Finite Group Databasebeta

← #10877417

Subgroups — #10877417

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#73517C211yesmaximal, characteristic
2#73493C211yesmaximal, characteristic
3C3×C8×C22⋊C4C211yesmaximal, characteristic
4#71972C211yesmaximal, characteristic
5C2×#4551C211yesmaximal, characteristic
6#69930C211yesmaximal, characteristic
7#69925C211yesmaximal, characteristic
8#69777C211yesmaximal, characteristic, complemented
9#69767C211yesmaximal, characteristic, complemented
10#69767C211yesmaximal, characteristic, complemented
11#69762C211yesmaximal, characteristic, complemented
12#68689C211yesmaximal, characteristic, complemented
13#68643C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68636C211yesmaximal, characteristic, complemented
16#27697—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#878—21no—
22C2×#869C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#4950—21no—
33#4950—21no—
34#4950—21no—
35#4935C2211yescharacteristic
36#4935C2211yescharacteristic
37#4935C2211yescharacteristic
38#4929C2211yescharacteristic
39#4929C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4661—21no—
43#4661—21no—
44#4661—21no—
45#4656—21no—
46C3×#336C2211yescharacteristic
47C3×#336C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
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#1824—31no—
73C8×C22⋊C4D311yescharacteristic
74C2×#340—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C23×Dic3—21no—
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—41no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113#924—22no—
114#924—22no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118#878—44no—
119#869—22no—
120#869—41no—
121C3×C22⋊C8—22no—
122C3×C4×C8—22no—
123C3×C4×C8—41no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133#816—22no—
134#816—22no—
135#816—41no—
136#816—41no—
137C23×C8C2×D311yescharacteristic
138C2×C4⋊Q8—31no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×C22⋊C8C2×D311yescharacteristic
148C2×C4×C8C2×D311yescharacteristic
149C2×C4×C8C2×D311yescharacteristic
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#380—31no—
165C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180C24×C3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—81no—
205C22×Dic3—81no—
206C22×Dic3—81no—
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—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—44no—
228C2×C3×C8#4112yes—
229C3×C22⋊C4SD1614yes—
230C3×C22⋊C4Q1614yes—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235#217—42no—
236#217—42no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3—42no—
253C4⋊Dic3—42no—
254C4⋊Dic3—81no—
255C4⋊Dic3—81no—
256C4⋊Dic3—81no—
257C4⋊Dic3—81no—
258C4⋊Dic3—81no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263C22×Q8—31no—
264C22×Q8—61no—
265C23×C4—61no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—41no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4⋊Q8—62no—
276C4⋊Q8—121no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—124no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
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—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—121no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4D1211yescharacteristic
306C2×C22⋊C4D1211yescharacteristic
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#103—62no—
310#103—62no—
311#103—121no—
312#103—121no—
313C22⋊C8—22no—
314C4×C8—22no—
315C4×C8—41no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95D1211yescharacteristic
324#95D1211yescharacteristic
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—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4#12412yes—
348C2×C3×C4C2×SD1612yes—
349C2×C3×C4C2×SD1612yes—
350C2×C3×C4C2×Q1612yes—
351C2×C3×C4C2×Q1612yes—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
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—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—84no—
383C2×Dic3—84no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—84no—
389C3×C8—22no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C24C2×D1211yescharacteristic
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—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—241no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C2×D1211yescharacteristic
426C22×C4#23512yes—
427C22×C4#23512yes—
428C22×C4C23×D311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—44no—
441C2×C8#23712yes—
442C4⋊C4—32no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
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—122no—
469C22⋊C4—122no—
470C22⋊C4#20414yes—
471C22⋊C4Dic1214yes—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
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—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4#52911yescharacteristic
496C3×C4#53911yescharacteristic
497C3×C4#55111yescharacteristic
498C3×C4#55211yescharacteristic
499Dic3—42no—
500Dic3—81no—
501Dic3—81no—
502Dic3—81no—
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23#87512yes—
512C23C2×#20412yes—
513C23C2×Dic1212yes—
514C23C22×D1211yescharacteristic
515Q8—121no—
516Q8—121no—
517Q8—121no—
518Q8—121no—
519Q8—244no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—62no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—244no—
559C2×C4—244no—
560C2×C4#87512yes—
561C2×C4C2×#20412yes—
562C2×C4C2×#20412yes—
563C2×C4C2×Dic1212yes—
564C2×C4C2×Dic1212yes—
565C2×C4C2×#23511yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×#23711yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4#101011yescharacteristic
571C8—22no—
572C8—41no—
573C8—41no—
574C8—41no—
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#468114yes—
590C22C2×#87511yescharacteristic
591C22#555411yescharacteristic
592C22#555511yescharacteristic
593C22C22×#20411yescharacteristic
594C22C22×Dic1211yescharacteristic
595C22#575111yescharacteristic
596C22#575211yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—122no—
603C4—241no—
604C4—241no—
605C4—241no—
606C4—241no—
607C4—241no—
608C4—241no—
609C4—241no—
610C4#549511yescharacteristic
611C4#553111yescharacteristic
612C4#553811yescharacteristic
613C4#556411yescharacteristic
614C3#2769711yescharacteristic, complemented, Sylow
615C2—24no—
616C2#7350411yescharacteristic
617C2C2×#468111yescharacteristic
618C2#7856311yescharacteristic
619C2#7941211yescharacteristic
620C2#7941511yescharacteristic
621C2#7943911yescharacteristic
622C2#7944811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C8×C22⋊C4C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC8×C22⋊C4D3
hypocenterC3#27697
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #69777C2semidirect
9. #69767C2semidirect
10. #69767C2semidirect
11. #69762C2semidirect
12. #68689C2semidirect
13. #68643C2semidirect
14. #68643C2semidirect
15. #68636C2semidirect
614. C3#27697semidirect

Maximal subgroups

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

subgroupindex
#735172
#734932
C3×C8×C22⋊C42
#719722
C2×#45512
#699302
#699252
#697772
#697672
#697622
#686892
#686432
#686362
#276973