Finite Group Databasebeta

← #13176688

Subgroups — #13176688

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⋊D3×#235C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723949C211yesmaximal, characteristic, complemented
4C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
5C32×#909C211yesmaximal, characteristic, complemented
6#11723381C211yesmaximal, characteristic, complemented
7#11723367C211yesmaximal, characteristic, complemented
8#11723341C212yesmaximal, complemented
9#11723338C212yesmaximal, complemented
10#11723329C212yesmaximal, complemented
11#11723319C212yesmaximal, complemented
12#10596248—31nomaximal
13#10594711—34nomaximal
14#89076C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C32×#235C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic, complemented
19C3⋊D3×#67—21no—
20C2×#6292—21no—
21C2×#6289—21no—
22#89028—21no—
23C3⋊D3×D12C2211yescharacteristic, complemented
24C4×D3×C3⋊D3—21no—
25#89020—21no—
26#89018—21no—
27#89018—21no—
28#89018C2212yes—
29#89018C2212yescomplemented
30#89018C2211yescharacteristic, complemented
31C3⋊D3×Dic6C2211yescharacteristic, complemented
32#89014C2211yescharacteristic, complemented
33C33×M16C2211yescharacteristic, complemented
34#88852C2212yescomplemented
35#88851C2212yescomplemented
36C3×#3748C2211yescharacteristic, complemented
37C3×#3743C2212yescomplemented
38C3×C8×C3⋊D3C2212yescomplemented
39C32×Dic12C2212yescomplemented
40C32×#204C2212yescomplemented
41#88799C2212yescomplemented
42#88795C2212yescomplemented
43#88795C2212yescomplemented
44#88790C2212yescomplemented
45#41×C3⋊D3—31no—
46C2×#3832—31no—
47D3×#235—34no—
48C2×#3799—34no—
49C32×#137—31no—
50#64158—31no—
51#64151—31no—
52#64128—32no—
53#64127—32no—
54#64125—32no—
55#64123—32no—
56#64117—31no—
57#64117—34no—
58#64117—64no—
59M16×C3⋊D3D311yescharacteristic, complemented
60C3×#909D314yescomplemented
61C3×D3×M16—34no—
62#63833—34no—
63#63825—34no—
64#63808—38no—
65#63806—38no—
66#63801—38no—
67#63794—38no—
68C22×C3⋊(C3⋊D3)—21no—
69C22×C3×C3⋊D3D411yescharacteristic
70C2×D3×C3⋊D3—21no—
71C2×C33×C4C2311yescharacteristic
72#6312—21no—
73C4⋊(C3⋊(C3⋊D3))—21no—
74C4⋊(C3⋊(C3⋊D3))C2312yes—
75C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
76C4×C3⋊(C3⋊D3)—21no—
77#6308C2311yescharacteristic
78C2×C3×C3⋊Dic3D411yescharacteristic
79C3×C4×C3⋊D3D411yescharacteristic
80C3×C4×C3⋊D3D411yescharacteristic
81C3×C4×C3⋊D3C2311yescharacteristic
82C3×C4×C3⋊D3C2311yescharacteristic
83C32×#67—21no—
84C2×C32×Dic3—21no—
85C32×D12C2311yescharacteristic
86C32×C4×D3—21no—
87C32×Dic6—21no—
88C32×Dic6C2311yescharacteristic
89C32×Dic6C2312yes—
90#6293—21no—
91#6292—22no—
92#6292—22no—
93#6292—21no—
94#6291—21no—
95#6290—21no—
96#6289—22no—
97#6289—22no—
98#6289—21no—
99Dic3×C3⋊D3—21no—
100D3×C3⋊Dic3—21no—
101C33×C8C2312yes—
102C3×C3⋊(C3⋊C8)C2312yes—
103#5777—94no—
104#5748—91no—
105C32×#41—31no—
106C2×C32×Q8—31no—
107#3832—32no—
108#3832—32no—
109#3832—31no—
110#3832—61no—
111#3832—61no—
112Q8×C3⋊D3—31no—
113#3830—31no—
114#3830—61no—
115D4×C3⋊D3—31no—
116D4×C3⋊D3—61no—
117#3828—31no—
118#3828—34no—
119#3828—61no—
120#3828—64no—
121C2×C4⋊(C3⋊D3)—31no—
122C2×C4⋊(C3⋊D3)—34no—
123C2×C4⋊(C3⋊D3)—61no—
124C2×C4⋊(C3⋊D3)—64no—
125C2×C4×C3⋊D3—61no—
126C2×C4×C3⋊D3—61no—
127C2×C4×C3⋊D3C2×D311yescharacteristic
128C3×#235C2×D314yes—
129C2×C3×C4×D3—34no—
130C2×C3×Dic6C2×D314yescomplemented
131D3×#67—64no—
132C2×#626—64no—
133C2×#624—64no—
134#3804—64no—
135D3×D12—34no—
136C4×D32—64no—
137#3799—38no—
138#3799—38no—
139#3799—34no—
140#3799—64no—
141#3799—64no—
142#3798—64no—
143#3795—34no—
144D3×Dic6—34no—
145C32×Q16—32no—
146C32×SD16—32no—
147C32×M16—24no—
148C32×M16C2×D314yescomplemented
149C32×M16C2×D311yescharacteristic, complemented
150#3756—32no—
151#3755—32no—
152#3755—32no—
153#3753—32no—
154#3748C2×D311yescharacteristic, complemented
155#3745—32no—
156#3745—38no—
157#3745—68no—
158#3744—32no—
159#3744—38no—
160#3744—68no—
161#3743C2×D312yescomplemented
162C8×C3⋊D3C2×D312yescomplemented
163C3×#208—34no—
164C3×Dic12C2×D318yescomplemented
165C3×#204C2×D318yescomplemented
166C3×ΓL(1,𝔽25)—38no—
167C3×D3×C8—38no—
168#3719—38no—
169#3717—38no—
170#3717—38no—
171#3714—38no—
172C22×C33C2×D411yescharacteristic
173C2×C3⋊(C3⋊D3)—22no—
174C2×C3⋊(C3⋊D3)—21no—
175C2×C3⋊(C3⋊D3)—42no—
176C2×C3×C3⋊D3—41no—
177C2×C3×C3⋊D3C2×D411yescharacteristic
178C2×C3×C3⋊D3C2×D411yescharacteristic
179C2×C32×D3—21no—
180D3×C3⋊D3—42no—
181C33×C4C2×D411yescharacteristic
182C33×C4C2411yescharacteristic
183C3⋊(C3⋊Dic3)—21no—
184C3×C3⋊Dic3C2×D411yescharacteristic
185C3×C3⋊Dic3C2×D411yescharacteristic
186C32×Dic3—21no—
187C32×Dic3—22no—
188D3×#41—94no—
189C2×#239—94no—
190C2×#235—91no—
191C2×#235—91no—
192C3×#137—34no—
193C2×C3×M16—91no—
194#949—94no—
195#942—94no—
196#919—98no—
197#918—98no—
198#916—98no—
199#914—98no—
200#909—91no—
201#909—92no—
202#909C3⋊D311yescharacteristic, complemented
203#908—91no—
204#908—94no—
205#908—91no—
206#908—92no—
207#908—184no—
208D3×M16—34no—
209#904—92no—
210#904—92no—
211C22×C3⋊D3—61no—
212C22×C3⋊D3—61no—
213C22×C3⋊D3—64no—
214C22×C3⋊D3—124no—
215C22×C3⋊D3D1211yescharacteristic
216C22×C3×D3—34no—
217C2×D32—64no—
218C32×Q8—31no—
219C32×Q8—32no—
220C32×Q8—61no—
221C32×D4—31no—
222C32×D4—61no—
223C2×C32×C4—24no—
224C2×C32×C4—61no—
225C2×C32×C4—61no—
226C2×C32×C4C22×D314yes—
227C2×C32×C4C22×D311yescharacteristic
228#638—61no—
229#638—61no—
230#638—61no—
231#638—64no—
232#638—124no—
233C2×C3⋊Dic3—61no—
234C2×C3⋊Dic3D1211yescharacteristic
235C4⋊(C3⋊D3)—32no—
236C4⋊(C3⋊D3)—31no—
237C4⋊(C3⋊D3)—38no—
238C4⋊(C3⋊D3)—34no—
239C4⋊(C3⋊D3)—68no—
240C4⋊(C3⋊D3)—64no—
241C4⋊(C3⋊D3)—62no—
242C4⋊(C3⋊D3)—62no—
243C4⋊(C3⋊D3)—61no—
244C4⋊(C3⋊D3)—61no—
245C4⋊(C3⋊D3)—64no—
246C4⋊(C3⋊D3)—124no—
247C4×C3⋊D3—61no—
248C4×C3⋊D3—61no—
249C4×C3⋊D3—61no—
250C4×C3⋊D3—62no—
251C4×C3⋊D3—62no—
252C4×C3⋊D3—64no—
253C4×C3⋊D3—124no—
254C4×C3⋊D3D1211yescharacteristic
255C4×C3⋊D3D1211yescharacteristic
256C4×C3⋊D3C22×D311yescharacteristic
257C4×C3⋊D3C22×D311yescharacteristic
258#634—31no—
259#634—34no—
260#634—64no—
261#634—61no—
262C3×#67—24no—
263C2×C3×Dic3—24no—
264C2×C3×Dic3—34no—
265C3×D12C22×D314yes—
266C3×C4×D3—24no—
267C3×C4×D3—34no—
268C3×C4×D3—34no—
269C3×C4×D3—34no—
270C3×C4×D3—34no—
271C3×Dic6—24no—
272C3×Dic6C22×D314yes—
273C3×Dic6C22×D318yes—
274#627—64no—
275#626—64no—
276#626—68no—
277#626—68no—
278#626—64no—
279#625—64no—
280#624—68no—
281#624—68no—
282#624—64no—
283D3×Dic3—64no—
284D3×Dic3—64no—
285C32×C8—28no—
286C32×C8C22×D318yes—
287C32×C8C22×D312yes—
288C3⋊(C3⋊C8)C22×D312yes—
289C3×C3⋊C8—38no—
290#575—271noSylow
291C2×C33—21no—
292C2×C33C22×D411yescharacteristic
293C3⋊(C3⋊D3)—41no—
294C3⋊(C3⋊D3)—42no—
295C3×C3⋊D3—21no—
296C3×C3⋊D3—21no—
297C32×D3—41no—
298C3×#41—34no—
299C2×C3×Q8—34no—
300C22×C3×C4—91no—
301C2×#67—181no—
302C2×#67—181no—
303#239—98no—
304#239—98no—
305#239—94no—
306#239—184no—
307#239—184no—
308D3×Q8—94no—
309#237—94no—
310#237—184no—
311D3×D4—94no—
312D3×D4—184no—
313#235—94no—
314#235—91no—
315#235—91no—
316#235—91no—
317#235—92no—
318#235—92no—
319#235—184no—
320#235—181no—
321#235—181no—
322#235—181no—
323#235—184no—
324#235—181no—
325#235C2×C3⋊D311yescharacteristic
326C2×D12—94no—
327C2×D12—91no—
328C2×D12—91no—
329C2×D12—184no—
330C2×D12—184no—
331C2×C4×D3—34no—
332C2×C4×D3—181no—
333C2×C4×D3—184no—
334C2×C4×D3—181no—
335C2×C4×D3—184no—
336C2×Dic6—91no—
337C2×Dic6C2×C3⋊D311yescharacteristic, complemented
338C3×Q16—38no—
339C3×SD16—38no—
340C3×M16—24no—
341C3×M16—91no—
342C3×M16—92no—
343C3×M16D3214yescomplemented
344C3×M16C2×C3⋊D311yescharacteristic, complemented
345C2×C3×C8—92no—
346#216—98no—
347#215—98no—
348#215—98no—
349#213—98no—
350#208—34no—
351Dic12—92no—
352Dic12C2×C3⋊D312yescomplemented
353D24—92no—
354D24—98no—
355D24—92no—
356D24—188no—
357#204—92no—
358#204—92no—
359#204—98no—
360#204—92no—
361#204—188no—
362#204C2×C3⋊D312yescomplemented
363ΓL(1,𝔽25)—38no—
364D3×C8—38no—
365C22×C32—24no—
366C22×C32—61no—
367C22×C32C2×D1211yescharacteristic
368C22×C32D3×D414yes—
369C2×C3⋊D3—41no—
370C2×C3⋊D3—62no—
371C2×C3⋊D3—61no—
372C2×C3⋊D3—68no—
373C2×C3⋊D3—64no—
374C2×C3⋊D3—124no—
375C2×C3⋊D3—128no—
376C2×C3⋊D3—122no—
377C2×C3⋊D3—122no—
378C2×C3⋊D3—128no—
379C2×C3⋊D3—248no—
380C2×C3⋊D3C2×D1211yescharacteristic
381C2×C3⋊D3C2×D1211yescharacteristic
382C2×C3×D3—24no—
383C2×C3×D3—34no—
384C2×C3×D3—34no—
385C2×C3×D3—124no—
386D32—128no—
387C32×C4—24no—
388C32×C4—24no—
389C32×C4—61no—
390C32×C4—62no—
391C32×C4C2×D1211yescharacteristic
392C32×C4D3×D414yes—
393C32×C4C23×D314yes—
394C32×C4C23×D311yescharacteristic
395C3⋊Dic3—61no—
396C3⋊Dic3—64no—
397C3⋊Dic3—124no—
398C3⋊Dic3C2×D1211yescharacteristic
399C3⋊Dic3C2×D1211yescharacteristic
400C3×Dic3—24no—
401C3×Dic3—28no—
402C3×Dic3—34no—
403C3×Dic3—34no—
404C2×#41—271no—
405C2×#41—271no—
406#137—31no—
407#137—272no—
408#137—271no—
409AGL(1,ℤ/8)—272no—
410AGL(1,ℤ/8)—271no—
411AGL(1,ℤ/8)—271no—
412#135—272no—
413#135—272no—
414C2×M16—91no—
415C33#57511yescharacteristic, complemented, Sylow
416C23×C3—91no—
417C22×D3—34no—
418C22×D3—181no—
419C22×D3—184no—
420C22×D3—184no—
421C22×D3—181no—
422C22×D3—364no—
423C3×Q8—34no—
424C3×Q8—38no—
425C3×Q8—64no—
426C3×D4—34no—
427C3×D4—64no—
428C2×C3×C4—24no—
429C2×C3×C4—64no—
430C2×C3×C4—64no—
431C2×C3×C4—91no—
432C2×C3×C4—91no—
433C2×C3×C4—91no—
434C2×C3×C4—91no—
435C2×C3×C4—91no—
436C2×C3×C4C2×D3214yes—
437C2×C3×C4C22×C3⋊D311yescharacteristic
438#67—21no—
439#67—182no—
440#67—182no—
441#67—181no—
442#67—184no—
443#67—181no—
444#67—184no—
445#67—184no—
446#67—181no—
447#67—364no—
448C2×Dic3—21no—
449C2×Dic3—34no—
450C2×Dic3—181no—
451C2×Dic3—184no—
452D12—98no—
453D12—94no—
454D12—92no—
455D12—91no—
456D12—188no—
457D12—184no—
458D12—188no—
459D12—188no—
460D12—184no—
461D12—181no—
462D12—184no—
463D12—181no—
464D12—364no—
465D12C22×C3⋊D311yescharacteristic
466C4×D3—21no—
467C4×D3—34no—
468C4×D3—34no—
469C4×D3—34no—
470C4×D3—34no—
471C4×D3—182no—
472C4×D3—182no—
473C4×D3—181no—
474C4×D3—184no—
475C4×D3—188no—
476C4×D3—188no—
477C4×D3—184no—
478C4×D3—181no—
479C4×D3—184no—
480C4×D3—181no—
481C4×D3—364no—
482Dic6—21no—
483Dic6—94no—
484Dic6—91no—
485Dic6—184no—
486Dic6—181no—
487Dic6—184no—
488Dic6C22×C3⋊D311yescharacteristic
489Dic6C22×C3⋊D312yes—
490C3×C8—28no—
491C3×C8—92no—
492C3×C8C2×D3218yes—
493C3×C8C22×C3⋊D312yes—
494C3⋊C8—38no—
495C2×C32—21no—
496C2×C32—24no—
497C2×C32—24no—
498C2×C32—44no—
499C2×C32—121no—
500C2×C32C22×D1211yescharacteristic
501C2×C32C2×D3×D414yes—
502C3⋊D3—21no—
503C3⋊D3—21no—
504C3⋊D3—121no—
505C3⋊D3—122no—
506C3⋊D3—124no—
507C3⋊D3—128no—
508C3⋊D3—244no—
509C3⋊D3—248no—
510C3×D3—44no—
511C3×D3—64no—
512C3×D3—64no—
513#41—31no—
514#41—271no—
515#41—272no—
516#41—271no—
517#41—272no—
518#41—271no—
519#41—541no—
520#41—541no—
521#41—541no—
522#41—541no—
523C2×Q8—31no—
524C2×Q8—271no—
525C2×D4—271no—
526C2×D4—271no—
527C2×D4—541no—
528C2×D4—541no—
529C22×C4—91no—
530C22×C4—541no—
531C22×C4—541no—
532Q16—32no—
533Q16—272no—
534SD16—32no—
535SD16—272no—
536SD16—272no—
537SD16—272no—
538D8—272no—
539D8—272no—
540M16—91no—
541M16—92no—
542M16D3×C3⋊D311yescharacteristic, complemented
543C2×C8—92no—
544C22×C3—24no—
545C22×C3—64no—
546C22×C3—91no—
547C22×C3—91no—
548C22×C3—361no—
549C22×C3D3×D1214yes—
550C22×C3D4×C3⋊D311yescharacteristic
551C2×D3—21no—
552C2×D3—34no—
553C2×D3—34no—
554C2×D3—124no—
555C2×D3—188no—
556C2×D3—184no—
557C2×D3—182no—
558C2×D3—181no—
559C2×D3—364no—
560C2×D3—368no—
561C2×D3—362no—
562C2×D3—368no—
563C2×D3—362no—
564C2×D3—368no—
565C2×D3—728no—
566C3×C4—24no—
567C3×C4—24no—
568C3×C4—64no—
569C3×C4—68no—
570C3×C4—91no—
571C3×C4—91no—
572C3×C4D3×D1214yes—
573C3×C4D4×C3⋊D311yescharacteristic
574C3×C4C22×D3214yes—
575C3×C4C23×C3⋊D311yescharacteristic
576Dic3—22no—
577Dic3—21no—
578Dic3—34no—
579Dic3—34no—
580Dic3—184no—
581Dic3—181no—
582Dic3—364no—
583C32—24no—
584C32#574811yescharacteristic, complemented
585C32#577714yescomplemented
586C23—91no—
587C23—541no—
588C23—541no—
589Q8—32no—
590Q8—31no—
591Q8—61no—
592Q8—271no—
593Q8—541no—
594D4—31no—
595D4—61no—
596D4—272no—
597D4—271no—
598D4—541no—
599D4—542no—
600D4—542no—
601D4—541no—
602D4—541no—
603D4—541no—
604D4—541no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—91no—
610C2×C4—91no—
611C2×C4—91no—
612C2×C4—541no—
613C2×C4—542no—
614C2×C4—542no—
615C2×C4—541no—
616C2×C4—541no—
617C2×C4—541no—
618C2×C4C2×D3×C3⋊D311yescharacteristic
619C8—92no—
620C8C2×D3×C3⋊D312yes—
621C2×C3—24no—
622C2×C3—21no—
623C2×C3—24no—
624C2×C3—44no—
625C2×C3—124no—
626C2×C3—181no—
627C2×C3—181no—
628C2×C3C2×D3×D1214yes—
629C2×C3C2×D4×C3⋊D311yescharacteristic
630D3—41no—
631D3—64no—
632D3—64no—
633D3—368no—
634D3—364no—
635D3—361no—
636D3—362no—
637D3—728no—
638D3—724no—
639C22—61no—
640C22—91no—
641C22—91no—
642C22—361no—
643C22—542no—
644C22—541no—
645C22—1082no—
646C22—1082no—
647C22C3⋊D3×D1211yescharacteristic
648C4—61no—
649C4—62no—
650C4—91no—
651C4—91no—
652C4—541no—
653C4C3⋊D3×D1211yescharacteristic
654C4C22×D3×C3⋊D311yescharacteristic
655C3—24no—
656C3#1059471114yescomplemented
657C3#1059624811yescharacteristic, complemented
658C2—21no—
659C2—121no—
660C2—181no—
661C2—181no—
662C2—1082no—
663C2—1081no—
664C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×D12
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×M16C22
socleC2×C33C22×D4
hypercenterM16D3×C3⋊D3
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×#235C2semidirect
2. C2×#89018C2semidirect
3. #11723949C2semidirect
4. C3×M16×C3⋊D3C2semidirect
5. C32×#909C2semidirect
6. #11723381C2semidirect
7. #11723367C2semidirect
8. #11723341C2semidirect
9. #11723338C2semidirect
10. #11723329C2semidirect
11. #11723319C2semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C2×C32×Dic6C22semidirect
23. C3⋊D3×D12C22semidirect
29. #89018C22semidirect
30. #89018C22semidirect
31. C3⋊D3×Dic6C22semidirect
32. #89014C22semidirect
33. C33×M16C22semidirect
34. #88852C22semidirect
35. #88851C22semidirect
36. C3×#3748C22semidirect
37. C3×#3743C22semidirect
38. C3×C8×C3⋊D3C22semidirect
39. C32×Dic12C22semidirect
40. C32×#204C22semidirect
41. #88799C22semidirect
42. #88795C22semidirect
43. #88795C22semidirect
44. #88790C22semidirect
59. M16×C3⋊D3D3semidirect
60. C3×#909D3semidirect
130. C2×C3×Dic6C2×D3semidirect
148. C32×M16C2×D3semidirect
149. C32×M16C2×D3semidirect
154. #3748C2×D3semidirect
161. #3743C2×D3semidirect
162. C8×C3⋊D3C2×D3semidirect
164. C3×Dic12C2×D3semidirect
165. C3×#204C2×D3semidirect
202. #909C3⋊D3semidirect
337. C2×Dic6C2×C3⋊D3semidirect
343. C3×M16D32semidirect
344. C3×M16C2×C3⋊D3semidirect
352. Dic12C2×C3⋊D3semidirect
362. #204C2×C3⋊D3semidirect
415. C33#575semidirect
542. M16D3×C3⋊D3semidirect
584. C32#5748semidirect
585. C32#5777semidirect
656. C3#10594711semidirect
657. C3#10596248semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
C2×#890182
#117239492
C3×M16×C3⋊D32
C32×#9092
#117233812
#117233672
#117233412
#117233382
#117233292
#117233192
#105962483
#105947113