Finite Group Databasebeta

← #10824826

Subgroups — #10824826

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82488C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C2×#4690C211yesmaximal, characteristic
4#73762C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70285C211yesmaximal, characteristic, complemented
7#70274C211yesmaximal, characteristic
8#70273C211yesmaximal, characteristic
9#70262C211yesmaximal, characteristic, complemented
10C4×#816C211yesmaximal, characteristic, complemented
11#69481C211yesmaximal, characteristic, complemented
12#68912C211yesmaximal, characteristic, complemented
13#68891C211yesmaximal, characteristic
14#68841C211yesmaximal, characteristic, complemented
15#68825C211yesmaximal, characteristic
16#23713—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#872C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#5030—24no—
36#4990—21no—
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4984C2211yescharacteristic
40#4976—21no—
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4940—21no—
48#4940—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50#4933—21no—
51#4928—21no—
52C2×#803C2211yescharacteristic
53#4690—24no—
54#4670—21no—
55#4653—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#441—31no—
63C2×#433—31no—
64#2323—31no—
65#2323—31no—
66#2088—31no—
67#2077—31no—
68#2068—31no—
69#2066—31no—
70#2056—31no—
71#2036—31no—
72#2035—31no—
73#2001—31no—
74#1776—31no—
75C4×#103—31no—
76C4×#103D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#946—22no—
92#946—22no—
93C2×#216—22no—
94C2×#216—22no—
95C2×#216—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4D411yescharacteristic
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#897—22no—
118#897—22no—
119#872—22no—
120#872—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×C4×C8—22no—
126C3×#95—21no—
127#835—22no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137(C2×D3)⋊C8—22no—
138(C2×D3)⋊C8—22no—
139#816—22no—
140#816—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#810—22no—
144#808—22no—
145#808—22no—
146#807—22no—
147#803D414yes—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×#117—31no—
154#441—64no—
155#433—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97—31no—
170C2×C4×C8C2×D311yescharacteristic
171#391—31no—
172#391—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#389—61no—
177#388—61no—
178#388—61no—
179#388—61no—
180#388—61no—
181#387—61no—
182#387—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6#4112yes—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4#4112yes—
239C3×C42—42no—
240C3×C42—81no—
241C3×C42C2×D412yes—
242C3×C42#4112yes—
243#216—28no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—22no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250C4⋊Dic3—24no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—81no—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—24no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—42no—
262Dic3⋊C4—81no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272C2×C3⋊C8C2×D412yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C23×C4—31no—
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C22⋊Q8—62no—
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—62no—
290C4×Q8—62no—
291#117—62no—
292#117—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—62no—
320#106—62no—
321#106—62no—
322C4⋊C8—62no—
323C4⋊C8—62no—
324#103—22no—
325#103—22no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330C22⋊C8—62no—
331C22⋊C8—62no—
332#97—34no—
333C4×C8—22no—
334#95—21no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4#12112yes—
373C2×C3×C4#12112yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—41no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3⋊C8—22no—
412C3⋊C8—22no—
413C24—31no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8#23512yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446Q16—68no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—32no—
454C2×C8—32no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—32no—
465C4⋊C4—42no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C4⋊C4#23512yes—
479C22⋊C4—64no—
480C22⋊C4—64no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—42no—
486C42—61no—
487C42—61no—
488C42—81no—
489C42—122no—
490C42—241no—
491C42#23512yes—
492C42D3×D412yes—
493C22×C3C2×#12111yescharacteristic
494C22×C3#52511yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#57711yescharacteristic
497C22×C3#57711yescharacteristic
498C22×C3#57811yescharacteristic
499C22×C3#57811yescharacteristic
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—41no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52511yescharacteristic
514C3×C4#52911yescharacteristic
515C3×C4#52911yescharacteristic
516C3×C4#53211yescharacteristic
517Dic3—42no—
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—62no—
537Q8—62no—
538Q8—121no—
539Q8—121no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4#88412yes—
582C2×C4#88412yes—
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×#23511yescharacteristic
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—41no—
590C8—41no—
591C8—62no—
592C8—62no—
593C2×C3#239211yescharacteristic
594C2×C3#306311yescharacteristic
595C2×C3#306311yescharacteristic
596C2×C3#312011yescharacteristic
597C2×C3#312111yescharacteristic
598C2×C3#312511yescharacteristic
599C2×C3#312711yescharacteristic
600D3—81no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22C2×#88411yescharacteristic
609C22#554711yescharacteristic
610C22#554711yescharacteristic
611C22#576211yescharacteristic
612C22#576611yescharacteristic
613C22#577011yescharacteristic
614C22#577111yescharacteristic
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—121no—
622C4—122no—
623C4—122no—
624C4—121no—
625C4—121no—
626C4—241no—
627C4—241no—
628C4#547911yescharacteristic
629C4#552811yescharacteristic
630C4#554711yescharacteristic
631C4#554811yescharacteristic
632C3#2371311yescharacteristic, complemented, Sylow
633C2—241no—
634C2#7358111yescharacteristic
635C2#7880011yescharacteristic
636C2#7884111yescharacteristic
637C2#7965811yescharacteristic
638C2#7966011yescharacteristic
639C2#7972411yescharacteristic
640C2#7974811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC4×#103D3
hypocenterC3#23713
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
5. C3×C4×#103C2semidirect
6. #70285C2semidirect
9. #70262C2semidirect
10. C4×#816C2semidirect
11. #69481C2semidirect
12. #68912C2semidirect
14. #68841C2semidirect
76. C4×#103D3semidirect
632. C3#23713semidirect

Maximal subgroups

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

subgroupindex
#824882
C2×#50302
C2×#46902
#737622
C3×C4×#1032
#702852
#702742
#702732
#702622
C4×#8162
#694812
#689122
#688912
#688412
#688252
#237133