Finite Group Databasebeta

← #11864515

Subgroups — #11864515

Order 1152.

1535 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64297C211yesmaximal, characteristic
2#10596287C211yesmaximal, characteristic, complemented
3C22×C3×#216C211yesmaximal, characteristic, complemented
4C2×C3×#942C211yesmaximal, characteristic, complemented
5#10591656C214yesmaximal, complemented
6#10591654C214yesmaximal, complemented
7#10591100C211yesmaximal, characteristic, complemented
8#10591084C211yesmaximal, characteristic, complemented
9#10591073C211yesmaximal, characteristic, complemented
10#83983—31nomaximal
11#83884—31nomaximal
12C22×C32×Q8C2211yescharacteristic
13C2×#3828C2211yescharacteristic
14C2×C3×#235C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×#3804—22no—
17C2×#3799—21no—
18C2×#3797C2212yes—
19#64297—24no—
20#64297—24no—
21C2×#3796C2212yes—
22C2×D3×Dic6—21no—
23#64156—21no—
24#64155C2214yes—
25#64153—21no—
26#64138—22no—
27#64135C2211yescharacteristic
28C2×C3×#216—24no—
29C2×C3×#216C2212yes—
30C2×C3×#216C2214yes—
31C3×#942—24no—
32C2×C3×#215C2212yes—
33C2×C3×#208C2211yescharacteristic
34C22×C3×C3⋊C8C2211yescharacteristic
35#63566C2212yes—
36#63566C2214yes—
37#63563C2212yes—
38#63557C2214yes—
39C2×#1010—31no—
40C2×#1007—31no—
41C2×C3×#137—31no—
42C22×C3×Q16—31no—
43#5816—31no—
44#5816—31no—
45#5816—61no—
46C22×#216D311yescharacteristic, complemented
47C2×#942D311yescharacteristic, complemented
48#5233—31no—
49#5225—31no—
50#5187—34no—
51#5186—34no—
52#5186—34no—
53#5173—34no—
54#5129—31no—
55#5120—31no—
56#5113—31no—
57#5110—31no—
58C2×C32×Q8—21no—
59C2×C32×Q8—22no—
60C2×C32×Q8—22no—
61C2×C32×Q8C2312yes—
62C22×C32×C4—21no—
63C22×C32×C4C2311yescharacteristic
64C2×#638D412yes—
65#3828—22no—
66#3828—22no—
67C2×C4⋊(C3⋊D3)D411yescharacteristic
68C2×C4×C3⋊D3C2312yes—
69C2×#634D411yescharacteristic
70C2×C3×#67—21no—
71C22×C3×Dic3—21no—
72C3×#235—22no—
73C3×#235D414yes—
74C2×C3×D12C2311yescharacteristic
75C2×C3×C4×D3—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6—22no—
78C2×C3×Dic6D414yes—
79C2×C3×Dic6C2312yes—
80C2×C3×Dic6C2311yescharacteristic
81C2×#627—22no—
82C2×#626—22no—
83C2×#624—22no—
84#3804—216no—
85C2×D3×Dic3—22no—
86#3799—28no—
87#3797—24no—
88#3797—24no—
89#3796—24no—
90#3796—24no—
91D3×Dic6—28no—
92#3757—21no—
93#3752—24no—
94C4⋊(C3⋊Dic3)C2312yes—
95#3750—24no—
96C4×C3⋊Dic3—21no—
97C3×#216—24no—
98C3×#216—28no—
99C3×#215—24no—
100C3×#208—22no—
101C2×C3×C3⋊C8—22no—
102C2×C3×C3⋊C8C2312yes—
103C2×C3×C3⋊C8C2312yes—
104#3011—91noSylow
105C2×C3×#41—31no—
106C22×C3×Q8—21no—
107C22×C3×Q8—31no—
108C22×C3×Q8C2×D311yescharacteristic
109C22×C3×Q8C2×D311yescharacteristic
110#1010—64no—
111#1010—64no—
112#1007—64no—
113#1007—64no—
114C2×#239—32no—
115C2×#239—61no—
116C2×D3×Q8—32no—
117C2×D3×Q8—32no—
118C2×D3×Q8—61no—
119C2×#237—32no—
120C2×#237—62no—
121C2×#235—31no—
122C2×#235—31no—
123C2×#235—61no—
124C2×#235—61no—
125C2×#235—62no—
126C2×#235C2×D311yescharacteristic
127C22×Dic6—61no—
128C22×Dic6C2×D311yescharacteristic
129C3×#137—64no—
130C2×C3×Q16—34no—
131C2×C3×Q16—32no—
132C2×C3×Q16—64no—
133C2×C3×SD16—32no—
134C2×C3×M16—31no—
135C22×C3×C8—31no—
136#947—61no—
137#947—61no—
138#947—62no—
139#946—34no—
140#946—34no—
141#946—64no—
142#944—61no—
143#944—61no—
144#944—62no—
145C2×#216—24no—
146C2×#216C2×D314yes—
147C2×#216C2×D312yes—
148#942—24no—
149C2×#215C2×D312yes—
150#929—62no—
151#929—62no—
152#929—64no—
153#926—31no—
154#926—31no—
155#926—61no—
156C2×#208C2×D311yescharacteristic
157C22×C3⋊C8C2×D311yescharacteristic
158#835—32no—
159#835—34no—
160#832—32no—
161#821—32no—
162(C2×D3)⋊C8—34no—
163(C2×D3)⋊C8—34no—
164#816—34no—
165#816—32no—
166C23×C32C2×D411yescharacteristic
167C22×C3⋊D3C2×D412yes—
168C22×C3×D3—21no—
169C32×Q8—22no—
170C32×Q8—22no—
171C32×Q8—42no—
172C2×C32×C4—22no—
173C2×C32×C4—42no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2411yescharacteristic
178#638—24no—
179C2×C3⋊Dic3—22no—
180C2×C3⋊Dic3C2×D412yes—
181C4⋊(C3⋊D3)—22no—
182C4×C3⋊D3—22no—
183C4×C3⋊D3—22no—
184#634—22no—
185C3×#67—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—21no—
189C3×D12—21no—
190C3×D12C2×D412yes—
191C3×C4×D3—24no—
192C3×Dic6—24no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6C2×D414yes—
196C3×Dic6C2×D412yes—
197#627—28no—
198#626—28no—
199#624—28no—
200D3×Dic3—28no—
201C3×C3⋊C8—22no—
202C3×C3⋊C8—22no—
203C2×2−1+4—91no—
204C2×#137—31no—
205C22×Q16—31no—
206#536—91no—
207#452—94no—
208#451—94no—
209#419—91no—
210#416—91no—
211#408—91no—
212C3×#41—34no—
213C3×#41—62no—
214C2×C3×Q8—22no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—22no—
218C2×C3×Q8—22no—
219C2×C3×Q8—22no—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—34no—
223C2×C3×Q8—31no—
224C2×C3×Q8—32no—
225C2×C3×Q8—42no—
226C2×C3×Q8—42no—
227C2×C3×Q8—61no—
228C2×C3×Q8—62no—
229C2×C3×Q8C22×D312yes—
230C2×C3×Q8C22×D312yes—
231C2×C3×D4—31no—
232C2×C3×D4—61no—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—61no—
238C22×C3×C4—61no—
239C22×C3×C4C22×D311yescharacteristic
240C22×C3×C4C22×D311yescharacteristic
241C2×#67—21no—
242C2×#67—32no—
243C2×#67—32no—
244C2×#67—62no—
245C2×#67—62no—
246C22×Dic3—21no—
247C22×Dic3—62no—
248#239—68no—
249#239—64no—
250#239—64no—
251D3×Q8—64no—
252D3×Q8—64no—
253D3×Q8—68no—
254D3×Q8—64no—
255D3×Q8—64no—
256#237—64no—
257#237—616no—
258#237—64no—
259#235—22no—
260#235—68no—
261#235—616no—
262#235—62no—
263#235—62no—
264#235—62no—
265#235—62no—
266#235—122no—
267#235—122no—
268#235#6714yes—
269C2×D12—31no—
270C2×D12—31no—
271C2×D12—61no—
272C2×D12—62no—
273C2×D12C22×D311yescharacteristic
274C2×C4×D3—21no—
275C2×C4×D3—32no—
276C2×C4×D3—32no—
277C2×C4×D3—62no—
278C2×C4×D3—62no—
279C2×C4×D3—62no—
280C2×C4×D3—62no—
281C2×Dic6—21no—
282C2×Dic6—22no—
283C2×Dic6—31no—
284C2×Dic6—31no—
285C2×Dic6—61no—
286C2×Dic6—68no—
287C2×Dic6—62no—
288C2×Dic6—62no—
289C2×Dic6#6714yes—
290C2×Dic6C22×D312yes—
291C2×Dic6C22×D311yescharacteristic
292C3×Q16—68no—
293C3×Q16—64no—
294C3×SD16—64no—
295C3×M16—62no—
296C2×C3×C8—32no—
297C2×C3×C8—32no—
298C2×C3×C8—62no—
299#217—62no—
300#217—61no—
301#217—61no—
302#216—24no—
303#216—28no—
304#215—24no—
305(C2×D3)⋊C4—64no—
306(C2×D3)⋊C4—64no—
307(C2×D3)⋊C4—68no—
308C4⋊Dic3—32no—
309C4⋊Dic3—32no—
310C4⋊Dic3—62no—
311Dic3⋊C4—64no—
312Dic3⋊C4—64no—
313Dic3⋊C4—68no—
314C4×Dic3—62no—
315C4×Dic3—61no—
316C4×Dic3—61no—
317#208—22no—
318C2×C3⋊C8—22no—
319C2×C3⋊C8C22×D312yes—
320C2×C3⋊C8C22×D312yes—
321C22×C32—22no—
322C22×C32#12012yes—
323C22×C32C22×D411yescharacteristic
324C2×C3⋊D3—22no—
325C2×C3⋊D3—24no—
326C2×C3×D3—22no—
327C2×C3×D3—42no—
328C32×C4—42no—
329C32×C4#12012yes—
330C32×C4C22×D411yescharacteristic
331C32×C4C22×D411yescharacteristic
332C3⋊Dic3—22no—
333C3⋊Dic3—42no—
334C3×Dic3—24no—
335C3×Dic3—22no—
3362−1+4—184no—
3372−1+4—184no—
338C2×#41—31no—
339C2×#41—91no—
340C2×#41—92no—
341C2×#41—182no—
342C2×#41—181no—
343C22×Q8—31no—
344C22×Q8—92no—
345C22×Q8—181no—
346C22×Q8D3211yescharacteristic
347#137—64no—
348C2×Q16—34no—
349C2×Q16—32no—
350C2×Q16—64no—
351C2×SD16—32no—
352C2×M16—31no—
353C22×C8—31no—
354C4⋊Q8—181no—
355#124—181no—
356#123—182no—
357C22⋊Q8—94no—
358#117—91no—
359#103—92no—
360#103—94no—
361#102—92no—
362C22⋊C8—94no—
363C23×C3—21no—
364C23×C3—61no—
365C23×C3D3×D411yescharacteristic
366C23×C3D3×D411yescharacteristic
367C22×D3—21no—
368C22×D3—32no—
369C22×D3—32no—
370C22×D3—62no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—32no—
376C3×Q8—34no—
377C3×Q8—42no—
378C3×Q8—42no—
379C3×Q8—44no—
380C3×Q8—42no—
381C3×Q8—42no—
382C3×Q8—64no—
383C3×Q8—61no—
384C3×Q8—62no—
385C3×D4—32no—
386C3×D4—64no—
387C3×D4—61no—
388C2×C3×C4—24no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—44no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—64no—
399C2×C3×C4—61no—
400C2×C3×C4—62no—
401C2×C3×C4—64no—
402C2×C3×C4D3×D412yes—
403C2×C3×C4D3×D412yes—
404C2×C3×C4D3×D411yescharacteristic
405C2×C3×C4D3×D411yescharacteristic
406C2×C3×C4C2×#6712yes—
407C2×C3×C4C2×#6712yes—
408C2×C3×C4C23×D311yescharacteristic
409C2×C3×C4C23×D311yescharacteristic
410#67—24no—
411#67—68no—
412#67—64no—
413#67—64no—
414#67—124no—
415C2×Dic3—24no—
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—32no—
419C2×Dic3—32no—
420C2×Dic3—62no—
421C2×Dic3—68no—
422C2×Dic3—64no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425D12—21no—
426D12—68no—
427D12—62no—
428D12—62no—
429D12—122no—
430D12C2×#6712yes—
431C4×D3—24no—
432C4×D3—68no—
433C4×D3—68no—
434C4×D3—68no—
435C4×D3—62no—
436C4×D3—62no—
437C4×D3—62no—
438C4×D3—62no—
439C4×D3—122no—
440C4×D3—122no—
441Dic6—24no—
442Dic6—22no—
443Dic6—21no—
444Dic6—68no—
445Dic6—68no—
446Dic6—62no—
447Dic6—62no—
448Dic6—122no—
449Dic6C2×#6714yes—
450Dic6C2×#6712yes—
451C3×C8—62no—
452C3×C8—62no—
453C3⋊C8—22no—
454C3⋊C8—22no—
455C2×C32—22no—
456C2×C32C2×#12011yescharacteristic
457C2×C32#57812yes—
458C3⋊D3—42no—
459C3×D3—42no—
460#41—34no—
461#41—62no—
462#41—1816no—
463#41—188no—
464#41—184no—
465#41—184no—
466#41—182no—
467#41—182no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—32no—
472C2×Q8—31no—
473C2×Q8—34no—
474C2×Q8—62no—
475C2×Q8—61no—
476C2×Q8—91no—
477C2×Q8—182no—
478C2×Q8—188no—
479C2×Q8—184no—
480C2×Q8—184no—
481C2×Q8C2×D3212yes—
482C2×D4—31no—
483C2×D4—61no—
484C2×D4—92no—
485C2×D4—91no—
486C2×D4—182no—
487C22×C4—21no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—92no—
491C22×C4—182no—
492C22×C4—182no—
493C22×C4C2×D3211yescharacteristic
494Q16—68no—
495Q16—64no—
496SD16—64no—
497M16—62no—
498C2×C8—32no—
499C2×C8—32no—
500C2×C8—62no—
501C4⋊C4—92no—
502C4⋊C4—184no—
503C22⋊C4—184no—
504C22⋊C4—181no—
505C42—181no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—22no—
510C22×C3—42no—
511C22×C3—62no—
512C22×C3—122no—
513C22×C3#93712yes—
514C22×C3#93712yes—
515C22×C3C2×D3×D411yescharacteristic
516C22×C3C2×D3×D411yescharacteristic
517C2×D3—22no—
518C2×D3—42no—
519C2×D3—64no—
520C2×D3—64no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—122no—
524C2×D3—124no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—22no—
528C3×C4—44no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—62no—
532C3×C4—64no—
533C3×C4#93712yes—
534C3×C4#93712yes—
535C3×C4C2×D3×D411yescharacteristic
536C3×C4C2×D3×D411yescharacteristic
537C3×C4C22×#6711yescharacteristic
538C3×C4C22×#6711yescharacteristic
539Dic3—24no—
540Dic3—22no—
541Dic3—62no—
542Dic3—62no—
543Dic3—122no—
544Dic3—124no—
545Dic3—122no—
546Dic3—122no—
547C32#301111yescharacteristic, complemented, Sylow
548C23—61no—
549C23—92no—
550C23#381111yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—34no—
554Q8—32no—
555Q8—42no—
556Q8—62no—
557Q8—61no—
558Q8—64no—
559Q8—188no—
560Q8—182no—
561D4—32no—
562D4—64no—
563D4—61no—
564D4—188no—
565D4—184no—
566D4—182no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—61no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—92no—
574C2×C4—188no—
575C2×C4—188no—
576C2×C4—182no—
577C2×C4—182no—
578C2×C4—182no—
579C2×C4D3×#6712yes—
580C2×C4D3×#6712yes—
581C2×C4#381111yescharacteristic
582C2×C4C22×D3211yescharacteristic
583C8—62no—
584C8—62no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—42no—
590C2×C3—122no—
591C2×C3#576612yes—
592C2×C3#577912yes—
593C2×C3C2×#93711yescharacteristic
594C2×C3C2×#93711yescharacteristic
595D3—42no—
596D3—122no—
597D3—122no—
598D3—242no—
599C22—22no—
600C22—62no—
601C22—122no—
602C22—184no—
603C22—182no—
604C22#6398312yes—
605C22C2×#381111yescharacteristic
606C4—42no—
607C4—62no—
608C4—64no—
609C4—182no—
610C4—362no—
611C4#6398312yes—
612C4C2×D3×#6711yescharacteristic
613C4C2×D3×#6711yescharacteristic
614C3—21no—
615C3#8388411yescharacteristic, complemented
616C3#8398311yescharacteristic, complemented
617C2—22no—
618C2—122no—
619C2—362no—
620C2#1059533712yes—
621C2C2×#6398311yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC22×C32C22×D4
prime radicalC2×#64297C2
hypercenterC22×Q8D32
hypocenterC32#3011
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #10596287C2semidirect
3. C22×C3×#216C2semidirect
4. C2×C3×#942C2semidirect
5. #10591656C2semidirect
6. #10591654C2semidirect
7. #10591100C2semidirect
8. #10591084C2semidirect
9. #10591073C2semidirect
46. C22×#216D3semidirect
47. C2×#942D3semidirect
547. C32#3011semidirect
615. C3#83884semidirect
616. C3#83983semidirect

Maximal subgroups

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

subgroupindex
C2×#642972
#105962872
C22×C3×#2162
C2×C3×#9422
#105916562
#105916542
#105911002
#105910842
#105910732
#839833
#838843