Finite Group Databasebeta

← #10780312

Subgroups — #10780312

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81565C211yesmaximal, characteristic, complemented
2#81067C211yesmaximal, characteristic, complemented
3#80344C211yesmaximal, characteristic
4#79403C211yesmaximal, characteristic, complemented
5C3×#1677C211yesmaximal, characteristic, complemented
6#71210C211yesmaximal, characteristic, complemented
7#71139C211yesmaximal, characteristic, complemented
8#71010C211yesmaximal, characteristic, complemented
9#70855C211yesmaximal, characteristic, complemented
10#68555C211yesmaximal, characteristic, complemented
11#68531C211yesmaximal, characteristic, complemented
12#68511C211yesmaximal, characteristic, complemented
13#68483C211yesmaximal, characteristic, complemented
14#68410C211yesmaximal, characteristic, complemented
15#68386C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17D3×C4⋊Q8C2211yescharacteristic
18#5716C2211yescharacteristic
19#5557C2211yescharacteristic
20C3×#486C2211yescharacteristic
21C3×#476C2211yescharacteristic
22C3×#459C2211yescharacteristic
23C3×C8×D4C2211yescharacteristic
24#5159—21no—
25#5153—21no—
26#5111—21no—
27#5101—21no—
28#5093C2211yescharacteristic
29#5091C2211yescharacteristic
30#5075C2211yescharacteristic
31#5070C2211yescharacteristic
32#5012C2211yescharacteristic
33#5008C2211yescharacteristic
34#5007C2211yescharacteristic
35#4862—21no—
36#4856—21no—
37#4851C2211yescharacteristic
38#4850C2211yescharacteristic
39#4837C2211yescharacteristic
40#4835C2211yescharacteristic
41#4833C2211yescharacteristic
42#4830C2211yescharacteristic
43#4780—21no—
44#4774—21no—
45#4771—21no—
46#4771—21no—
47#4765—21no—
48#4765—21no—
49#4765—21no—
50#4763—21no—
51#4729—21no—
52#4720—21no—
53C4×ΓL(1,𝔽25)C2211yescharacteristic
54#4681C2211yescharacteristic
55#4679C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4546C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3251—31no—
69#3238—31no—
70#3141—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1698—31no—
75#1697—31no—
76#1684—31no—
77#1683—31no—
78#1678—31no—
79#1677D311yescharacteristic, complemented
80#1569—31no—
81#1553—31no—
82#1548—31no—
83C2×D3×Q8—21no—
84C2×#237—21no—
85C2×C3×D8—21no—
86C22×C3×C8—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127C2311yescharacteristic
89C3×C4×D4C2311yescharacteristic
90#944—21no—
91#938—21no—
92#938—21no—
93#938D411yescharacteristic
94#936—21no—
95#936D411yescharacteristic
96D4×Dic3—21no—
97D4×Dic3—21no—
98C2×#214—21no—
99C4×#67—21no—
100#926—21no—
101C2×#208—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)D411yescharacteristic
104C2×ΓL(1,𝔽25)D411yescharacteristic
105#897D411yescharacteristic
106D3×C4⋊C4—21no—
107D3×C4⋊C4—21no—
108#890D411yescharacteristic
109#888—21no—
110#888—21no—
111#883—21no—
112#879—21no—
113C42×D3C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6C2311yescharacteristic
117C3×#106—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×#102—21no—
122C3×#102—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×C22⋊C8—21no—
126C3×C4×C8C2311yescharacteristic
127#832—21no—
128#832—21no—
129#832—21no—
130#832—21no—
131#832—21no—
132#830—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135#818—21no—
136#817—21no—
137#816—21no—
138#816—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143#815D411yescharacteristic
144#815D411yescharacteristic
145Dic3⋊C8—21no—
146Dic3⋊C8—21no—
147Dic3⋊C8—21no—
148C8×Dic3—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#808—21no—
154#807—21no—
155#807—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#489—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486C2×D311yescharacteristic
171#483—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#476—31no—
177#476C2×D311yescharacteristic
178#475—31no—
179#474—31no—
180#469—61no—
181#468—61no—
182#463—61no—
183#460—31no—
184#459C2×D311yescharacteristic
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4C2×D311yescharacteristic
191#433—61no—
192#432—31no—
193#431—31no—
194#423—31no—
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#333—31no—
206#333C2×D311yescharacteristic
207#332—31no—
208#332—31no—
209#331—31no—
210#331C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C2×#67—21no—
218C2×#67—41no—
219C22×Dic3—41no—
220D3×Q8—82no—
221#237—42no—
222#237—42no—
223C2×C4×D3—41no—
224C2×C4×D3C2×D411yescharacteristic
225C2×C4×D3C2×D411yescharacteristic
226C2×C4×D3#4111yescharacteristic
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C2×Dic6C2×D411yescharacteristic
231C3×D8—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—42no—
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C22⋊C4—21no—
242C3×C42C2411yescharacteristic
243#217—21no—
244#217—21no—
245#217—21no—
246#217—41no—
247#217—41no—
248#214—81no—
249(C2×D3)⋊C4—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3#4111yescharacteristic
265#208—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271ΓL(1,𝔽25)—41no—
272ΓL(1,𝔽25)—41no—
273ΓL(1,𝔽25)—81no—
274C2×#41—61no—
275C22×Q8—61no—
276C2×SD16—61no—
277C2×D8—21no—
278C2×M16—31no—
279C2×M16—31no—
280C2×M16—61no—
281C2×M16—61no—
282C22×C8—21no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—61no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288C4⋊Q8C22×D311yescharacteristic
289#127C22×D311yescharacteristic
290#126—61no—
291#125—31no—
292#124—31no—
293#124—61no—
294#123—61no—
295C22⋊Q8—31no—
296#121—31no—
297#121—61no—
298#121—61no—
299C4×Q8—31no—
300C4×D4—61no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4C22×D311yescharacteristic
304#117—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C42—31no—
308#107—61no—
309#107—61no—
310#107—61no—
311#106—21no—
312#106—61no—
313#106—61no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—61no—
324#103—61no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—21no—
331#102—21no—
332#102—21no—
333#102—21no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—31no—
344#97—31no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C23×C3—21no—
352C23×C3—41no—
353C22×D3#12111yescharacteristic
354C3×Q8—81no—
355C3×D4—21no—
356C3×D4—21no—
357C3×D4—41no—
358C3×D4—41no—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—81no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369#67—81no—
370#67—82no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—82no—
376C2×Dic3#12111yescharacteristic
377C2×Dic3#12111yescharacteristic
378C2×Dic3#12111yescharacteristic
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—41no—
384C4×D3—82no—
385Dic6—21no—
386Dic6—21no—
387Dic6—41no—
388Dic6—41no—
389Dic6—82no—
390Dic6—81no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—21no—
404C2×Q8—31no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—121no—
408C2×Q8—242no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—61no—
413C2×D4—121no—
414C2×D4D3×D411yescharacteristic
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—121no—
420C22×C4—121no—
421SD16—241no—
422D8—81no—
423M16—121no—
424M16—121no—
425M16—122no—
426M16—241no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—42no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8D3×D411yescharacteristic
437C2×C8D3×D411yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—31no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4D3×D411yescharacteristic
452C22⋊C4—21no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—31no—
460C42—31no—
461C42—31no—
462C42—61no—
463C42C23×D311yescharacteristic
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3C2×#12111yescharacteristic
471C2×D3—21no—
472C2×D3—21no—
473C2×D3—21no—
474C3×C4—21no—
475C3×C4—41no—
476C3×C4—81no—
477C3×C4#57511yescharacteristic
478C3×C4#57511yescharacteristic
479C3×C4#57711yescharacteristic
480C3×C4#57811yescharacteristic
481Dic3—21no—
482Dic3—21no—
483Dic3—21no—
484Dic3—41no—
485Dic3—41no—
486Dic3—81no—
487Dic3—81no—
488C23—21no—
489C23—31no—
490C23—41no—
491Q8—61no—
492Q8—61no—
493Q8—81no—
494Q8—121no—
495Q8—121no—
496Q8—241no—
497Q8—242no—
498D4—21no—
499D4—21no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—81no—
505D4—241no—
506D4—242no—
507C2×C4—21no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—242no—
524C2×C4—242no—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4D3×#4111yescharacteristic
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—61no—
534C8—61no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3—81no—
539C2×C3—161no—
540C2×C3#304011yescharacteristic
541C2×C3#306311yescharacteristic
542C2×C3#306811yescharacteristic
543D3—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—61no—
548C22—61no—
549C22—61no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22D3×#12111yescharacteristic
554C4—21no—
555C4—41no—
556C4—61no—
557C4—61no—
558C4—61no—
559C4—81no—
560C4—121no—
561C4—121no—
562C4—241no—
563C4—241no—
564C4#576211yescharacteristic
565C4#576811yescharacteristic
566C4#577111yescharacteristic
567C4#577311yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—81no—
570C2—121no—
571C2—161no—
572C2#8053511yescharacteristic
573C2#8066211yescharacteristic
574C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1677C2
socleC22×C3C2×#121
prime radical#80344C2
hypercenter#1677D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1677

Complements

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

NH ≅ G/Ntype
1. #81565C2semidirect
2. #81067C2semidirect
4. #79403C2semidirect
5. C3×#1677C2semidirect
6. #71210C2semidirect
7. #71139C2semidirect
8. #71010C2semidirect
9. #70855C2semidirect
10. #68555C2semidirect
11. #68531C2semidirect
12. #68511C2semidirect
13. #68483C2semidirect
14. #68410C2semidirect
15. #68386C2semidirect
79. #1677D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
#815652
#810672
#803442
#794032
C3×#16772
#712102
#711392
#710102
#708552
#685552
#685312
#685112
#684832
#684102
#683862
#193823