Finite Group Databasebeta

← #13178010

Subgroups — #13178010

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
1C3×#41×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89014C211yesmaximal, characteristic, complemented
3#11723992C211yesmaximal, characteristic, complemented
4C32×#949C211yesmaximal, characteristic, complemented
5#11723584C212yesmaximal, complemented
6#11723583C212yesmaximal, complemented
7#11723560C212yesmaximal, complemented
8#11723559C212yesmaximal, complemented
9#11723388C211yesmaximal, characteristic, complemented
10#11723364C211yesmaximal, characteristic, complemented
11C3⋊D3×#208C211yesmaximal, characteristic, complemented
12#10596244—31nomaximal
13#10595469—34nomaximal
14C33×#41C2211yescharacteristic
15C2×#6308C2211yescharacteristic, 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×D12C2211yescharacteristic, complemented
27C2×#6290—21no—
28C2×D3×C3⋊Dic3—21no—
29#89014—21no—
30#89014—21no—
31#89014C2212yes—
32#89014C2212yescomplemented
33#88863C2212yescomplemented
34#88861C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C32×#215C2212yescomplemented
37C32×#213C2212yescomplemented
38C32×#208C2211yescharacteristic, complemented
39#88800C2212yescomplemented
40#88796C2212yescomplemented
41#88792C2212yescomplemented
42#88789C2212yescomplemented
43#88787C2212yescomplemented
44C3⋊D3×C3⋊C8C2212yescomplemented
45#41×C3⋊D3D311yescharacteristic, complemented
46C2×#3830—31no—
47C3×D3×#41—34no—
48C2×#3795—34no—
49C32×AGL(1,ℤ/8)—31no—
50#64160—31no—
51#64160—34no—
52#64160—64no—
53#64141—31no—
54#64125—32no—
55#64124—32no—
56#64121—32no—
57#64120—32no—
58#64118—31no—
59M16×C3⋊D3—31no—
60C3×#949D314yescomplemented
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×C3×C3⋊D3D411yescharacteristic
70C22×C32×D3—21no—
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×#638—21no—
80C3×#638—21no—
81C3×#638—21no—
82C2×C3×C3⋊Dic3—21no—
83C2×C3×C3⋊Dic3D411yescharacteristic
84C3×C4⋊(C3⋊D3)—21no—
85C3×C4⋊(C3⋊D3)C2311yescharacteristic
86C3×C4×C3⋊D3—21no—
87C3×C4×C3⋊D3—21no—
88C3×C4×C3⋊D3—21no—
89C3×C4×C3⋊D3D411yescharacteristic
90C3×C4×C3⋊D3D411yescharacteristic
91C3×C4×C3⋊D3C2311yescharacteristic
92C3×C4×C3⋊D3C2311yescharacteristic
93C3×#634—21no—
94C3×#634C2311yescharacteristic
95C32×D12—21no—
96C32×D12C2312yes—
97#6290—22no—
98#6290—22no—
99D3×C3⋊Dic3—22no—
100D3×C3⋊Dic3—22no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C32×C3⋊C8C2312yes—
103#5822—91no—
104#5773—94no—
105C32×#41—24no—
106C32×#41C2×D314yes—
107C32×#41C2×D311yescharacteristic
108C2×C32×D4—31no—
109C2×#638—61no—
110C22×C3⋊Dic3—61no—
111#3832C2×D311yescharacteristic, complemented
112Q8×C3⋊D3C2×D311yescharacteristic, complemented
113#3830—21no—
114#3830—32no—
115#3830—32no—
116#3830—61no—
117#3830—61no—
118#3830C2×D311yescharacteristic, complemented
119D4×C3⋊D3—21no—
120D4×C3⋊D3C2×D311yescharacteristic, complemented
121#3828—21no—
122#3828C2×D311yescharacteristic
123C2×C4×C3⋊D3—21no—
124C2×C4×C3⋊D3C2×D311yescharacteristic
125C2×#634—31no—
126C2×#634—34no—
127C2×#634—64no—
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×D12C2×D314yescomplemented
137C2×C3×C4×D3—34no—
138C2×C3×C4×D3—64no—
139C2×#625—64no—
140C2×D3×Dic3—64no—
141#3795—38no—
142#3795—38no—
143#3795—64no—
144#3795—64no—
145C32×SD16—32no—
146C32×D8—32no—
147C32×M16—31no—
148#3756—32no—
149#3756—38no—
150#3756—68no—
151#3754—32no—
152#3754—38no—
153#3754—32no—
154#3754—68no—
155#3753—32no—
156#3748—31no—
157#3748—34no—
158#3748—64no—
159#3746—32no—
160#3744—32no—
161#3743—32no—
162C8×C3⋊D3—32no—
163C3×#215C2×D318yescomplemented
164C3×#213C2×D318yescomplemented
165C3×#208C2×D314yescomplemented
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—21no—
175C2×C3×C3⋊D3—41no—
176C2×C3×C3⋊D3—42no—
177C2×C3×C3⋊D3C2×D411yescharacteristic
178C2×C3×C3⋊D3C2×D411yescharacteristic
179C2×C32×D3—22no—
180C2×C32×D3—42no—
181C33×C4—21no—
182C33×C4C2×D411yescharacteristic
183C33×C4C2411yescharacteristic
184C3⋊(C3⋊Dic3)—22no—
185C3×C3⋊Dic3—21no—
186C3×C3⋊Dic3C2×D411yescharacteristic
187C3×C3⋊Dic3C2×D411yescharacteristic
188C2×C3×#41—91no—
189D3×#41—34no—
190C2×#237—94no—
191C2×#235—91no—
192C3×AGL(1,ℤ/8)—34no—
193#951—94no—
194#951—91no—
195#951—91no—
196#951—184no—
197#950—92no—
198#950—92no—
199#949—91no—
200#949C3⋊D311yescharacteristic, complemented
201#942—92no—
202#932—94no—
203#932—92no—
204C2×#208—91no—
205#916—98no—
206#915—98no—
207#912—98no—
208#911—98no—
209#909—94no—
210D3×M16—94no—
211C23×C32—61no—
212C22×C3⋊D3—21no—
213C22×C3⋊D3#6711yescharacteristic
214C22×C3×D3—24no—
215C22×C3×D3—34no—
216C22×C3×D3—64no—
217C32×Q8—24no—
218C32×Q8C22×D314yes—
219C32×Q8C22×D311yescharacteristic
220C32×D4—21no—
221C32×D4—24no—
222C32×D4—24no—
223C32×D4—32no—
224C32×D4—44no—
225C32×D4—61no—
226C32×D4C22×D314yes—
227C32×D4C22×D311yescharacteristic
228C2×C32×C4—21no—
229C2×C32×C4—24no—
230C2×C32×C4—24no—
231C2×C32×C4—44no—
232C2×C32×C4C22×D314yes—
233C2×C32×C4C22×D311yescharacteristic
234#638—21no—
235#638—21no—
236#638—21no—
237#638—62no—
238#638—62no—
239C2×C3⋊Dic3—21no—
240C2×C3⋊Dic3—61no—
241C2×C3⋊Dic3—62no—
242C2×C3⋊Dic3—62no—
243C2×C3⋊Dic3—64no—
244C2×C3⋊Dic3—124no—
245C2×C3⋊Dic3#6711yescharacteristic
246C4⋊(C3⋊D3)—21no—
247C4⋊(C3⋊D3)C22×D311yescharacteristic
248C4×C3⋊D3—21no—
249C4×C3⋊D3—21no—
250C4×C3⋊D3—21no—
251C4×C3⋊D3#6711yescharacteristic
252C4×C3⋊D3#6711yescharacteristic
253C4×C3⋊D3C22×D311yescharacteristic
254C4×C3⋊D3C22×D311yescharacteristic
255#634—21no—
256#634—32no—
257#634—38no—
258#634—68no—
259#634—61no—
260#634—64no—
261#634—124no—
262#634C22×D311yescharacteristic
263C3×#67—64no—
264C3×#67—64no—
265C3×#67—64no—
266C2×C3×Dic3—34no—
267C2×C3×Dic3—64no—
268C3×D12—24no—
269C3×D12—34no—
270C3×D12—64no—
271C3×D12C22×D318yes—
272C3×C4×D3—34no—
273C3×C4×D3—34no—
274C3×C4×D3—34no—
275C3×C4×D3—34no—
276C3×C4×D3—64no—
277C3×C4×D3—64no—
278C3×C4×D3—64no—
279C3×Dic6—34no—
280C3×Dic6—64no—
281#625—68no—
282#625—68no—
283D3×Dic3—68no—
284D3×Dic3—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—21no—
295C3×C3⋊D3—21no—
296C3×C3⋊D3—41no—
297C32×D3—42no—
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—91no—
307C2×C3×D4—34no—
308C2×C3×D4—91no—
309C2×C3×D4—181no—
310C22×C3×C4—91no—
311C22×C3×C4—181no—
312C2×#67—181no—
313C2×#67—184no—
314C22×Dic3—184no—
315#239—34no—
316D3×Q8—34no—
317#237—34no—
318#237—64no—
319#237—98no—
320#237—98no—
321#237—184no—
322#237—184no—
323D3×D4—34no—
324D3×D4—64no—
325#235—34no—
326#235—64no—
327#235—92no—
328#235—92no—
329#235—181no—
330#235—181no—
331C2×D12C2×C3⋊D311yescharacteristic, complemented
332C2×C4×D3—34no—
333C2×C4×D3—64no—
334C2×C4×D3—181no—
335C2×Dic6—94no—
336C2×Dic6—91no—
337C2×Dic6—184no—
338C3×SD16—38no—
339C3×D8—38no—
340C3×M16—34no—
341#216—98no—
342#216—92no—
343#216—92no—
344#216—188no—
345#215—92no—
346#215C2×C3⋊D312yescomplemented
347#214—98no—
348#214—98no—
349#214—92no—
350#214—92no—
351#214—188no—
352#213—98no—
353#213—92no—
354#213C2×C3⋊D312yescomplemented
355#208—94no—
356#208—91no—
357#208—92no—
358#208—184no—
359#208C2×C3⋊D311yescharacteristic, complemented
360C2×C3⋊C8—92no—
361Dic12—98no—
362#204—98no—
363ΓL(1,𝔽25)—98no—
364D3×C8—98no—
365C22×C32—21no—
366C22×C32—24no—
367C22×C32—24no—
368C22×C32—44no—
369C22×C32—62no—
370C22×C32—122no—
371C22×C32D3×D414yes—
372C22×C32C2×#6711yescharacteristic
373C2×C3⋊D3—21no—
374C2×C3⋊D3—41no—
375C2×C3⋊D3—42no—
376C2×C3⋊D3C2×#6711yescharacteristic
377C2×C3⋊D3C2×#6711yescharacteristic
378C2×C3×D3—28no—
379C2×C3×D3—34no—
380C2×C3×D3—34no—
381C2×C3×D3—48no—
382C2×C3×D3—64no—
383C2×C3×D3—124no—
384C2×C3×D3—128no—
385C32×C4—21no—
386C32×C4—24no—
387C32×C4—24no—
388C32×C4—24no—
389C32×C4—44no—
390C32×C4D3×D414yes—
391C32×C4C2×#6711yescharacteristic
392C32×C4C23×D314yes—
393C32×C4C23×D311yescharacteristic
394C3⋊Dic3—21no—
395C3⋊Dic3—62no—
396C3⋊Dic3—68no—
397C3⋊Dic3—128no—
398C3⋊Dic3C2×#6711yescharacteristic
399C3⋊Dic3C2×#6711yescharacteristic
400C3×Dic3—34no—
401C3×Dic3—34no—
402C3×Dic3—64no—
403C2×#41—91no—
404C2×#41—271no—
405#137—271no—
406#137—272no—
407#137—271no—
408AGL(1,ℤ/8)—31no—
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—64no—
416C23×C3—91no—
417C23×C3—181no—
418C22×D3—21no—
419C22×D3—34no—
420C22×D3—64no—
421C3×Q8—24no—
422C3×Q8—91no—
423C3×Q8—181no—
424C3×Q8C2×D3214yes—
425C3×Q8C22×C3⋊D311yescharacteristic
426C3×D4—24no—
427C3×D4—21no—
428C3×D4—24no—
429C3×D4—38no—
430C3×D4—44no—
431C3×D4—64no—
432C3×D4—91no—
433C3×D4—181no—
434C3×D4—181no—
435C3×D4—181no—
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—44no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—181no—
449C2×C3×C4—181no—
450C2×C3×C4—181no—
451C2×C3×C4—181no—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C22×C3⋊D311yescharacteristic
454#67—64no—
455#67—64no—
456#67—64no—
457#67—182no—
458#67—182no—
459#67—188no—
460#67—188no—
461C2×Dic3—34no—
462C2×Dic3—64no—
463C2×Dic3—188no—
464C2×Dic3—188no—
465C2×Dic3—184no—
466C2×Dic3—181no—
467C2×Dic3—364no—
468D12—21no—
469D12—34no—
470D12—64no—
471D12C22×C3⋊D312yes—
472C4×D3—34no—
473C4×D3—34no—
474C4×D3—34no—
475C4×D3—34no—
476C4×D3—64no—
477C4×D3—64no—
478C4×D3—64no—
479C4×D3—182no—
480C4×D3—182no—
481Dic6—34no—
482Dic6—64no—
483Dic6—98no—
484Dic6—92no—
485Dic6—188no—
486Dic6—184no—
487Dic6—181no—
488Dic6—364no—
489C3×C8—38no—
490C3⋊C8—98no—
491C3⋊C8—92no—
492C3⋊C8—188no—
493C3⋊C8C22×C3⋊D312yes—
494C2×C32—21no—
495C2×C32—24no—
496C2×C32—24no—
497C2×C32—44no—
498C2×C32—41no—
499C2×C32—44no—
500C2×C32—84no—
501C2×C32—122no—
502C2×C32C2×D3×D414yes—
503C2×C32C22×#6711yescharacteristic
504C3⋊D3—21no—
505C3⋊D3—21no—
506C3⋊D3—41no—
507C3×D3—48no—
508C3×D3—64no—
509C3×D3—64no—
510C3×D3—124no—
511#41—91no—
512#41—91no—
513#41—91no—
514#41—181no—
515#41—181no—
516#41—272no—
517#41—272no—
518#41—541no—
519#41—541no—
520#41D3×C3⋊D311yescharacteristic
521C2×Q8—91no—
522C2×Q8—271no—
523C2×D4—31no—
524C2×D4—91no—
525C2×D4—181no—
526C2×D4—541no—
527C22×C4—91no—
528C22×C4—181no—
529C22×C4—541no—
530Q16—272no—
531Q16—272no—
532SD16—32no—
533SD16—272no—
534SD16—272no—
535SD16—272no—
536D8—32no—
537D8—272no—
538M16—31no—
539M16—272no—
540M16—271no—
541C2×C8—272no—
542C22×C3—24no—
543C22×C3—21no—
544C22×C3—24no—
545C22×C3—44no—
546C22×C3—68no—
547C22×C3—91no—
548C22×C3—91no—
549C22×C3—128no—
550C22×C3—181no—
551C22×C3—361no—
552C22×C3—362no—
553C22×C3D3×#6714yes—
554C22×C3D4×C3⋊D311yescharacteristic
555C2×D3—22no—
556C2×D3—34no—
557C2×D3—34no—
558C2×D3—42no—
559C2×D3—64no—
560C2×D3—124no—
561C2×D3—128no—
562C3×C4—24no—
563C3×C4—21no—
564C3×C4—24no—
565C3×C4—24no—
566C3×C4—44no—
567C3×C4—91no—
568C3×C4—91no—
569C3×C4—181no—
570C3×C4D3×#6714yes—
571C3×C4D4×C3⋊D311yescharacteristic
572C3×C4C22×D3214yes—
573C3×C4C23×C3⋊D311yescharacteristic
574Dic3—34no—
575Dic3—34no—
576Dic3—64no—
577Dic3—188no—
578Dic3—182no—
579Dic3—368no—
580C32—24no—
581C32#577314yescomplemented
582C32#582211yescharacteristic, complemented
583C23—61no—
584C23—91no—
585C23—181no—
586Q8—91no—
587Q8—181no—
588Q8—272no—
589Q8—541no—
590Q8C2×D3×C3⋊D311yescharacteristic
591D4—21no—
592D4—32no—
593D4—61no—
594D4—91no—
595D4—181no—
596D4—181no—
597D4—181no—
598D4—181no—
599D4—542no—
600D4—542no—
601D4C2×D3×C3⋊D311yescharacteristic
602C2×C4—21no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—181no—
609C2×C4—181no—
610C2×C4—181no—
611C2×C4—181no—
612C2×C4—542no—
613C2×C4—542no—
614C2×C4—541no—
615C2×C4C2×D3×C3⋊D311yescharacteristic
616C8—32no—
617C8—272no—
618C2×C3—24no—
619C2×C3—21no—
620C2×C3—24no—
621C2×C3—44no—
622C2×C3—44no—
623C2×C3—41no—
624C2×C3—84no—
625C2×C3—128no—
626C2×C3—181no—
627C2×C3—181no—
628C2×C3—361no—
629C2×C3C2×D3×#6714yes—
630C2×C3C2×D4×C3⋊D311yescharacteristic
631D3—42no—
632D3—64no—
633D3—64no—
634D3—124no—
635C22—21no—
636C22—62no—
637C22—91no—
638C22—91no—
639C22—122no—
640C22—181no—
641C22—361no—
642C22—362no—
643C22C3⋊D3×#6711yescharacteristic
644C4—21no—
645C4—91no—
646C4—91no—
647C4—181no—
648C4—542no—
649C4C3⋊D3×#6711yescharacteristic
650C4C22×D3×C3⋊D311yescharacteristic
651C3—24no—
652C3#1059546914yescomplemented
653C3#1059624411yescharacteristic, complemented
654C2—21no—
655C2—41no—
656C2—122no—
657C2—181no—
658C2—181no—
659C2—361no—
660C2C2×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×#89014C2semidirect
3. #11723992C2semidirect
4. C32×#949C2semidirect
5. #11723584C2semidirect
6. #11723583C2semidirect
7. #11723560C2semidirect
8. #11723559C2semidirect
9. #11723388C2semidirect
10. #11723364C2semidirect
11. C3⋊D3×#208C2semidirect
15. C2×#6308C22semidirect
16. C3×#3832C22semidirect
17. C3×Q8×C3⋊D3C22semidirect
19. C3×#3830C22semidirect
21. C3×D4×C3⋊D3C22semidirect
26. C2×C32×D12C22semidirect
32. #89014C22semidirect
33. #88863C22semidirect
34. #88861C22semidirect
35. #88855C22semidirect
36. C32×#215C22semidirect
37. C32×#213C22semidirect
38. C32×#208C22semidirect
39. #88800C22semidirect
40. #88796C22semidirect
41. #88792C22semidirect
42. #88789C22semidirect
43. #88787C22semidirect
44. C3⋊D3×C3⋊C8C22semidirect
45. #41×C3⋊D3D3semidirect
60. C3×#949D3semidirect
111. #3832C2×D3semidirect
112. Q8×C3⋊D3C2×D3semidirect
118. #3830C2×D3semidirect
120. D4×C3⋊D3C2×D3semidirect
136. C2×C3×D12C2×D3semidirect
163. C3×#215C2×D3semidirect
164. C3×#213C2×D3semidirect
165. C3×#208C2×D3semidirect
200. #949C3⋊D3semidirect
331. C2×D12C2×C3⋊D3semidirect
346. #215C2×C3⋊D3semidirect
354. #213C2×C3⋊D3semidirect
359. #208C2×C3⋊D3semidirect
414. C33#575semidirect
581. C32#5773semidirect
582. C32#5822semidirect
652. C3#10595469semidirect
653. C3#10596244semidirect

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×#890142
#117239922
C32×#9492
#117235842
#117235832
#117235602
#117235592
#117233882
#117233642
C3⋊D3×#2082
#105962443
#105954693