Finite Group Databasebeta

← #10944273

Subgroups — #10944273

Order 768.

1580 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
2#82834C211yesmaximal, characteristic, complemented
3C2×#5048C211yesmaximal, characteristic, complemented
4C2×#4812C212yesmaximal, complemented
5#74048C211yesmaximal, characteristic
6C3×#1892C211yesmaximal, characteristic, complemented
7#71973C211yesmaximal, characteristic, complemented
8#70271C212yesmaximal, complemented
9#69512C211yesmaximal, characteristic, complemented
10#68885C212yesmaximal, complemented
11#68870C212yesmaximal, complemented
12#32935—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#946C2212yes—
17C22×#215C2211yescharacteristic, complemented
18C2×#930C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×ΓL(1,𝔽25)—24no—
21C22×ΓL(1,𝔽25)—28no—
22C22×ΓL(1,𝔽25)C2212yes—
23C22×ΓL(1,𝔽25)C2212yes—
24C2×#895C2212yes—
25C2×#882—21no—
26C2×C3×#103C2212yescomplemented
27C3×#380C2211yescharacteristic
28C2×#835C2211yescharacteristic, complemented
29#5212C2211yescharacteristic
30C2×#821C2212yescomplemented
31#5048—24no—
32#4989—22no—
33#4987C2212yes—
34#4987C2212yes—
35C2×#809C2211yescharacteristic, complemented
36C2×#808C2211yescharacteristic, complemented
37C2×#807C2212yescomplemented
38#4958—21no—
39#4955—21no—
40#4946—21no—
41#4812—28no—
42#4665—22no—
43C3×#340C2212yescomplemented
44#4551C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#468—32no—
48C2×#465—31no—
49#2848—31no—
50#2847—31no—
51#1895—32no—
52#1894—32no—
53#1893—31no—
54#1892D311yescharacteristic, complemented
55#1809—32no—
56#1808—31no—
57C24×D3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C23×Dic3D411yescharacteristic
62C22×D12—22no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3D412yes—
69C22×C4×D3C2312yes—
70C22×C3×C8—22no—
71C22×C3×C8—24no—
72C22×C3×C8C2312yes—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×#217—21no—
81#946—24no—
82#946—24no—
83C2×#215—22no—
84C2×#215—24no—
85#930—22no—
86#930—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—22no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8D414yescomplemented
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C2×ΓL(1,𝔽25)—232no—
100C2×ΓL(1,𝔽25)—24no—
101C2×ΓL(1,𝔽25)—24no—
102C2×ΓL(1,𝔽25)—24no—
103C2×ΓL(1,𝔽25)—24no—
104C2×ΓL(1,𝔽25)—24no—
105C2×ΓL(1,𝔽25)D418yescomplemented
106#895—24no—
107#895—24no—
108#882—44no—
109C3×#103—24no—
110C3×#95—21no—
111#835—22no—
112C3⋊#95—21no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115#821—24no—
116#809—22no—
117#808—22no—
118#807—24no—
119C24×C4—31no—
120C22×SD16—31no—
121C22×M16—32no—
122C22×M16—32no—
123C22×M16—64no—
124C22×M16—68no—
125C23×C8—31no—
126C23×C8C2×D311yescharacteristic
127C2×C22⋊Q8—32no—
128C2×C22⋊Q8C2×D311yescharacteristic
129C2×#121—31no—
130C2×#121—32no—
131C2×#120—61no—
132#468—68no—
133#465—64no—
134C2×#107—32no—
135C2×#106—31no—
136C2×#103—31no—
137C2×#103C2×D312yescomplemented
138C2×#102—32no—
139C2×#102—31no—
140#386—62no—
141#386—61no—
142#385—61no—
143#385—62no—
144#380—31no—
145#380—32no—
146#380—32no—
147#380C2×D311yescharacteristic
148#379—61no—
149#340—32no—
150#340—32no—
151#340C2×D312yescomplemented
152C24×C3C2×D411yescharacteristic
153C23×D3—22no—
154C23×D3—24no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—21no—
158C23×D3—42no—
159C23×D3C2×D412yes—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—42no—
174C2×#67—42no—
175C2×#67—42no—
176C22×Dic3—22no—
177C22×Dic3—24no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3C2×D412yes—
181C2×D12—21no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—21no—
185C2×D12—44no—
186C2×D12—44no—
187C2×D12—44no—
188C2×C4×D3—28no—
189C2×C4×D3—216no—
190C2×C4×D3—28no—
191C2×C4×D3—24no—
192C2×C4×D3—24no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—24no—
198C2×C4×D3C2×D414yes—
199C2×C4×D3#4114yes—
200C2×C3×C8—28no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8C2×D414yes—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211#217—42no—
212#215—44no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—44no—
216(C2×D3)⋊C4—42no—
217(C2×D3)⋊C4—44no—
218C4⋊Dic3—22no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222Dic3⋊C4—44no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8C2×D412yes—
226C2×C3⋊C8C2×D414yes—
227C2×C3⋊C8C2×D412yes—
228ΓL(1,𝔽25)—216no—
229ΓL(1,𝔽25)—216no—
230C25—31no—
231C22×Q8C22×D311yescharacteristic
232C22×D4—31no—
233C22×D4—62no—
234C22×D4—61no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—64no—
239C23×C4—64no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4C22×D311yescharacteristic
243C2×SD16—62no—
244C2×SD16—64no—
245C2×M16—38no—
246C2×M16—64no—
247C2×M16—64no—
248C2×M16—64no—
249C2×M16—64no—
250C2×M16—632no—
251C2×M16—64no—
252C22×C8—22no—
253C22×C8—24no—
254C22×C8—31no—
255C22×C8—34no—
256C22×C8—31no—
257C22×C8—64no—
258C22×C8C22×D312yes—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—64no—
262C22⋊Q8—64no—
263#121—64no—
264#121—64no—
265#121—62no—
266#121—62no—
267#120—124no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282#107—64no—
283#106—62no—
284#103—24no—
285#103—62no—
286#102—62no—
287#102—64no—
288#95—21no—
289#95—62no—
290#95—62no—
291#95—61no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#12112yes—
295C23×C3AGL(1,ℤ/8)12yes—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C22×D3—28no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—48no—
306C22×D3—48no—
307C22×D3—44no—
308C22×D3—44no—
309C22×D3—42no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—42no—
314C22×D3#12114yes—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4#12112yes—
330C2×C3×C4C22×D412yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4112yes—
334#67—84no—
335C2×Dic3—28no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3#12114yes—
345D12—41no—
346D12—41no—
347D12—42no—
348D12—88no—
349C4×D3—216no—
350C4×D3—24no—
351C4×D3—24no—
352C4×D3—24no—
353C4×D3—24no—
354C3×C8—24no—
355C3⋊C8—22no—
356C3⋊C8#12014yescomplemented
357C24—32no—
358C24—64no—
359C24—62no—
360C24—62no—
361C24—62no—
362C24—61no—
363C24—122no—
364C24D3×D411yescharacteristic
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—122no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—32no—
385C22×C4—34no—
386C22×C4—34no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—68no—
390C22×C4—68no—
391C22×C4—616no—
392C22×C4—64no—
393C22×C4—62no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4#23512yes—
402C22×C4D3×D411yescharacteristic
403C22×C4D3×D412yes—
404C22×C4C23×D311yescharacteristic
405SD16—124no—
406M16—616no—
407M16—616no—
408C2×C8—28no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—32no—
412C2×C8—34no—
413C2×C8—32no—
414C2×C8—68no—
415C2×C8—62no—
416C2×C8D3×D414yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—61no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—62no—
426C4⋊C4—124no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—124no—
430C22⋊C4—124no—
431C22⋊C4—124no—
432C22⋊C4—124no—
433C22⋊C4—122no—
434C22⋊C4—122no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3#46814yescomplemented
440C22×C3C2×#12111yescharacteristic
441C22×C3C2×#12112yes—
442C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
443C22×C3C2×#13711yescharacteristic
444C22×C3#57512yes—
445C2×D3—24no—
446C2×D3—24no—
447C2×D3—24no—
448C2×D3—416no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—81no—
454C2×D3—81no—
455C2×D3—81no—
456C2×D3—82no—
457C2×D3—82no—
458C3×C4—22no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4—82no—
462C3×C4C2×#12011yescharacteristic
463C3×C4#52511yescharacteristic
464C3×C4#52512yes—
465Dic3—24no—
466Dic3—81no—
467Dic3—82no—
468C23—22no—
469C23—22no—
470C23—34no—
471C23—68no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—128no—
479C23—128no—
480C23—124no—
481C23—124no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—122no—
486C23—122no—
487C23#88412yes—
488C23#91412yes—
489C23#91812yes—
490C23C2×D3×D411yescharacteristic
491Q8—41no—
492Q8—42no—
493Q8—41no—
494D4—121no—
495D4—122no—
496D4—121no—
497D4—248no—
498D4—244no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—34no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—616no—
512C2×C4—68no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4#88412yes—
526C2×C4C2×#23511yescharacteristic
527C2×C4C2×D3×D411yescharacteristic
528C2×C4C2×D3×D412yes—
529C2×C4D3×#4112yes—
530C8—24no—
531C8—34no—
532C8—62no—
533C2×C3—24no—
534C2×C3#284711yescharacteristic
535C2×C3C2×#46811yescharacteristic
536C2×C3#305411yescharacteristic
537C2×C3#305512yes—
538C2×C3#305612yes—
539D3—44no—
540D3—161no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—64no—
546C22—64no—
547C22—64no—
548C22—1216no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—242no—
554C22—242no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22#481214yescomplemented
559C22C2×#88411yescharacteristic
560C22D3×#12112yes—
561C22C2×#91411yescharacteristic
562C22C2×#91811yescharacteristic
563C22#576812yes—
564C4—22no—
565C4—64no—
566C4—82no—
567C4—81no—
568C4—81no—
569C4—242no—
570C4—241no—
571C4#554712yes—
572C4D3×#12011yescharacteristic
573C4#565611yescharacteristic
574C3#3293511yescharacteristic, complemented, Sylow
575C2—24no—
576C2—124no—
577C2—481no—
578C2#7493511yescharacteristic
579C2C2×#481211yescharacteristic
580C2#7883911yescharacteristic
581C2#8065712yes—
582C2#8075912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1892C2
socleC23×C3C22×D4
prime radical#74048C2
hypercenter#1892D3
hypocenterC3#32935
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
1. C23×ΓL(1,𝔽25)C2semidirect
2. #82834C2semidirect
3. C2×#5048C2semidirect
4. C2×#4812C2semidirect
6. C3×#1892C2semidirect
7. #71973C2semidirect
8. #70271C2semidirect
9. #69512C2semidirect
10. #68885C2semidirect
11. #68870C2semidirect
17. C22×#215C22semidirect
26. C2×C3×#103C22semidirect
28. C2×#835C22semidirect
30. C2×#821C22semidirect
35. C2×#809C22semidirect
36. C2×#808C22semidirect
37. C2×#807C22semidirect
43. C3×#340C22semidirect
44. #4551C22semidirect
45. #4529C22semidirect
54. #1892D3semidirect
96. C22×C3⋊C8D4semidirect
105. C2×ΓL(1,𝔽25)D4semidirect
137. C2×#103C2×D3semidirect
151. #340C2×D3semidirect
356. C3⋊C8#120semidirect
439. C22×C3#468semidirect
558. C22#4812semidirect
574. C3#32935semidirect

Maximal subgroups

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

subgroupindex
C23×ΓL(1,𝔽25)2
#828342
C2×#50482
C2×#48122
#740482
C3×#18922
#719732
#702712
#695122
#688852
#688702
#329353