Finite Group Databasebeta

← #10945141

Subgroups — #10945141

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4857C211yesmaximal, characteristic, complemented
2C2×#4767C211yesmaximal, characteristic
3#76426C211yesmaximal, characteristic
4C4×#887C211yesmaximal, characteristic, complemented
5C3×#1926C211yesmaximal, characteristic
6#70570C211yesmaximal, characteristic, complemented
7#70213C211yesmaximal, characteristic
8#70173C211yesmaximal, characteristic
9#70162C211yesmaximal, characteristic
10#69660C211yesmaximal, characteristic
11#69465C211yesmaximal, characteristic, complemented
12#68908C211yesmaximal, characteristic, complemented
13#68801C211yesmaximal, characteristic, complemented
14#68685C211yesmaximal, characteristic, complemented
15#68605C211yesmaximal, characteristic, complemented
16#33029—31nomaximal, Sylow
17C2×#936C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C22×Dic12C2211yescharacteristic
20C2×#890C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4951—21no—
43#4951—21no—
44#4944—21no—
45#4944—21no—
46C4×Dic3⋊C4—21no—
47C42×Dic3—21no—
48C42×Dic3C2211yescharacteristic
49#4929C2211yescharacteristic
50#4857—24no—
51#4767—24no—
52#4649—21no—
53#4649—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#464—31no—
64#2861—31no—
65C42×Q8—31no—
66#1989—31no—
67#1983—31no—
68#1928—31no—
69#1927—31no—
70#1926D311yescharacteristic
71#1925—31no—
72#1880—31no—
73#1880—31no—
74#1878—31no—
75#1878—31no—
76#1831—31no—
77C22×C3×D4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#936—22no—
94#936—22no—
95C2×#214—22no—
96C2×#214—22no—
97C2×#214—22no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×Dic12—22no—
119C2×Dic12—22no—
120C2×Dic12—22no—
121#890—22no—
122#890—22no—
123#887—22no—
124#887—22no—
125#887—22no—
126#887D414yes—
127C4×Dic6—22no—
128C4×Dic6—22no—
129C3×#106—22no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#95—21no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#816—22no—
141#816—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#810—22no—
145#810—22no—
146#810—22no—
147#807—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#125—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155#479—64no—
156#464—64no—
157C2×#107—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#102C2×D311yescharacteristic
169#391—31no—
170#391—31no—
171#390—31no—
172#390C2×D311yescharacteristic
173#388—61no—
174#388—61no—
175#388—61no—
176#388—61no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—31no—
179C4×C4⋊C4—61no—
180C4×C4⋊C4—61no—
181C4×C4⋊C4—61no—
182C4×C4⋊C4C2×D311yescharacteristic
183C43—31no—
184C43—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6C2×D412yes—
224C2×Dic6#4112yes—
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⋊C4C2×D412yes—
237C3×C4⋊C4#4112yes—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—42no—
243#217—42no—
244#217—42no—
245#217—42no—
246#217—42no—
247#214—44no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—24no—
255Dic3⋊C4—24no—
256Dic3⋊C4—42no—
257C4×Dic3—24no—
258C4×Dic3—24no—
259C4×Dic3—22no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—24no—
263C4×Dic3—22no—
264C4×Dic3—41no—
265C4×Dic3—41no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3C2×D412yes—
269C4×Dic3#4112yes—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276Dic12—44no—
277C22×Q8—31no—
278C22×Q8—31no—
279C22×D4C22×D311yescharacteristic
280C2×Q16—62no—
281C2×Q16—62no—
282C2×Q16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—62no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290#125—62no—
291#125—62no—
292#124—62no—
293#124—62no—
294C4×Q8—34no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×Q8—62no—
299C4×Q8—62no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—21no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—62no—
330#106—22no—
331C4⋊C8—62no—
332C4⋊C8—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#103—62no—
338#102—22no—
339#102—22no—
340#102—62no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3C22×D411yescharacteristic
356C3×D4—41no—
357C3×D4—41no—
358C3×D4—41no—
359C3×D4—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4#13712yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3#12112yes—
403C2×Dic3#12112yes—
404C2×Dic3#13512yes—
405Dic6—22no—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6#12414yes—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—21no—
418C2×Q8—32no—
419C2×Q8—32no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—124no—
451SD16—124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4#23512yes—
478C4⋊C4D3×D412yes—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—21no—
486C42—21no—
487C42—32no—
488C42—32no—
489C42—42no—
490C42—62no—
491C42—64no—
492C42—64no—
493C42—64no—
494C42—62no—
495C42—61no—
496C42—61no—
497C42—121no—
498C42—121no—
499C42—122no—
500C42—122no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3C2×#12111yescharacteristic
509C22×C3#52511yescharacteristic
510C22×C3#52511yescharacteristic
511C22×C3C2×#13511yescharacteristic
512C22×C3C2×#13711yescharacteristic
513C22×C3#57511yescharacteristic
514C22×C3#57511yescharacteristic
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—81no—
520C3×C4C2×#12411yescharacteristic
521C3×C4#52911yescharacteristic
522C3×C4#52911yescharacteristic
523C3×C4#53211yescharacteristic
524Dic3—24no—
525Dic3—22no—
526Dic3—22no—
527Dic3—22no—
528Dic3—42no—
529Dic3—41no—
530Dic3—81no—
531Dic3—81no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C2×D3×D411yescharacteristic
540Q8—34no—
541Q8—62no—
542Q8—62no—
543Q8—62no—
544Q8—121no—
545Q8—121no—
546Q8—121no—
547Q8—121no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—32no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4#91512yes—
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C8—41no—
598C8—41no—
599C8—121no—
600C8—121no—
601C2×C3—161no—
602C2×C3#285011yescharacteristic
603C2×C3#303511yescharacteristic
604C2×C3#303711yescharacteristic
605C2×C3#310211yescharacteristic
606C2×C3#310811yescharacteristic
607C2×C3#311111yescharacteristic
608C2×C3#311711yescharacteristic
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22#554711yescharacteristic
617C22D3×#12111yescharacteristic
618C22#565611yescharacteristic
619C22C2×#91511yescharacteristic
620C22D3×#13511yescharacteristic
621C22#577311yescharacteristic
622C22#577311yescharacteristic
623C4—22no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—62no—
628C4—62no—
629C4—64no—
630C4—62no—
631C4—81no—
632C4—122no—
633C4—121no—
634C4—241no—
635C4—241no—
636C4#552811yescharacteristic
637C4#554811yescharacteristic
638C4D3×#12411yescharacteristic
639C4#570611yescharacteristic
640C3#3302911yescharacteristic, complemented, Sylow
641C2—161no—
642C2#7595711yescharacteristic
643C2#7965611yescharacteristic
644C2#7987011yescharacteristic
645C2#8038311yescharacteristic
646C2#8053311yescharacteristic
647C2#8131111yescharacteristic
648C2#8134811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1926C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1926D3
hypocenterC3#33029
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
1. C2×#4857C2semidirect
4. C4×#887C2semidirect
6. #70570C2semidirect
11. #69465C2semidirect
12. #68908C2semidirect
13. #68801C2semidirect
14. #68685C2semidirect
15. #68605C2semidirect
640. C3#33029semidirect

Maximal subgroups

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

subgroupindex
C2×#48572
C2×#47672
#764262
C4×#8872
C3×#19262
#705702
#702132
#701732
#701622
#696602
#694652
#689082
#688012
#686852
#686052
#330293