Finite Group Databasebeta

← #10786113

Subgroups — #10786113

Order 768.

632 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#83524C211yesmaximal, characteristic
2#83517C211yesmaximal, characteristic, complemented
3#80990C211yesmaximal, characteristic, complemented
4#80415C211yesmaximal, characteristic, complemented
5C3×#1647C211yesmaximal, characteristic, complemented
6#71370C211yesmaximal, characteristic, complemented
7#71368C211yesmaximal, characteristic, complemented
8#71103C211yesmaximal, characteristic, complemented
9#71037C211yesmaximal, characteristic, complemented
10#68465C211yesmaximal, characteristic, complemented
11#68450C211yesmaximal, characteristic, complemented
12#68206C211yesmaximal, characteristic, complemented
13#68181C211yesmaximal, characteristic, complemented
14#68152C211yesmaximal, characteristic, complemented
15#68150C211yesmaximal, characteristic, complemented
16#19860—31nomaximal, Sylow
17C3×#547C2211yescharacteristic
18C2×#950—21no—
19#5732C2211yescharacteristic
20#5551C2211yescharacteristic
21#5543C2211yescharacteristic
22C3×#500C2211yescharacteristic
23C3×#474C2211yescharacteristic
24C3×#461C2211yescharacteristic
25#5239—21no—
26#5165—21no—
27Dic3×SD16—21no—
28#5094C2211yescharacteristic
29#5092C2211yescharacteristic
30#5083C2211yescharacteristic
31#5080C2211yescharacteristic
32#5079C2211yescharacteristic
33#5078C2211yescharacteristic
34#5051—21no—
35#5047—21no—
36#5041—21no—
37#5034—21no—
38#5019C2211yescharacteristic
39C4×#214C2211yescharacteristic
40C4×#213C2211yescharacteristic
41D4×C3⋊C8C2211yescharacteristic
42D4×C3⋊C8C2211yescharacteristic
43#5008C2211yescharacteristic
44#4971—21no—
45C2×#808—21no—
46#4884—21no—
47#4881—21no—
48#4851C2211yescharacteristic
49#4842C2211yescharacteristic
50#4838C2211yescharacteristic
51#4830C2211yescharacteristic
52#4800—21no—
53#4799—21no—
54#4780—21no—
55#4779—21no—
56#4681C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×D4⋊C8C2211yescharacteristic
59C3×D4⋊C8C2211yescharacteristic
60#4547C2211yescharacteristic
61#4482C2211yescharacteristic
62#4481C2211yescharacteristic
63#4479C2211yescharacteristic
64#4479C2211yescharacteristic
65#4459C2211yescharacteristic
66#4456C2211yescharacteristic
67#4454C2211yescharacteristic
68#3360—31no—
69#3353—31no—
70#3335—31no—
71#3283—31no—
72#1723—31no—
73#1691—31no—
74#1683—31no—
75#1673—31no—
76#1670—31no—
77#1650—31no—
78#1647D311yescharacteristic, complemented
79#1643—31no—
80#1635—31no—
81#1581—31no—
82SD16⋊C8—31no—
83C2×C3×#41—21no—
84C2×#235—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#123—41no—
89C3×C22⋊Q8—21no—
90C3×#121—21no—
91C3×C4×D4C2311yescharacteristic
92C3×C4×D4C2311yescharacteristic
93C2×C3×C4⋊C4—21no—
94#950—44no—
95#947—21no—
96Q8×Dic3—21no—
97C2×#216—21no—
98C2×#216—21no—
99C2×#215—21no—
100#938—21no—
101D4×Dic3—21no—
102C2×#214—21no—
103C2×#214D411yescharacteristic
104C2×#213D411yescharacteristic
105C2×C4⋊Dic3—21no—
106#924—21no—
107#924—21no—
108C22×C3⋊C8—21no—
109C22×C3⋊C8—21no—
110C2×ΓL(1,𝔽25)—21no—
111#898—41no—
112#897—21no—
113#892—21no—
114D3×C4⋊C4—21no—
115#890—21no—
116#883—41no—
117#878—41no—
118C4×D12C2311yescharacteristic
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121C4×Dic6C2311yescharacteristic
122C3×#107—21no—
123C3×#107—41no—
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×#103—21no—
127C3×#102—21no—
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#835—21no—
131#835—21no—
132#832—21no—
133#832D411yescharacteristic
134#832D411yescharacteristic
135#830—21no—
136#830—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139#817—21no—
140#817—21no—
141#816—21no—
142#816—41no—
143Dic3⋊C8—21no—
144C8×Dic3—21no—
145#810—21no—
146#810—21no—
147#810D411yescharacteristic
148#809D411yescharacteristic
149#808—21no—
150#808—21no—
151#808—21no—
152#808—21no—
153#808D411yescharacteristic
154#807—21no—
155#807—21no—
156#807D411yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#556—31no—
163#554—31no—
164#547—31no—
165#547C2×D311yescharacteristic
166#500—31no—
167#500C2×D311yescharacteristic
168#498—31no—
169#494—31no—
170#488—61no—
171#487—31no—
172#487—31no—
173#484—61no—
174#483—61no—
175#482—61no—
176#481—61no—
177#479—61no—
178#476—31no—
179#476—31no—
180#475—31no—
181#475—31no—
182#474C2×D311yescharacteristic
183#468—61no—
184#467—61no—
185#465—61no—
186#463—61no—
187#461—31no—
188#461—31no—
189#461C2×D311yescharacteristic
190#452—61no—
191#452—61no—
192C8×Q8—31no—
193C4×SD16—31no—
194C4×SD16—61no—
195C4×D8—31no—
196#436—31no—
197C8×D4—31no—
198C8×D4—31no—
199#428—61no—
200C2×#106—61no—
201#335—31no—
202#335—31no—
203#333—31no—
204#332—31no—
205#332—31no—
206#332C2×D311yescharacteristic
207Q8⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8C2×D311yescharacteristic
210D4⋊C8C2×D311yescharacteristic
211#322—31no—
212C3×#41—42no—
213C3×#41—81no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×D4—41no—
217C2×C3×D4C2×D411yescharacteristic
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—41no—
222C2×#67—41no—
223C22×Dic3—41no—
224#235—42no—
225#235—81no—
226C2×D12C2×D411yescharacteristic
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C2×D411yescharacteristic
232C3×SD16—42no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—41no—
246C3×C42C2411yescharacteristic
247#217—41no—
248#217—41no—
249#216—42no—
250#216—81no—
251#215—42no—
252#214—22no—
253#214—42no—
254#213—22no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—41no—
257(C2×D3)⋊C4—41no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—41no—
263C4⋊Dic3—41no—
264C4⋊Dic3C2×D411yescharacteristic
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—21no—
267Dic3⋊C4—41no—
268Dic3⋊C4—41no—
269C4×Dic3—21no—
270C4×Dic3—41no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274C2×C3⋊C8—42no—
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277C2×C3⋊C8C2×D411yescharacteristic
278C2×C3⋊C8C2×D411yescharacteristic
279ΓL(1,𝔽25)—42no—
280C2×#41—21no—
281C2×#41—61no—
282#135—124no—
283C2×Q16—61no—
284C2×Q16—61no—
285C2×SD16—21no—
286C2×SD16—31no—
287C2×SD16—61no—
288C2×SD16—61no—
289C2×D8—31no—
290C2×M16—61no—
291C22×C8—61no—
292C22×C8—61no—
293C4⋊Q8—31no—
294C4⋊Q8—31no—
295C4⋊Q8C22×D311yescharacteristic
296C4⋊Q8C22×D311yescharacteristic
297#126—121no—
298#125—61no—
299#124—61no—
300#123—41no—
301#123—121no—
302C22⋊Q8—21no—
303C22⋊Q8—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306C22⋊Q8—121no—
307#121—21no—
308#121—61no—
309C4×Q8—31no—
310C4×Q8—61no—
311C4×D4—31no—
312C4×D4—61no—
313C4×D4C22×D311yescharacteristic
314C4×D4C22×D311yescharacteristic
315#117—61no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319#107—21no—
320#107—31no—
321#107—41no—
322#107—61no—
323#107—61no—
324#106—31no—
325#106—61no—
326#106—61no—
327#106—61no—
328#106—61no—
329#106—61no—
330#106—61no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—21no—
337#103—31no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#103—61no—
344#103—121no—
345#102—21no—
346#102—31no—
347#102—31no—
348#102—31no—
349#102—61no—
350#102—61no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353C22⋊C8—61no—
354C4×C8—31no—
355C4×C8—31no—
356C4×C8—61no—
357C4×C8C22×D311yescharacteristic
358C23×C3—21no—
359C23×C3—21no—
360C22×D3—21no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×D4—21no—
366C3×D4—21no—
367C3×D4—21no—
368C3×D4—82no—
369C3×D4AGL(1,ℤ/8)12yes—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—82no—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—82no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—161no—
389D12—21no—
390D12—21no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—82no—
394Dic6—21no—
395Dic6—21no—
396Dic6—41no—
397Dic6—41no—
398Dic6—81no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8#13512yescomplemented
408#41—42no—
409#41—81no—
410#41—122no—
411#41—241no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—31no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×D4—31no—
418C2×D4—41no—
419C2×D4—121no—
420C2×D4D3×D411yescharacteristic
421C2×D4D3×D411yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—61no—
426C22×C4—121no—
427C22×C4—121no—
428Q16—122no—
429Q16—241no—
430SD16—42no—
431SD16—62no—
432SD16—122no—
433SD16—122no—
434D8—62no—
435M16—122no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—31no—
440C2×C8—31no—
441C2×C8—31no—
442C2×C8—31no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—122no—
446C2×C8—122no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—31no—
452C4⋊C4—31no—
453C4⋊C4—41no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—121no—
460C4⋊C4—121no—
461C4⋊C4—121no—
462C4⋊C4—121no—
463C4⋊C4D3×D411yescharacteristic
464C4⋊C4D3×D411yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—41no—
468C22⋊C4—61no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C42—61no—
474C42—121no—
475C42C23×D311yescharacteristic
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3D4211yescharacteristic
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C3×C4—21no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4C2×#13511yescharacteristic
492C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C3×C4#57711yescharacteristic
494C3×C4#57811yescharacteristic
495Dic3—41no—
496Dic3—41no—
497Dic3—41no—
498Dic3—81no—
499Dic3—81no—
500C23—21no—
501C23—21no—
502C23—61no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—61no—
508Q8—61no—
509Q8—121no—
510Q8—121no—
511Q8—241no—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—61no—
516D4—61no—
517D4—82no—
518D4—242no—
519D4#91112yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—82no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—242no—
537C2×C4—481no—
538C2×C4C2×D3×D411yescharacteristic
539C2×C4C2×D3×D411yescharacteristic
540C2×C4#100411yescharacteristic
541C8—32no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—61no—
546C8—61no—
547C8—61no—
548C8—121no—
549C8—121no—
550C2×C3—42no—
551C2×C3—81no—
552C2×C3#326511yescharacteristic
553C2×C3#328311yescharacteristic
554C2×C3#328911yescharacteristic
555D3—81no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22#559011yescharacteristic
566C4—21no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—121no—
572C4—121no—
573C4—121no—
574C4—241no—
575C4—241no—
576C4C2×#91111yescharacteristic
577C4D3×#13511yescharacteristic
578C4#577511yescharacteristic
579C4#577911yescharacteristic
580C3#1986011yescharacteristic, complemented, Sylow
581C2—42no—
582C2—81no—
583C2—241no—
584C2#8002511yescharacteristic
585C2#8026911yescharacteristic
586C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1647C2
socleC22×C3D42
prime radical#83524C2
hypercenter#1647D3
hypocenterC3#19860
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1647

Complements

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

NH ≅ G/Ntype
2. #83517C2semidirect
3. #80990C2semidirect
4. #80415C2semidirect
5. C3×#1647C2semidirect
6. #71370C2semidirect
7. #71368C2semidirect
8. #71103C2semidirect
9. #71037C2semidirect
10. #68465C2semidirect
11. #68450C2semidirect
12. #68206C2semidirect
13. #68181C2semidirect
14. #68152C2semidirect
15. #68150C2semidirect
78. #1647D3semidirect
407. C3⋊C8#135semidirect
580. C3#19860semidirect

Maximal subgroups

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

subgroupindex
#835242
#835172
#809902
#804152
C3×#16472
#713702
#713682
#711032
#710372
#684652
#684502
#682062
#681812
#681522
#681502
#198603