Finite Group Databasebeta

← #13178011

Subgroups — #13178011

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
1C3×#41×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723990C211yesmaximal, characteristic, complemented
4C32×#951C211yesmaximal, characteristic, complemented
5#11723586C212yesmaximal, complemented
6#11723585C212yesmaximal, complemented
7#11723562C212yesmaximal, complemented
8#11723561C212yesmaximal, complemented
9#11723375C211yesmaximal, characteristic, complemented
10#11723371C211yesmaximal, characteristic, complemented
11C3⋊D3×#208C211yesmaximal, characteristic, complemented
12#10596248—31nomaximal
13#10595470—34nomaximal
14C33×#41C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C3×#3832C2211yescharacteristic, complemented
17C3×Q8×C3⋊D3C2211yescharacteristic, complemented
18C3×#3830—21no—
19C3×#3830C2211yescharacteristic, complemented
20C3×D4×C3⋊D3—21no—
21C3×D4×C3⋊D3C2211yescharacteristic, complemented
22C3×#3828—21no—
23C3×#3828C2211yescharacteristic
24C2×C3×C4×C3⋊D3—21no—
25C2×C3×C4×C3⋊D3C2211yescharacteristic
26C2×C32×Dic6C2211yescharacteristic, complemented
27C2×#6292—21no—
28C2×#6289—21no—
29#89018—21no—
30#89018—21no—
31#89018C2212yes—
32#89018C2212yescomplemented
33#88862C2212yescomplemented
34#88860C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C32×#216C2212yescomplemented
37C32×#214C2212yescomplemented
38C32×#208C2211yescharacteristic, complemented
39#88798C2212yescomplemented
40#88797C2212yescomplemented
41#88793C2212yescomplemented
42#88791C2212yescomplemented
43#88787C2212yescomplemented
44C3⋊D3×C3⋊C8C2212yescomplemented
45#41×C3⋊D3D311yescharacteristic, complemented
46C2×#3832—31no—
47C3×D3×#41—34no—
48C2×#3799—34no—
49C32×#137—31no—
50#64158—31no—
51#64158—34no—
52#64158—64no—
53#64151—31no—
54#64128—32no—
55#64127—32no—
56#64125—32no—
57#64123—32no—
58#64117—31no—
59M16×C3⋊D3—31no—
60C3×#951D314yescomplemented
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×C3×C3⋊D3—21no—
70C22×C3×C3⋊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—
78C3×#638—21no—
79C3×#638—21no—
80C3×#638—21no—
81C2×C3×C3⋊Dic3—21no—
82C2×C3×C3⋊Dic3D411yescharacteristic
83C3×C4⋊(C3⋊D3)—21no—
84C3×C4⋊(C3⋊D3)C2311yescharacteristic
85C3×C4×C3⋊D3—21no—
86C3×C4×C3⋊D3—21no—
87C3×C4×C3⋊D3—21no—
88C3×C4×C3⋊D3D411yescharacteristic
89C3×C4×C3⋊D3D411yescharacteristic
90C3×C4×C3⋊D3C2311yescharacteristic
91C3×C4×C3⋊D3C2311yescharacteristic
92C3×#634—21no—
93C3×#634C2311yescharacteristic
94C2×C32×Dic3—21no—
95C32×Dic6—21no—
96C32×Dic6C2312yes—
97#6292—22no—
98#6292—22no—
99#6289—22no—
100#6289—22no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C32×C3⋊C8C2312yes—
103#5822—91no—
104#5777—94no—
105C32×#41—24no—
106C32×#41C2×D314yes—
107C32×#41C2×D311yescharacteristic
108C2×C32×Q8—31no—
109#3832—32no—
110#3832—32no—
111#3832—61no—
112#3832—61no—
113#3832C2×D311yescharacteristic, complemented
114Q8×C3⋊D3C2×D311yescharacteristic, complemented
115#3830—21no—
116#3830C2×D311yescharacteristic, complemented
117D4×C3⋊D3—21no—
118D4×C3⋊D3C2×D311yescharacteristic, complemented
119#3828—21no—
120#3828C2×D311yescharacteristic
121C2×C4⋊(C3⋊D3)—31no—
122C2×C4⋊(C3⋊D3)—34no—
123C2×C4⋊(C3⋊D3)—61no—
124C2×C4⋊(C3⋊D3)—64no—
125C2×C4×C3⋊D3—21no—
126C2×C4×C3⋊D3—61no—
127C2×C4×C3⋊D3C2×D311yescharacteristic
128C3×#239—34no—
129C3×D3×Q8—34no—
130C3×#237—34no—
131C3×#237—64no—
132C3×D3×D4—34no—
133C3×D3×D4—64no—
134C3×#235—34no—
135C3×#235—64no—
136C2×C3×C4×D3—34no—
137C2×C3×C4×D3—64no—
138C2×C3×Dic6C2×D314yescomplemented
139C2×#626—64no—
140C2×#624—64no—
141#3799—38no—
142#3799—38no—
143#3799—64no—
144#3799—64no—
145C32×Q16—32no—
146C32×SD16—32no—
147C32×M16—31no—
148#3756—32no—
149#3755—32no—
150#3755—38no—
151#3755—32no—
152#3755—68no—
153#3753—32no—
154#3753—38no—
155#3753—68no—
156#3748—31no—
157#3748—34no—
158#3748—64no—
159#3745—32no—
160#3744—32no—
161#3743—32no—
162C8×C3⋊D3—32no—
163C3×#216C2×D318yescomplemented
164C3×#214C2×D318yescomplemented
165C3×#208C2×D314yescomplemented
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×C3×C3⋊D3—21no—
177C2×C3×C3⋊D3—41no—
178C2×C3×C3⋊D3—42no—
179C2×C3×C3⋊D3C2×D411yescharacteristic
180C2×C3×C3⋊D3C2×D411yescharacteristic
181C33×C4—21no—
182C33×C4C2×D411yescharacteristic
183C33×C4C2411yescharacteristic
184C3×C3⋊Dic3—21no—
185C3×C3⋊Dic3C2×D411yescharacteristic
186C3×C3⋊Dic3C2×D411yescharacteristic
187C32×Dic3—22no—
188C2×C3×#41—91no—
189D3×#41—34no—
190C2×#239—94no—
191C2×#235—91no—
192C3×#137—34no—
193#951—91no—
194#951C3⋊D311yescharacteristic, complemented
195#950—92no—
196#950—92no—
197#949—94no—
198#949—91no—
199#949—91no—
200#949—184no—
201#942—94no—
202#942—92no—
203#932—92no—
204C2×#208—91no—
205#919—98no—
206#918—98no—
207#916—98no—
208#914—98no—
209#908—94no—
210D3×M16—94no—
211C22×C3⋊D3—21no—
212C22×C3⋊D3—61no—
213C22×C3⋊D3—64no—
214C22×C3⋊D3—124no—
215C22×C3⋊D3#6711yescharacteristic
216C22×C3×D3—34no—
217C22×C3×D3—64no—
218C32×Q8—24no—
219C32×Q8—32no—
220C32×Q8—61no—
221C32×Q8C22×D314yes—
222C32×Q8C22×D311yescharacteristic
223C32×D4—21no—
224C32×D4—24no—
225C32×D4—24no—
226C32×D4—44no—
227C32×D4C22×D314yes—
228C32×D4C22×D311yescharacteristic
229C2×C32×C4—21no—
230C2×C32×C4—24no—
231C2×C32×C4—24no—
232C2×C32×C4—44no—
233C2×C32×C4—61no—
234C2×C32×C4C22×D314yes—
235C2×C32×C4C22×D311yescharacteristic
236#638—21no—
237#638—21no—
238#638—21no—
239C2×C3⋊Dic3—21no—
240C2×C3⋊Dic3#6711yescharacteristic
241C4⋊(C3⋊D3)—21no—
242C4⋊(C3⋊D3)—32no—
243C4⋊(C3⋊D3)—38no—
244C4⋊(C3⋊D3)—68no—
245C4⋊(C3⋊D3)—62no—
246C4⋊(C3⋊D3)—62no—
247C4⋊(C3⋊D3)—61no—
248C4⋊(C3⋊D3)—64no—
249C4⋊(C3⋊D3)—124no—
250C4⋊(C3⋊D3)C22×D311yescharacteristic
251C4×C3⋊D3—21no—
252C4×C3⋊D3—21no—
253C4×C3⋊D3—21no—
254C4×C3⋊D3—62no—
255C4×C3⋊D3—62no—
256C4×C3⋊D3#6711yescharacteristic
257C4×C3⋊D3#6711yescharacteristic
258C4×C3⋊D3C22×D311yescharacteristic
259C4×C3⋊D3C22×D311yescharacteristic
260#634—21no—
261#634C22×D311yescharacteristic
262C3×#67—64no—
263C3×#67—64no—
264C3×#67—64no—
265C2×C3×Dic3—24no—
266C2×C3×Dic3—34no—
267C2×C3×Dic3—64no—
268C3×D12—34no—
269C3×D12—64no—
270C3×C4×D3—34no—
271C3×C4×D3—34no—
272C3×C4×D3—34no—
273C3×C4×D3—34no—
274C3×C4×D3—64no—
275C3×C4×D3—64no—
276C3×C4×D3—64no—
277C3×Dic6—24no—
278C3×Dic6—34no—
279C3×Dic6—64no—
280C3×Dic6C22×D318yes—
281#626—68no—
282#626—68no—
283#624—68no—
284#624—68no—
285C32×C8—32no—
286C3⋊(C3⋊C8)—32no—
287C3⋊(C3⋊C8)—38no—
288C3⋊(C3⋊C8)—68no—
289C3×C3⋊C8C22×D318yes—
290#575—271noSylow
291C2×C33—21no—
292C2×C33—41no—
293C2×C33C22×D411yescharacteristic
294C3⋊(C3⋊D3)—42no—
295C3×C3⋊D3—21no—
296C3×C3⋊D3—21no—
297C3×C3⋊D3—41no—
298C3×#41—24no—
299C3×#41—91no—
300C3×#41—91no—
301C3×#41—91no—
302C3×#41—181no—
303C3×#41—181no—
304C3×#41D3214yes—
305C3×#41C2×C3⋊D311yescharacteristic
306C2×C3×Q8—34no—
307C2×C3×Q8—91no—
308C2×C3×D4—91no—
309C2×C3×D4—181no—
310C22×C3×C4—91no—
311C22×C3×C4—181no—
312C2×#67—181no—
313#239—34no—
314#239—98no—
315#239—98no—
316#239—184no—
317#239—184no—
318D3×Q8—34no—
319#237—34no—
320#237—64no—
321D3×D4—34no—
322D3×D4—64no—
323#235—34no—
324#235—64no—
325#235—92no—
326#235—92no—
327#235—181no—
328#235—181no—
329C2×D12—94no—
330C2×D12—91no—
331C2×D12—184no—
332C2×D12—184no—
333C2×C4×D3—34no—
334C2×C4×D3—64no—
335C2×C4×D3—181no—
336C2×C4×D3—184no—
337C2×Dic6C2×C3⋊D311yescharacteristic, complemented
338C3×Q16—38no—
339C3×SD16—38no—
340C3×M16—34no—
341#216—98no—
342#216—92no—
343#216C2×C3⋊D312yescomplemented
344#215—98no—
345#215—98no—
346#215—92no—
347#215—92no—
348#215—188no—
349#214—92no—
350#214C2×C3⋊D312yescomplemented
351#213—98no—
352#213—92no—
353#213—92no—
354#213—188no—
355#208—94no—
356#208—91no—
357#208—92no—
358#208—184no—
359#208C2×C3⋊D311yescharacteristic, complemented
360C2×C3⋊C8—92no—
361D24—98no—
362#204—98no—
363ΓL(1,𝔽25)—98no—
364D3×C8—98no—
365C22×C32—21no—
366C22×C32—24no—
367C22×C32—24no—
368C22×C32—44no—
369C22×C32D3×D414yes—
370C22×C32C2×#6711yescharacteristic
371C2×C3⋊D3—21no—
372C2×C3⋊D3—41no—
373C2×C3⋊D3—42no—
374C2×C3⋊D3—62no—
375C2×C3⋊D3—68no—
376C2×C3⋊D3—128no—
377C2×C3⋊D3—122no—
378C2×C3⋊D3—128no—
379C2×C3⋊D3—248no—
380C2×C3⋊D3C2×#6711yescharacteristic
381C2×C3⋊D3C2×#6711yescharacteristic
382C2×C3×D3—34no—
383C2×C3×D3—34no—
384C2×C3×D3—64no—
385C2×C3×D3—124no—
386C2×C3×D3—128no—
387C32×C4—21no—
388C32×C4—24no—
389C32×C4—24no—
390C32×C4—24no—
391C32×C4—44no—
392C32×C4—62no—
393C32×C4D3×D414yes—
394C32×C4C2×#6711yescharacteristic
395C32×C4C23×D314yes—
396C32×C4C23×D311yescharacteristic
397C3⋊Dic3—21no—
398C3⋊Dic3C2×#6711yescharacteristic
399C3⋊Dic3C2×#6711yescharacteristic
400C3×Dic3—28no—
401C3×Dic3—34no—
402C3×Dic3—34no—
403C3×Dic3—64no—
404C2×#41—91no—
405C2×#41—271no—
406#137—31no—
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—91no—
417C23×C3—181no—
418C22×D3—34no—
419C22×D3—64no—
420C22×D3—184no—
421C22×D3—181no—
422C22×D3—364no—
423C3×Q8—24no—
424C3×Q8—38no—
425C3×Q8—64no—
426C3×Q8—91no—
427C3×Q8—181no—
428C3×Q8C2×D3214yes—
429C3×Q8C22×C3⋊D311yescharacteristic
430C3×D4—24no—
431C3×D4—21no—
432C3×D4—24no—
433C3×D4—44no—
434C3×D4—91no—
435C3×D4—181no—
436C3×D4—181no—
437C3×D4—181no—
438C3×D4—181no—
439C3×D4C2×D3214yes—
440C3×D4C22×C3⋊D311yescharacteristic
441C2×C3×C4—24no—
442C2×C3×C4—21no—
443C2×C3×C4—24no—
444C2×C3×C4—44no—
445C2×C3×C4—64no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—181no—
452C2×C3×C4—181no—
453C2×C3×C4—181no—
454C2×C3×C4—181no—
455C2×C3×C4C2×D3214yes—
456C2×C3×C4C22×C3⋊D311yescharacteristic
457#67—64no—
458#67—64no—
459#67—64no—
460#67—182no—
461#67—182no—
462C2×Dic3—21no—
463C2×Dic3—34no—
464C2×Dic3—64no—
465D12—34no—
466D12—64no—
467D12—98no—
468D12—92no—
469D12—188no—
470D12—188no—
471D12—188no—
472D12—184no—
473D12—181no—
474D12—364no—
475C4×D3—34no—
476C4×D3—34no—
477C4×D3—34no—
478C4×D3—34no—
479C4×D3—64no—
480C4×D3—64no—
481C4×D3—64no—
482C4×D3—182no—
483C4×D3—182no—
484C4×D3—188no—
485C4×D3—188no—
486Dic6—21no—
487Dic6—34no—
488Dic6—64no—
489Dic6C22×C3⋊D312yes—
490C3×C8—38no—
491C3⋊C8—98no—
492C3⋊C8—92no—
493C3⋊C8—188no—
494C3⋊C8C22×C3⋊D312yes—
495C2×C32—21no—
496C2×C32—24no—
497C2×C32—24no—
498C2×C32—44no—
499C2×C32—41no—
500C2×C32—44no—
501C2×C32—84no—
502C2×C32C2×D3×D414yes—
503C2×C32C22×#6711yescharacteristic
504C3⋊D3—21no—
505C3⋊D3—21no—
506C3⋊D3—41no—
507C3⋊D3—122no—
508C3⋊D3—128no—
509C3⋊D3—248no—
510C3×D3—64no—
511C3×D3—64no—
512C3×D3—124no—
513#41—91no—
514#41—91no—
515#41—91no—
516#41—181no—
517#41—181no—
518#41—272no—
519#41—272no—
520#41—541no—
521#41—541no—
522#41D3×C3⋊D311yescharacteristic
523C2×Q8—31no—
524C2×Q8—91no—
525C2×D4—91no—
526C2×D4—181no—
527C2×D4—271no—
528C2×D4—541no—
529C22×C4—91no—
530C22×C4—181no—
531C22×C4—541no—
532Q16—32no—
533Q16—272no—
534SD16—32no—
535SD16—272no—
536SD16—272no—
537SD16—272no—
538D8—272no—
539D8—272no—
540M16—31no—
541M16—272no—
542M16—271no—
543C2×C8—272no—
544C22×C3—24no—
545C22×C3—21no—
546C22×C3—24no—
547C22×C3—44no—
548C22×C3—91no—
549C22×C3—91no—
550C22×C3—181no—
551C22×C3—361no—
552C22×C3—362no—
553C22×C3D3×#6714yes—
554C22×C3D4×C3⋊D311yescharacteristic
555C2×D3—34no—
556C2×D3—34no—
557C2×D3—64no—
558C2×D3—124no—
559C2×D3—128no—
560C2×D3—188no—
561C2×D3—182no—
562C2×D3—368no—
563C2×D3—362no—
564C2×D3—368no—
565C2×D3—728no—
566C3×C4—24no—
567C3×C4—21no—
568C3×C4—24no—
569C3×C4—24no—
570C3×C4—44no—
571C3×C4—68no—
572C3×C4—91no—
573C3×C4—91no—
574C3×C4—181no—
575C3×C4D3×#6714yes—
576C3×C4D4×C3⋊D311yescharacteristic
577C3×C4C22×D3214yes—
578C3×C4C23×C3⋊D311yescharacteristic
579Dic3—22no—
580Dic3—34no—
581Dic3—34no—
582Dic3—64no—
583C32—24no—
584C32#577714yescomplemented
585C32#582211yescharacteristic, complemented
586C23—91no—
587C23—181no—
588C23—541no—
589Q8—32no—
590Q8—61no—
591Q8—91no—
592Q8—181no—
593Q8C2×D3×C3⋊D311yescharacteristic
594D4—21no—
595D4—91no—
596D4—181no—
597D4—181no—
598D4—181no—
599D4—181no—
600D4—272no—
601D4—542no—
602D4—542no—
603D4—541no—
604D4C2×D3×C3⋊D311yescharacteristic
605C2×C4—21no—
606C2×C4—61no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—91no—
610C2×C4—91no—
611C2×C4—91no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4—181no—
615C2×C4—181no—
616C2×C4—542no—
617C2×C4—542no—
618C2×C4C2×D3×C3⋊D311yescharacteristic
619C8—32no—
620C8—272no—
621C2×C3—24no—
622C2×C3—21no—
623C2×C3—24no—
624C2×C3—44no—
625C2×C3—44no—
626C2×C3—41no—
627C2×C3—84no—
628C2×C3—181no—
629C2×C3—181no—
630C2×C3—361no—
631C2×C3C2×D3×#6714yes—
632C2×C3C2×D4×C3⋊D311yescharacteristic
633D3—64no—
634D3—64no—
635D3—124no—
636D3—368no—
637D3—362no—
638D3—728no—
639C22—21no—
640C22—91no—
641C22—91no—
642C22—181no—
643C22—361no—
644C22—362no—
645C22—542no—
646C22—1082no—
647C22C3⋊D3×#6711yescharacteristic
648C4—21no—
649C4—62no—
650C4—91no—
651C4—91no—
652C4—181no—
653C4C3⋊D3×#6711yescharacteristic
654C4C22×D3×C3⋊D311yescharacteristic
655C3—24no—
656C3#1059547014yescomplemented
657C3#1059624811yescharacteristic, complemented
658C2—21no—
659C2—41no—
660C2—181no—
661C2—181no—
662C2—361no—
663C2—1082no—
664C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
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. C3×#41×C3⋊D3C2semidirect
2. C2×#89018C2semidirect
3. #11723990C2semidirect
4. C32×#951C2semidirect
5. #11723586C2semidirect
6. #11723585C2semidirect
7. #11723562C2semidirect
8. #11723561C2semidirect
9. #11723375C2semidirect
10. #11723371C2semidirect
11. C3⋊D3×#208C2semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C3×#3832C22semidirect
17. C3×Q8×C3⋊D3C22semidirect
19. C3×#3830C22semidirect
21. C3×D4×C3⋊D3C22semidirect
26. C2×C32×Dic6C22semidirect
32. #89018C22semidirect
33. #88862C22semidirect
34. #88860C22semidirect
35. #88855C22semidirect
36. C32×#216C22semidirect
37. C32×#214C22semidirect
38. C32×#208C22semidirect
39. #88798C22semidirect
40. #88797C22semidirect
41. #88793C22semidirect
42. #88791C22semidirect
43. #88787C22semidirect
44. C3⋊D3×C3⋊C8C22semidirect
45. #41×C3⋊D3D3semidirect
60. C3×#951D3semidirect
113. #3832C2×D3semidirect
114. Q8×C3⋊D3C2×D3semidirect
116. #3830C2×D3semidirect
118. D4×C3⋊D3C2×D3semidirect
138. C2×C3×Dic6C2×D3semidirect
163. C3×#216C2×D3semidirect
164. C3×#214C2×D3semidirect
165. C3×#208C2×D3semidirect
194. #951C3⋊D3semidirect
337. C2×Dic6C2×C3⋊D3semidirect
343. #216C2×C3⋊D3semidirect
350. #214C2×C3⋊D3semidirect
359. #208C2×C3⋊D3semidirect
415. C33#575semidirect
584. C32#5777semidirect
585. C32#5822semidirect
656. C3#10595470semidirect
657. C3#10596248semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
C2×#890182
#117239902
C32×#9512
#117235862
#117235852
#117235622
#117235612
#117233752
#117233712
C3⋊D3×#2082
#105962483
#105954703