Finite Group Databasebeta

← #11858097

Subgroups — #11858097

Order 1152.

1531 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64309C211yesmaximal, characteristic
2C2×#63805C211yesmaximal, characteristic, complemented
3#10594509C211yesmaximal, characteristic, complemented
4#10592471C211yesmaximal, characteristic, complemented
5C3×#4771C212yesmaximal, complemented
6#10591420C212yesmaximal, complemented
7#10591418C211yesmaximal, characteristic, complemented
8#10590948C212yesmaximal, complemented
9#10590672C211yesmaximal, characteristic, complemented
10#10590602C211yesmaximal, characteristic, complemented
11D3×#810C212yesmaximal, complemented
12#78883—32nomaximal
13C2×#3830C2211yescharacteristic
14C2×C3×#237C2212yes—
15C22×#627—21no—
16C2×#3805—21no—
17C2×#3804—22no—
18#64309—24no—
19#64309C418yescomplemented
20C2×#3798C2211yescharacteristic
21C2×#3797C2211yescharacteristic
22C2×D3×Dic6C2212yes—
23C2×C8×C3⋊D3C2211yescharacteristic
24#64101C2211yescharacteristic
25C2×C3×ΓL(1,𝔽25)C2212yes—
26C3×D3×C4⋊C4C2212yes—
27#63904—22no—
28C4×#625—21no—
29#63900C2211yescharacteristic
30#63841C2212yes—
31#63838—21no—
32C2×#3712C2211yescharacteristic
33C2×D3×C3⋊C8C2212yes—
34#63805—24no—
35C32×#102C2211yescharacteristic
36#63659C2211yescharacteristic
37#63643C2211yescharacteristic
38#63637C2211yescharacteristic
39C3×#832C2212yes—
40C3×#816C2212yes—
41C3×#810C2212yes—
42#63567C2212yes—
43#63566C2212yes—
44#63565C2212yes—
45C2×#1010—32no—
46C2×#907—32no—
47#5504—32no—
48C3×#417—32no—
49#5233—32no—
50#5120—32no—
51C2×#810—32no—
52#4802—32no—
53#4802—32no—
54D3×#103—32no—
55D3×#103—32no—
56#4772—32no—
57#4772—32no—
58#4772—61no—
59#4771—32no—
60#4771D312yescomplemented
61C2×C32×D4C2311yescharacteristic
62C2×#638—21no—
63C22×C3⋊Dic3—21no—
64#3830—22no—
65#3830C2×C414yescomplemented
66C2×C4×C3⋊D3C2311yescharacteristic
67C2×#634C2311yescharacteristic
68C2×C3×#67—22no—
69C22×C3×Dic3—22no—
70C3×#237—24no—
71C3×#237C2×C418yescomplemented
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3C2312yes—
74C2×C3×Dic6C2312yes—
75C2×#627—21no—
76C2×#627—22no—
77C2×#627—28no—
78C2×#627D411yescharacteristic
79C2×#626D412yes—
80C2×#625D411yescharacteristic
81C2×#624—21no—
82C2×#624—21no—
83#3805—28no—
84#3804—216no—
85C2×D3×Dic3—22no—
86#3798C2×C414yes—
87#3798D414yes—
88#3797—22no—
89#3797C2×C414yescomplemented
90D3×Dic6—24no—
91D3×Dic6C2×C418yescomplemented
92C2×C32×C8C2311yescharacteristic
93C32×C4⋊C4C2311yescharacteristic
94#3752—21no—
95C4⋊(C3⋊Dic3)C2311yescharacteristic
96C4×C3⋊Dic3—21no—
97C2×C3⋊(C3⋊C8)C2311yescharacteristic
98C8×C3⋊D3—22no—
99C3×C4⋊Dic3C2312yes—
100C3×Dic3⋊C4—22no—
101C2×C3×C3⋊C8C2312yes—
102C3×ΓL(1,𝔽25)—24no—
103#3724—21no—
104(C2×D3)⋊Dic3—22no—
105Dic32—22no—
106#3712—22no—
107D3×C3⋊C8—24no—
108#2898—91noSylow
109C2×C3×#41—32no—
110#1010—316no—
111#1010—68no—
112C2×D3×Q8—32no—
113C2×D3×Q8—32no—
114C2×#237—32no—
115C2×#237—61no—
116C2×#237—62no—
117C2×#237—62no—
118C2×#237C2×D312yes—
119C2×#235—32no—
120C2×#235—62no—
121C22×Dic6—32no—
122C22×Dic6—62no—
123C2×C3×M16—32no—
124C2×C3×C4⋊C4—32no—
125C4×#67—62no—
126C4×#67—62no—
127#926—32no—
128C22×C3⋊C8—32no—
129#907—68no—
130C2×ΓL(1,𝔽25)—32no—
131C2×ΓL(1,𝔽25)C2×D312yes—
132C2×D3×C8—32no—
133C2×D3×C8—32no—
134C2×D3×C8—61no—
135#893—62no—
136#892—32no—
137#892—32no—
138#892—61no—
139D3×C4⋊C4—32no—
140D3×C4⋊C4C2×D312yes—
141#887—62no—
142C3×#103—32no—
143C3×#103—32no—
144C3×#102—21no—
145C3×#102—32no—
146C3×#102C2×D312yes—
147#835—32no—
148#835—32no—
149#832—32no—
150#832—61no—
151#832C2×D312yes—
152#816—32no—
153#816—32no—
154#816—32no—
155#816—61no—
156#816C2×D312yes—
157#810—32no—
158#810—32no—
159#810—32no—
160#810—61no—
161#810C2×D312yes—
162C23×C32—21no—
163C22×C3⋊D3C2×D411yescharacteristic
164C22×C3×D3C2×D412yes—
165C32×D4—21no—
166C32×D4C22×C412yes—
167C2×C32×C4—21no—
168C2×C32×C4C2411yescharacteristic
169#638—24no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—24no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3C2×D411yescharacteristic
174C4×C3⋊D3C22×C412yes—
175C4×C3⋊D3C2×D412yes—
176#634—21no—
177#634C22×C412yes—
178C3×#67—28no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—28no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3C2×D412yes—
183C3×C4×D3—82no—
184C3×C4×D3C22×C414yes—
185C3×C4×D3C2×D414yes—
186C3×Dic6—22no—
187C3×Dic6C22×C414yes—
188#627—24no—
189#627—28no—
190#627C22⋊C414yes—
191#626C22⋊C418yes—
192#625C22⋊C414yes—
193#624—24no—
194#624—81no—
195D3×Dic3—28no—
196C32×C8—21no—
197C3⋊(C3⋊C8)—21no—
198C3×C3⋊C8—22no—
199C2×2−1+4—91no—
200C2×#131—91no—
201#520—91no—
202#419—91no—
203#419—92no—
204#417—32no—
205#417—91no—
206#416—92no—
207#416—91no—
208C2×#103—91no—
209C2×#103—92no—
210C3×#41—38no—
211C3×#41—64no—
212C2×C3×Q8—32no—
213C2×C3×D4—21no—
214C2×C3×D4—62no—
215C2×C3×D4C22×D312yes—
216C22×C3×C4—32no—
217C22×C3×C4—62no—
218C22×C3×C4—62no—
219C2×#67—22no—
220C2×#67—32no—
221C2×#67—32no—
222C2×#67—62no—
223C2×#67—121no—
224C22×Dic3—22no—
225C22×Dic3—62no—
226C22×Dic3—62no—
227C22×Dic3—121no—
228D3×Q8—38no—
229D3×Q8—38no—
230D3×Q8—64no—
231D3×Q8—64no—
232#237—24no—
233#237—38no—
234#237—64no—
235#237—64no—
236#237—616no—
237#237—616no—
238#237—122no—
239#237C4×D318yescomplemented
240#235—38no—
241#235—38no—
242#235—616no—
243C2×D12—32no—
244C2×C4×D3—22no—
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×C4×D3—62no—
248C2×C4×D3—62no—
249C2×C4×D3—62no—
250C2×C4×D3C22×D312yes—
251C2×Dic6—32no—
252C2×Dic6—32no—
253C2×Dic6—38no—
254C2×Dic6—61no—
255C2×Dic6—64no—
256C2×Dic6—616no—
257C2×Dic6—62no—
258C2×Dic6—62no—
259C2×Dic6—62no—
260C2×Dic6C22×D312yes—
261C3×M16—64no—
262C2×C3×C8—21no—
263C2×C3×C8—32no—
264C2×C3×C8C22×D312yes—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—32no—
267C3×C4⋊C4—62no—
268C3×C4⋊C4C22×D312yes—
269#217—62no—
270(C2×D3)⋊C4—62no—
271(C2×D3)⋊C4—62no—
272(C2×D3)⋊C4—121no—
273C4⋊Dic3—32no—
274C4⋊Dic3—61no—
275C4⋊Dic3C22×D312yes—
276Dic3⋊C4—22no—
277Dic3⋊C4—62no—
278C4×Dic3—62no—
279C4×Dic3—62no—
280C4×Dic3—62no—
281C4×Dic3—121no—
282C2×C3⋊C8—32no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8—64no—
285C2×C3⋊C8C22×D312yes—
286ΓL(1,𝔽25)—24no—
287ΓL(1,𝔽25)—64no—
288D3×C8—64no—
289D3×C8—64no—
290D3×C8—122no—
291C22×C32—22no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32C22×D411yescharacteristic
295C2×C3⋊D3—21no—
296C2×C3⋊D3—21no—
297C2×C3⋊D3C2×C22⋊C412yes—
298C2×C3×D3—22no—
299C2×C3×D3—22no—
300C2×C3×D3C2×C22⋊C414yes—
301C32×C4—41no—
302C32×C4C23×C411yescharacteristic
303C32×C4C22×D411yescharacteristic
304C3⋊Dic3—22no—
305C3⋊Dic3—41no—
306C3⋊Dic3C2×C22⋊C412yes—
307C3×Dic3—24no—
308C3×Dic3—42no—
309C3×Dic3C2×C22⋊C414yes—
3102−1+4—98no—
3112−1+4—184no—
312C2×#41—32no—
313C2×#41—91no—
314C2×#41—91no—
315C2×#41—182no—
316C2×#41—181no—
317C22×Q8—92no—
318C22×Q8—91no—
319C22×Q8—181no—
320#131—184no—
321C2×M16—32no—
322C2×M16—91no—
323C22×C8—92no—
324C22×C8—91no—
325C4×Q8—181no—
326C4×D4—182no—
327C4×D4—181no—
328#117—92no—
329#117—91no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—91no—
332#103—32no—
333#103—32no—
334#103—92no—
335#103—92no—
336#103—92no—
337#103—91no—
338#103—91no—
339#102—32no—
340#102—91no—
341#102D3211yescharacteristic
342C23×C3—22no—
343C23×C3—32no—
344C23×C3—41no—
345C22×D3—32no—
346C22×D3—61no—
347C22×D3D3×D412yes—
348C3×Q8—34no—
349C3×Q8—62no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—41no—
353C3×D4—68no—
354C3×D4C2×C4×D314yes—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—32no—
358C2×C3×C4—34no—
359C2×C3×C4—34no—
360C2×C3×C4—41no—
361C2×C3×C4—68no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—242no—
365C2×C3×C4C23×D312yes—
366#67—28no—
367#67—38no—
368#67—38no—
369#67—68no—
370#67—124no—
371C2×Dic3—22no—
372C2×Dic3—28no—
373C2×Dic3—22no—
374C2×Dic3—32no—
375C2×Dic3—61no—
376C2×Dic3—68no—
377C2×Dic3—68no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—124no—
381C2×Dic3—121no—
382C2×Dic3—121no—
383C2×Dic3D3×D412yes—
384D12—38no—
385C4×D3—34no—
386C4×D3—34no—
387C4×D3—62no—
388C4×D3—62no—
389C4×D3—68no—
390C4×D3—68no—
391C4×D3—82no—
392C4×D3—242no—
393C4×D3C2×C4×D314yes—
394C4×D3D3×D414yes—
395Dic6—22no—
396Dic6—38no—
397Dic6—34no—
398Dic6—62no—
399Dic6—68no—
400Dic6—68no—
401Dic6—62no—
402Dic6—68no—
403Dic6—121no—
404Dic6C2×C4×D314yes—
405C3×C8—22no—
406C3×C8—41no—
407C3×C8—62no—
408C3⋊C8—22no—
409C3⋊C8—62no—
410C3⋊C8—121no—
411C2×C32—42no—
412C2×C32C22×C22⋊C411yescharacteristic
413C2×C32#57711yescharacteristic
414C2×C32#57811yescharacteristic
415C3⋊D3—22no—
416C3×D3—24no—
417#41—38no—
418#41—64no—
419#41—94no—
420#41—94no—
421#41—94no—
422#41—1816no—
423#41—182no—
424#41—188no—
425C2×Q8—32no—
426C2×Q8—91no—
427C2×Q8—98no—
428C2×Q8—94no—
429C2×Q8—91no—
430C2×Q8—182no—
431C2×Q8—184no—
432C2×Q8—181no—
433C2×Q8—188no—
434C2×Q8—182no—
435C2×D4—62no—
436C2×D4—92no—
437C2×D4—91no—
438C2×D4—181no—
439C2×D4C2×D3211yescharacteristic
440C22×C4—32no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—91no—
444C22×C4—181no—
445C22×C4—181no—
446C22×C4—182no—
447C22×C4—181no—
448M16—64no—
449M16—182no—
450C2×C8—32no—
451C2×C8—91no—
452C2×C8—182no—
453C2×C8—184no—
454C2×C8C2×D3211yescharacteristic
455C4⋊C4—32no—
456C4⋊C4—62no—
457C4⋊C4—91no—
458C4⋊C4—181no—
459C4⋊C4C2×D3211yescharacteristic
460C22⋊C4—181no—
461C22⋊C4—182no—
462C42—181no—
463C42—182no—
464C22×C3—24no—
465C22×C3—21no—
466C22×C3—34no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—62no—
471C22×C3—62no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3C2×D3×D412yes—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—34no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—121no—
482C2×D3—121no—
483C2×D3D3×C22⋊C414yes—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—34no—
487C3×C4—42no—
488C3×C4—64no—
489C3×C4—81no—
490C3×C4—122no—
491C3×C4C22×C4×D312yes—
492C3×C4C2×D3×D412yes—
493Dic3—24no—
494Dic3—34no—
495Dic3—42no—
496Dic3—62no—
497Dic3—64no—
498Dic3—122no—
499Dic3—122no—
500Dic3—241no—
501Dic3D3×C22⋊C414yes—
502C32#289811yescharacteristic, complemented, Sylow
503C23—21no—
504C23—32no—
505C23—91no—
506Q8—34no—
507Q8—62no—
508Q8—94no—
509Q8—92no—
510Q8—188no—
511Q8—184no—
512Q8—181no—
513D4—21no—
514D4—68no—
515D4—98no—
516D4—94no—
517D4—184no—
518D4C4×D3212yes—
519C2×C4—21no—
520C2×C4—32no—
521C2×C4—34no—
522C2×C4—34no—
523C2×C4—68no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—91no—
527C2×C4—92no—
528C2×C4—92no—
529C2×C4—184no—
530C2×C4—188no—
531C2×C4—184no—
532C2×C4—181no—
533C2×C4—181no—
534C2×C4—242no—
535C2×C4—721no—
536C2×C4C22×D3211yescharacteristic
537C8—21no—
538C8—62no—
539C8—181no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—44no—
544C2×C3—64no—
545C2×C3—82no—
546C2×C3C2×D3×C22⋊C412yes—
547C2×C3#577512yes—
548C2×C3#577912yes—
549D3—24no—
550D3—64no—
551D3—122no—
552C22—22no—
553C22—34no—
554C22—41no—
555C22—41no—
556C22—62no—
557C22—62no—
558C22—92no—
559C22—181no—
560C22—181no—
561C22D32×D411yescharacteristic
562C4—34no—
563C4—41no—
564C4—64no—
565C4—92no—
566C4—122no—
567C4—182no—
568C4—361no—
569C4C2×C4×D3211yescharacteristic
570C4D32×D411yescharacteristic
571C3—21no—
572C3#7888312yescomplemented
573C2—42no—
574C2—64no—
575C2—182no—
576C2D32×C22⋊C411yescharacteristic
577C2#1059473311yescharacteristic
578C2#1059475211yescharacteristic
subgroupG/H
centerC22D32×D4
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC32×#102C22
socleC22×C32C22×D4
prime radicalC2×#64309C2
cosocleC2×C32×C4C24
hypercenter#102D32
hypocenterC32#2898
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#63805C2semidirect
3. #10594509C2semidirect
4. #10592471C2semidirect
5. C3×#4771C2semidirect
6. #10591420C2semidirect
7. #10591418C2semidirect
8. #10590948C2semidirect
9. #10590672C2semidirect
10. #10590602C2semidirect
11. D3×#810C2semidirect
19. #64309C4semidirect
60. #4771D3semidirect
65. #3830C2×C4semidirect
71. C3×#237C2×C4semidirect
89. #3797C2×C4semidirect
91. D3×Dic6C2×C4semidirect
239. #237C4×D3semidirect
502. C32#2898semidirect
572. C3#78883semidirect

Maximal subgroups

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

subgroupindex
C2×#643092
C2×#638052
#105945092
#105924712
C3×#47712
#105914202
#105914182
#105909482
#105906722
#105906022
D3×#8102
#788833