Finite Group Databasebeta

← #13178015

Subgroups — #13178015

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×#41C211yesmaximal, characteristic, complemented
2C2×#89022C211yesmaximal, characteristic, complemented
3#11723990C211yesmaximal, characteristic, complemented
4C3×#64160C211yesmaximal, characteristic, complemented
5#11723582C212yesmaximal, complemented
6#11723581C212yesmaximal, complemented
7#11723558C212yesmaximal, complemented
8#11723557C212yesmaximal, complemented
9#11723373C211yesmaximal, characteristic, complemented
10#11723367C211yesmaximal, characteristic, complemented
11D3×#3748C211yesmaximal, characteristic, complemented
12#10596293—31nomaximal
13#10595470—34nomaximal
14C33×#41C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C2×C3×#634C2211yescharacteristic, complemented
17C32×#239C2211yescharacteristic, complemented
18C32×D3×Q8C2211yescharacteristic, complemented
19C32×#237—21no—
20C32×#237C2211yescharacteristic, complemented
21C32×D3×D4—21no—
22C32×D3×D4C2211yescharacteristic, complemented
23C32×#235—21no—
24C32×#235C2211yescharacteristic
25C2×C32×C4×D3—21no—
26C2×C32×C4×D3C2211yescharacteristic
27C2×#6291—21no—
28C2×#6289—21no—
29#89022—21no—
30#89022—21no—
31#89022C2212yes—
32#89022C2212yescomplemented
33#88862C2212yescomplemented
34#88860C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C3×#3756C2212yescomplemented
37C3×#3754C2212yescomplemented
38C3×#3748C2211yescharacteristic, complemented
39#88798C2212yescomplemented
40#88795C2212yescomplemented
41#88792C2212yescomplemented
42#88790C2212yescomplemented
43#88786C2212yescomplemented
44D3×C3⋊(C3⋊C8)C2212yescomplemented
45C2×C32×#41—31no—
46C2×#3828—31no—
47C3×D3×#41D314yescomplemented
48C2×#3799—34no—
49#64160—31no—
50#64160D311yescharacteristic, complemented
51#64159—32no—
52#64159—32no—
53#64158—31no—
54#64158—34no—
55#64158—31no—
56#64158—64no—
57#64151—32no—
58#64141—32no—
59C2×#3748—31no—
60C3×#951—34no—
61#63949—38no—
62#63948—38no—
63#63935—38no—
64#63934—38no—
65#63827—34no—
66#63825—34no—
67D3×#208—34no—
68C22×C3⋊(C3⋊D3)—21no—
69C22×C32×D3—21no—
70C22×C32×D3D411yescharacteristic
71C33×Q8C2311yescharacteristic
72C33×D4—21no—
73C33×D4C2311yescharacteristic
74C2×C33×C4—21no—
75C2×C33×C4C2311yescharacteristic
76C4⋊(C3⋊(C3⋊D3))—21no—
77C4⋊(C3⋊(C3⋊D3))C2312yes—
78C2×C3×C3⋊Dic3—21no—
79C3×#634—21no—
80C3×#634C2312yes—
81C32×#67—21no—
82C32×#67—21no—
83C32×#67—21no—
84C2×C32×Dic3—21no—
85C2×C32×Dic3D411yescharacteristic
86C32×D12—21no—
87C32×D12C2311yescharacteristic
88C32×C4×D3—21no—
89C32×C4×D3—21no—
90C32×C4×D3—21no—
91C32×C4×D3D411yescharacteristic
92C32×C4×D3D411yescharacteristic
93C32×C4×D3C2311yescharacteristic
94C32×C4×D3C2311yescharacteristic
95C32×Dic6—21no—
96C32×Dic6C2311yescharacteristic
97#6291—22no—
98#6291—22no—
99#6289—22no—
100#6289—22no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C3×C3⋊(C3⋊C8)C2312yes—
103#5822—94no—
104#5777—91no—
105C32×#41—24no—
106C32×#41—31no—
107C32×#41—31no—
108C32×#41—31no—
109C32×#41—61no—
110C32×#41—61no—
111C32×#41C2×D314yes—
112C32×#41C2×D311yescharacteristic
113C2×C32×Q8—31no—
114C2×C32×D4—31no—
115C2×C32×D4—61no—
116C22×C32×C4—31no—
117C22×C32×C4—61no—
118C2×#638—61no—
119#3828—32no—
120#3828—32no—
121#3828—61no—
122#3828—61no—
123C2×C4⋊(C3⋊D3)—31no—
124C2×C4⋊(C3⋊D3)—34no—
125C2×C4⋊(C3⋊D3)—64no—
126C2×C4×C3⋊D3—61no—
127C2×#634C2×D311yescharacteristic, complemented
128C3×#239C2×D314yescomplemented
129C3×D3×Q8C2×D314yescomplemented
130C3×#237—24no—
131C3×#237C2×D314yescomplemented
132C3×D3×D4—24no—
133C3×D3×D4C2×D314yescomplemented
134C3×#235—24no—
135C3×#235C2×D314yes—
136C2×C3×C4×D3—24no—
137C2×C3×C4×D3C2×D314yes—
138C2×C3×Dic6—34no—
139C2×#626—64no—
140C2×#624—64no—
141#3799—38no—
142#3799—38no—
143#3799—64no—
144#3799—64no—
145#3756—32no—
146#3756C2×D312yescomplemented
147#3755—32no—
148#3755—38no—
149#3755—32no—
150#3755—68no—
151#3754—32no—
152#3754C2×D312yescomplemented
153#3753—32no—
154#3753—38no—
155#3753—32no—
156#3753—68no—
157#3748—31no—
158#3748—34no—
159#3748—32no—
160#3748—64no—
161#3748C2×D311yescharacteristic, complemented
162C2×C3⋊(C3⋊C8)—32no—
163C3×#216—38no—
164C3×#214—38no—
165C3×#208—34no—
166#3718—38no—
167#3717—38no—
168#3715—38no—
169#3714—38no—
170#3711—38no—
171D3×C3⋊C8—38no—
172C22×C33—21no—
173C22×C33C2×D411yescharacteristic
174C2×C3⋊(C3⋊D3)—22no—
175C2×C3⋊(C3⋊D3)—42no—
176C2×C32×D3—21no—
177C2×C32×D3—41no—
178C2×C32×D3—42no—
179C2×C32×D3C2×D411yescharacteristic
180C2×C32×D3C2×D411yescharacteristic
181C33×C4—21no—
182C33×C4C2×D411yescharacteristic
183C33×C4C2411yescharacteristic
184C3×C3⋊Dic3—22no—
185C32×Dic3—21no—
186C32×Dic3C2×D411yescharacteristic
187C32×Dic3C2×D411yescharacteristic
188C2×C3×#41—34no—
189D3×#41C3⋊D311yescharacteristic, complemented
190C2×#239—91no—
191C2×#235—94no—
192C3×#137—91no—
193#951—34no—
194#951—94no—
195#950—98no—
196#950—98no—
197#949—94no—
198#949—94no—
199#949—91no—
200#949—184no—
201#942—98no—
202#942—91no—
203#932—98no—
204C2×#208—94no—
205#919—92no—
206#918—92no—
207#916—92no—
208#914—92no—
209#908—91no—
210D3×M16—91no—
211C23×C32—31no—
212C23×C32—61no—
213C22×C3⋊D3—61no—
214C22×C3⋊D3—64no—
215C22×C3⋊D3—124no—
216C22×C3×D3—24no—
217C22×C3×D3#6714yes—
218C32×Q8—24no—
219C32×Q8—31no—
220C32×Q8—61no—
221C32×Q8C22×D314yes—
222C32×Q8C22×D311yescharacteristic
223C32×D4—21no—
224C32×D4—24no—
225C32×D4—24no—
226C32×D4—31no—
227C32×D4—44no—
228C32×D4—61no—
229C32×D4—61no—
230C32×D4—61no—
231C32×D4—61no—
232C32×D4C22×D314yes—
233C32×D4C22×D311yescharacteristic
234C2×C32×C4—21no—
235C2×C32×C4—24no—
236C2×C32×C4—24no—
237C2×C32×C4—31no—
238C2×C32×C4—31no—
239C2×C32×C4—31no—
240C2×C32×C4—31no—
241C2×C32×C4—31no—
242C2×C32×C4—44no—
243C2×C32×C4—61no—
244C2×C32×C4—61no—
245C2×C32×C4—61no—
246C2×C32×C4—61no—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D311yescharacteristic
249#638—62no—
250#638—62no—
251C2×C3⋊Dic3—21no—
252C4⋊(C3⋊D3)—32no—
253C4⋊(C3⋊D3)—38no—
254C4⋊(C3⋊D3)—68no—
255C4⋊(C3⋊D3)—61no—
256C4⋊(C3⋊D3)—64no—
257C4⋊(C3⋊D3)—124no—
258C4×C3⋊D3—62no—
259C4×C3⋊D3—62no—
260#634—21no—
261#634C22×D312yes—
262C3×#67—24no—
263C3×#67—24no—
264C3×#67—24no—
265C2×C3×Dic3—24no—
266C2×C3×Dic3—64no—
267C2×C3×Dic3#6714yes—
268C3×D12—24no—
269C3×D12C22×D314yes—
270C3×C4×D3—24no—
271C3×C4×D3—24no—
272C3×C4×D3—24no—
273C3×C4×D3#6714yes—
274C3×C4×D3#6714yes—
275C3×C4×D3C22×D314yes—
276C3×C4×D3C22×D314yes—
277C3×Dic6—24no—
278C3×Dic6—38no—
279C3×Dic6—64no—
280C3×Dic6C22×D314yes—
281#626—68no—
282#626—68no—
283#624—68no—
284#624—68no—
285C3⋊(C3⋊C8)—32no—
286C3⋊(C3⋊C8)—38no—
287C3⋊(C3⋊C8)—68no—
288C3⋊(C3⋊C8)C22×D312yes—
289C3×C3⋊C8—38no—
290#575—271noSylow
291C2×C33—21no—
292C2×C33—41no—
293C2×C33C22×D411yescharacteristic
294C3⋊(C3⋊D3)—42no—
295C32×D3—21no—
296C32×D3—21no—
297C32×D3—41no—
298C3×#41—24no—
299C3×#41—34no—
300C3×#41—34no—
301C3×#41—34no—
302C3×#41—64no—
303C3×#41—64no—
304C3×#41D3214yes—
305C3×#41C2×C3⋊D311yescharacteristic
306C2×C3×Q8—34no—
307C2×C3×Q8—91no—
308C2×C3×D4—34no—
309C2×C3×D4—64no—
310C22×C3×C4—34no—
311C22×C3×C4—64no—
312C2×#67—184no—
313#239—92no—
314#239—92no—
315#239—181no—
316#239—181no—
317#239C2×C3⋊D311yescharacteristic, complemented
318D3×Q8C2×C3⋊D311yescharacteristic, complemented
319#237—21no—
320#237C2×C3⋊D311yescharacteristic, complemented
321D3×D4—21no—
322D3×D4C2×C3⋊D311yescharacteristic, complemented
323#235—21no—
324#235—98no—
325#235—98no—
326#235—184no—
327#235—184no—
328#235C2×C3⋊D311yescharacteristic
329C2×D12—94no—
330C2×D12—91no—
331C2×D12—184no—
332C2×D12—181no—
333C2×C4×D3—21no—
334C2×C4×D3—184no—
335C2×C4×D3—181no—
336C2×C4×D3C2×C3⋊D311yescharacteristic
337C2×Dic6—34no—
338C3×Q16—92no—
339C3×SD16—92no—
340C3×M16—91no—
341#216—38no—
342#216—98no—
343#216—92no—
344#215—98no—
345#215—98no—
346#215—92no—
347#215—92no—
348#215—188no—
349#214—38no—
350#214—98no—
351#213—98no—
352#213—98no—
353#213—92no—
354#213—188no—
355#208—34no—
356#208—98no—
357#208—94no—
358#208—91no—
359#208—184no—
360C2×C3⋊C8—98no—
361D24—92no—
362#204—92no—
363ΓL(1,𝔽25)—92no—
364D3×C8—92no—
365C22×C32—21no—
366C22×C32—24no—
367C22×C32—24no—
368C22×C32—31no—
369C22×C32—31no—
370C22×C32—44no—
371C22×C32—61no—
372C22×C32—121no—
373C22×C32—122no—
374C22×C32D3×D411yescharacteristic
375C22×C32C2×#6714yes—
376C2×C3⋊D3—62no—
377C2×C3⋊D3—68no—
378C2×C3⋊D3—128no—
379C2×C3⋊D3—122no—
380C2×C3⋊D3—128no—
381C2×C3⋊D3—248no—
382C2×C3×D3—24no—
383C2×C3×D3—44no—
384C2×C3×D3—48no—
385C2×C3×D3C2×#6714yes—
386C2×C3×D3C2×#6714yes—
387C32×C4—21no—
388C32×C4—24no—
389C32×C4—24no—
390C32×C4—24no—
391C32×C4—31no—
392C32×C4—31no—
393C32×C4—44no—
394C32×C4—61no—
395C32×C4D3×D411yescharacteristic
396C32×C4C2×#6714yes—
397C32×C4C23×D314yes—
398C32×C4C23×D311yescharacteristic
399C3⋊Dic3—22no—
400C3×Dic3—24no—
401C3×Dic3—68no—
402C3×Dic3C2×#6714yes—
403C3×Dic3C2×#6714yes—
404C2×#41—31no—
405C2×#41—271no—
406#137—91no—
407#137—272no—
408#137—271no—
409AGL(1,ℤ/8)—271no—
410AGL(1,ℤ/8)—272no—
411AGL(1,ℤ/8)—271no—
412#135—272no—
413#135—272no—
414C2×M16—271no—
415C33#57511yescharacteristic, complemented, Sylow
416C23×C3—34no—
417C23×C3—64no—
418C22×D3—21no—
419C22×D3—184no—
420C22×D3—181no—
421C22×D3—364no—
422C22×D3#63811yescharacteristic
423C3×Q8—24no—
424C3×Q8—34no—
425C3×Q8—64no—
426C3×Q8—92no—
427C3×Q8—181no—
428C3×Q8C2×D3214yes—
429C3×Q8C22×C3⋊D311yescharacteristic
430C3×D4—24no—
431C3×D4—21no—
432C3×D4—24no—
433C3×D4—34no—
434C3×D4—44no—
435C3×D4—64no—
436C3×D4—64no—
437C3×D4—64no—
438C3×D4—64no—
439C3×D4C2×D3214yes—
440C3×D4C22×C3⋊D311yescharacteristic
441C2×C3×C4—24no—
442C2×C3×C4—21no—
443C2×C3×C4—24no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—34no—
447C2×C3×C4—34no—
448C2×C3×C4—34no—
449C2×C3×C4—44no—
450C2×C3×C4—64no—
451C2×C3×C4—64no—
452C2×C3×C4—64no—
453C2×C3×C4—64no—
454C2×C3×C4—181no—
455C2×C3×C4C2×D3214yes—
456C2×C3×C4C22×C3⋊D311yescharacteristic
457#67—21no—
458#67—21no—
459#67—21no—
460#67—188no—
461#67—188no—
462C2×Dic3—21no—
463C2×Dic3—64no—
464C2×Dic3#63811yescharacteristic
465D12—21no—
466D12—98no—
467D12—92no—
468D12—188no—
469D12—182no—
470D12—182no—
471D12—184no—
472D12—181no—
473D12—364no—
474D12C22×C3⋊D311yescharacteristic
475C4×D3—21no—
476C4×D3—21no—
477C4×D3—21no—
478C4×D3—188no—
479C4×D3—188no—
480C4×D3—182no—
481C4×D3—182no—
482C4×D3#63811yescharacteristic
483C4×D3#63811yescharacteristic
484C4×D3C22×C3⋊D311yescharacteristic
485C4×D3C22×C3⋊D311yescharacteristic
486Dic6—21no—
487Dic6—38no—
488Dic6—64no—
489Dic6C22×C3⋊D311yescharacteristic
490C3×C8—92no—
491C3⋊C8—38no—
492C3⋊C8—98no—
493C3⋊C8—92no—
494C3⋊C8—188no—
495C2×C32—21no—
496C2×C32—24no—
497C2×C32—24no—
498C2×C32—44no—
499C2×C32—41no—
500C2×C32—44no—
501C2×C32—61no—
502C2×C32—61no—
503C2×C32—84no—
504C2×C32—121no—
505C2×C32C2×D3×D411yescharacteristic
506C2×C32C22×#6714yes—
507C3⋊D3—122no—
508C3⋊D3—128no—
509C3⋊D3—248no—
510C3×D3—24no—
511C3×D3—24no—
512C3×D3—44no—
513#41—31no—
514#41—31no—
515#41—31no—
516#41—61no—
517#41—61no—
518#41—272no—
519#41—272no—
520#41—541no—
521#41—541no—
522#41D3×C3⋊D311yescharacteristic
523C2×Q8—31no—
524C2×Q8—91no—
525C2×D4—31no—
526C2×D4—61no—
527C2×D4—271no—
528C2×D4—541no—
529C22×C4—31no—
530C22×C4—61no—
531C22×C4—541no—
532Q16—92no—
533Q16—272no—
534SD16—92no—
535SD16—272no—
536SD16—272no—
537SD16—272no—
538D8—272no—
539D8—272no—
540M16—91no—
541M16—272no—
542M16—271no—
543C2×C8—272no—
544C22×C3—24no—
545C22×C3—21no—
546C22×C3—24no—
547C22×C3—34no—
548C22×C3—34no—
549C22×C3—44no—
550C22×C3—64no—
551C22×C3—124no—
552C22×C3—128no—
553C22×C3D3×#6714yes—
554C22×C3C2×#63811yescharacteristic
555C2×D3—21no—
556C2×D3—41no—
557C2×D3—42no—
558C2×D3—188no—
559C2×D3—182no—
560C2×D3—368no—
561C2×D3—362no—
562C2×D3—368no—
563C2×D3—728no—
564C2×D3C2×#63811yescharacteristic
565C2×D3C2×#63811yescharacteristic
566C3×C4—24no—
567C3×C4—21no—
568C3×C4—24no—
569C3×C4—24no—
570C3×C4—34no—
571C3×C4—34no—
572C3×C4—44no—
573C3×C4—64no—
574C3×C4—182no—
575C3×C4D3×#6714yes—
576C3×C4C2×#63811yescharacteristic
577C3×C4C22×D3214yes—
578C3×C4C23×C3⋊D311yescharacteristic
579Dic3—21no—
580Dic3—68no—
581Dic3C2×#63811yescharacteristic
582Dic3C2×#63811yescharacteristic
583C32—24no—
584C32#577711yescharacteristic, complemented
585C32#582214yescomplemented
586C23—31no—
587C23—61no—
588C23—541no—
589Q8—31no—
590Q8—61no—
591Q8—92no—
592Q8—181no—
593Q8C2×D3×C3⋊D311yescharacteristic
594D4—21no—
595D4—31no—
596D4—61no—
597D4—61no—
598D4—61no—
599D4—61no—
600D4—272no—
601D4—542no—
602D4—542no—
603D4—541no—
604D4C2×D3×C3⋊D311yescharacteristic
605C2×C4—21no—
606C2×C4—31no—
607C2×C4—31no—
608C2×C4—31no—
609C2×C4—31no—
610C2×C4—31no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—181no—
616C2×C4—542no—
617C2×C4—542no—
618C2×C4C2×D3×C3⋊D311yescharacteristic
619C8—92no—
620C8—272no—
621C2×C3—24no—
622C2×C3—21no—
623C2×C3—24no—
624C2×C3—44no—
625C2×C3—44no—
626C2×C3—41no—
627C2×C3—64no—
628C2×C3—64no—
629C2×C3—84no—
630C2×C3—124no—
631C2×C3C2×D3×#6714yes—
632C2×C3C22×#63811yescharacteristic
633D3—21no—
634D3—21no—
635D3—41no—
636D3—368no—
637D3—362no—
638D3—728no—
639C22—21no—
640C22—31no—
641C22—31no—
642C22—61no—
643C22—121no—
644C22—122no—
645C22—542no—
646C22—1082no—
647C22D3×#63811yescharacteristic
648C4—21no—
649C4—31no—
650C4—31no—
651C4—61no—
652C4—182no—
653C4D3×#63811yescharacteristic
654C4C22×D3×C3⋊D311yescharacteristic
655C3—24no—
656C3#1059547014yescomplemented
657C3#1059629311yescharacteristic, complemented
658C2—21no—
659C2—41no—
660C2—61no—
661C2—61no—
662C2—121no—
663C2—1082no—
664C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C32×D3×#41C2semidirect
2. C2×#89022C2semidirect
3. #11723990C2semidirect
4. C3×#64160C2semidirect
5. #11723582C2semidirect
6. #11723581C2semidirect
7. #11723558C2semidirect
8. #11723557C2semidirect
9. #11723373C2semidirect
10. #11723367C2semidirect
11. D3×#3748C2semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C2×C3×#634C22semidirect
17. C32×#239C22semidirect
18. C32×D3×Q8C22semidirect
20. C32×#237C22semidirect
22. C32×D3×D4C22semidirect
32. #89022C22semidirect
33. #88862C22semidirect
34. #88860C22semidirect
35. #88855C22semidirect
36. C3×#3756C22semidirect
37. C3×#3754C22semidirect
38. C3×#3748C22semidirect
39. #88798C22semidirect
40. #88795C22semidirect
41. #88792C22semidirect
42. #88790C22semidirect
43. #88786C22semidirect
44. D3×C3⋊(C3⋊C8)C22semidirect
47. C3×D3×#41D3semidirect
50. #64160D3semidirect
127. C2×#634C2×D3semidirect
128. C3×#239C2×D3semidirect
129. C3×D3×Q8C2×D3semidirect
131. C3×#237C2×D3semidirect
133. C3×D3×D4C2×D3semidirect
146. #3756C2×D3semidirect
152. #3754C2×D3semidirect
161. #3748C2×D3semidirect
189. D3×#41C3⋊D3semidirect
317. #239C2×C3⋊D3semidirect
318. D3×Q8C2×C3⋊D3semidirect
320. #237C2×C3⋊D3semidirect
322. D3×D4C2×C3⋊D3semidirect
415. C33#575semidirect
584. C32#5777semidirect
585. C32#5822semidirect
656. C3#10595470semidirect
657. C3#10596293semidirect

Maximal subgroups

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

subgroupindex
C32×D3×#412
C2×#890222
#117239902
C3×#641602
#117235822
#117235812
#117235582
#117235572
#117233732
#117233672
D3×#37482
#105962933
#105954703