Finite Group Databasebeta

← #10824815

Subgroups — #10824815

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
1C2×#4853C211yesmaximal, characteristic, complemented
2C2×#4720C211yesmaximal, characteristic
3#73871C211yesmaximal, characteristic
4#73871C211yesmaximal, characteristic
5C3×#2077C211yesmaximal, characteristic
6#70568C211yesmaximal, characteristic, complemented
7#69964C211yesmaximal, characteristic
8#69527C211yesmaximal, characteristic
9#69526C211yesmaximal, characteristic
10#68913C211yesmaximal, characteristic, complemented
11#68912C211yesmaximal, characteristic, complemented
12#68864C211yesmaximal, characteristic, complemented
13#68786C211yesmaximal, characteristic, complemented
14#68781C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#23713—31nomaximal, Sylow
17C2×#926C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×Dic12C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4945—21no—
47#4945—21no—
48#4944—21no—
49#4944—21no—
50#4853—24no—
51#4720—24no—
52#4655—21no—
53#4655—21no—
54#4647—21no—
55#4647—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#441—31no—
63C2×#433—31no—
64#2323—31no—
65#2323—31no—
66#2088—31no—
67#2077D311yescharacteristic
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×#103—31no—
77C23×C3×C4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217D411yescharacteristic
91C2×#217D411yescharacteristic
92#926—22no—
93#926—22no—
94C2×C4⋊Dic3D411yescharacteristic
95C2×C4⋊Dic3D411yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98#924—22no—
99#924—22no—
100#924—22no—
101#924—22no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×Dic12—22no—
117C2×Dic12—22no—
118C2×Dic12—22no—
119#887—22no—
120#887—22no—
121#887—22no—
122#887—22no—
123C3×#106—22no—
124C3×C22⋊C8—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#830—22no—
136#816—22no—
137#816—22no—
138#815D414yes—
139Dic3⋊C8—22no—
140Dic3⋊C8—22no—
141C8×Dic3—22no—
142#810—22no—
143#810—22no—
144#810—22no—
145#810—22no—
146#808—22no—
147#807—22no—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×#117—31no—
154#441—64no—
155#433—64no—
156C2×#107—31no—
157C2×#106—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×#103—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×#97—31no—
170C2×C4×C8—31no—
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#380C2×D311yescharacteristic
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
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×Dic6—21no—
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×C8C2×D412yes—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C22⋊C4—81no—
241C3×C22⋊C4—81no—
242#217—24no—
243#217—24no—
244#217—22no—
245#217—22no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—22no—
249C4⋊Dic3—22no—
250C4⋊Dic3#4112yes—
251C4⋊Dic3#4112yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—81no—
257Dic3⋊C4—81no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3—81no—
263C4×Dic3—81no—
264C4×Dic3C2×D412yes—
265C4×Dic3#4112yes—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272Dic12—28no—
273C22×Q8—31no—
274C22×Q8—31no—
275C23×C4C22×D311yescharacteristic
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—62no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291#117—62no—
292#117—62no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42—61no—
319#107—62no—
320#106—22no—
321#106—62no—
322C4⋊C8—62no—
323C4⋊C8—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330C22⋊C8—22no—
331C22⋊C8—62no—
332#97—34no—
333C4×C8—62no—
334#95—21no—
335#95—21no—
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×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3C22×D411yescharacteristic
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
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#12112yes—
400C2×Dic3#12112yes—
401Dic6—22no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409C3×C8—22no—
410C3×C8—41no—
411C3⋊C8—22no—
412C3⋊C8—41no—
413C24D3×D411yescharacteristic
414C2×Q8—32no—
415C2×Q8—32no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
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—32no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8D3×D412yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4—241no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C42—32no—
486C42—32no—
487C42—61no—
488C42—61no—
489C42—122no—
490C42—122no—
491C42—241no—
492C42—241no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3C2×#12111yescharacteristic
501C22×C3#52511yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#57711yescharacteristic
504C22×C3#57711yescharacteristic
505C22×C3#57811yescharacteristic
506C22×C3#57811yescharacteristic
507C3×C4—41no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4#52511yescharacteristic
513C3×C4#52911yescharacteristic
514C3×C4#52911yescharacteristic
515C3×C4#53211yescharacteristic
516Dic3—22no—
517Dic3—22no—
518Dic3—42no—
519Dic3—42no—
520Dic3—42no—
521Dic3—42no—
522Dic3—41no—
523Dic3—41no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23C2×D3×D411yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—62no—
535Q8—62no—
536Q8—121no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—41no—
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—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4C2×D3×D411yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—22no—
590C8—41no—
591C8—62no—
592C8—121no—
593C2×C3—81no—
594C2×C3#239211yescharacteristic
595C2×C3#306311yescharacteristic
596C2×C3#306311yescharacteristic
597C2×C3#312011yescharacteristic
598C2×C3#312111yescharacteristic
599C2×C3#312511yescharacteristic
600C2×C3#312711yescharacteristic
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22D3×#12111yescharacteristic
609C22#565611yescharacteristic
610C22#565611yescharacteristic
611C22#576211yescharacteristic
612C22#577111yescharacteristic
613C22#577511yescharacteristic
614C22#577911yescharacteristic
615C4—41no—
616C4—62no—
617C4—62no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—122no—
623C4—121no—
624C4—122no—
625C4—122no—
626C4—122no—
627C4—121no—
628C4#565611yescharacteristic
629C4#566811yescharacteristic
630C4#570611yescharacteristic
631C4#570611yescharacteristic
632C3#2371311yescharacteristic, complemented, Sylow
633C2—81no—
634C2#7745711yescharacteristic
635C2#8066211yescharacteristic
636C2#8081011yescharacteristic
637C2#8131311yescharacteristic
638C2#8135311yescharacteristic
639C2#8141611yescharacteristic
640C2#8149911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2077C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2077D3
hypocenterC3#23713
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
1. C2×#4853C2semidirect
6. #70568C2semidirect
10. #68913C2semidirect
11. #68912C2semidirect
12. #68864C2semidirect
13. #68786C2semidirect
14. #68781C2semidirect
15. #68781C2semidirect
632. C3#23713semidirect

Maximal subgroups

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

subgroupindex
C2×#48532
C2×#47202
#738712
C3×#20772
#705682
#699642
#695272
#695262
#689132
#689122
#688642
#687862
#687812
#237133