16×C3⋊D3—31no—
45C3×#907—31no—
46C3×#907D311yescharacteristic
47C2×#3712—31no—
48#63814—31no—
49#63814—31no—
50#63805—32no—
51#63803—32no—
52D3×ΓL(1,𝔽25)—32no—
53D3×ΓL(1,𝔽25)—32no—
54#41×D9—31no—
55C9×#131—31no—
56#63510—31no—
57#63469—32no—
58#63469D311yescharacteristic, complemented
59M16×D9—31no—
60M16×D9—32no—
61M16×D9—61no—
62#63464—31no—
63#63464—32no—
64C2×#3663—31no—
65C2×#3663—32no—
66C22×C3⋊D9C2×C411yescharacteristic
67C2×D3×D9C2×C412yes—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3⋊Dic9C2×C411yescharacteristic
70C4×C3⋊D9C2×C411yescharacteristic
71C4×C3⋊D9C2×C411yescharacteristic
72C4×C3⋊D9C2311yescharacteristic
73C4×C3⋊D9C2311yescharacteristic
74C9×#67C2×C412yes—
75C3×#611C2×C412yes—
76C9×D12C2×C411yescharacteristic
77C4×D3×C9C2312yes—
78C3×D36C2×C411yescharacteristic
79C3×C4×D9C2312yes—
80C9×Dic6C2×C411yescharacteristic
81C3×Dic18C2×C411yescharacteristic
82#6195C2×C412yes—
83#6194C2×C412yes—
84D3×Dic9C2×C412yes—
85#6192C2×C412yes—
86(C3⋊D9)⋊C4C2×C412yes—
87Dic3×D9C2×C412yes—
88#6189C2×C412yes—
89C3×C8×C9C2312yes—
90C3⋊(C9⋊C8)C2312yes—
91C9×C3⋊C8C2311yescharacteristic
92C9×C3⋊C8C2311yescharacteristic
93C3×C9⋊C8C2311yescharacteristic
94C3×C9⋊C8C2311yescharacteristic
95#5750—91no—
96#5750—91no—
97C2×C4×C3⋊D3—31no—
98C3×#235—31no—
99C3×#235C2×D311yescharacteristic
100#3811—32no—
101#3805—32no—
102#3802—31no—
103C4×D32—32no—
104#3798—32no—
105#3797—31no—
106#3796—31no—
107#3796—31no—
108C32×M16C2×D311yescharacteristic
109#3748—31no—
110#3743—32no—
111C8×C3⋊D3—32no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C3×ΓL(1,𝔽25)—32no—
115C3×ΓL(1,𝔽25)C2×D312yes—
116C3×D3×C8—32no—
117C3×D3×C8C2×D312yes—
118#3712—31no—
119#3712—31no—
120#3712—31no—
121#3712—31no—
122#3711—32no—
123#3711—32no—
124D3×C3⋊C8—32no—
125D3×C3⋊C8—32no—
126C9×#41—31no—
127#3701—31no—
128Q8×D9—31no—
129#3699—32no—
130#3699—31no—
131D4×D9—32no—
132D4×D9—31no—
133#3697—32no—
134#3697C2×D311yescharacteristic
135C2×C4×D9—31no—
136C2×C4×D9—32no—
137C2×C4×D9—61no—
138C9×M16—21no—
139C9×M16—32no—
140C9×M16C2×D311yescharacteristic, complemented
141C2×C8×C9—31no—
142C2×C8×C9—32no—
143#3667—31no—
144#3667—32no—
145#3667—61no—
146C2×C9⋊C8—32no—
147C2×C9⋊C8C2×D311yescharacteristic, complemented
148#3663—32no—
149#3663—32no—
150#3663—31no—
151#3663—31no—
152#3663—31no—
153#3663—31no—
154#3663—62no—
155#3663C2×D312yescomplemented
156C8×D9—32no—
157C8×D9—32no—
158C8×D9—62no—
159C8×D9C2×D312yescomplemented
160C22×C3×C9C22×C411yescharacteristic
161C2×C3⋊D9—41no—
162C2×C3⋊D9C22×C411yescharacteristic
163C2×C3⋊D9C22×C411yescharacteristic
164C2×D3×C9C22×C412yes—
165C2×C3×D9C22×C412yes—
166D3×D9—42no—
167C3×C4×C9C22×C411yescharacteristic
168C3×C4×C9C2411yescharacteristic
169C3⋊Dic9C22×C411yescharacteristic
170C3⋊Dic9C22×C411yescharacteristic
171C9×Dic3C22×C412yes—
172C3×Dic9C22×C412yes—
173D3×#41—91no—
174D3×#41—91no—
175C3×#131—31no—
176C3×#131—91no—
177#948—91no—
178#948—91no—
179#907—31no—
180#907—92no—
181#907—92no—
182#907D911yescharacteristic, complemented
183D3×M16—92no—
184D3×M16—91no—
185D3×M16—91no—
186D3×M16—92no—
187D3×M16—181no—
188#901—92no—
189#901—91no—
190#901—91no—
191#901—92no—
192C2×ΓL(1,𝔽25)—92no—
193C2×ΓL(1,𝔽25)—91no—
194C2×ΓL(1,𝔽25)—91no—
195C2×ΓL(1,𝔽25)—92no—
196C22×C3⋊D3—31no—
197C2×D32—32no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C3⋊Dic3—31no—
200C4×C3⋊D3—31no—
201C4×C3⋊D3—31no—
202C4×C3⋊D3—31no—
203C4×C3⋊D3—31no—
204C3×#67—32no—
205C3×#67C4×D312yes—
206C3×D12—31no—
207C3×D12C4×D311yescharacteristic
208C3×C4×D3—32no—
209C3×C4×D3C22×D312yes—
210C3×Dic6—31no—
211C3×Dic6C4×D311yescharacteristic
212#627—32no—
213#626—32no—
214#626—32no—
215#625—32no—
216#624—32no—
217D3×Dic3—32no—
218D3×Dic3—32no—
219C22×D9—31no—
220C22×D9—32no—
221C22×D9—61no—
222C32×C8C22×D312yes—
223C3⋊(C3⋊C8)—32no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—31no—
226C3×C3⋊C8C22×D311yescharacteristic
227C3×C3⋊C8C22×D311yescharacteristic
228Q8×C9—31no—
229D4×C9—32no—
230D4×C9—31no—
231C2×C4×C9—21no—
232C2×C4×C9—32no—
233C2×C4×C9C22×D311yescharacteristic
234#611—32no—
235#611—32no—
236#611C4×D312yes—
237C2×Dic9—31no—
238C2×Dic9—32no—
239C2×Dic9—61no—
240D36—32no—
241D36C4×D311yescharacteristic
242C4×D9—32no—
243C4×D9—31no—
244C4×D9—31no—
245C4×D9—31no—
246C4×D9—31no—
247C4×D9—32no—
248C4×D9—61no—
249C4×D9—61no—
250C4×D9—61no—
251C4×D9—61no—
252C4×D9C22×D312yes—
253Dic18—32no—
254Dic18C4×D311yescharacteristic
255C8×C9—22no—
256C8×C9—31no—
257C8×C9—31no—
258C8×C9C22×D312yes—
259C9⋊C8—32no—
260C9⋊C8—62no—
261C9⋊C8C22×D311yescharacteristic
262C9⋊C8C22×D311yescharacteristic
263#568—271noSylow
264C2×C3×C9—21no—
265C2×C3×C9C23×C411yescharacteristic
266C3⋊D9—21no—
267C3⋊D9—21no—
268D3×C9—22no—
269C3×D9—22no—
270C3×#41—31no—
271C3×#41—91no—
272#239—91no—
273#239—91no—
274D3×Q8—91no—
275D3×Q8—91no—
276#237—91no—
277#237—92no—
278#237—92no—
279#237—91no—
280D3×D4—91no—
281D3×D4—92no—
282D3×D4—92no—
283D3×D4—91no—
284#235—31no—
285#235—92no—
286#235—92no—
287#235C2×D911yescharacteristic
288C2×C4×D3—92no—
289C2×C4×D3—91no—
290C2×C4×D3—91no—
291C2×C4×D3—92no—
292C2×C4×D3—181no—
293C3×M16—21no—
294C3×M16—32no—
295C3×M16—92no—
296C3×M16C2×D911yescharacteristic, complemented
297C3×M16D3211yescharacteristic
298C2×C3×C8—31no—
299C2×C3×C8—32no—
300C2×C3×C8—91no—
301C2×C3×C8—92no—
302#208—92no—
303#208—91no—
304#208—91no—
305#208—92no—
306#208—181no—
307C2×C3⋊C8—31no—
308C2×C3⋊C8—92no—
309C2×C3⋊C8—92no—
310C2×C3⋊C8C2×D911yescharacteristic, complemented
311ΓL(1,𝔽25)—32no—
312ΓL(1,𝔽25)—91no—
313ΓL(1,𝔽25)—91no—
314ΓL(1,𝔽25)—91no—
315ΓL(1,𝔽25)—91no—
316ΓL(1,𝔽25)—92no—
317ΓL(1,𝔽25)—92no—
318ΓL(1,𝔽25)—92no—
319ΓL(1,𝔽25)—92no—
320ΓL(1,𝔽25)—91no—
321ΓL(1,𝔽25)—91no—
322ΓL(1,𝔽25)—91no—
323ΓL(1,𝔽25)—91no—
324ΓL(1,𝔽25)—182no—
325ΓL(1,𝔽25)C2×D912yescomplemented
326D3×C8—32no—
327D3×C8—92no—
328D3×C8—92no—
329D3×C8—92no—
330D3×C8—92no—
331D3×C8—182no—
332D3×C8C2×D912yescomplemented
333C22×C32C2×C4×D311yescharacteristic
334C2×C3⋊D3—31no—
335C2×C3⋊D3—31no—
336C2×C3⋊D3—121no—
337C2×C3×D3—32no—
338C2×C3×D3C2×C4×D312yes—
339D32—122no—
340C32×C4C2×C4×D311yescharacteristic
341C32×C4C23×D311yescharacteristic
342C3⋊Dic3—31no—
343C3⋊Dic3—31no—
344C3×Dic3—32no—
345C3×Dic3C2×C4×D312yes—
346C22×C9—21no—
347C22×C9—32no—
348C22×C9C2×C4×D311yescharacteristic
349C2×D9—31no—
350C2×D9—31no—
351C2×D9—61no—
352C2×D9—61no—
353C2×D9—122no—
354C2×D9—121no—
355C2×D9—122no—
356C2×D9C2×C4×D312yes—
357C4×C9—21no—
358C4×C9—21no—
359C4×C9—32no—
360C4×C9C2×C4×D311yescharacteristic
361C4×C9C23×D311yescharacteristic
362Dic9—31no—
363Dic9—31no—
364Dic9—61no—
365Dic9—61no—
366Dic9C2×C4×D312yes—
367C2×#41—271no—
368#131—31no—
369#131—91no—
370#131—272no—
371#131—271no—
372#131—271no—
373#131—272no—
374C2×M16—271no—
375C2×M16—272no—
376C2×M16—272no—
377C2×M16—271no—
378C3×C9#56811yescharacteristic, complemented, Sylow
379C22×D3—92no—
380C22×D3—91no—
381C22×D3—91no—
382C22×D3—92no—
383C22×D3—181no—
384C3×Q8—31no—
385C3×Q8—91no—
386C3×D4—32no—
387C3×D4—31no—
388C3×D4—92no—
389C3×D4—91no—
390C2×C3×C4—21no—
391C2×C3×C4—32no—
392C2×C3×C4—92no—
393C2×C3×C4C22×D911yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395#67—32no—
396#67—92no—
397#67—92no—
398#67—92no—
399#67—92no—
400#67C4×D912yes—
401C2×Dic3—92no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—92no—
405C2×Dic3—181no—
406D12—31no—
407D12—92no—
408D12—92no—
409D12C4×D911yescharacteristic
410C4×D3—32no—
411C4×D3—92no—
412C4×D3—92no—
413C4×D3—92no—
414C4×D3—91no—
415C4×D3—91no—
416C4×D3—91no—
417C4×D3—91no—
418C4×D3—91no—
419C4×D3—91no—
420C4×D3—91no—
421C4×D3—91no—
422C4×D3—92no—
423C4×D3—181no—
424C4×D3—181no—
425C4×D3—181no—
426C4×D3—181no—
427C4×D3C22×D912yes—
428Dic6—31no—
429Dic6—92no—
430Dic6—92no—
431Dic6C4×D911yescharacteristic
432C3×C8—22no—
433C3×C8—31no—
434C3×C8—31no—
435C3×C8—91no—
436C3×C8—91no—
437C3×C8C22×D912yes—
438C3×C8C2×D3212yes—
439C3⋊C8—31no—
440C3⋊C8—31no—
441C3⋊C8—92no—
442C3⋊C8—92no—
443C3⋊C8—182no—
444C3⋊C8C22×D911yescharacteristic
445C3⋊C8C22×D911yescharacteristic
446C2×C32—21no—
447C2×C32C22×C4×D311yescharacteristic
448C3⋊D3—61no—
449C3⋊D3—61no—
450C3×D3—22no—
451C3×D3—62no—
452C2×C9—21no—
453C2×C9—21no—
454C2×C9—41no—
455C2×C9—62no—
456C2×C9C22×C4×D311yescharacteristic
457D9—22no—
458D9—62no—
459D9—61no—
460D9—61no—
461D9—121no—
462#41—31no—
463#41—91no—
464#41—272no—
465#41—272no—
466#41—271no—
467#41—271no—
468C2×Q8—271no—
469C2×D4—272no—
470C2×D4—271no—
471C22×C4—272no—
472C22×C4—271no—
473M16—32no—
474M16—92no—
475M16—271no—
476M16—271no—
477M16—271no—
478M16—271no—
479M16—272no—
480M16—272no—
481M16—271no—
482M16—272no—
483M16D3×D911yescharacteristic, complemented
484C2×C8—31no—
485C2×C8—32no—
486C2×C8—91no—
487C2×C8—92no—
488C2×C8—272no—
489C2×C8—272no—
490C2×C8—272no—
491C22×C3—21no—
492C22×C3—32no—
493C22×C3—92no—
494C22×C3C2×C4×D911yescharacteristic
495C22×C3C4×D3211yescharacteristic
496C2×D3—32no—
497C2×D3—91no—
498C2×D3—91no—
499C2×D3—91no—
500C2×D3—91no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—362no—
504C2×D3—362no—
505C2×D3—361no—
506C2×D3—361no—
507C2×D3—362no—
508C2×D3C2×C4×D912yes—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—32no—
512C3×C4—92no—
513C3×C4C2×C4×D911yescharacteristic
514C3×C4C23×D911yescharacteristic
515C3×C4C4×D3211yescharacteristic
516C3×C4C22×D3211yescharacteristic
517Dic3—32no—
518Dic3—91no—
519Dic3—91no—
520Dic3—91no—
521Dic3—91no—
522Dic3—181no—
523Dic3—181no—
524Dic3C2×C4×D912yes—
525C32#575011yescharacteristic
526C9—21no—
527C9#575011yescharacteristic, complemented
528C23—272no—
529C23—271no—
530Q8—31no—
531Q8—91no—
532Q8—272no—
533D4—32no—
534D4—31no—
535D4—92no—
536D4—91no—
537D4—272no—
538D4—272no—
539D4—272no—
540C2×C4—32no—
541C2×C4—92no—
542C2×C4—272no—
543C2×C4—272no—
544C2×C4—272no—
545C2×C4—271no—
546C2×C4—271no—
547C2×C4—271no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4C2×D3×D911yescharacteristic
551C8—31no—
552C8—31no—
553C8—91no—
554C8—91no—
555C8—272no—
556C8C2×D3×D912yes—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—41no—
561C2×C3—62no—
562C2×C3—182no—
563C2×C3C22×C4×D911yescharacteristic
564C2×C3C2×C4×D3211yescharacteristic
565D3—22no—
566D3—62no—
567D3—182no—
568D3—181no—
569D3—181no—
570D3—181no—
571D3—181no—
572D3—361no—
573C22—32no—
574C22—92no—
575C22—271no—
576C22—271no—
577C22—1081no—
578C22—1082no—
579C22C4×D3×D911yescharacteristic
580C4—32no—
581C4—92no—
582C4—271no—
583C4—271no—
584C4C4×D3×D911yescharacteristic
585C4C22×D3×D911yescharacteristic
586C3—21no—
587C3#1058985211yescharacteristic, complemented
588C3#1059470811yescharacteristic
589C2—21no—
590C2—62no—
591C2—182no—
592C2—541no—
593C2—541no—
594C2C2×C4×D3×D911yescharacteristic
subgroupG/H
centerC4C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC3×C4C22×D32
FittingC3×C9×M16C22
socleC2×C32C22×C4×D3
prime radical#11722475C2
cosocleC3×C4×C9C24
hypercenterM16D3×D9
hypocenterC3×C9#568
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D9C2semidirect
3. C9×#907C2semidirect
4. C3×#63469C2semidirect
5. #11720877C2semidirect
6. C2×#88414C2semidirect
7. #11720858C2semidirect
8. #11720809C2semidirect
9. #11720806C2semidirect
10. D3×#3663C2semidirect
11. D9×ΓL(1,𝔽25)C2semidirect
25. C3×C9×M16C22semidirect
26. #88532C22semidirect
28. C8×C3⋊D9C22semidirect
29. C2×C9×C3⋊C8C22semidirect
30. C2×C3×C9⋊C8C22semidirect
31. C9×ΓL(1,𝔽25)C22semidirect
32. D3×C8×C9C22semidirect
33. C3×#3663C22semidirect
34. C3×C8×D9C22semidirect
35. #88420C22semidirect
36. D3×C9⋊C8C22semidirect
39. #88414C22semidirect
40. #88414C22semidirect
41. #88413C22semidirect
42. D9×C3⋊C8C22semidirect
58. #63469D3semidirect
140. C9×M16C2×D3semidirect
147. C2×C9⋊C8C2×D3semidirect
155. #3663C2×D3semidirect
159. C8×D9C2×D3semidirect
182. #907D9semidirect
296. C3×M16C2×D9semidirect
310. C2×C3⋊C8C2×D9semidirect
325. ΓL(1,𝔽25)C2×D9semidirect
332. D3×C8C2×D9semidirect
378. C3×C9#568semidirect
483. M16D3×D9semidirect
527. C9#5750semidirect
587. C3#10589852semidirect

Maximal subgroups

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

subgroupindex
#117224752
M16×C3⋊D92
C9×#9072
C3×#634692
#117208772
C2×#884142
#117208582
#117208092
#117208062
D3×#36632
D9×ΓL(1,𝔽25)2
#105947083
#105898523