Finite Group Databasebeta

← #10944286

Subgroups — #10944286

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×ΓL(1,𝔽25)C211yesmaximal, characteristic
2#82834C211yesmaximal, characteristic
3C2×#5051C211yesmaximal, characteristic, complemented
4C2×#4816C212yesmaximal, complemented
5#74049C211yesmaximal, characteristic
6C3×#1892C211yesmaximal, characteristic, complemented
7#71972C211yesmaximal, characteristic, complemented
8#70271C212yesmaximal, complemented
9#69511C211yesmaximal, characteristic, complemented
10#68884C212yesmaximal, complemented
11#68870C212yesmaximal, complemented
12#32937—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#946C2212yes—
17C22×#216C2211yescharacteristic, complemented
18C2×#924C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×ΓL(1,𝔽25)—24no—
21C22×ΓL(1,𝔽25)—28no—
22C22×ΓL(1,𝔽25)C2212yes—
23C22×ΓL(1,𝔽25)C2212yes—
24C2×#897C2212yes—
25C2×C3×#103C2212yescomplemented
26C3×#380C2211yescharacteristic
27C2×#835C2211yescharacteristic, complemented
28#5212C2211yescharacteristic
29C2×#816C2212yescomplemented
30#5051—24no—
31#4989—22no—
32#4987C2212yes—
33#4987C2212yes—
34C2×#810C2211yescharacteristic, complemented
35C2×#808C2212yescomplemented
36C2×#807C2211yescharacteristic, complemented
37#4955—21no—
38#4955—21no—
39#4950—21no—
40#4946—21no—
41#4816—28no—
42#4668—22no—
43C3×#340C2212yescomplemented
44#4551C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#470—32no—
48C2×#467—31no—
49#2848—31no—
50#2848—31no—
51#1894—32no—
52#1894—32no—
53#1892—31no—
54#1892D311yescharacteristic, complemented
55#1809—32no—
56#1807—31no—
57C24×D3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×Dic3D411yescharacteristic
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—24no—
64C22×C4×D3—24no—
65C22×C4×D3D412yes—
66C22×C4×D3C2312yes—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8—22no—
69C22×C3×C8—24no—
70C22×C3×C8C2312yes—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×#217—21no—
79C2×#217—21no—
80#946—24no—
81#946—24no—
82C2×#216—22no—
83C2×#216—24no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91#924—22no—
92#924—22no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—22no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8D414yes—
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)D418yes—
106#897—24no—
107#897—24no—
108C3×#103—24no—
109C3×#95—21no—
110#835—22no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#816—24no—
115#810—22no—
116#808—24no—
117#807—22no—
118C24×C4—31no—
119C22×Q16—31no—
120C22×M16—32no—
121C22×M16—32no—
122C22×M16—64no—
123C22×M16—68no—
124C23×C8—31no—
125C23×C8C2×D311yescharacteristic
126C2×C22⋊Q8—32no—
127C2×C22⋊Q8—31no—
128C2×C22⋊Q8—32no—
129C2×C22⋊Q8C2×D311yescharacteristic
130#470—68no—
131#467—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103C2×D312yescomplemented
138#385—62no—
139#385—61no—
140#385—61no—
141#385—62no—
142#380—31no—
143#380—32no—
144#380—32no—
145#380C2×D311yescharacteristic
146#379—61no—
147#379—61no—
148#340—32no—
149#340—32no—
150#340C2×D312yescomplemented
151C24×C3C2×D411yescharacteristic
152C23×D3—22no—
153C23×D3—24no—
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—42no—
157C23×D3C2×D412yes—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4#4112yes—
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3C2×D412yes—
177C2×C4×D3—28no—
178C2×C4×D3—216no—
179C2×C4×D3—28no—
180C2×C4×D3—24no—
181C2×C4×D3—24no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—24no—
187C2×C4×D3C2×D414yes—
188C2×C4×D3#4114yes—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×C3×C8—28no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8C2×D414yes—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—42no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—42no—
204#217—42no—
205#217—42no—
206#216—44no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—44no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—44no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—44no—
216Dic3⋊C4—42no—
217Dic3⋊C4—44no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—28no—
220C2×C3⋊C8C2×D412yes—
221C2×C3⋊C8C2×D412yes—
222C2×C3⋊C8C2×D414yes—
223ΓL(1,𝔽25)—216no—
224ΓL(1,𝔽25)—216no—
225C25—31no—
226C22×Q8—31no—
227C22×Q8C22×D311yescharacteristic
228C23×C4—32no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—64no—
232C23×C4—64no—
233C23×C4—62no—
234C23×C4—62no—
235C23×C4C22×D311yescharacteristic
236C2×Q16—62no—
237C2×Q16—64no—
238C2×M16—38no—
239C2×M16—64no—
240C2×M16—64no—
241C2×M16—64no—
242C2×M16—64no—
243C2×M16—632no—
244C2×M16—64no—
245C22×C8—22no—
246C22×C8—24no—
247C22×C8—31no—
248C22×C8—34no—
249C22×C8—31no—
250C22×C8—64no—
251C22×C8C22×D312yes—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—64no—
255C22⋊Q8—64no—
256C22⋊Q8—64no—
257C22⋊Q8—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276#107—62no—
277#106—64no—
278#103—24no—
279#103—62no—
280#103—62no—
281#103—64no—
282#95—21no—
283#95—62no—
284#95—62no—
285#95—61no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3#12112yes—
289C23×C3#13712yes—
290C23×C3#13712yes—
291C23×C3C22×D411yescharacteristic
292C22×D3—28no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—48no—
300C22×D3—48no—
301C22×D3—44no—
302C22×D3—44no—
303C22×D3#12114yes—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4#12112yes—
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×Dic3—28no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3#12114yes—
334C4×D3—216no—
335C4×D3—24no—
336C4×D3—24no—
337C4×D3—24no—
338C4×D3—24no—
339Dic6—41no—
340Dic6—41no—
341Dic6—42no—
342C3×C8—24no—
343C3⋊C8—22no—
344C3⋊C8#12014yes—
345C24—32no—
346C24—64no—
347C24—62no—
348C24—62no—
349C24—62no—
350C24—122no—
351C24D3×D411yescharacteristic
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—62no—
359C2×Q8—62no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—32no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—68no—
371C22×C4—68no—
372C22×C4—616no—
373C22×C4—64no—
374C22×C4—62no—
375C22×C4—64no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4#23512yes—
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D412yes—
386C22×C4C23×D311yescharacteristic
387Q16—124no—
388M16—616no—
389M16—616no—
390C2×C8—28no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—34no—
394C2×C8—32no—
395C2×C8—32no—
396C2×C8—68no—
397C2×C8—62no—
398C2×C8D3×D414yes—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—124no—
409C4⋊C4—124no—
410C4⋊C4—122no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—124no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—122no—
418C22⋊C4—122no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3#47014yescomplemented
424C22×C3C2×#12111yescharacteristic
425C22×C3C2×#12112yes—
426C22×C3C2×#13711yescharacteristic
427C22×C3C2×#13711yescharacteristic
428C22×C3#57512yes—
429C2×D3—24no—
430C2×D3—24no—
431C2×D3—24no—
432C2×D3—416no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C3×C4—22no—
438C3×C4—81no—
439C3×C4—81no—
440C3×C4—82no—
441C3×C4C2×#12011yescharacteristic
442C3×C4#52511yescharacteristic
443C3×C4#52512yes—
444Dic3—24no—
445Dic3—81no—
446Dic3—81no—
447Dic3—82no—
448C23—22no—
449C23—22no—
450C23—34no—
451C23—68no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—128no—
459C23—128no—
460C23—124no—
461C23—124no—
462C23#88412yes—
463C23#91512yes—
464C23#91812yes—
465C23C2×D3×D411yescharacteristic
466Q8—41no—
467Q8—42no—
468Q8—41no—
469Q8—121no—
470Q8—122no—
471Q8—121no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—34no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—616no—
485C2×C4—68no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4#88412yes—
500C2×C4C2×#23511yescharacteristic
501C2×C4C2×D3×D411yescharacteristic
502C2×C4C2×D3×D412yes—
503C2×C4D3×#4112yes—
504C8—24no—
505C8—34no—
506C8—62no—
507C2×C3—24no—
508C2×C3#284711yescharacteristic
509C2×C3C2×#47011yescharacteristic
510C2×C3#305411yescharacteristic
511C2×C3#305612yes—
512C2×C3#305612yes—
513D3—44no—
514C22—24no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—64no—
519C22—64no—
520C22—64no—
521C22—1216no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22#481614yescomplemented
527C22C2×#88411yescharacteristic
528C22D3×#12112yes—
529C22C2×#91511yescharacteristic
530C22C2×#91811yescharacteristic
531C22#576812yes—
532C4—22no—
533C4—64no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4—242no—
538C4—241no—
539C4—241no—
540C4#554712yes—
541C4D3×#12011yescharacteristic
542C4#565611yescharacteristic
543C3#3293711yescharacteristic, complemented, Sylow
544C2—24no—
545C2—124no—
546C2#7493511yescharacteristic
547C2C2×#481611yescharacteristic
548C2#7883911yescharacteristic
549C2#8065812yes—
550C2#8075912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1892C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenter#1892D3
hypocenterC3#32937
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
3. C2×#5051C2semidirect
4. C2×#4816C2semidirect
6. C3×#1892C2semidirect
7. #71972C2semidirect
8. #70271C2semidirect
9. #69511C2semidirect
10. #68884C2semidirect
11. #68870C2semidirect
17. C22×#216C22semidirect
25. C2×C3×#103C22semidirect
27. C2×#835C22semidirect
29. C2×#816C22semidirect
34. C2×#810C22semidirect
35. C2×#808C22semidirect
36. C2×#807C22semidirect
43. C3×#340C22semidirect
44. #4551C22semidirect
45. #4529C22semidirect
54. #1892D3semidirect
137. C2×#103C2×D3semidirect
150. #340C2×D3semidirect
423. C22×C3#470semidirect
526. C22#4816semidirect
543. C3#32937semidirect

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×#50512
C2×#48162
#740492
C3×#18922
#719722
#702712
#695112
#688842
#688702
#329373