Finite Group Databasebeta

← #13178012

Subgroups — #13178012

Order 1728.

1812 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×#41C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723992C211yesmaximal, characteristic, complemented
4C3×#64158C211yesmaximal, characteristic, complemented
5#11723580C212yesmaximal, complemented
6#11723579C212yesmaximal, complemented
7#11723556C212yesmaximal, complemented
8#11723555C212yesmaximal, complemented
9#11723379C211yesmaximal, characteristic, complemented
10#11723365C211yesmaximal, characteristic, complemented
11D3×#3748C211yesmaximal, characteristic, complemented
12#10596293—31nomaximal
13#10595469—34nomaximal
14C33×#41C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, 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×#6290—21no—
28C2×Dic3×C3⋊D3—21no—
29#89021—21no—
30#89021—21no—
31#89021C2212yes—
32#89021C2212yescomplemented
33#88863C2212yescomplemented
34#88861C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C3×#3755C2212yescomplemented
37C3×#3753C2212yescomplemented
38C3×#3748C2211yescharacteristic, complemented
39#88799C2212yescomplemented
40#88794C2212yescomplemented
41#88793C2212yescomplemented
42#88789C2212yescomplemented
43#88786C2212yescomplemented
44D3×C3⋊(C3⋊C8)C2212yescomplemented
45C2×C32×#41—31no—
46C2×#3828—31no—
47C3×D3×#41D314yescomplemented
48C2×#3795—34no—
49#64160—31no—
50#64160—34no—
51#64160—31no—
52#64160—64no—
53#64159—32no—
54#64159—32no—
55#64158—31no—
56#64158D311yescharacteristic, complemented
57#64151—32no—
58#64141—32no—
59C2×#3748—31no—
60C3×#949—34no—
61#63947—38no—
62#63946—38no—
63#63933—38no—
64#63932—38no—
65#63831—34no—
66#63823—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
76C2×C3⋊(C3⋊Dic3)—21no—
77#6308—21no—
78#6308C2312yes—
79C3×C4⋊(C3⋊D3)—21no—
80C3×C4⋊(C3⋊D3)C2312yes—
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#6290—22no—
98#6290—22no—
99Dic3×C3⋊D3—22no—
100Dic3×C3⋊D3—22no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C3×C3⋊(C3⋊C8)C2312yes—
103#5822—94no—
104#5773—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)C2×D311yescharacteristic, complemented
124C2×C4×C3⋊D3—61no—
125C2×#634—31no—
126C2×#634—34no—
127C2×#634—64no—
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×D12—34no—
137C2×C3×C4×D3—24no—
138C2×C3×C4×D3C2×D314yes—
139C2×#625—64no—
140C2×D3×Dic3—64no—
141#3795—38no—
142#3795—38no—
143#3795—64no—
144#3795—64no—
145#3756—32no—
146#3756—38no—
147#3756—32no—
148#3756—68no—
149#3755—32no—
150#3755C2×D312yescomplemented
151#3754—32no—
152#3754—38no—
153#3754—32no—
154#3754—68no—
155#3753—32no—
156#3753C2×D312yescomplemented
157#3748—31no—
158#3748—34no—
159#3748—32no—
160#3748—64no—
161#3748C2×D311yescharacteristic, complemented
162C2×C3⋊(C3⋊C8)—32no—
163C3×#215—38no—
164C3×#213—38no—
165C3×#208—34no—
166#3719—38no—
167#3716—38no—
168#3715—38no—
169#3713—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×#237—91no—
191C2×#235—94no—
192C3×AGL(1,ℤ/8)—91no—
193#951—94no—
194#951—94no—
195#951—91no—
196#951—184no—
197#950—98no—
198#950—98no—
199#949—34no—
200#949—94no—
201#942—98no—
202#932—98no—
203#932—91no—
204C2×#208—94no—
205#916—92no—
206#915—92no—
207#912—92no—
208#911—92no—
209#909—91no—
210D3×M16—91no—
211C23×C32—31no—
212C23×C32—61no—
213C22×C3⋊D3—21no—
214C22×C3×D3—24no—
215C22×C3×D3—64no—
216C22×C3×D3#6714yes—
217C32×Q8—24no—
218C32×Q8—31no—
219C32×Q8—61no—
220C32×Q8C22×D314yes—
221C32×Q8C22×D311yescharacteristic
222C32×D4—21no—
223C32×D4—24no—
224C32×D4—24no—
225C32×D4—31no—
226C32×D4—44no—
227C32×D4—61no—
228C32×D4—61no—
229C32×D4—61no—
230C32×D4—61no—
231C32×D4C22×D314yes—
232C32×D4C22×D311yescharacteristic
233C2×C32×C4—21no—
234C2×C32×C4—24no—
235C2×C32×C4—24no—
236C2×C32×C4—31no—
237C2×C32×C4—31no—
238C2×C32×C4—31no—
239C2×C32×C4—31no—
240C2×C32×C4—31no—
241C2×C32×C4—44no—
242C2×C32×C4—61no—
243C2×C32×C4—61no—
244C2×C32×C4—61no—
245C2×C32×C4—61no—
246C2×C32×C4C22×D314yes—
247C2×C32×C4C22×D311yescharacteristic
248#638—62no—
249#638—62no—
250C2×C3⋊Dic3—61no—
251C2×C3⋊Dic3—64no—
252C2×C3⋊Dic3—124no—
253C4⋊(C3⋊D3)—21no—
254C4⋊(C3⋊D3)C22×D312yes—
255C4×C3⋊D3—62no—
256C4×C3⋊D3—62no—
257#634—32no—
258#634—38no—
259#634—68no—
260#634—61no—
261#634—64no—
262#634—124no—
263C3×#67—24no—
264C3×#67—24no—
265C3×#67—24no—
266C2×C3×Dic3—24no—
267C2×C3×Dic3#6714yes—
268C3×D12—24no—
269C3×D12—38no—
270C3×D12—64no—
271C3×D12C22×D314yes—
272C3×C4×D3—24no—
273C3×C4×D3—24no—
274C3×C4×D3—24no—
275C3×C4×D3#6714yes—
276C3×C4×D3#6714yes—
277C3×C4×D3C22×D314yes—
278C3×C4×D3C22×D314yes—
279C3×Dic6—24no—
280C3×Dic6C22×D314yes—
281#625—68no—
282#625—68no—
283D3×Dic3—68no—
284D3×Dic3—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×D4—34no—
308C2×C3×D4—64no—
309C2×C3×D4—91no—
310C22×C3×C4—34no—
311C22×C3×C4—64no—
312C2×#67—184no—
313C2×#67—181no—
314C22×Dic3—181no—
315#239C2×C3⋊D311yescharacteristic, complemented
316D3×Q8C2×C3⋊D311yescharacteristic, complemented
317#237—21no—
318#237—92no—
319#237—92no—
320#237—181no—
321#237—181no—
322#237C2×C3⋊D311yescharacteristic, complemented
323D3×D4—21no—
324D3×D4C2×C3⋊D311yescharacteristic, complemented
325#235—21no—
326#235—98no—
327#235—98no—
328#235—184no—
329#235—184no—
330#235C2×C3⋊D311yescharacteristic
331C2×D12—34no—
332C2×C4×D3—21no—
333C2×C4×D3—184no—
334C2×C4×D3C2×C3⋊D311yescharacteristic
335C2×Dic6—94no—
336C2×Dic6—91no—
337C2×Dic6—184no—
338C3×SD16—92no—
339C3×D8—92no—
340C3×M16—91no—
341#216—98no—
342#216—98no—
343#216—92no—
344#216—188no—
345#215—38no—
346#215—98no—
347#214—98no—
348#214—98no—
349#214—92no—
350#214—92no—
351#214—188no—
352#213—38no—
353#213—98no—
354#213—92no—
355#208—34no—
356#208—98no—
357#208—94no—
358#208—91no—
359#208—184no—
360C2×C3⋊C8—98no—
361Dic12—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—22no—
377C2×C3⋊D3—42no—
378C2×C3×D3—24no—
379C2×C3×D3—44no—
380C2×C3×D3—48no—
381C2×C3×D3—68no—
382C2×C3×D3—128no—
383C2×C3×D3C2×#6714yes—
384C2×C3×D3C2×#6714yes—
385C32×C4—21no—
386C32×C4—24no—
387C32×C4—24no—
388C32×C4—24no—
389C32×C4—31no—
390C32×C4—31no—
391C32×C4—44no—
392C32×C4—61no—
393C32×C4D3×D411yescharacteristic
394C32×C4C2×#6714yes—
395C32×C4C23×D314yes—
396C32×C4C23×D311yescharacteristic
397C3⋊Dic3—62no—
398C3⋊Dic3—68no—
399C3⋊Dic3—128no—
400C3×Dic3—24no—
401C3×Dic3C2×#6714yes—
402C3×Dic3C2×#6714yes—
403C2×#41—31no—
404C2×#41—271no—
405#137—271no—
406#137—272no—
407#137—271no—
408AGL(1,ℤ/8)—91no—
409AGL(1,ℤ/8)—272no—
410AGL(1,ℤ/8)—271no—
411#135—272no—
412#135—272no—
413C2×M16—271no—
414C33#57511yescharacteristic, complemented, Sylow
415C23×C3—34no—
416C23×C3—64no—
417C23×C3—181no—
418C22×D3—21no—
419C22×D3—64no—
420C22×D3#63811yescharacteristic
421C3×Q8—24no—
422C3×Q8—34no—
423C3×Q8—64no—
424C3×Q8C2×D3214yes—
425C3×Q8C22×C3⋊D311yescharacteristic
426C3×D4—24no—
427C3×D4—21no—
428C3×D4—24no—
429C3×D4—34no—
430C3×D4—44no—
431C3×D4—64no—
432C3×D4—64no—
433C3×D4—64no—
434C3×D4—64no—
435C3×D4—92no—
436C3×D4—181no—
437C3×D4C2×D3214yes—
438C3×D4C22×C3⋊D311yescharacteristic
439C2×C3×C4—24no—
440C2×C3×C4—21no—
441C2×C3×C4—24no—
442C2×C3×C4—34no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—34no—
447C2×C3×C4—44no—
448C2×C3×C4—64no—
449C2×C3×C4—64no—
450C2×C3×C4—64no—
451C2×C3×C4—64no—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C22×C3⋊D311yescharacteristic
454#67—21no—
455#67—21no—
456#67—21no—
457#67—188no—
458#67—188no—
459#67—182no—
460#67—182no—
461C2×Dic3—21no—
462C2×Dic3—184no—
463C2×Dic3—181no—
464C2×Dic3—182no—
465C2×Dic3—182no—
466C2×Dic3—364no—
467C2×Dic3#63811yescharacteristic
468D12—21no—
469D12—38no—
470D12—64no—
471D12C22×C3⋊D311yescharacteristic
472C4×D3—21no—
473C4×D3—21no—
474C4×D3—21no—
475C4×D3—188no—
476C4×D3—188no—
477C4×D3#63811yescharacteristic
478C4×D3#63811yescharacteristic
479C4×D3C22×C3⋊D311yescharacteristic
480C4×D3C22×C3⋊D311yescharacteristic
481Dic6—21no—
482Dic6—98no—
483Dic6—92no—
484Dic6—188no—
485Dic6—184no—
486Dic6—181no—
487Dic6—364no—
488Dic6C22×C3⋊D311yescharacteristic
489C3×C8—92no—
490C3⋊C8—38no—
491C3⋊C8—98no—
492C3⋊C8—92no—
493C3⋊C8—188no—
494C2×C32—21no—
495C2×C32—24no—
496C2×C32—24no—
497C2×C32—44no—
498C2×C32—41no—
499C2×C32—44no—
500C2×C32—61no—
501C2×C32—61no—
502C2×C32—84no—
503C2×C32—121no—
504C2×C32C2×D3×D411yescharacteristic
505C2×C32C22×#6714yes—
506C3⋊D3—42no—
507C3×D3—24no—
508C3×D3—24no—
509C3×D3—44no—
510C3×D3—128no—
511#41—31no—
512#41—31no—
513#41—31no—
514#41—61no—
515#41—61no—
516#41—272no—
517#41—272no—
518#41—541no—
519#41—541no—
520#41D3×C3⋊D311yescharacteristic
521C2×Q8—31no—
522C2×Q8—271no—
523C2×D4—31no—
524C2×D4—61no—
525C2×D4—91no—
526C2×D4—541no—
527C22×C4—31no—
528C22×C4—61no—
529C22×C4—541no—
530Q16—272no—
531Q16—272no—
532SD16—92no—
533SD16—272no—
534SD16—272no—
535SD16—272no—
536D8—92no—
537D8—272no—
538M16—91no—
539M16—272no—
540M16—271no—
541C2×C8—272no—
542C22×C3—24no—
543C22×C3—21no—
544C22×C3—24no—
545C22×C3—34no—
546C22×C3—34no—
547C22×C3—44no—
548C22×C3—64no—
549C22×C3—124no—
550C22×C3—128no—
551C22×C3—182no—
552C22×C3—362no—
553C22×C3D3×#6714yes—
554C22×C3C2×#63811yescharacteristic
555C2×D3—21no—
556C2×D3—41no—
557C2×D3—42no—
558C2×D3—68no—
559C2×D3—128no—
560C2×D3C2×#63811yescharacteristic
561C2×D3C2×#63811yescharacteristic
562C3×C4—24no—
563C3×C4—21no—
564C3×C4—24no—
565C3×C4—24no—
566C3×C4—34no—
567C3×C4—34no—
568C3×C4—44no—
569C3×C4—64no—
570C3×C4D3×#6714yes—
571C3×C4C2×#63811yescharacteristic
572C3×C4C22×D3214yes—
573C3×C4C23×C3⋊D311yescharacteristic
574Dic3—21no—
575Dic3—188no—
576Dic3—182no—
577Dic3—368no—
578Dic3C2×#63811yescharacteristic
579Dic3C2×#63811yescharacteristic
580C32—24no—
581C32#577311yescharacteristic, complemented
582C32#582214yescomplemented
583C23—31no—
584C23—61no—
585C23—181no—
586Q8—31no—
587Q8—61no—
588Q8—272no—
589Q8—541no—
590Q8C2×D3×C3⋊D311yescharacteristic
591D4—21no—
592D4—31no—
593D4—61no—
594D4—61no—
595D4—61no—
596D4—61no—
597D4—92no—
598D4—181no—
599D4—542no—
600D4—542no—
601D4C2×D3×C3⋊D311yescharacteristic
602C2×C4—21no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—31no—
606C2×C4—31no—
607C2×C4—31no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—542no—
613C2×C4—542no—
614C2×C4—541no—
615C2×C4C2×D3×C3⋊D311yescharacteristic
616C8—92no—
617C8—272no—
618C2×C3—24no—
619C2×C3—21no—
620C2×C3—24no—
621C2×C3—44no—
622C2×C3—44no—
623C2×C3—41no—
624C2×C3—64no—
625C2×C3—64no—
626C2×C3—84no—
627C2×C3—124no—
628C2×C3—362no—
629C2×C3C2×D3×#6714yes—
630C2×C3C22×#63811yescharacteristic
631D3—21no—
632D3—21no—
633D3—41no—
634D3—128no—
635C22—21no—
636C22—31no—
637C22—31no—
638C22—61no—
639C22—121no—
640C22—122no—
641C22—182no—
642C22—362no—
643C22D3×#63811yescharacteristic
644C4—21no—
645C4—31no—
646C4—31no—
647C4—61no—
648C4—542no—
649C4D3×#63811yescharacteristic
650C4C22×D3×C3⋊D311yescharacteristic
651C3—24no—
652C3#1059546914yescomplemented
653C3#1059629311yescharacteristic, complemented
654C2—21no—
655C2—41no—
656C2—61no—
657C2—61no—
658C2—121no—
659C2—362no—
660C2C2×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×#89021C2semidirect
3. #11723992C2semidirect
4. C3×#64158C2semidirect
5. #11723580C2semidirect
6. #11723579C2semidirect
7. #11723556C2semidirect
8. #11723555C2semidirect
9. #11723379C2semidirect
10. #11723365C2semidirect
11. D3×#3748C2semidirect
15. C2×#6308C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C32×#239C22semidirect
18. C32×D3×Q8C22semidirect
20. C32×#237C22semidirect
22. C32×D3×D4C22semidirect
32. #89021C22semidirect
33. #88863C22semidirect
34. #88861C22semidirect
35. #88855C22semidirect
36. C3×#3755C22semidirect
37. C3×#3753C22semidirect
38. C3×#3748C22semidirect
39. #88799C22semidirect
40. #88794C22semidirect
41. #88793C22semidirect
42. #88789C22semidirect
43. #88786C22semidirect
44. D3×C3⋊(C3⋊C8)C22semidirect
47. C3×D3×#41D3semidirect
56. #64158D3semidirect
123. C2×C4⋊(C3⋊D3)C2×D3semidirect
128. C3×#239C2×D3semidirect
129. C3×D3×Q8C2×D3semidirect
131. C3×#237C2×D3semidirect
133. C3×D3×D4C2×D3semidirect
150. #3755C2×D3semidirect
156. #3753C2×D3semidirect
161. #3748C2×D3semidirect
189. D3×#41C3⋊D3semidirect
315. #239C2×C3⋊D3semidirect
316. D3×Q8C2×C3⋊D3semidirect
322. #237C2×C3⋊D3semidirect
324. D3×D4C2×C3⋊D3semidirect
414. C33#575semidirect
581. C32#5773semidirect
582. C32#5822semidirect
652. C3#10595469semidirect
653. 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×#890212
#117239922
C3×#641582
#117235802
#117235792
#117235562
#117235552
#117233792
#117233652
D3×#37482
#105962933
#105954693