Finite Group Databasebeta

← #13177537

Subgroups — #13177537

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
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C3⋊D3×#235C211yesmaximal, characteristic, complemented
3#11723973C211yesmaximal, characteristic, complemented
4C32×#932C211yesmaximal, characteristic, complemented
5#11723561C212yesmaximal, complemented
6#11723559C212yesmaximal, complemented
7#11723552C212yesmaximal, complemented
8#11723546C212yesmaximal, complemented
9#11723387C211yesmaximal, characteristic, complemented
10#11723373C211yesmaximal, characteristic, complemented
11C3⋊D3×#208C211yesmaximal, characteristic, complemented
12#10596244—31nomaximal
13#10595095—34nomaximal
14C2×C33×D4C2211yescharacteristic, complemented
15#89076C2211yescharacteristic
16C2×C3×#638—21no—
17C22×C3×C3⋊Dic3—21no—
18C3×#3830—21no—
19C3×#3830—21no—
20C3×#3830C2212yes—
21C3×#3830C2212yescomplemented
22C2×C3×C4×C3⋊D3C2211yescharacteristic
23C2×C3×#634C2211yescharacteristic, complemented
24C32×#235C2211yescharacteristic
25C3⋊D3×#67—21no—
26#89028—21no—
27C3⋊D3×D12C2211yescharacteristic, complemented
28C4×D3×C3⋊D3—21no—
29#89020—21no—
30#89018C2211yescharacteristic, complemented
31C3⋊D3×Dic6C2211yescharacteristic, complemented
32#89014C2211yescharacteristic, complemented
33#88861C2212yescomplemented
34#88860C2212yescomplemented
35#88855C2211yescharacteristic, complemented
36C32×#214C2212yescomplemented
37C32×#213C2212yescomplemented
38C32×#208C2211yescharacteristic, complemented
39#88800C2212yescomplemented
40#88798C2212yescomplemented
41#88797C2212yescomplemented
42#88792C2212yescomplemented
43#88787C2212yescomplemented
44C3⋊D3×C3⋊C8C2212yescomplemented
45#41×C3⋊D3—31no—
46C2×#3830D311yescharacteristic, complemented
47C2×C3×#237—34no—
48D3×#235—34no—
49C32×AGL(1,ℤ/8)—31no—
50#64160—31no—
51#64141—31no—
52#64141—34no—
53#64141—64no—
54#64125—32no—
55#64124—32no—
56#64121—32no—
57#64120—32no—
58#64118—31no—
59M16×C3⋊D3—31no—
60C3×#932D314yescomplemented
61#63935—38no—
62#63933—38no—
63#63929—38no—
64#63925—38no—
65#63833—34no—
66#63827—34no—
67D3×#208—34no—
68C23×C33—21no—
69C22×C3×C3⋊D3D411yescharacteristic
70C2×D3×C3⋊D3—21no—
71C33×D4—21no—
72C33×D4C2312yes—
73C2×C33×C4C2311yescharacteristic
74#6312—21no—
75C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
76C4×C3⋊(C3⋊D3)—21no—
77#6308C2311yescharacteristic
78C3×#638—22no—
79C3×#638—22no—
80C2×C3×C3⋊Dic3—21no—
81C2×C3×C3⋊Dic3—22no—
82C2×C3×C3⋊Dic3—22no—
83C2×C3×C3⋊Dic3D411yescharacteristic
84C3×C4×C3⋊D3D411yescharacteristic
85C3×C4×C3⋊D3D411yescharacteristic
86C3×C4×C3⋊D3C2311yescharacteristic
87C3×C4×C3⋊D3C2311yescharacteristic
88C3×#634—21no—
89C3×#634C2312yes—
90C32×#67—21no—
91C32×D12C2311yescharacteristic
92C32×C4×D3—21no—
93C32×Dic6C2311yescharacteristic
94#6293—21no—
95#6292—21no—
96#6291—21no—
97#6290—21no—
98#6289—21no—
99Dic3×C3⋊D3—21no—
100D3×C3⋊Dic3—21no—
101C3⋊(C3⋊(C3⋊C8))C2312yes—
102C32×C3⋊C8C2312yes—
103#5822—91no—
104#5773—94no—
105C32×#41—31no—
106C2×C32×D4—24no—
107C2×C32×D4C2×D314yescomplemented
108C2×C32×D4C2×D311yescharacteristic, complemented
109C2×#638—21no—
110C22×C3⋊Dic3—21no—
111#3832—31no—
112Q8×C3⋊D3—31no—
113#3830—21no—
114#3830—21no—
115#3830—31no—
116#3830—61no—
117#3830C2×D312yes—
118#3830C2×D312yescomplemented
119D4×C3⋊D3—31no—
120D4×C3⋊D3—61no—
121#3828—31no—
122#3828—34no—
123#3828—61no—
124#3828—64no—
125C2×C4×C3⋊D3—61no—
126C2×C4×C3⋊D3C2×D311yescharacteristic
127C2×#634C2×D311yescharacteristic, complemented
128C2×C3×#67—64no—
129C22×C3×Dic3—64no—
130C3×#237—38no—
131C3×#237—38no—
132C3×#237—64no—
133C3×#237—64no—
134C3×#235C2×D314yes—
135C2×C3×C4×D3—34no—
136C2×C3×Dic6—34no—
137D3×#67—64no—
138#3804—64no—
139D3×D12—34no—
140C4×D32—64no—
141#3799—34no—
142#3798—64no—
143#3795—34no—
144D3×Dic6—34no—
145C32×SD16—32no—
146C32×D8—32no—
147C32×M16—31no—
148#3756—32no—
149#3754—32no—
150#3754—38no—
151#3754—32no—
152#3754—68no—
153#3753—32no—
154#3753—38no—
155#3753—68no—
156#3748—31no—
157#3748—34no—
158#3748—64no—
159#3746—32no—
160#3744—32no—
161#3743—32no—
162C8×C3⋊D3—32no—
163C3×#214C2×D318yescomplemented
164C3×#213C2×D318yescomplemented
165C3×#208C2×D314yescomplemented
166#3719—38no—
167#3718—38no—
168#3717—38no—
169#3715—38no—
170#3711—38no—
171D3×C3⋊C8—38no—
172C22×C33—22no—
173C22×C33—42no—
174C22×C33C2×D411yescharacteristic
175C2×C3⋊(C3⋊D3)—21no—
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⋊Dic3—22no—
185C3×C3⋊Dic3C2×D411yescharacteristic
186C3×C3⋊Dic3C2×D411yescharacteristic
187C32×Dic3—21no—
188C2×C3×#41—91no—
189D3×#41—94no—
190C2×#237—34no—
191C2×#235—91no—
192C3×AGL(1,ℤ/8)—34no—
193#951—94no—
194#951—92no—
195#950—92no—
196#950—92no—
197#949—92no—
198#942—91no—
199#942—91no—
200#932—94no—
201#932—91no—
202#932—184no—
203#932C3⋊D311yescharacteristic, complemented
204C2×#208—91no—
205#916—98no—
206#915—98no—
207#912—98no—
208#911—98no—
209#909—94no—
210D3×M16—94no—
211C23×C32—21no—
212C23×C32—24no—
213C23×C32—44no—
214C22×C3⋊D3—61no—
215C22×C3⋊D3#6711yescharacteristic
216C22×C3×D3—34no—
217C2×D32—64no—
218C32×Q8—31no—
219C32×D4—21no—
220C32×D4—28no—
221C32×D4—24no—
222C32×D4—31no—
223C32×D4—44no—
224C32×D4—61no—
225C32×D4C22×D318yes—
226C32×D4C22×D312yes—
227C2×C32×C4—24no—
228C2×C32×C4—61no—
229C2×C32×C4C22×D314yes—
230C2×C32×C4C22×D311yescharacteristic
231#638—22no—
232#638—22no—
233#638—61no—
234#638—61no—
235#638—61no—
236#638—64no—
237#638—124no—
238C2×C3⋊Dic3—21no—
239C2×C3⋊Dic3—22no—
240C2×C3⋊Dic3—22no—
241C2×C3⋊Dic3—61no—
242C2×C3⋊Dic3#6711yescharacteristic
243C4⋊(C3⋊D3)—31no—
244C4⋊(C3⋊D3)—34no—
245C4⋊(C3⋊D3)—64no—
246C4⋊(C3⋊D3)—61no—
247C4×C3⋊D3—61no—
248C4×C3⋊D3—61no—
249C4×C3⋊D3—61no—
250C4×C3⋊D3—64no—
251C4×C3⋊D3—124no—
252C4×C3⋊D3#6711yescharacteristic
253C4×C3⋊D3#6711yescharacteristic
254C4×C3⋊D3C22×D311yescharacteristic
255C4×C3⋊D3C22×D311yescharacteristic
256#634—21no—
257#634—31no—
258#634—34no—
259#634—64no—
260#634—61no—
261#634C22×D312yes—
262C3×#67—24no—
263C3×#67—68no—
264C3×#67—68no—
265C2×C3×Dic3—34no—
266C2×C3×Dic3—68no—
267C2×C3×Dic3—68no—
268C2×C3×Dic3—64no—
269C3×D12C22×D314yes—
270C3×C4×D3—24no—
271C3×C4×D3—34no—
272C3×C4×D3—34no—
273C3×C4×D3—34no—
274C3×C4×D3—34no—
275C3×Dic6—38no—
276C3×Dic6—64no—
277C3×Dic6C22×D314yes—
278#627—64no—
279#626—64no—
280#626—64no—
281#625—64no—
282#624—64no—
283D3×Dic3—64no—
284D3×Dic3—64no—
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—42no—
293C2×C33C22×D411yescharacteristic
294C3⋊(C3⋊D3)—41no—
295C3×C3⋊D3—21no—
296C3×C3⋊D3—21no—
297C32×D3—41no—
298C3×#41—34no—
299C3×#41—92no—
300C3×#41—92no—
301C3×#41—181no—
302C3×#41—181no—
303C2×C3×Q8—91no—
304C2×C3×D4—24no—
305C2×C3×D4—181no—
306C2×C3×D4D3214yescomplemented
307C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
308C22×C3×C4—91no—
309C22×C3×C4—181no—
310C2×#67—64no—
311C2×#67—181no—
312C22×Dic3—64no—
313#239—94no—
314D3×Q8—94no—
315#237—38no—
316#237—38no—
317#237—64no—
318#237—64no—
319#237—94no—
320#237—184no—
321D3×D4—94no—
322D3×D4—184no—
323#235—94no—
324#235—91no—
325#235—91no—
326#235—91no—
327#235—184no—
328#235—181no—
329#235—184no—
330#235—181no—
331#235C2×C3⋊D311yescharacteristic
332C2×D12—91no—
333C2×C4×D3—34no—
334C2×C4×D3—181no—
335C2×C4×D3—184no—
336C2×Dic6—34no—
337C2×Dic6—91no—
338C3×SD16—38no—
339C3×D8—38no—
340C3×M16—34no—
341#216—98no—
342#216—92no—
343#216—92no—
344#215—92no—
345#215—92no—
346#214—98no—
347#214—98no—
348#214—92no—
349#214—188no—
350#214C2×C3⋊D312yescomplemented
351#213—98no—
352#213—92no—
353#213—188no—
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—22no—
366C22×C32—24no—
367C22×C32—28no—
368C22×C32—48no—
369C22×C32—42no—
370C22×C32—48no—
371C22×C32—61no—
372C22×C32—88no—
373C22×C32D3×D414yes—
374C22×C32C2×#6711yescharacteristic
375C2×C3⋊D3—41no—
376C2×C3⋊D3—61no—
377C2×C3⋊D3—64no—
378C2×C3⋊D3—124no—
379C2×C3⋊D3—122no—
380C2×C3⋊D3C2×#6711yescharacteristic
381C2×C3⋊D3C2×#6711yescharacteristic
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×C4D3×D414yes—
391C32×C4C2×#6711yescharacteristic
392C32×C4C23×D314yes—
393C32×C4C23×D311yescharacteristic
394C3⋊Dic3—22no—
395C3⋊Dic3—61no—
396C3⋊Dic3—64no—
397C3⋊Dic3—124no—
398C3⋊Dic3C2×#6711yescharacteristic
399C3⋊Dic3C2×#6711yescharacteristic
400C3×Dic3—24no—
401C3×Dic3—34no—
402C3×Dic3—34no—
403C3×Dic3—68no—
404C2×#41—91no—
405C2×#41—271no—
406#137—271no—
407#137—272no—
408#137—271no—
409AGL(1,ℤ/8)—31no—
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—24no—
417C23×C3—21no—
418C23×C3—44no—
419C23×C3—91no—
420C22×D3—34no—
421C22×D3—181no—
422C22×D3—184no—
423C3×Q8—34no—
424C3×Q8—92no—
425C3×Q8—181no—
426C3×D4—24no—
427C3×D4—21no—
428C3×D4—28no—
429C3×D4—34no—
430C3×D4—44no—
431C3×D4—64no—
432C3×D4—182no—
433C3×D4—182no—
434C3×D4C2×D3218yes—
435C3×D4C22×C3⋊D312yes—
436C2×C3×C4—24no—
437C2×C3×C4—64no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—182no—
444C2×C3×C4—182no—
445C2×C3×C4—181no—
446C2×C3×C4C2×D3214yes—
447C2×C3×C4C22×C3⋊D311yescharacteristic
448#67—21no—
449#67—68no—
450#67—68no—
451#67—181no—
452#67—184no—
453#67—181no—
454#67—184no—
455#67—184no—
456#67—181no—
457#67—364no—
458C2×Dic3—34no—
459C2×Dic3—68no—
460C2×Dic3—68no—
461C2×Dic3—64no—
462C2×Dic3—181no—
463C2×Dic3—184no—
464D12—94no—
465D12—91no—
466D12—184no—
467D12—184no—
468D12—181no—
469D12C22×C3⋊D311yescharacteristic
470C4×D3—21no—
471C4×D3—34no—
472C4×D3—34no—
473C4×D3—34no—
474C4×D3—34no—
475C4×D3—181no—
476C4×D3—184no—
477C4×D3—184no—
478C4×D3—181no—
479C4×D3—184no—
480C4×D3—181no—
481C4×D3—364no—
482Dic6—38no—
483Dic6—64no—
484Dic6—94no—
485Dic6—91no—
486Dic6—184no—
487Dic6—181no—
488Dic6—184no—
489Dic6C22×C3⋊D311yescharacteristic
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—42no—
500C2×C32—48no—
501C2×C32—88no—
502C2×C32—121no—
503C2×C32C2×D3×D414yes—
504C2×C32C22×#6711yescharacteristic
505C3⋊D3—21no—
506C3⋊D3—21no—
507C3⋊D3—121no—
508C3⋊D3—124no—
509C3⋊D3—244no—
510C3×D3—44no—
511C3×D3—64no—
512C3×D3—64no—
513#41—31no—
514#41—92no—
515#41—92no—
516#41—181no—
517#41—181no—
518#41—271no—
519#41—271no—
520#41—271no—
521#41—541no—
522#41—541no—
523C2×Q8—91no—
524C2×Q8—271no—
525C2×D4—181no—
526C2×D4—271no—
527C2×D4—541no—
528C2×D4D3×C3⋊D311yescharacteristic, complemented
529C22×C4—91no—
530C22×C4—181no—
531C22×C4—541no—
532Q16—272no—
533Q16—272no—
534SD16—32no—
535SD16—272no—
536SD16—272no—
537SD16—272no—
538D8—32no—
539D8—272no—
540M16—31no—
541M16—272no—
542M16—271no—
543C2×C8—272no—
544C22×C3—28no—
545C22×C3—22no—
546C22×C3—24no—
547C22×C3—48no—
548C22×C3—42no—
549C22×C3—48no—
550C22×C3—64no—
551C22×C3—88no—
552C22×C3—91no—
553C22×C3—91no—
554C22×C3—361no—
555C22×C3D3×#6714yes—
556C22×C3D4×C3⋊D311yescharacteristic
557C2×D3—21no—
558C2×D3—34no—
559C2×D3—34no—
560C2×D3—124no—
561C2×D3—184no—
562C2×D3—181no—
563C2×D3—364no—
564C2×D3—362no—
565C2×D3—368no—
566C3×C4—24no—
567C3×C4—24no—
568C3×C4—64no—
569C3×C4—91no—
570C3×C4—91no—
571C3×C4—182no—
572C3×C4D3×#6714yes—
573C3×C4D4×C3⋊D311yescharacteristic
574C3×C4C22×D3214yes—
575C3×C4C23×C3⋊D311yescharacteristic
576Dic3—21no—
577Dic3—34no—
578Dic3—34no—
579Dic3—68no—
580Dic3—184no—
581Dic3—181no—
582Dic3—364no—
583C32—24no—
584C32#577314yescomplemented
585C32#582211yescharacteristic, complemented
586C23—21no—
587C23—91no—
588C23—541no—
589Q8—31no—
590Q8—92no—
591Q8—181no—
592Q8—271no—
593Q8—541no—
594D4—21no—
595D4—31no—
596D4—61no—
597D4—182no—
598D4—182no—
599D4—271no—
600D4—541no—
601D4—541no—
602D4—541no—
603D4—541no—
604D4C2×D3×C3⋊D312yes—
605C2×C4—61no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—91no—
610C2×C4—91no—
611C2×C4—182no—
612C2×C4—182no—
613C2×C4—181no—
614C2×C4—541no—
615C2×C4—541no—
616C2×C4—541no—
617C2×C4—541no—
618C2×C4C2×D3×C3⋊D311yescharacteristic
619C8—32no—
620C8—272no—
621C2×C3—24no—
622C2×C3—21no—
623C2×C3—24no—
624C2×C3—44no—
625C2×C3—48no—
626C2×C3—42no—
627C2×C3—88no—
628C2×C3—124no—
629C2×C3—181no—
630C2×C3—181no—
631C2×C3C2×D3×#6714yes—
632C2×C3C2×D4×C3⋊D311yescharacteristic
633D3—41no—
634D3—64no—
635D3—64no—
636D3—364no—
637D3—361no—
638D3—724no—
639C22—22no—
640C22—42no—
641C22—61no—
642C22—91no—
643C22—91no—
644C22—361no—
645C22—541no—
646C22—1082no—
647C22C3⋊D3×#6711yescharacteristic
648C4—61no—
649C4—91no—
650C4—91no—
651C4—182no—
652C4—541no—
653C4C3⋊D3×#6711yescharacteristic
654C4C22×D3×C3⋊D311yescharacteristic
655C3—24no—
656C3#1059509514yescomplemented
657C3#1059624411yescharacteristic, complemented
658C2—21no—
659C2—42no—
660C2—121no—
661C2—181no—
662C2—181no—
663C2—1081no—
664C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C22×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C3⋊D3×#235C2semidirect
3. #11723973C2semidirect
4. C32×#932C2semidirect
5. #11723561C2semidirect
6. #11723559C2semidirect
7. #11723552C2semidirect
8. #11723546C2semidirect
9. #11723387C2semidirect
10. #11723373C2semidirect
11. C3⋊D3×#208C2semidirect
14. C2×C33×D4C22semidirect
21. C3×#3830C22semidirect
23. C2×C3×#634C22semidirect
27. C3⋊D3×D12C22semidirect
30. #89018C22semidirect
31. C3⋊D3×Dic6C22semidirect
32. #89014C22semidirect
33. #88861C22semidirect
34. #88860C22semidirect
35. #88855C22semidirect
36. C32×#214C22semidirect
37. C32×#213C22semidirect
38. C32×#208C22semidirect
39. #88800C22semidirect
40. #88798C22semidirect
41. #88797C22semidirect
42. #88792C22semidirect
43. #88787C22semidirect
44. C3⋊D3×C3⋊C8C22semidirect
46. C2×#3830D3semidirect
60. C3×#932D3semidirect
107. C2×C32×D4C2×D3semidirect
108. C2×C32×D4C2×D3semidirect
118. #3830C2×D3semidirect
127. C2×#634C2×D3semidirect
163. C3×#214C2×D3semidirect
164. C3×#213C2×D3semidirect
165. C3×#208C2×D3semidirect
203. #932C3⋊D3semidirect
306. C2×C3×D4D32semidirect
307. C2×C3×D4C2×C3⋊D3semidirect
350. #214C2×C3⋊D3semidirect
354. #213C2×C3⋊D3semidirect
359. #208C2×C3⋊D3semidirect
415. C33#575semidirect
528. C2×D4D3×C3⋊D3semidirect
584. C32#5773semidirect
585. C32#5822semidirect
656. C3#10595095semidirect
657. C3#10596244semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C3⋊D3×#2352
#117239732
C32×#9322
#117235612
#117235592
#117235522
#117235462
#117233872
#117233732
C3⋊D3×#2082
#105962443
#105950953