Finite Group Databasebeta

← #11864378

Subgroups — #11864378

Order 1152.

1599 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64308C211yesmaximal, characteristic
2C2×C3×#909C211yesmaximal, characteristic, complemented
3#10594855C211yesmaximal, characteristic, complemented
4C22×#3717C211yesmaximal, characteristic, complemented
5#10592830C211yesmaximal, characteristic, complemented
6C3×#5129C211yesmaximal, characteristic, complemented
7#10591084C211yesmaximal, characteristic, complemented
8#10590966C214yesmaximal, complemented
9#10590942C214yesmaximal, complemented
10#83982—31nomaximal
11#78671—31nomaximal
12C22×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C3×#235C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic, complemented
16C2×#3811—22no—
17#64308—24no—
18#64308—24no—
19C2×D3×D12—21no—
20C2×D3×D12C2212yes—
21C2×#3799—21no—
22C2×#3799C2212yes—
23C2×C32×M16C2211yescharacteristic, complemented
24#64135C2211yescharacteristic, complemented
25C22×C3×C3⋊C8C2211yescharacteristic, complemented
26C3×#909—24no—
27C2×C3×Dic12C2212yescomplemented
28C2×C3×#204C2212yescomplemented
29#63961—22no—
30#63873—21no—
31#63863—21no—
32#63856C2214yescomplemented
33C2×#3717—24no—
34C2×#3717C2214yescomplemented
35#63648C2214yescomplemented
36C3×(C2×D3)⋊C8C2214yescomplemented
37#63566C2212yescomplemented
38#63563C2212yescomplemented
39C2×#1009—31no—
40C2×#1009—31no—
41C2×C3×#137—31no—
42#5825—31no—
43#5816—31no—
44C22×#215—31no—
45C2×#909D311yescharacteristic, complemented
46C22×#204—31no—
47C3×#408—31no—
48#5233—31no—
49#5187—34no—
50#5169—34no—
51#5131—31no—
52#5131—31no—
53#5131—61no—
54#5129D311yescharacteristic, complemented
55#5113—31no—
56#4733—34no—
57#4731—34no—
58C23×C3⋊D3—21no—
59C22×D32—22no—
60C22×C32×C4C2311yescharacteristic
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4⋊(C3⋊D3)—22no—
63C2×C4⋊(C3⋊D3)D414yes—
64C2×C4⋊(C3⋊D3)C2312yes—
65C2×C3×#67—21no—
66C2×C3×#67D412yes—
67C22×C3×Dic3—21no—
68C3×#235—22no—
69C3×#235—22no—
70C3×#235—22no—
71C3×#235D414yes—
72C2×C3×D12D411yescharacteristic
73C2×C3×D12C2311yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×C4×D3C2312yes—
76C2×C3×Dic6—22no—
77C2×C3×Dic6—22no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6D411yescharacteristic
80C2×C3×Dic6C2312yes—
81C2×C3×Dic6C2311yescharacteristic
82#3811—216no—
83C2×#626—22no—
84C2×#626—22no—
85C2×#624—22no—
86D3×D12—28no—
87D3×D12—24no—
88D3×D12—24no—
89#3799—28no—
90#3799—24no—
91#3799—24no—
92C32×M16—22no—
93C2×C32×C8C2312yes—
94#3757—21no—
95C4⋊(C3⋊Dic3)C2312yes—
96C4×C3⋊Dic3—21no—
97C2×C3×C3⋊C8—22no—
98C2×C3×C3⋊C8C2312yes—
99C3×Dic12—24no—
100C3×#204—24no—
101#3724—24no—
102(C2×D3)⋊Dic3—24no—
103#3717—28no—
104#3009—91noSylow
105C2×C3×#41—31no—
106C2×C3×#41—31no—
107C22×C3×Q8—31no—
108#1009—64no—
109#1009—64no—
110#1009—64no—
111#1009—64no—
112C2×#239—32no—
113C2×#239—61no—
114C2×D3×D4—32no—
115C2×D3×D4—61no—
116C2×D3×D4—62no—
117C2×D3×D4—62no—
118C2×#235—32no—
119C2×#235—61no—
120C2×#235C2×D311yescharacteristic
121C2×#235C2×D311yescharacteristic
122C22×D12—31no—
123C22×D12—31no—
124C22×D12—32no—
125C22×D12—61no—
126C22×D12—61no—
127C22×Dic6C2×D311yescharacteristic, complemented
128C3×#137—64no—
129C2×C3×Q16—32no—
130C2×C3×SD16—32no—
131C2×C3×M16—21no—
132C2×C3×M16C2×D311yescharacteristic, complemented
133C2×C3×M16C2×D311yescharacteristic, complemented
134C22×C3×C8—31no—
135#947—61no—
136#946—34no—
137#944—61no—
138#944—61no—
139C2×#215—34no—
140C2×#215—64no—
141#936—61no—
142#935—62no—
143#929—62no—
144#926—31no—
145#926—31no—
146#926—61no—
147#924—34no—
148C22×C3⋊C8C2×D311yescharacteristic, complemented
149#909—24no—
150C2×Dic12C2×D312yescomplemented
151C2×#204—34no—
152C2×#204—64no—
153C2×#204C2×D312yescomplemented
154C3×C22⋊C8—34no—
155#835—32no—
156#832—32no—
157#821—32no—
158#821—34no—
159#821—34no—
160#821—64no—
161(C2×D3)⋊C8C2×D314yescomplemented
162#816—32no—
163C23×C32C2×D411yescharacteristic
164C22×C3⋊D3—22no—
165C22×C3⋊D3—24no—
166C22×C3⋊D3—82no—
167C22×C3×D3—21no—
168C22×C3×D3C2×D412yes—
169C2×D32—28no—
170C2×D32—84no—
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D412yes—
173C2×C32×C4C2×D412yes—
174C2×C32×C4C2411yescharacteristic
175C2×C3⋊Dic3—22no—
176C4⋊(C3⋊D3)—24no—
177C4⋊(C3⋊D3)—22no—
178C4⋊(C3⋊D3)C2×D414yes—
179C3×#67—24no—
180C3×#67—24no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3C2×D412yes—
185C3×D12—22no—
186C3×D12—21no—
187C3×D12C2×D412yes—
188C3×C4×D3—24no—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×Dic6—22no—
192C3×Dic6—21no—
193C3×Dic6—22no—
194C3×Dic6—22no—
195C3×Dic6—42no—
196C3×Dic6C2×D412yes—
197#626—28no—
198#626—28no—
199#624—28no—
200C32×C8—22no—
201C3×C3⋊C8—22no—
202C2×H(5,𝔽2)—91no—
203C2×#137—31no—
204C22×SD16—91no—
205#536—91no—
206#452—94no—
207#450—94no—
208#418—91no—
209#416—91no—
210#408—31no—
211C23×D3—62no—
212C23×D3—61no—
213C23×D3—61no—
214C23×D3—62no—
215C23×D3—121no—
216C3×#41—34no—
217C3×#41—62no—
218C3×#41—62no—
219C3×#41—62no—
220C2×C3×Q8—31no—
221C2×C3×Q8—31no—
222C2×C3×Q8—32no—
223C2×C3×Q8—61no—
224C2×C3×Q8—62no—
225C2×C3×Q8—62no—
226C2×C3×D4—32no—
227C2×C3×D4—31no—
228C2×C3×D4—31no—
229C2×C3×D4—61no—
230C22×C3×C4—21no—
231C22×C3×C4—32no—
232C22×C3×C4—61no—
233C22×C3×C4—61no—
234C22×C3×C4C22×D311yescharacteristic
235C22×C3×C4C22×D311yescharacteristic
236C2×#67—21no—
237C2×#67—62no—
238C2×#67—62no—
239C2×#67D1212yes—
240C22×Dic3—21no—
241#239—64no—
242#239—64no—
243#239—68no—
244D3×D4—64no—
245D3×D4—64no—
246D3×D4—68no—
247D3×D4—616no—
248D3×D4—616no—
249#235—22no—
250#235—22no—
251#235—22no—
252#235—64no—
253#235—68no—
254#235—64no—
255#235#6714yes—
256C2×D12—34no—
257C2×D12—34no—
258C2×D12—32no—
259C2×D12—32no—
260C2×D12—62no—
261C2×D12—64no—
262C2×D12—64no—
263C2×D12—68no—
264C2×D12—62no—
265C2×D12—62no—
266C2×D12—61no—
267C2×D12—61no—
268C2×D12—62no—
269C2×D12—62no—
270C2×D12—64no—
271C2×D12—121no—
272C2×D12—122no—
273C2×D12D1211yescharacteristic
274C2×D12C22×D311yescharacteristic
275C2×C4×D3—21no—
276C2×C4×D3—62no—
277C2×C4×D3—62no—
278C2×C4×D3C22×D312yes—
279C2×Dic6—22no—
280C2×Dic6—22no—
281C2×Dic6—21no—
282C2×Dic6D1211yescharacteristic
283C2×Dic6C22×D312yes—
284C2×Dic6C22×D311yescharacteristic
285C3×Q16—64no—
286C3×SD16—64no—
287C3×M16—22no—
288C3×M16—22no—
289C3×M16—42no—
290C2×C3×C8—22no—
291C2×C3×C8—32no—
292C2×C3×C8—62no—
293C2×C3×C8C22×D312yes—
294C2×C3×C8C22×D312yes—
295#217—64no—
296#217—61no—
297#217—61no—
298#217—121no—
299#215—68no—
300(C2×D3)⋊C4—64no—
301C4⋊Dic3—32no—
302C4⋊Dic3—32no—
303C4⋊Dic3—62no—
304Dic3⋊C4—64no—
305Dic3⋊C4—64no—
306C4×Dic3—61no—
307C4×Dic3—61no—
308C4×Dic3—121no—
309C2×C3⋊C8—22no—
310C2×C3⋊C8C22×D312yes—
311Dic12—24no—
312#204—24no—
313#204—68no—
314C22×C32—22no—
315C22×C32#12012yes—
316C22×C32C22×D411yescharacteristic
317C2×C3⋊D3—24no—
318C2×C3⋊D3—44no—
319C2×C3⋊D3—84no—
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—24no—
323C2×C3×D3—42no—
324D32—88no—
325C32×C4#12012yes—
326C32×C4C22×D411yescharacteristic
327C32×C4C22×D411yescharacteristic
328C3⋊Dic3—42no—
329C3×Dic3—22no—
330C3×Dic3—22no—
331C3×Dic3—42no—
332H(5,𝔽2)—184no—
333H(5,𝔽2)—184no—
334C2×#41—31no—
335C2×#41—31no—
336C2×#41—92no—
337C2×#41—181no—
338C22×Q8—31no—
339C22×D4—92no—
340C22×D4—91no—
341C22×D4—181no—
342C22×D4—182no—
343#137—64no—
344C2×Q16—32no—
345C2×SD16—32no—
346C2×SD16—94no—
347C2×SD16—184no—
348C2×M16D3211yescharacteristic, complemented
349C22×C8—31no—
350C4⋊Q8—181no—
351#124—181no—
352#123—182no—
353C22⋊Q8—94no—
354#117—91no—
355#103—92no—
356#102—92no—
357#102—94no—
358C22⋊C8—34no—
359C23×C3—21no—
360C23×C3—32no—
361C23×C3—61no—
362C23×C3C2×D1211yescharacteristic
363C23×C3D3×D411yescharacteristic
364C22×D3—21no—
365C22×D3—68no—
366C22×D3—68no—
367C22×D3—62no—
368C22×D3—62no—
369C22×D3—64no—
370C22×D3—64no—
371C22×D3—122no—
372C22×D3—124no—
373C22×D3—244no—
374C22×D3—244no—
375C22×D3—242no—
376C22×D3—242no—
377C22×D3—244no—
378C22×D3C2×D1212yes—
379C3×Q8—32no—
380C3×Q8—62no—
381C3×Q8—61no—
382C3×Q8—62no—
383C3×Q8—62no—
384C3×Q8—122no—
385C3×D4—32no—
386C3×D4—64no—
387C3×D4—64no—
388C3×D4—62no—
389C3×D4—61no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—32no—
395C2×C3×C4—61no—
396C2×C3×C4—64no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—122no—
401C2×C3×C4C2×D1211yescharacteristic
402C2×C3×C4D3×D411yescharacteristic
403C2×C3×C4D3×D412yes—
404C2×C3×C4D3×D412yes—
405C2×C3×C4D3×D412yes—
406C2×C3×C4C2×#6712yes—
407C2×C3×C4C23×D311yescharacteristic
408C2×C3×C4C23×D311yescharacteristic
409#67—24no—
410#67—24no—
411#67—68no—
412#67—68no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—42no—
416C2×Dic3—62no—
417C2×Dic3—62no—
418C2×Dic3—122no—
419C2×Dic3C2×D1212yes—
420D12—22no—
421D12—21no—
422D12—34no—
423D12—34no—
424D12—64no—
425D12—68no—
426D12—68no—
427D12—64no—
428D12—64no—
429D12—62no—
430D12—62no—
431D12—122no—
432D12—124no—
433D12C2×#6712yes—
434C4×D3—24no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—68no—
438C4×D3—68no—
439Dic6—22no—
440Dic6—21no—
441Dic6—22no—
442Dic6—22no—
443Dic6—42no—
444Dic6C2×#6712yes—
445C3×C8—22no—
446C3×C8—22no—
447C3×C8—42no—
448C3×C8—62no—
449C3⋊C8—22no—
450C2×C32—22no—
451C2×C32C2×#12011yescharacteristic
452C2×C32#57712yes—
453C3⋊D3—44no—
454C3×D3—42no—
455C3×D3—42no—
456C24—182no—
457C24—181no—
458#41—34no—
459#41—62no—
460#41—62no—
461#41—62no—
462#41—184no—
463#41—184no—
464#41—188no—
465C2×Q8—31no—
466C2×Q8—31no—
467C2×Q8—32no—
468C2×Q8—61no—
469C2×Q8—62no—
470C2×Q8—62no—
471C2×D4—32no—
472C2×D4—31no—
473C2×D4—31no—
474C2×D4—61no—
475C2×D4—92no—
476C2×D4—94no—
477C2×D4—182no—
478C2×D4—184no—
479C2×D4—184no—
480C2×D4—188no—
481C2×D4—182no—
482C2×D4—1816no—
483C2×D4—181no—
484C2×D4—182no—
485C22×C4—32no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—182no—
489C22×C4C2×D3211yescharacteristic
490Q16—64no—
491SD16—64no—
492SD16—188no—
493M16—22no—
494C2×C8—32no—
495C2×C8—62no—
496C2×C8C2×D3212yes—
497C4⋊C4—92no—
498C4⋊C4—184no—
499C22⋊C4—184no—
500C22⋊C4—181no—
501C42—181no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—21no—
505C22×C3—22no—
506C22×C3—42no—
507C22×C3—62no—
508C22×C3—64no—
509C22×C3—62no—
510C22×C3—122no—
511C22×C3#88212yes—
512C22×C3#93712yes—
513C22×C3C22×D1211yescharacteristic
514C22×C3C2×D3×D411yescharacteristic
515C2×D3—22no—
516C2×D3—24no—
517C2×D3—22no—
518C2×D3—42no—
519C2×D3—64no—
520C2×D3—64no—
521C2×D3—124no—
522C2×D3—124no—
523C2×D3—124no—
524C2×D3—244no—
525C2×D3—248no—
526C2×D3—248no—
527C2×D3—244no—
528C2×D3—244no—
529C2×D3—248no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—22no—
533C3×C4—62no—
534C3×C4—62no—
535C3×C4—122no—
536C3×C4#88212yes—
537C3×C4#93712yes—
538C3×C4C2×D3×D411yescharacteristic
539C3×C4C2×D3×D411yescharacteristic
540C3×C4C2×D3×D411yescharacteristic
541C3×C4C22×#6711yescharacteristic
542Dic3—22no—
543Dic3—22no—
544Dic3—42no—
545Dic3—122no—
546Dic3—122no—
547Dic3—242no—
548C32#300911yescharacteristic, complemented, Sylow
549C23—32no—
550C23—61no—
551C23—188no—
552C23—182no—
553C23—184no—
554C23—722no—
555C23—724no—
556C23D3×D1211yescharacteristic
557Q8—32no—
558Q8—61no—
559Q8—62no—
560Q8—62no—
561Q8—62no—
562Q8—122no—
563D4—32no—
564D4—64no—
565D4—64no—
566D4—61no—
567D4—62no—
568D4—94no—
569D4—188no—
570D4—188no—
571D4—182no—
572D4—184no—
573C2×C4—32no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—64no—
578C2×C4—62no—
579C2×C4—122no—
580C2×C4—188no—
581C2×C4—182no—
582C2×C4D3×D1211yescharacteristic
583C2×C4#380212yes—
584C2×C4D3×#6712yes—
585C2×C4C22×D3211yescharacteristic
586C8—22no—
587C8—62no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—42no—
593C2×C3—122no—
594C2×C3—122no—
595C2×C3C2×#88211yescharacteristic
596C2×C3#575112yes—
597C2×C3#577812yes—
598C2×C3C2×#93711yescharacteristic
599D3—42no—
600D3—42no—
601D3—124no—
602D3—124no—
603D3—244no—
604C22—22no—
605C22—64no—
606C22—62no—
607C22—62no—
608C22—122no—
609C22—184no—
610C22—364no—
611C22—724no—
612C22—728no—
613C22#6392212yes—
614C22C2×D3×D1211yescharacteristic
615C4—62no—
616C4—62no—
617C4—122no—
618C4—362no—
619C4#6392212yes—
620C4C2×#380211yescharacteristic
621C4C2×D3×#6711yescharacteristic
622C3—21no—
623C3#7867111yescharacteristic, complemented
624C3#8398211yescharacteristic, complemented
625C2—22no—
626C2—122no—
627C2—122no—
628C2—364no—
629C2#1059471512yes—
630C2C2×#6392211yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radicalC2×#64308C2
hypercenterC2×M16D32
hypocenterC32#3009
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
2. C2×C3×#909C2semidirect
3. #10594855C2semidirect
4. C22×#3717C2semidirect
5. #10592830C2semidirect
6. C3×#5129C2semidirect
7. #10591084C2semidirect
8. #10590966C2semidirect
9. #10590942C2semidirect
15. C22×C3×Dic6C22semidirect
23. C2×C32×M16C22semidirect
24. #64135C22semidirect
25. C22×C3×C3⋊C8C22semidirect
27. C2×C3×Dic12C22semidirect
28. C2×C3×#204C22semidirect
32. #63856C22semidirect
34. C2×#3717C22semidirect
35. #63648C22semidirect
36. C3×(C2×D3)⋊C8C22semidirect
37. #63566C22semidirect
38. #63563C22semidirect
45. C2×#909D3semidirect
54. #5129D3semidirect
127. C22×Dic6C2×D3semidirect
132. C2×C3×M16C2×D3semidirect
133. C2×C3×M16C2×D3semidirect
148. C22×C3⋊C8C2×D3semidirect
150. C2×Dic12C2×D3semidirect
153. C2×#204C2×D3semidirect
161. (C2×D3)⋊C8C2×D3semidirect
348. C2×M16D32semidirect
548. C32#3009semidirect
623. C3#78671semidirect
624. C3#83982semidirect

Maximal subgroups

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

subgroupindex
C2×#643082
C2×C3×#9092
#105948552
C22×#37172
#105928302
C3×#51292
#105910842
#105909662
#105909422
#839823
#786713