Finite Group Databasebeta

← #11867805

Subgroups — #11867805

Order 1152.

1003 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic, complemented
2C2×C3×#907C211yesmaximal, characteristic, complemented
3#10595044C211yesmaximal, characteristic
4#10594855C211yesmaximal, characteristic, complemented
5#10592820C211yesmaximal, characteristic, complemented
6C2×#63566C211yesmaximal, characteristic, complemented
7#10591084C211yesmaximal, characteristic, complemented
8#10590854C212yesmaximal, complemented
9#10590851C212yesmaximal, complemented
10#10590833C212yesmaximal, complemented
11#10590829C212yesmaximal, complemented
12#83986—31nomaximal
13#83808—31nomaximal
14C2×C3×#235C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C22×C3×Dic6C2211yescharacteristic
17C2×C32×M16C2211yescharacteristic
18#64135C2211yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C3×#909—24no—
22C3×#907—24no—
23C2×C3×Dic12C2212yes—
24C2×C3×#204C2212yes—
25C2×C3×ΓL(1,𝔽25)C2212yes—
26C2×C3×D3×C8C2212yes—
27#63980—22no—
28#63961—22no—
29#63908—21no—
30#63908C2212yes—
31#63873—21no—
32#63863—21no—
33#63856—21no—
34#63856C2212yes—
35#63856C2212yes—
36#63856C2212yes—
37#63645C2212yes—
38#63644C2212yes—
39#63566C2211yescharacteristic
40#63566C2211yescharacteristic
41#63566C2212yes—
42#63566C2212yes—
43#63563C2211yescharacteristic
44#63563C2211yescharacteristic
45C2×C3×#137—31no—
46C2×C3×#131—31no—
47#5833—31no—
48#5833—31no—
49#5825—31no—
50#5816—31no—
51C2×#909D311yescharacteristic, complemented
52C2×#907D311yescharacteristic, complemented
53#5233—31no—
54C2×#835—31no—
55#5189—32no—
56#5188—32no—
57#5174—32no—
58#5171—32no—
59#5137—32no—
60#5134—32no—
61#5121—31no—
62#5121—31no—
63#5121—61no—
64#5116—32no—
65#5114—32no—
66#5113—31no—
67C2×#816—31no—
68C22×C32×C4C2311yescharacteristic
69C22×C3⋊Dic3—21no—
70C2×C3×#67—21no—
71C2×C3×#67D412yes—
72C22×C3×Dic3—21no—
73C3×#235—22no—
74C3×#235—22no—
75C3×#235—22no—
76C3×#235—22no—
77C2×C3×D12D411yescharacteristic
78C2×C3×D12C2311yescharacteristic
79C2×C3×C4×D3—21no—
80C2×C3×C4×D3C2312yes—
81C2×C3×Dic6—22no—
82C2×C3×Dic6—22no—
83C2×C3×Dic6—21no—
84C2×C3×Dic6D411yescharacteristic
85C2×C3×Dic6C2312yes—
86C2×C3×Dic6C2311yescharacteristic
87C2×#625—22no—
88C32×M16D414yescomplemented
89C2×C32×C8C2312yes—
90#3757—21no—
91C4⋊(C3⋊Dic3)—21no—
92C4⋊(C3⋊Dic3)C2311yescharacteristic
93C4⋊(C3⋊Dic3)C2311yescharacteristic
94C4⋊(C3⋊Dic3)C2312yes—
95C4×C3⋊Dic3—21no—
96C2×C3×C3⋊C8—22no—
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2311yescharacteristic
99C3×Dic12—24no—
100C3×#204—24no—
101C3×ΓL(1,𝔽25)—24no—
102C3×D3×C8—24no—
103#3724—22no—
104#3724—22no—
105#3724—22no—
106(C2×D3)⋊Dic3—22no—
107(C2×D3)⋊Dic3—22no—
108(C2×D3)⋊Dic3—22no—
109(C2×D3)⋊Dic3—22no—
110#3063—91noSylow
111C2×C3×#41—31no—
112C2×C3×#41—31no—
113C22×C3×Q8—31no—
114C2×#235C2×D311yescharacteristic
115C2×#235C2×D311yescharacteristic
116C22×Dic6C2×D311yescharacteristic
117C3×#137—64no—
118C2×C3×Q16—32no—
119C2×C3×SD16—32no—
120C3×#131—64no—
121C2×C3×M16—21no—
122C2×C3×M16—32no—
123C2×C3×M16C2×D311yescharacteristic
124C2×C3×M16C2×D311yescharacteristic
125C22×C3×C8—31no—
126C22×C3×C8—32no—
127#947—61no—
128#946—32no—
129#946—32no—
130#946—32no—
131#946—61no—
132#944—61no—
133#944—61no—
134#939—62no—
135#939—62no—
136#938—32no—
137#938—61no—
138#936—61no—
139#935—62no—
140#930—32no—
141#930—61no—
142#929—62no—
143#926—31no—
144#926—31no—
145#926—61no—
146C2×C4⋊Dic3—31no—
147C2×C4⋊Dic3—31no—
148C2×C4⋊Dic3—61no—
149#924—32no—
150#924—32no—
151#924—32no—
152#924—61no—
153C22×C3⋊C8C2×D311yescharacteristic
154#909—24no—
155#907—24no—
156C2×Dic12C2×D312yes—
157C2×#204C2×D312yes—
158C2×ΓL(1,𝔽25)C2×D312yes—
159C2×D3×C8C2×D312yes—
160#835—32no—
161#835—32no—
162#835—31no—
163#835—31no—
164#832—31no—
165#832—31no—
166#821—31no—
167#821—31no—
168#818—32no—
169#818—32no—
170#818—62no—
171#817—32no—
172#817—32no—
173#817—62no—
174#816—32no—
175#816—31no—
176#816—32no—
177#816—31no—
178C23×C32C2×D411yescharacteristic
179C22×C3×D3—21no—
180C22×C3×D3C2×D412yes—
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2×D412yes—
183C2×C32×C4#4112yes—
184C2×C32×C4C2411yescharacteristic
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—81no—
189C3×#67—24no—
190C3×#67—42no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3C2×D412yes—
195C3×D12—22no—
196C3×D12—21no—
197C3×D12—21no—
198C3×C4×D3—22no—
199C3×C4×D3—22no—
200C3×C4×D3—42no—
201C3×Dic6—22no—
202C3×Dic6—22no—
203C3×Dic6—21no—
204C3×Dic6—22no—
205C3×Dic6—21no—
206C3×Dic6—42no—
207#625—82no—
208C32×C8C2×D414yes—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C2×#137—31no—
212C2×#131—31no—
213#537—91no—
214#536—91no—
215#470—92no—
216#468—92no—
217#467—92no—
218#465—92no—
219#428—91no—
220#416—91no—
221C2×#103—91no—
222C3×#41—62no—
223C3×#41—62no—
224C3×#41—62no—
225C3×#41—62no—
226C2×C3×Q8—31no—
227C2×C3×Q8—32no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—62no—
231C2×C3×Q8—62no—
232C2×C3×D4—32no—
233C2×C3×D4—31no—
234C2×C3×D4—31no—
235C2×C3×D4—61no—
236C22×C3×C4—21no—
237C22×C3×C4—32no—
238C22×C3×C4—61no—
239C22×C3×C4—61no—
240C22×C3×C4C22×D311yescharacteristic
241C22×C3×C4C22×D311yescharacteristic
242C2×#67—21no—
243C2×#67—62no—
244C2×#67—62no—
245C2×#67D1212yes—
246C22×Dic3—21no—
247C22×Dic3—61no—
248C22×Dic3—61no—
249C22×Dic3—121no—
250#235—22no—
251#235—22no—
252#235—22no—
253#235—22no—
254C2×D12D1211yescharacteristic
255C2×D12C22×D311yescharacteristic
256C2×C4×D3—21no—
257C2×C4×D3C22×D312yes—
258C2×Dic6—22no—
259C2×Dic6—22no—
260C2×Dic6—21no—
261C2×Dic6D1211yescharacteristic
262C2×Dic6C22×D312yes—
263C2×Dic6C22×D311yescharacteristic
264C3×Q16—64no—
265C3×SD16—64no—
266C3×M16—24no—
267C3×M16—64no—
268C3×M16#6714yescomplemented
269C3×M16#6714yescomplemented
270C2×C3×C8—22no—
271C2×C3×C8—31no—
272C2×C3×C8—31no—
273C2×C3×C8—62no—
274C2×C3×C8—64no—
275C2×C3×C8C22×D312yes—
276C2×C3×C8C22×D312yes—
277#217—62no—
278#217—62no—
279#217—62no—
280#217—61no—
281#217—61no—
282#217—62no—
283#217—121no—
284(C2×D3)⋊C4—62no—
285(C2×D3)⋊C4—62no—
286(C2×D3)⋊C4—62no—
287(C2×D3)⋊C4—62no—
288C4⋊Dic3—32no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292C4⋊Dic3—32no—
293C4⋊Dic3—31no—
294C4⋊Dic3—61no—
295C4⋊Dic3—62no—
296C4⋊Dic3—61no—
297C4⋊Dic3—61no—
298C4⋊Dic3—61no—
299C4⋊Dic3—121no—
300Dic3⋊C4—62no—
301Dic3⋊C4—62no—
302Dic3⋊C4—62no—
303Dic3⋊C4—62no—
304Dic3⋊C4—62no—
305Dic3⋊C4—62no—
306C4×Dic3—61no—
307C4×Dic3—61no—
308C4×Dic3—121no—
309C2×C3⋊C8—22no—
310C2×C3⋊C8C22×D311yescharacteristic
311C2×C3⋊C8C22×D311yescharacteristic
312Dic12—24no—
313#204—24no—
314ΓL(1,𝔽25)—24no—
315D3×C8—24no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32#12112yes—
319C22×C32C22×D411yescharacteristic
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—22no—
323C2×C3×D3—41no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C32×C4#12112yes—
327C32×C4C22×D411yescharacteristic
328C32×C4C2×#4111yescharacteristic
329C3⋊Dic3—42no—
330C3⋊Dic3—41no—
331C3⋊Dic3—41no—
332C3×Dic3—22no—
333C3×Dic3—42no—
334C3×Dic3—41no—
335C2×#41—31no—
336C2×#41—31no—
337C22×Q8—31no—
338#137—64no—
339C2×Q16—32no—
340C2×SD16—32no—
341#131—64no—
342C2×M16—32no—
343C2×M16D3211yescharacteristic
344C22×C8—31no—
345C22×C8—32no—
346C4⋊Q8—181no—
347#124—181no—
348#123—182no—
349C22⋊Q8—92no—
350C22⋊Q8—92no—
351C22⋊Q8—92no—
352C22⋊Q8—181no—
353#121—92no—
354#121—181no—
355#121—182no—
356#117—91no—
357C2×C4⋊C4—91no—
358#107—92no—
359#106—92no—
360#103—92no—
361#103—92no—
362#103—91no—
363#103—91no—
364#102—91no—
365#102—91no—
366C23×C3—21no—
367C23×C3—32no—
368C23×C3—61no—
369C23×C3C2×D1211yescharacteristic
370C23×C3D3×D411yescharacteristic
371C22×D3—21no—
372C22×D3C2×D1212yes—
373C3×Q8—62no—
374C3×Q8—62no—
375C3×Q8—61no—
376C3×Q8—62no—
377C3×Q8—61no—
378C3×Q8—122no—
379C3×D4—64no—
380C3×D4—62no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—122no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—32no—
389C2×C3×C4—62no—
390C2×C3×C4—62no—
391C2×C3×C4—61no—
392C2×C3×C4—62no—
393C2×C3×C4—122no—
394C2×C3×C4—122no—
395C2×C3×C4C2×D1211yescharacteristic
396C2×C3×C4#23512yes—
397C2×C3×C4D3×D411yescharacteristic
398C2×C3×C4#23712yes—
399C2×C3×C4C2×#6712yes—
400C2×C3×C4C2×#6712yes—
401C2×C3×C4C23×D311yescharacteristic
402C2×C3×C4C23×D311yescharacteristic
403#67—24no—
404#67—42no—
405#67—242no—
406#67—242no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—42no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—62no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—62no—
416C2×Dic3—121no—
417C2×Dic3—121no—
418C2×Dic3—122no—
419C2×Dic3—241no—
420C2×Dic3—241no—
421C2×Dic3—242no—
422C2×Dic3C2×D1212yes—
423D12—22no—
424D12—21no—
425D12—21no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—42no—
429Dic6—22no—
430Dic6—22no—
431Dic6—21no—
432Dic6—22no—
433Dic6—21no—
434Dic6—42no—
435C3×C8—24no—
436C3×C8—61no—
437C3×C8—61no—
438C3×C8C2×#6714yes—
439C3×C8C2×#6714yes—
440C3⋊C8—21no—
441C3⋊C8—21no—
442C2×C32—22no—
443C2×C32C2×#12111yescharacteristic
444C2×C32#57711yescharacteristic
445C2×C32#57811yescharacteristic
446C3×D3—42no—
447C3×D3—81no—
448#41—62no—
449#41—62no—
450#41—62no—
451#41—62no—
452C2×Q8—31no—
453C2×Q8—32no—
454C2×Q8—31no—
455C2×Q8—61no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×D4—32no—
459C2×D4—31no—
460C2×D4—31no—
461C2×D4—61no—
462C2×D4—182no—
463C22×C4—32no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—181no—
467C22×C4C2×D3211yescharacteristic
468Q16—64no—
469SD16—64no—
470M16—64no—
471M16#62514yescomplemented
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—62no—
475C2×C8—64no—
476C2×C8C2×D3212yes—
477C4⋊C4—91no—
478C4⋊C4—91no—
479C4⋊C4—92no—
480C4⋊C4—182no—
481C4⋊C4—182no—
482C4⋊C4—182no—
483C4⋊C4—181no—
484C22⋊C4—182no—
485C22⋊C4—182no—
486C22⋊C4—182no—
487C22⋊C4—182no—
488C22⋊C4—181no—
489C42—181no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—22no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—62no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—121no—
502C22×C3—121no—
503C22×C3—121no—
504C22×C3#93012yes—
505C22×C3#93812yes—
506C22×C3C22×D1211yescharacteristic
507C22×C3C2×D3×D411yescharacteristic
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—41no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—22no—
517C3×C4—62no—
518C3×C4—121no—
519C3×C4—122no—
520C3×C4#93012yes—
521C3×C4#93812yes—
522C3×C4C2×#23511yescharacteristic
523C3×C4C2×#23711yescharacteristic
524C3×C4C22×#6711yescharacteristic
525C3×C4C22×#6711yescharacteristic
526Dic3—22no—
527Dic3—42no—
528Dic3—41no—
529Dic3—122no—
530Dic3—121no—
531Dic3—121no—
532Dic3—122no—
533Dic3—121no—
534Dic3—121no—
535Dic3—242no—
536Dic3—241no—
537Dic3—241no—
538C32#306311yescharacteristic, complemented, Sylow
539C23—32no—
540C23—61no—
541C23D3×D1211yescharacteristic
542Q8—62no—
543Q8—62no—
544Q8—61no—
545Q8—62no—
546Q8—61no—
547Q8—122no—
548D4—64no—
549D4—61no—
550D4—61no—
551D4—62no—
552D4—122no—
553D4—722no—
554C2×C4—32no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—181no—
562C2×C4—181no—
563C2×C4—182no—
564C2×C4—721no—
565C2×C4#379512yes—
566C2×C4D3×D1211yescharacteristic
567C2×C4C2×#62512yes—
568C2×C4C22×D3211yescharacteristic
569C8—61no—
570C8—61no—
571C8C2×#62514yes—
572C2×C3—22no—
573C2×C3—22no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—42no—
578C2×C3—122no—
579C2×C3—241no—
580C2×C3#575111yescharacteristic
581C2×C3#575311yescharacteristic
582C2×C3#577811yescharacteristic
583C2×C3#577911yescharacteristic
584C2×C3C2×#93011yescharacteristic
585C2×C3C2×#93811yescharacteristic
586D3—42no—
587D3—81no—
588C22—21no—
589C22—21no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22#6390812yes—
597C22C2×D3×D1211yescharacteristic
598C4—62no—
599C4—121no—
600C4—122no—
601C4—362no—
602C4—361no—
603C4—361no—
604C4#6390812yes—
605C4C2×#379511yescharacteristic
606C4C22×#62511yescharacteristic
607C3—21no—
608C3#8380811yescharacteristic, complemented
609C3#8398611yescharacteristic, complemented
610C2—22no—
611C2—122no—
612C2—241no—
613C2#1059471511yescharacteristic
614C2#1059472111yescharacteristic
615C2C2×#6390811yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radical#10595044C2
hypercenterC2×M16D32
hypocenterC32#3063
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×C3×#909C2semidirect
2. C2×C3×#907C2semidirect
4. #10594855C2semidirect
5. #10592820C2semidirect
6. C2×#63566C2semidirect
7. #10591084C2semidirect
8. #10590854C2semidirect
9. #10590851C2semidirect
10. #10590833C2semidirect
11. #10590829C2semidirect
51. C2×#909D3semidirect
52. C2×#907D3semidirect
88. C32×M16D4semidirect
268. C3×M16#67semidirect
269. C3×M16#67semidirect
471. M16#625semidirect
538. C32#3063semidirect
608. C3#83808semidirect
609. C3#83986semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C2×C3×#9072
#105950442
#105948552
#105928202
C2×#635662
#105910842
#105908542
#105908512
#105908332
#105908292
#839863
#838083