Finite Group Databasebeta

← #10859107

Subgroups — #10859107

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2362C211yesmaximal, characteristic
2#73495C211yesmaximal, characteristic
3#71697C211yesmaximal, characteristic
4#71692C211yesmaximal, characteristic
5#71628C211yesmaximal, characteristic, complemented
6#71627C211yesmaximal, characteristic, complemented
7#71581C211yesmaximal, characteristic
8#71576C211yesmaximal, characteristic
9#69682C211yesmaximal, characteristic, complemented
10#69669C211yesmaximal, characteristic, complemented
11#69613C211yesmaximal, characteristic, complemented
12#69610C211yesmaximal, characteristic, complemented
13#69491C211yesmaximal, characteristic, complemented
14#69461C211yesmaximal, characteristic, complemented
15#69450C211yesmaximal, characteristic
16#26433—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#388—21no—
26C3×#388—21no—
27C3×#387—21no—
28C3×C4×C4⋊C4—21no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4946—21no—
40#4942—21no—
41#4936C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4660—21no—
47#4657—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2442—31no—
61#2362D311yescharacteristic
62#2101—31no—
63#2088—31no—
64#2066—31no—
65#2026—31no—
66#1989—31no—
67#1921—31no—
68#1894—31no—
69#1872—31no—
70#1863—31no—
71#1854—31no—
72#1851—31no—
73#1805—31no—
74#1736—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×Dic3—21no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8Q814yes—
80C3×C4×Q8—22no—
81C3×C4×Q8—41no—
82C3×#117—22no—
83C3×#117—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—41no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×#208—22no—
112C2×#208—41no—
113C2×#208—41no—
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#870—22no—
119#870—41no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126#835—22no—
127#835—41no—
128#835—41no—
129#835—41no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#830—22no—
136#808—41no—
137#807—22no—
138C4⋊(C3⋊C8)—41no—
139C4⋊(C3⋊C8)—41no—
140C22×M16—31no—
141C2×#125—31no—
142C2×C22⋊Q8C2×D311yescharacteristic
143C2×C4×Q8C2×D311yescharacteristic
144C2×#117C2×D311yescharacteristic
145C22×C4⋊C4—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8—31no—
155#400—61no—
156#398—61no—
157#391C2×D311yescharacteristic
158#389—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#388—21no—
162#388—21no—
163#387—21no—
164#385—61no—
165#382—31no—
166#380—31no—
167#380—61no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4C2×D311yescharacteristic
170C4×C22⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×Q812yes—
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3—81no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4C2×Q812yes—
221C3×C4⋊C4#4112yes—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—41no—
224C3×C22⋊C4—41no—
225C3×C22⋊C4—42no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—42no—
231C3×C42—42no—
232C3×C42—42no—
233#217—42no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—42no—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247#208—44no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8C22×D311yescharacteristic
261C23×C4—61no—
262C23×C4C22×D311yescharacteristic
263C2×M16—62no—
264C2×M16—121no—
265C2×M16—121no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—31no—
269C22×C8—31no—
270#125—62no—
271#125—121no—
272C22⋊Q8—22no—
273C22⋊Q8Dic614yes—
274C4×Q8—22no—
275C4×Q8—41no—
276#117—22no—
277#117—41no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—121no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4#6711yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—62no—
306#106—121no—
307C4⋊C8—121no—
308C4⋊C8—121no—
309#103—62no—
310#103—121no—
311#103—121no—
312#103—121no—
313C22⋊C8—62no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95#6711yescharacteristic
324#95#6711yescharacteristic
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
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×C4C22⋊Q812yes—
368C2×C3×C4C22×Q811yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—84no—
387C3⋊C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24C2×#6711yescharacteristic
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8C2×Dic612yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—241no—
413C22×C4C2×Dic612yes—
414C22×C4#23512yes—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421M16—124no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—32no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—122no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4C2×Dic612yes—
455C4⋊C4#23712yes—
456C22⋊C4—24no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—42no—
460C22⋊C4—122no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—42no—
467C42—42no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3C2×C22⋊Q811yescharacteristic
476C22×C3#54611yescharacteristic
477C22×C3#54611yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57511yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57811yescharacteristic
482C3×C4—22no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4#52911yescharacteristic
492C3×C4#53811yescharacteristic
493C3×C4#55411yescharacteristic
494C3×C4#55411yescharacteristic
495Dic3—42no—
496Dic3—81no—
497Dic3—81no—
498Dic3—81no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#92412yes—
506C23C22×#6711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
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—21no—
531C2×C4—21no—
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—62no—
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—244no—
554C2×C4#92412yes—
555C2×C4C22×Dic611yescharacteristic
556C2×C4C2×#23511yescharacteristic
557C2×C4C2×#23711yescharacteristic
558C2×C4#100411yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4#100911yescharacteristic
561C8—121no—
562C8—121no—
563C8—121no—
564C8—121no—
565C2×C3—42no—
566C2×C3#245011yescharacteristic
567C2×C3#308011yescharacteristic
568C2×C3#308211yescharacteristic
569C2×C3#330711yescharacteristic
570C2×C3#331111yescharacteristic
571C2×C3#332711yescharacteristic
572C2×C3#332711yescharacteristic
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C2×#92411yescharacteristic
581C22#580611yescharacteristic
582C22#582211yescharacteristic
583C22#582211yescharacteristic
584C22#582911yescharacteristic
585C22#583711yescharacteristic
586C22#583811yescharacteristic
587C4—22no—
588C4—41no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—122no—
597C4—241no—
598C4—241no—
599C4—241no—
600C4#553811yescharacteristic
601C4#553911yescharacteristic
602C4#554311yescharacteristic
603C4#556011yescharacteristic
604C3#2643311yescharacteristic, complemented, Sylow
605C2—42no—
606C2#8236111yescharacteristic
607C2#8250911yescharacteristic
608C2#8255011yescharacteristic
609C2#8351211yescharacteristic
610C2#8354111yescharacteristic
611C2#8360511yescharacteristic
612C2#8366511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2362C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2362D3
hypocenterC3#26433
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
5. #71628C2semidirect
6. #71627C2semidirect
9. #69682C2semidirect
10. #69669C2semidirect
11. #69613C2semidirect
12. #69610C2semidirect
13. #69491C2semidirect
14. #69461C2semidirect
604. C3#26433semidirect

Maximal subgroups

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

subgroupindex
C3×#23622
#734952
#716972
#716922
#716282
#716272
#715812
#715762
#696822
#696692
#696132
#696102
#694912
#694612
#694502
#264333