Finite Group Databasebeta

← #11866912

Subgroups — #11866912

Order 1152.

1237 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×C3⋊D3C211yesmaximal, characteristic, complemented
2#10595081C211yesmaximal, characteristic
3#10594925C211yesmaximal, characteristic, complemented
4C2×#63796C211yesmaximal, characteristic, complemented
5C3×#5134C212yesmaximal, complemented
6#10591180C212yesmaximal, complemented
7#10590940C211yesmaximal, characteristic, complemented
8#10590845C212yesmaximal, complemented
9#10590842C212yesmaximal, complemented
10#10590687C211yesmaximal, characteristic, complemented
11#10590672C211yesmaximal, characteristic, complemented
12D3×#468—32nomaximal
13C22×C4×C3⋊D3C2211yescharacteristic
14C2×#3811—21no—
15C2×#3802C2211yescharacteristic
16C2×#3797C2211yescharacteristic
17C2×C32×M16C2211yescharacteristic, complemented
18C2×#3748C2211yescharacteristic, complemented
19M16×C3⋊D3—24no—
20C2×#3743C2211yescharacteristic, complemented
21C2×#3743C2211yescharacteristic, complemented
22C2×C8×C3⋊D3C2211yescharacteristic
23C2×C8×C3⋊D3C2211yescharacteristic
24C3×#930C2212yes—
25C3×#924C2212yescomplemented
26C2×C3×#204C2212yescomplemented
27#63982—21no—
28#63980—21no—
29#63974—21no—
30#63921—21no—
31#63916C2212yes—
32#63912—21no—
33#63900C2211yescharacteristic
34#63900C2211yescharacteristic
35#63883—21no—
36#63871C2212yescomplemented
37#63865—21no—
38C2×#3715C2212yescomplemented
39#63796—24no—
40C3×#821C2212yescomplemented
41C3×#818C2212yescomplemented
42C3×#816C2212yescomplemented
43#63576C2212yescomplemented
44#63565C2212yescomplemented
45#63562C2212yescomplemented
46C2×D3×SD16—32no—
47C2×D3×M16—32no—
48C2×D3×M16—61no—
49D3×C22⋊Q8—32no—
50D3×#121—32no—
51C3×#468—32no—
52#5174—32no—
53#5134D312yescomplemented
54#5049—32no—
55#5042—32no—
56#4886—32no—
57D3×#107—32no—
58#4812—32no—
59D3×#103—32no—
60#4780—32no—
61D3×#102—32no—
62C23×C3⋊D3D411yescharacteristic
63C22×D32—21no—
64C22×C32×C4C2311yescharacteristic
65C22×C3⋊Dic3D411yescharacteristic
66C2×C4×C3⋊D3—22no—
67C2×C4×C3⋊D3—22no—
68C2×C4×C3⋊D3D411yescharacteristic
69C2×C4×C3⋊D3D411yescharacteristic
70C2×C4×C3⋊D3C2311yescharacteristic
71C2×C4×C3⋊D3C2311yescharacteristic
72C2×C3×#67—22no—
73C2×C3×D12C2312yes—
74C2×C3×Dic6C2312yes—
75#3811—44no—
76C2×#627—21no—
77C2×#626—22no—
78C2×#625—21no—
79C2×#624—21no—
80C2×#624—21no—
81C2×#624—21no—
82#3802—22no—
83#3802—22no—
84#3797—22no—
85#3797—22no—
86C32×M16—22no—
87C2×C32×C8C2311yescharacteristic
88C2×C32×C8C2311yescharacteristic
89#3748—22no—
90C2×C3⋊(C3⋊C8)C2311yescharacteristic
91C2×C3⋊(C3⋊C8)C2311yescharacteristic
92#3743—22no—
93#3743—22no—
94C8×C3⋊D3—22no—
95C8×C3⋊D3D414yescomplemented
96C3×#217—22no—
97C3×(C2×D3)⋊C4—22no—
98C3×C4⋊Dic3C2312yes—
99C3×C4⋊Dic3C2312yes—
100C3×Dic3⋊C4—22no—
101C3×#204—24no—
102#3724—21no—
103#3724—21no—
104Dic3⋊Dic3—22no—
105#3722—22no—
106(C2×D3)⋊Dic3—22no—
107#3715—24no—
108C2×#468—91noSylow
109C2×D3×Q8—32no—
110C2×D3×D4—32no—
111C2×D3×D4—62no—
112C22×C4×D3—32no—
113C22×C4×D3—61no—
114C2×C3×SD16—32no—
115C2×C3×M16—21no—
116C2×C3×M16C2×D312yescomplemented
117C3×C22⋊Q8—32no—
118C3×#121—32no—
119#946—32no—
120C2×#215—32no—
121#939—62no—
122#938—32no—
123C2×#214—32no—
124#930C2×D312yes—
125#924C2×D312yescomplemented
126C2×#208—32no—
127C2×#208—61no—
128D3×SD16—68no—
129D3×M16—68no—
130D3×M16—68no—
131C2×#204C2×D312yescomplemented
132C2×ΓL(1,𝔽25)—32no—
133C2×ΓL(1,𝔽25)—32no—
134C2×ΓL(1,𝔽25)—61no—
135C2×ΓL(1,𝔽25)—61no—
136C2×D3×C8—32no—
137C2×D3×C8—32no—
138C2×D3×C8—61no—
139C2×D3×C8—61no—
140#897—32no—
141#896—62no—
142#895—32no—
143D3×C4⋊C4—32no—
144D3×C4⋊C4—32no—
145D3×C4⋊C4—62no—
146#884—62no—
147D3×C22⋊C4—62no—
148D3×C22⋊C4—62no—
149#878—62no—
150C3×#107—32no—
151C3×#103—32no—
152C3×#102—32no—
153#835—32no—
154#832—32no—
155#821C2×D312yescomplemented
156#818C2×D312yescomplemented
157#816C2×D312yescomplemented
158#810—32no—
159#809—32no—
160#807—32no—
161#807—32no—
162C23×C32C2×D411yescharacteristic
163C22×C3⋊D3—22no—
164C22×C3⋊D3—22no—
165C22×C3⋊D3—22no—
166C22×C3⋊D3C2×D411yescharacteristic
167C22×C3⋊D3C2×D411yescharacteristic
168C22×C3×D3—22no—
169C2×D32—42no—
170C2×D32—42no—
171C2×D32—42no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2411yescharacteristic
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3C2×D411yescharacteristic
178C2×C3⋊Dic3C2×D411yescharacteristic
179C4×C3⋊D3—22no—
180C4×C3⋊D3—21no—
181C4×C3⋊D3—21no—
182C4×C3⋊D3—22no—
183C4×C3⋊D3C2×D412yes—
184C4×C3⋊D3#4112yes—
185C3×#67—44no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C3×D12—22no—
190C3×D12—22no—
191C3×Dic6—22no—
192C3×Dic6—22no—
193#627—42no—
194#626—44no—
195#625—42no—
196#624—42no—
197#624—42no—
198#624—42no—
199C32×C8—21no—
200C32×C8C2×D412yescomplemented
201C3⋊(C3⋊C8)—21no—
202C3⋊(C3⋊C8)C2×D412yescomplemented
203C22×SD16—91no—
204C22×M16—91no—
205C2×C22⋊Q8—91no—
206C2×#121—91no—
207#468—32no—
208#468—92no—
209#468—92no—
210#468—92no—
211C2×#107—91no—
212C2×#103—91no—
213C2×#102—91no—
214C23×D3—32no—
215C23×D3—61no—
216C23×D3—62no—
217C2×C3×Q8—32no—
218C2×C3×D4—32no—
219C2×C3×D4—62no—
220C22×C3×C4—21no—
221C22×C3×C4C22×D312yes—
222C2×#67—22no—
223C2×#67—62no—
224C2×#67—62no—
225C22×Dic3—32no—
226C22×Dic3—61no—
227D3×Q8—64no—
228D3×Q8—64no—
229D3×D4—64no—
230D3×D4—64no—
231D3×D4—128no—
232C2×D12—62no—
233C2×D12C22×D312yes—
234C2×C4×D3—32no—
235C2×C4×D3—32no—
236C2×C4×D3—32no—
237C2×C4×D3—32no—
238C2×C4×D3—61no—
239C2×C4×D3—61no—
240C2×C4×D3—61no—
241C2×C4×D3—61no—
242C2×C4×D3—64no—
243C2×C4×D3—64no—
244C2×C4×D3—64no—
245C2×C4×D3—64no—
246C2×C4×D3—62no—
247C2×C4×D3—62no—
248C2×C4×D3—62no—
249C2×Dic6—62no—
250C2×Dic6C22×D312yes—
251C3×SD16—64no—
252C3×M16—24no—
253C3×M16—24no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8C22×D312yes—
257C2×C3×C8C22×D312yes—
258C3×C4⋊C4—32no—
259C3×C4⋊C4—32no—
260C3×C4⋊C4—62no—
261C3×C22⋊C4—62no—
262C3×C22⋊C4—62no—
263#217—22no—
264#217—62no—
265#215—64no—
266#214—64no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—62no—
269(C2×D3)⋊C4—62no—
270(C2×D3)⋊C4—62no—
271C4⋊Dic3—62no—
272C4⋊Dic3C22×D312yes—
273C4⋊Dic3C22×D312yes—
274Dic3⋊C4—22no—
275Dic3⋊C4—62no—
276Dic3⋊C4—62no—
277Dic3⋊C4—62no—
278#208—64no—
279#208—64no—
280C2×C3⋊C8—32no—
281C2×C3⋊C8—32no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284#204—24no—
285ΓL(1,𝔽25)—64no—
286ΓL(1,𝔽25)—64no—
287ΓL(1,𝔽25)—64no—
288ΓL(1,𝔽25)—64no—
289D3×C8—38no—
290D3×C8—64no—
291D3×C8—64no—
292D3×C8—64no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32C22×D411yescharacteristic
297C2×C3⋊D3—21no—
298C2×C3⋊D3—21no—
299C2×C3⋊D3—21no—
300C2×C3⋊D3—44no—
301C2×C3⋊D3#12112yes—
302C2×C3⋊D3AGL(1,ℤ/8)12yes—
303C2×C3⋊D3#13712yes—
304C2×C3×D3—42no—
305C2×C3×D3—42no—
306C2×C3×D3—42no—
307D32—84no—
308C32×C4—21no—
309C32×C4C22×D411yescharacteristic
310C32×C4C2×#4111yescharacteristic
311C3⋊Dic3—21no—
312C3⋊Dic3#12112yes—
313C3×Dic3—42no—
314C3×Dic3—42no—
315C3×Dic3—42no—
316C22×Q8—91no—
317C22×D4—91no—
318C22×D4—181no—
319C23×C4—91no—
320C2×SD16—32no—
321C2×SD16—92no—
322C2×SD16—184no—
323C2×M16—91no—
324C2×M16—91no—
325C2×M16—91no—
326C2×M16—184no—
327C2×M16D3211yescharacteristic, complemented
328C22×C8—91no—
329C22×C8—91no—
330C22⋊Q8—32no—
331C22⋊Q8—92no—
332C22⋊Q8—181no—
333C22⋊Q8—181no—
334#121—32no—
335#121—92no—
336#121—181no—
337#121—181no—
338C2×C4⋊C4—91no—
339C2×C4⋊C4—91no—
340C2×C4⋊C4—181no—
341C2×C22⋊C4—181no—
342C2×C22⋊C4—181no—
343#107—32no—
344#107—92no—
345#103—32no—
346#103—92no—
347#102—32no—
348#102—92no—
349C23×C3—21no—
350C23×C3—62no—
351C23×C3D3×D412yes—
352C22×D3—22no—
353C22×D3—32no—
354C22×D3—32no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—64no—
358C22×D3—64no—
359C22×D3—64no—
360C22×D3—64no—
361C22×D3—122no—
362C22×D3—122no—
363C22×D3—124no—
364C22×D3—124no—
365C22×D3—124no—
366C3×Q8—62no—
367C3×Q8—62no—
368C3×D4—62no—
369C3×D4—62no—
370C3×D4—124no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4D3×D412yes—
381C2×C3×C4C23×D312yes—
382#67—44no—
383#67—124no—
384#67—124no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—64no—
393C2×Dic3—64no—
394D12—22no—
395D12—22no—
396D12—124no—
397C4×D3—34no—
398C4×D3—34no—
399C4×D3—62no—
400C4×D3—62no—
401C4×D3—64no—
402C4×D3—64no—
403C4×D3—62no—
404C4×D3—64no—
405C4×D3—64no—
406C4×D3—62no—
407C4×D3—62no—
408C4×D3—62no—
409C4×D3—124no—
410C4×D3—124no—
411C4×D3—124no—
412Dic6—22no—
413Dic6—22no—
414Dic6—124no—
415C3×C8—22no—
416C3×C8—22no—
417C3×C8—22no—
418C3×C8D3×D414yescomplemented
419C3⋊C8—34no—
420C3⋊C8—62no—
421C3⋊C8—62no—
422C3⋊C8—62no—
423C2×C32—41no—
424C2×C32C2×#12111yescharacteristic
425C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
426C2×C32C2×#13711yescharacteristic
427C3⋊D3—41no—
428C3⋊D3#46814yescomplemented
429C3×D3—82no—
430C24—91no—
431C24—181no—
432C2×Q8—32no—
433C2×Q8—181no—
434C2×Q8—182no—
435C2×Q8—182no—
436C2×D4—32no—
437C2×D4—62no—
438C2×D4—182no—
439C2×D4—182no—
440C2×D4—182no—
441C2×D4—181no—
442C2×D4—364no—
443C22×C4—91no—
444C22×C4—91no—
445C22×C4—91no—
446C22×C4—91no—
447C22×C4—91no—
448C22×C4—182no—
449C22×C4—182no—
450C22×C4—181no—
451C22×C4—181no—
452C22×C4—181no—
453C22×C4C2×D3211yescharacteristic
454SD16—64no—
455SD16—184no—
456M16—22no—
457M16—182no—
458M16—182no—
459M16—182no—
460C2×C8—91no—
461C2×C8—94no—
462C2×C8—91no—
463C2×C8—182no—
464C2×C8C2×D3211yescharacteristic
465C2×C8C2×D3211yescharacteristic
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—62no—
469C4⋊C4—181no—
470C4⋊C4—181no—
471C4⋊C4—182no—
472C22⋊C4—62no—
473C22⋊C4—62no—
474C22⋊C4—182no—
475C22⋊C4—182no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3—122no—
486C22×C3C2×D3×D412yes—
487C2×D3—34no—
488C2×D3—34no—
489C2×D3—34no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—62no—
500C2×D3—121no—
501C2×D3—121no—
502C2×D3—128no—
503C2×D3—128no—
504C2×D3—248no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—122no—
510C3×C4—122no—
511C3×C4—122no—
512C3×C4C2×D3×D412yes—
513C3×C4D3×#4112yes—
514Dic3—34no—
515Dic3—42no—
516Dic3—42no—
517Dic3—42no—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521C32C2×#46811yescharacteristic, complemented, Sylow
522C23—62no—
523C23—91no—
524C23—91no—
525C23—182no—
526C23—182no—
527C23—182no—
528C23—362no—
529C23—362no—
530C23—362no—
531C23#380211yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—362no—
535D4—62no—
536D4—62no—
537D4—124no—
538D4—362no—
539D4—364no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—91no—
546C2×C4—92no—
547C2×C4—92no—
548C2×C4—91no—
549C2×C4—182no—
550C2×C4—182no—
551C2×C4—182no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—362no—
555C2×C4—362no—
556C2×C4—362no—
557C2×C4#380211yescharacteristic
558C2×C4C22×D3211yescharacteristic
559C8—21no—
560C8—92no—
561C8—181no—
562C8#381112yescomplemented
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—42no—
567C2×C3—42no—
568C2×C3—242no—
569C2×C3D3×#12112yes—
570C2×C3D3×AGL(1,ℤ/8)12yes—
571C2×C3D3×#13712yes—
572D3—38no—
573D3—64no—
574D3—82no—
575D3—122no—
576D3—122no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—92no—
581C22—92no—
582C22—92no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—181no—
587C22—181no—
588C22—181no—
589C22—364no—
590C22—724no—
591C22C2×#380211yescharacteristic
592C4—21no—
593C4—92no—
594C4—122no—
595C4—122no—
596C4—122no—
597C4—181no—
598C4#6430611yescharacteristic
599C4C2×#381111yescharacteristic
600C3—21no—
601C3D3×#46812yescomplemented
602C2—41no—
603C2—94no—
604C2—242no—
605C2—361no—
606C2#1059471611yescharacteristic
607C2#1059471711yescharacteristic
608C2#1059508111yescharacteristic
subgroupG/H
centerC22C2×#3802
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radical#10595081C2
hypercenterC2×M16D32
hypocenterC32C2×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×M16×C3⋊D3C2semidirect
3. #10594925C2semidirect
4. C2×#63796C2semidirect
5. C3×#5134C2semidirect
6. #10591180C2semidirect
7. #10590940C2semidirect
8. #10590845C2semidirect
9. #10590842C2semidirect
10. #10590687C2semidirect
11. #10590672C2semidirect
17. C2×C32×M16C22semidirect
18. C2×#3748C22semidirect
20. C2×#3743C22semidirect
21. C2×#3743C22semidirect
25. C3×#924C22semidirect
26. C2×C3×#204C22semidirect
36. #63871C22semidirect
38. C2×#3715C22semidirect
40. C3×#821C22semidirect
41. C3×#818C22semidirect
42. C3×#816C22semidirect
43. #63576C22semidirect
44. #63565C22semidirect
45. #63562C22semidirect
53. #5134D3semidirect
95. C8×C3⋊D3D4semidirect
116. C2×C3×M16C2×D3semidirect
125. #924C2×D3semidirect
131. C2×#204C2×D3semidirect
155. #821C2×D3semidirect
156. #818C2×D3semidirect
157. #816C2×D3semidirect
200. C32×C8C2×D4semidirect
202. C3⋊(C3⋊C8)C2×D4semidirect
327. C2×M16D32semidirect
418. C3×C8D3×D4semidirect
428. C3⋊D3#468semidirect
521. C32C2×#468semidirect
562. C8#3811semidirect
601. C3D3×#468semidirect

Maximal subgroups

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

subgroupindex
C2×M16×C3⋊D32
#105950812
#105949252
C2×#637962
C3×#51342
#105911802
#105909402
#105908452
#105908422
#105906872
#105906722
D3×#4683