Finite Group Databasebeta

← #10877419

Subgroups — #10877419

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82178C211yesmaximal, characteristic
2#73899C211yesmaximal, characteristic
3C3×#1824C211yesmaximal, characteristic
4C2×#4529C211yesmaximal, characteristic
5#69947C211yesmaximal, characteristic
6#69930C211yesmaximal, characteristic
7#69514C211yesmaximal, characteristic
8#68914C211yesmaximal, characteristic, complemented
9#68910C211yesmaximal, characteristic, complemented
10#68851C211yesmaximal, characteristic, complemented
11#68850C211yesmaximal, characteristic, complemented
12#68847C211yesmaximal, characteristic, complemented
13#68843C211yesmaximal, characteristic, complemented
14#68811C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#27697—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×#878—21no—
23C2×C3×C22⋊C8C2211yescharacteristic
24C3×#380C2211yescharacteristic
25#5210—21no—
26#5210—21no—
27C4×#217C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4984C2211yescharacteristic
33#4983—21no—
34#4983—21no—
35#4975C2211yescharacteristic
36#4975C2211yescharacteristic
37#4975C2211yescharacteristic
38#4973C2211yescharacteristic
39#4973C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43#4946—21no—
44#4657—21no—
45#4656—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2502—31no—
61#2440—31no—
62#2100—31no—
63#2100—31no—
64#2072—31no—
65#2024—31no—
66#1990—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1872—31no—
71#1834—31no—
72#1824D311yescharacteristic
73C8×C22⋊C4—31no—
74C2×#340—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×#217—21no—
91C2×#217D411yescharacteristic
92C2×#217D411yescharacteristic
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—41no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
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×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#888—22no—
119#888—41no—
120#878—44no—
121C3×C22⋊C8—22no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#816—22no—
132#816—41no—
133C8×Dic3—22no—
134C8×Dic3—41no—
135#810—22no—
136#810—41no—
137C23×C8—31no—
138C2×C4⋊Q8—31no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4C2×D311yescharacteristic
143C2×#103—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×C22⋊C8C2×D311yescharacteristic
148C2×C4×C8—31no—
149C2×C4×C8—31no—
150#398—61no—
151#398—61no—
152#398—61no—
153#395—61no—
154#391—31no—
155#391—31no—
156#389—31no—
157#389—31no—
158#385—61no—
159#385—61no—
160#385—61no—
161#384—31no—
162#384—31no—
163#384—31no—
164#380C2×D311yescharacteristic
165C4×C22⋊C4—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#336—31no—
179#336—31no—
180C24×C3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—42no—
190C22×C3×C4—81no—
191C22×C3×C4—81no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4#4112yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—81no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C2×Dic6—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C22⋊C4—42no—
232#217—42no—
233#217SD1614yes—
234#217Q1614yes—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—42no—
241C4⋊Dic3—42no—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—44no—
262C2×C3⋊C8#4112yes—
263C22×Q8—31no—
264C22×Q8—61no—
265C23×C4—21no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—62no—
271C22×C8—62no—
272C22×C8—121no—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4⋊Q8—62no—
276C4⋊Q8—121no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—124no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—121no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—61no—
307C2×C42—31no—
308C2×C42—31no—
309#103—62no—
310#103—62no—
311#103—121no—
312#103—121no—
313C22⋊C8—22no—
314C4×C8—62no—
315C4×C8—121no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3#12412yes—
330C23×C3C2×SD1612yes—
331C23×C3C2×Q1612yes—
332C23×C3C22×D411yescharacteristic
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—84no—
352C2×C3×C4#12412yes—
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—84no—
380C2×Dic3C2×SD1612yes—
381C2×Dic3C2×SD1612yes—
382C2×Dic3C2×Q1612yes—
383C2×Dic3C2×Q1612yes—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—84no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—22no—
392C3⋊C8—41no—
393C24D3×D411yescharacteristic
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—122no—
401C2×Q8—122no—
402C2×Q8—241no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—42no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—241no—
425C22×C4#23512yes—
426C22×C4D3×D411yescharacteristic
427C22×C4#23712yes—
428C22×C4C23×D311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—32no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—124no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—32no—
451C4⋊C4—42no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C22⋊C4—34no—
469C22⋊C4—34no—
470C22⋊C4—42no—
471C22⋊C4—122no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—61no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3#48714yes—
483C22×C3C2×#12411yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54711yescharacteristic
486C22×C3C22×SD1611yescharacteristic
487C22×C3C22×Q1611yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57711yescharacteristic
490C3×C4—22no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4#52911yescharacteristic
496C3×C4#53911yescharacteristic
497C3×C4#55111yescharacteristic
498C3×C4#55211yescharacteristic
499Dic3—22no—
500Dic3—22no—
501Dic3—22no—
502Dic3—22no—
503Dic3—42no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23#88512yes—
512C23D3×SD1612yes—
513C23D3×Q1612yes—
514C23C2×D3×D411yescharacteristic
515Q8—121no—
516Q8—121no—
517Q8—121no—
518Q8—121no—
519Q8—244no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—84no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—244no—
564C2×C4#88512yes—
565C2×C4C2×#23511yescharacteristic
566C2×C4C2×#23711yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4#100911yescharacteristic
570C2×C4#101011yescharacteristic
571C8—41no—
572C8—41no—
573C8—62no—
574C8—121no—
575C2×C3—24no—
576C2×C3#248811yescharacteristic
577C2×C3C2×#48711yescharacteristic
578C2×C3#313611yescharacteristic
579C2×C3#329511yescharacteristic
580C2×C3#329911yescharacteristic
581C2×C3#333511yescharacteristic
582C2×C3#333811yescharacteristic
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22#479214yes—
590C22C2×#88511yescharacteristic
591C22#565411yescharacteristic
592C22#565511yescharacteristic
593C22C2×D3×SD1611yescharacteristic
594C22C2×D3×Q1611yescharacteristic
595C22#577411yescharacteristic
596C22#577511yescharacteristic
597C4—22no—
598C4—62no—
599C4—62no—
600C4—62no—
601C4—62no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—122no—
607C4—241no—
608C4—241no—
609C4—241no—
610C4#552711yescharacteristic
611C4#553111yescharacteristic
612C4#564911yescharacteristic
613C4#566511yescharacteristic
614C3#2769711yescharacteristic, complemented, Sylow
615C2—24no—
616C2#7407611yescharacteristic
617C2C2×#479211yescharacteristic
618C2#7892011yescharacteristic
619C2#8132011yescharacteristic
620C2#8134211yescharacteristic
621C2#8147611yescharacteristic
622C2#8149011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1824C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1824D3
hypocenterC3#27697
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
8. #68914C2semidirect
9. #68910C2semidirect
10. #68851C2semidirect
11. #68850C2semidirect
12. #68847C2semidirect
13. #68843C2semidirect
14. #68811C2semidirect
15. #68803C2semidirect
614. C3#27697semidirect

Maximal subgroups

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

subgroupindex
#821782
#738992
C3×#18242
C2×#45292
#699472
#699302
#695142
#689142
#689102
#688512
#688502
#688472
#688432
#688112
#688032
#276973