Finite Group Databasebeta

← #10853240

Subgroups — #10853240

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82249C211yesmaximal, characteristic
2#82160C211yesmaximal, characteristic
3C3×#1736C211yesmaximal, characteristic
4#71995C211yesmaximal, characteristic
5#71988C211yesmaximal, characteristic
6#71576C211yesmaximal, characteristic
7#71553C211yesmaximal, characteristic
8#68843C211yesmaximal, characteristic, complemented
9#68840C211yesmaximal, characteristic, complemented
10#68829C211yesmaximal, characteristic, complemented
11#68818C211yesmaximal, characteristic, complemented
12#68817C211yesmaximal, characteristic, complemented
13#68807C211yesmaximal, characteristic, complemented
14#68801C211yesmaximal, characteristic, complemented
15#68795C211yesmaximal, characteristic, complemented
16#26018—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#890C2211yescharacteristic
21#5212—21no—
22#5212C2211yescharacteristic
23#5212C2211yescharacteristic
24C4×#217—21no—
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#818C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4984C2211yescharacteristic
32#4984C2211yescharacteristic
33#4984C2211yescharacteristic
34#4980—21no—
35#4980—21no—
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4949—21no—
43#4948—21no—
44#4933—21no—
45#4928—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#2441—31no—
61#2345—31no—
62#2100—31no—
63#2088—31no—
64#2086—31no—
65#2059—31no—
66#1989—31no—
67#1947—31no—
68#1920—31no—
69#1919—31no—
70#1854—31no—
71#1851—31no—
72#1823—31no—
73#1809—31no—
74#1736D311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#117—22no—
83C3×#117—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4Q814yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217D411yescharacteristic
96C2×#217D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#890—22no—
116#890—41no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#830—22no—
128#818—41no—
129#817—22no—
130Dic3⋊C8—22no—
131Dic3⋊C8—41no—
132#808—22no—
133#807—41no—
134C22×M16C2×D311yescharacteristic
135C2×#125—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
138C2×#107—31no—
139C2×#107—31no—
140C2×#106—31no—
141C2×#106—31no—
142C2×C4⋊C8—31no—
143C2×C4⋊C8—31no—
144C2×C22⋊C8—31no—
145#399—61no—
146#396—61no—
147#389—31no—
148#389—31no—
149#389—31no—
150#389—31no—
151#389—61no—
152#387—61no—
153#387—61no—
154#387—61no—
155#384—31no—
156#380—31no—
157#380—31no—
158#380—61no—
159C4×C4⋊C4—31no—
160C4×C4⋊C4—31no—
161C4×C22⋊C4—31no—
162C4×C22⋊C4—61no—
163#340—31no—
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340C2×D311yescharacteristic
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177C24×C3C2×D411yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—81no—
188C22×C3×C4—81no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×Q812yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C3×M16—44no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8#4112yes—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×Q812yes—
215C3×C4⋊C4C2×Q812yes—
216C3×C4⋊C4#4112yes—
217C3×C22⋊C4—42no—
218C3×C42—42no—
219#217—22no—
220#217—41no—
221#217—42no—
222#217—81no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—42no—
232C4⋊Dic3—42no—
233C4⋊Dic3—81no—
234C4⋊Dic3—81no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—81no—
238Dic3⋊C4—81no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—81no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C23×C4—21no—
252C23×C4C22×D311yescharacteristic
253C2×M16—22no—
254C2×M16—22no—
255C2×M16—41no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#125—62no—
261#125—121no—
262#117—22no—
263#117—41no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4Dic614yes—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C42—21no—
289C2×C42—31no—
290C2×C42—31no—
291C2×C42—61no—
292C2×C42—61no—
293#107—121no—
294#107—121no—
295#106—62no—
296#106—62no—
297C4⋊C8—62no—
298C4⋊C8—121no—
299C22⋊C8—62no—
300#95—31no—
301#95—31no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3C22⋊Q812yes—
317C23×C3C22×D411yescharacteristic
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—84no—
337C2×C3×C4C22⋊Q812yes—
338C2×C3×C4C22×Q811yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
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#13712yes—
368C3×C8—22no—
369C3×C8—41no—
370C3⋊C8—41no—
371C3⋊C8—41no—
372C24D3×D411yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4C2×Dic612yes—
393C22×C4D3×D411yescharacteristic
394C22×C4#23712yes—
395C22×C4C23×D311yescharacteristic
396M16—44no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8#23512yes—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—122no—
424C4⋊C4—122no—
425C4⋊C4—122no—
426C4⋊C4—122no—
427C4⋊C4—241no—
428C4⋊C4—241no—
429C4⋊C4—241no—
430C4⋊C4—241no—
431C4⋊C4C2×Dic612yes—
432C4⋊C4C2×Dic612yes—
433C4⋊C4#23512yes—
434C22⋊C4—42no—
435C22⋊C4—62no—
436C22⋊C4—121no—
437C22⋊C4—122no—
438C22⋊C4—241no—
439C42—42no—
440C42—61no—
441C42—61no—
442C42—61no—
443C42—61no—
444C42—241no—
445C42—241no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3C2×C22⋊Q811yescharacteristic
454C22×C3#54611yescharacteristic
455C22×C3#54711yescharacteristic
456C22×C3C2×#13711yescharacteristic
457C22×C3#57511yescharacteristic
458C22×C3#57611yescharacteristic
459C22×C3#57711yescharacteristic
460C3×C4—22no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—81no—
465C3×C4#53311yescharacteristic
466C3×C4#54411yescharacteristic
467C3×C4#55111yescharacteristic
468C3×C4#55211yescharacteristic
469Dic3—22no—
470Dic3—41no—
471Dic3—41no—
472Dic3—41no—
473Dic3—81no—
474Dic3—81no—
475Dic3—81no—
476Dic3—81no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23#87812yes—
484C23C2×D3×D411yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—32no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—84no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4#87812yes—
529C2×C4C22×Dic611yescharacteristic
530C2×C4C2×#23511yescharacteristic
531C2×C4C2×#23511yescharacteristic
532C2×C4C2×#23711yescharacteristic
533C2×C4#100411yescharacteristic
534C2×C4#100711yescharacteristic
535C8—22no—
536C8—41no—
537C8—121no—
538C8—121no—
539C2×C3—42no—
540C2×C3#243811yescharacteristic
541C2×C3#307811yescharacteristic
542C2×C3#308111yescharacteristic
543C2×C3#330911yescharacteristic
544C2×C3#333111yescharacteristic
545C2×C3#334611yescharacteristic
546C2×C3#335111yescharacteristic
547C22—22no—
548C22—22no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22C2×#87811yescharacteristic
555C22#549011yescharacteristic
556C22#550711yescharacteristic
557C22#577411yescharacteristic
558C22#577511yescharacteristic
559C22D3×#13711yescharacteristic
560C22#577711yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—62no—
566C4—81no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4—241no—
571C4—241no—
572C4—241no—
573C4—241no—
574C4#547011yescharacteristic
575C4#549511yescharacteristic
576C4#550011yescharacteristic
577C4#551311yescharacteristic
578C3#2601811yescharacteristic, complemented, Sylow
579C2—42no—
580C2#7335011yescharacteristic
581C2#7889711yescharacteristic
582C2#7890711yescharacteristic
583C2#7891811yescharacteristic
584C2#7894911yescharacteristic
585C2#7922611yescharacteristic
586C2#7923211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1736C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1736D3
hypocenterC3#26018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
8. #68843C2semidirect
9. #68840C2semidirect
10. #68829C2semidirect
11. #68818C2semidirect
12. #68817C2semidirect
13. #68807C2semidirect
14. #68801C2semidirect
15. #68795C2semidirect
578. C3#26018semidirect

Maximal subgroups

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

subgroupindex
#822492
#821602
C3×#17362
#719952
#719882
#715762
#715532
#688432
#688402
#688292
#688182
#688172
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