Finite Group Databasebeta

← #10944191

Subgroups — #10944191

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1004C211yesmaximal, characteristic, complemented
2C3×#2544C211yesmaximal, characteristic, complemented
3C2×#5077C211yesmaximal, characteristic
4C2×#5061C211yesmaximal, characteristic, complemented
5#82945C212yesmaximal, complemented
6#71763C211yesmaximal, characteristic, complemented
7#71723C211yesmaximal, characteristic, complemented
8#71617C211yesmaximal, characteristic, complemented
9#69693C212yesmaximal, complemented
10#69663C212yesmaximal, complemented
11#69494C212yesmaximal, complemented
12#32924—31nomaximal, Sylow
13C2×#1004C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic, complemented
17C2×C3×C4×D4C2211yescharacteristic
18C2×#942C2212yes—
19#5798C2211yescharacteristic
20C2×#932C2212yes—
21C2×C4×#67—22no—
22#5546—22no—
23#5545—22no—
24C4×D3×D4—22no—
25C4×#237—22no—
26#5504C2212yes—
27#5486—22no—
28C4×#235C2212yes—
29C3×#394—22no—
30C3×#391C2211yescharacteristic, complemented
31C3×#390C2211yescharacteristic, complemented
32C3×#386—22no—
33C3×C4×C4⋊C4C2211yescharacteristic, complemented
34#5225C2212yescomplemented
35C2×#833C2211yescharacteristic
36C2×#833C2211yescharacteristic
37#5215C2212yescomplemented
38#5077—24no—
39#5061—24no—
40#5057—22no—
41#5057—22no—
42#5054—22no—
43#5054—22no—
44C2×#805C2212yes—
45C2×#805C2212yes—
46#4924C2212yescomplemented
47#4530C2212yescomplemented
48#4523C2212yescomplemented
49#4523C2212yescomplemented
50C4×H(5,𝔽2)—31no—
51#3040—32no—
52C2×#455—31no—
53C2×#453—31no—
54#2544D311yescharacteristic, complemented
55#1887—31no—
56#1885—31no—
57#1883—32no—
58#1882—32no—
59#1806—32no—
60#1800—31no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4D412yes—
66C22×#67—22no—
67#1004—22no—
68#1004—24no—
69#1004D414yes—
70C2×#237—22no—
71C2×D3×D4—22no—
72C2×#235C2312yes—
73C22×C4×D3—22no—
74C3×#127—22no—
75C3×#127—22no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217—22no—
93C2×#216—22no—
94#942—24no—
95#942—24no—
96C2×#215—22no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99C2×#214—22no—
100#932—24no—
101#932—24no—
102C2×#213—22no—
103C4×#67—24no—
104C4×#67—24no—
105C4×#67—22no—
106C4×#67—22no—
107C4×#67—24no—
108C2×(C2×D3)⋊C4—22no—
109#926C2312yes—
110C2×Dic3⋊C4—22no—
111C2×C4×Dic3—22no—
112C2×#208C2312yes—
113C2×#208C2312yes—
114#893—22no—
115#892—22no—
116D3×C4⋊C4—22no—
117#887—22no—
118#67⋊C4—24no—
119#67⋊C4—24no—
120D3×C22⋊C4—24no—
121#877—24no—
122C4×D12D412yescomplemented
123C4×D12D412yescomplemented
124#872—22no—
125C42×D3—22no—
126C4×Dic6D412yescomplemented
127C4×Dic6D412yescomplemented
128C3×#95—21no—
129C3×#95—22no—
130#835—22no—
131#835—22no—
132#833—22no—
133#833—22no—
134#832—22no—
135#832—22no—
136#805—24no—
137#805—24no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×H(5,𝔽2)—31no—
141C2×#137—32no—
142C2×AGL(1,ℤ/8)—32no—
143C2×#127C2×D311yescharacteristic
144C2×#124C2×D311yescharacteristic
145C2×#124C2×D311yescharacteristic, complemented
146#520—32no—
147#520—62no—
148#518—31no—
149#518—62no—
150#518—62no—
151C4×#41—32no—
152C4×#41—62no—
153C2×C4×D4—62no—
154C2×C4×D4—62no—
155C2×C4×D4C2×D311yescharacteristic
156#464—62no—
157#464—62no—
158#463—62no—
159#463—62no—
160#455—64no—
161#453—64no—
162#423—32no—
163C2×#104—32no—
164C2×#104—32no—
165#419—32no—
166#418—32no—
167C2×#100—31no—
168C2×#100—31no—
169#394—22no—
170#391C2×D311yescharacteristic, complemented
171#390C2×D311yescharacteristic, complemented
172#386—22no—
173C4×C4⋊C4C2×D311yescharacteristic, complemented
174#343—32no—
175#339—32no—
176#339—32no—
177C24×C3—21no—
178C24×C3C2×D412yes—
179C23×D3—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—82no—
193C2×C3×D4C2×D412yes—
194C2×C3×D4#4112yes—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—82no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—28no—
207C2×#67—28no—
208C2×#67—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212#237—28no—
213#237—28no—
214D3×D4—28no—
215D3×D4—28no—
216#235—22no—
217#235—22no—
218#235C2×D414yes—
219#235#4114yes—
220C2×D12C2×D412yes—
221C2×C4×D3—22no—
222C2×C4×D3—22no—
223C2×C4×D3—24no—
224C2×C4×D3—22no—
225C2×C4×D3—84no—
226C2×Dic6C2×D412yes—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—42no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—44no—
231C3×C22⋊C4—41no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—41no—
234C3×C22⋊C4—82no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42C2×D412yes—
240C3×C42C2×D412yes—
241#217—24no—
242#217—22no—
243#216—44no—
244#215—44no—
245#214—44no—
246#213—44no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250C4⋊Dic3C2×D412yes—
251C4⋊Dic3C2×D412yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255C4×Dic3—24no—
256C4×Dic3—22no—
257C4×Dic3—22no—
258C4×Dic3—22no—
259#208—22no—
260#208—22no—
261#208—22no—
262#208—22no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—22no—
265H(5,𝔽2)—34no—
266H(5,𝔽2)—64no—
267H(5,𝔽2)—62no—
268C2×#41—32no—
269C2×#41—62no—
270C22×Q8C22×D311yescharacteristic
271C22×D4—22no—
272C22×D4—62no—
273C22×D4—62no—
274C22×D4C22×D311yescharacteristic
275C22×D4C22×D311yescharacteristic
276C23×C4—62no—
277C23×C4#6712yes—
278#137—64no—
279#137—64no—
280AGL(1,ℤ/8)—64no—
281AGL(1,ℤ/8)—64no—
282C2×Q16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×D8—62no—
286C2×M16—32no—
287C2×M16—32no—
288#127—22no—
289#127—22no—
290#124—22no—
291#124—22no—
292#124—22no—
293#124—22no—
294C4×Q8—32no—
295C4×Q8—32no—
296C4×Q8—62no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—32no—
300C4×D4—32no—
301C4×D4—64no—
302C4×D4—64no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—64no—
307C4×D4—64no—
308C4×D4—62no—
309C4×D4—64no—
310C4×D4—62no—
311#117—32no—
312#117—64no—
313#117—62no—
314#117—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—62no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—64no—
324C2×C22⋊C4—62no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4#6712yes—
327C2×C42—62no—
328C2×C42—62no—
329C2×C42C22×D311yescharacteristic
330C2×C42C22×D311yescharacteristic
331C2×C42C22×D311yescharacteristic
332C4⋊C8—62no—
333C4⋊C8—62no—
334#104—64no—
335#104—64no—
336#103—62no—
337#103—62no—
338#102—62no—
339#102—62no—
340#100—62no—
341#100—62no—
342#95—21no—
343#95—22no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—42no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3#12114yes—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—28no—
357C22×D3—82no—
358C22×D3—82no—
359C3×Q8—42no—
360C3×Q8—42no—
361C3×D4—24no—
362C3×D4—24no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—88no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—44no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—82no—
381C2×C3×C4#12011yescharacteristic
382C2×C3×C4#12012yes—
383C2×C3×C4#12011yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D412yes—
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4112yes—
388#67—216no—
389#67—24no—
390#67—24no—
391C2×Dic3—28no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—82no—
396C2×Dic3—82no—
397D12—22no—
398D12#12114yes—
399C4×D3—24no—
400C4×D3—24no—
401C4×D3—44no—
402C4×D3—82no—
403C4×D3—84no—
404Dic6—22no—
405Dic6#12114yes—
406C3⋊C8—42no—
407C3⋊C8—42no—
408C24—21no—
409C24—62no—
410C24C2×#6712yes—
411#41—34no—
412#41—34no—
413#41—68no—
414#41—68no—
415#41—62no—
416#41—62no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—32no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—32no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—68no—
432C2×D4—68no—
433C2×D4—64no—
434C2×D4—68no—
435C2×D4—68no—
436C2×D4—62no—
437C2×D4—82no—
438C2×D4D3×D412yes—
439C2×D4#23712yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—64no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—82no—
453C22×C4—244no—
454C22×C4C2×#6712yes—
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—124no—
459SD16—124no—
460SD16—124no—
461D8—124no—
462M16—62no—
463M16—62no—
464M16—62no—
465M16—62no—
466C2×C8—62no—
467C2×C8—62no—
468C4⋊C4—22no—
469C4⋊C4—32no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—64no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C22⋊C4—24no—
476C22⋊C4—44no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—82no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—64no—
491C42—62no—
492C42—62no—
493C42—62no—
494C42D3×D412yes—
495C42D3×D412yes—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—24no—
499C22×C3—44no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—82no—
503C22×C3—82no—
504C22×C3—82no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3C2×#12011yescharacteristic
509C22×C3C2×#12111yescharacteristic
510C22×C3#52511yescharacteristic
511C2×D3—24no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—88no—
515C3×C4—22no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4C2×#12112yes—
523C3×C4#52512yes—
524Dic3—24no—
525Dic3—42no—
526Dic3—42no—
527C23—22no—
528C23—22no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—68no—
537C23—62no—
538C23—242no—
539C23—242no—
540C23#93814yes—
541C23C22×#6711yescharacteristic
542Q8—34no—
543Q8—42no—
544Q8—42no—
545Q8—62no—
546D4—24no—
547D4—24no—
548D4—34no—
549D4—42no—
550D4—42no—
551D4—616no—
552D4—64no—
553D4—64no—
554D4—62no—
555D4—88no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—68no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—82no—
577C2×C4—124no—
578C2×C4—242no—
579C2×C4—244no—
580C2×C4—242no—
581C2×C4—242no—
582C2×C4#93711yescharacteristic
583C2×C4#93712yes—
584C2×C4#93711yescharacteristic
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×D3×D412yes—
587C2×C4C2×#23711yescharacteristic
588C2×C4D3×#4112yes—
589C8—122no—
590C8—122no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—44no—
594C2×C3—161no—
595C2×C3#284711yescharacteristic
596C2×C3#301811yescharacteristic
597C2×C3#301911yescharacteristic
598D3—44no—
599C22—21no—
600C22—21no—
601C22—24no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—64no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—122no—
613C22—122no—
614C22—248no—
615C22C2×#93711yescharacteristic
616C22C2×#93811yescharacteristic
617C22#582811yescharacteristic
618C4—22no—
619C4—21no—
620C4—21no—
621C4—42no—
622C4—42no—
623C4—64no—
624C4—81no—
625C4—81no—
626C4—122no—
627C4—122no—
628C4D3×#12112yes—
629C4#560612yes—
630C3#3292411yescharacteristic, complemented, Sylow
631C2—21no—
632C2—21no—
633C2—44no—
634C2—124no—
635C2—161no—
636C2#8272311yescharacteristic
637C2#8332211yescharacteristic
638C2#8339411yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2544C2
socleC22×C3C2×#120
prime radicalC2×#5077C2
hypercenter#2544D3
hypocenterC3#32924
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. C4×#1004C2semidirect
2. C3×#2544C2semidirect
4. C2×#5061C2semidirect
5. #82945C2semidirect
6. #71763C2semidirect
7. #71723C2semidirect
8. #71617C2semidirect
9. #69693C2semidirect
10. #69663C2semidirect
11. #69494C2semidirect
16. C2×C3×#124C22semidirect
30. C3×#391C22semidirect
31. C3×#390C22semidirect
33. C3×C4×C4⋊C4C22semidirect
34. #5225C22semidirect
37. #5215C22semidirect
46. #4924C22semidirect
47. #4530C22semidirect
48. #4523C22semidirect
49. #4523C22semidirect
54. #2544D3semidirect
122. C4×D12D4semidirect
123. C4×D12D4semidirect
126. C4×Dic6D4semidirect
127. C4×Dic6D4semidirect
145. C2×#124C2×D3semidirect
170. #391C2×D3semidirect
171. #390C2×D3semidirect
173. C4×C4⋊C4C2×D3semidirect
630. C3#32924semidirect

Maximal subgroups

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

subgroupindex
C4×#10042
C3×#25442
C2×#50772
C2×#50612
#829452
#717632
#717232
#716172
#696932
#696632
#694942
#329243