Finite Group Databasebeta

← #29828656

Subgroups — #29828656

Order 15616.

4004 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839907C214yesmaximal, complemented
2C2×#16839901C211yesmaximal, characteristic, complemented
3C2×#16839896C211yesmaximal, characteristic, complemented
4#21454101C211yesmaximal, characteristic, complemented
5#21445759C214yesmaximal, complemented
6#21445754C211yesmaximal, characteristic, complemented
7#21445750C211yesmaximal, characteristic, complemented
8#21445729C211yesmaximal, characteristic, complemented
9C23×#41×C61C211yesmaximal, characteristic, complemented
10C22×#11758201C2212yescomplemented
11C22×#11758199C2212yescomplemented
12C2×#14519751C2211yescharacteristic
13C2×#14519751C2212yescomplemented
14C2×#14519746—21no—
15#16839907—216no—
16#16839901—24no—
17#16839896—24no—
18C2×#14519701C2214yescomplemented
19C2×#14519701C2212yescomplemented
20C2×#14519698C2214yescomplemented
21C2×#14519698C2212yescomplemented
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2214yescomplemented
25#16839641—21no—
26#16839641—22no—
27#16839622C2211yescharacteristic
28#16839622C2212yescomplemented
29C22×#41×C61—216no—
30C22×#41×C61—24no—
31C22×#41×C61—24no—
32C22×#41×C61—22no—
33C22×#41×C61C2214yes—
34C23×Q8×C61C2211yescharacteristic
35C23×D4×C61—21no—
36C23×D4×C61C2211yescharacteristic
37C24×C4×C61—21no—
38C24×C4×C61C2211yescharacteristic
39C22×D244C2311yescharacteristic
40C22×C22⋊D61—21no—
41C22×C22⋊D61—22no—
42C2×#11758201—24no—
43C2×#11758201—28no—
44C2×#11758199—24no—
45C2×#11758199—28no—
46C22×#92418C2312yes—
47C22×#92418C2312yes—
48#14519751—22no—
49#14519751—22no—
50#14519751—28no—
51#14519746—44no—
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2312yes—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—22no—
56C2×#11758198—21no—
57C2×#11758198—22no—
58#14519701—24no—
59#14519701—28no—
60#14519698—24no—
61#14519698—28no—
62C22⋊#92418—22no—
63C22⋊#92418—22no—
64C22⋊#92418—28no—
65#14519684—21no—
66#14519684—22no—
67C25×C61—21no—
68C25×C61D411yescharacteristic
69C2×#41×C61—264no—
70C2×#41×C61—264no—
71C2×#41×C61—216no—
72C2×#41×C61—216no—
73C2×#41×C61—216no—
74C2×#41×C61—28no—
75C2×#41×C61—28no—
76C2×#41×C61—28no—
77C2×#41×C61—28no—
78C2×#41×C61—28no—
79C2×#41×C61D4116yescomplemented
80C22×Q8×C61—28no—
81C22×Q8×C61—22no—
82C22×Q8×C61—21no—
83C22×Q8×C61D414yescomplemented
84C22×Q8×C61C2312yes—
85C22×D4×C61—216no—
86C22×D4×C61—28no—
87C22×D4×C61—24no—
88C22×D4×C61—28no—
89C22×D4×C61—21no—
90C22×D4×C61—22no—
91C22×D4×C61D414yescomplemented
92C22×D4×C61C2312yes—
93C23×C4×C61—216no—
94C23×C4×C61—28no—
95C23×C4×C61—21no—
96C23×C4×C61—24no—
97C23×C4×C61—22no—
98C23×C4×C61—22no—
99C23×C4×C61—28no—
100C23×C4×C61D411yescharacteristic
101C23×C4×C61D412yes—
102C23×C4×C61C2311yescharacteristic
103C23×C4×C61C2312yes—
104C24×C61—28no—
105C24×C61—24no—
106C24×C61—22no—
107C24×C61—24no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—42no—
112C24×C61—44no—
113C24×C61—42no—
114C24×C61—42no—
115C24×C61C2×D412yes—
116C24×C61C2×D411yescharacteristic
117C23×D61—21no—
118#41×C61—2128no—
119#41×C61—232no—
120#41×C61—232no—
121#41×C61—232no—
122#41×C61—232no—
123C2×Q8×C61—216no—
124C2×Q8×C61—28no—
125C2×Q8×C61—216no—
126C2×Q8×C61—24no—
127C2×Q8×C61—24no—
128C2×Q8×C61—22no—
129C2×Q8×C61C2×D418yes—
130C2×Q8×C61C2×D414yes—
131C2×D4×C61—264no—
132C2×D4×C61—216no—
133C2×D4×C61—216no—
134C2×D4×C61—28no—
135C2×D4×C61—216no—
136C2×D4×C61—216no—
137C2×D4×C61—216no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—24no—
141C2×D4×C61C2×D418yes—
142C2×D4×C61C2×D414yes—
143C22×C4×C61—28no—
144C22×C4×C61—264no—
145C22×C4×C61—216no—
146C22×C4×C61—216no—
147C22×C4×C61—216no—
148C22×C4×C61—216no—
149C22×C4×C61—22no—
150C22×C4×C61—24no—
151C22×C4×C61—24no—
152C22×C4×C61—22no—
153C22×C4×C61—24no—
154C22×C4×C61—216no—
155C22×C4×C61—24no—
156C22×C4×C61—24no—
157C22×C4×C61—24no—
158C22×C4×C61—22no—
159C22×C4×C61—24no—
160C22×C4×C61—22no—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D412yes—
165C22×C4×C61C2×D418yes—
166C22×C4×C61C2411yescharacteristic
167C2×C22⋊D61—44no—
168C2×C22⋊D61—44no—
169C2×C22⋊D61—44no—
170C2×C22⋊D61—42no—
171C2×C22⋊D61—42no—
172C2×C22⋊D61—42no—
173C22×Dic61—21no—
174C22×Dic61—22no—
175C2×D244—21no—
176C2×D244—24no—
177C2×D244—21no—
178C22⋊Dic61—44no—
179C22⋊Dic61—42no—
180#11758201—48no—
181#11758199—48no—
182#11758198—44no—
183#11758198—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—24no—
186C4⋊Dic61—21no—
187C2×#92418—22no—
188C2×#92418—24no—
189C2×#92418—24no—
190C2×#92418—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—28no—
194C23×C61—216no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—28no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—48no—
204C23×C61—48no—
205C23×C61—416no—
206C23×C61—44no—
207C23×C61—48no—
208C23×C61—416no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—44no—
212C23×C61—44no—
213C23×C61—48no—
214C23×C61#12014yes—
215C23×C61#12012yes—
216C23×C61#13512yes—
217C23×C61AGL(1,ℤ/8)12yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—41no—
220C22×D61—42no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—42no—
224Q8×C61—216no—
225Q8×C61—24no—
226Q8×C61—28no—
227Q8×C61#12018yescomplemented
228D4×C61—264no—
229D4×C61—216no—
230D4×C61—28no—
231D4×C61—24no—
232D4×C61#12018yescomplemented
233C2×C4×C61—28no—
234C2×C4×C61—216no—
235C2×C4×C61—264no—
236C2×C4×C61—216no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—24no—
240C2×C4×C61—28no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—28no—
245C2×C4×C61—28no—
246C2×C4×C61#12014yes—
247C2×C4×C61#12012yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D414yes—
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—88no—
252C22⋊D61—84no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259D244—42no—
260D244—42no—
261#92418—42no—
262#92418—42no—
263#32880—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—28no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—416no—
273C22×C61—432no—
274C22×C61—42no—
275C22×C61—44no—
276C22×C61—42no—
277C22×C61—48no—
278C22×C61—42no—
279C22×C61—44no—
280C22×C61—42no—
281C22×C61—48no—
282C22×C61#44914yescomplemented
283C22×C61C2×#12011yescharacteristic
284C22×C61C2×#12012yes—
285C22×C61C2×#13511yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61#57512yes—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—82no—
291C2×D61—81no—
292C2×D61—82no—
293C4×C61—24no—
294C4×C61—28no—
295C4×C61—22no—
296C4×C61C2×#12012yes—
297C4×C61C2×#12012yes—
298Dic61—81no—
299Dic61—82no—
300C23×#41D6111yescharacteristic, complemented
301C2×#449—614no—
302C2×#448—611no—
303C2×#447—611no—
304#2847—611no—
305#1788—614no—
306#1787—611no—
307#1786—611no—
308#1779—611no—
309C2×C61—24no—
310C2×C61—44no—
311C2×C61—48no—
312C2×C61#284611yescharacteristic
313C2×C61C2×#44911yescharacteristic
314C2×C61#300212yes—
315C2×C61#300511yescharacteristic
316C2×C61#300612yes—
317D61—161no—
318C22×#41—216no—
319C22×#41—24no—
320C22×#41—24no—
321C22×#41—22no—
322C22×#41C2×D6114yes—
323C23×Q8C2×D6111yescharacteristic
324C23×D4—21no—
325C23×D4C2×D6111yescharacteristic
326C24×C4—21no—
327C24×C4C2×D6111yescharacteristic
328C22×SD16—612no—
329C22×D8—612no—
330C2×#121—611no—
331C2×#121—612no—
332C2×#120—1221no—
333#449—12216no—
334#448—1224no—
335#447—1224no—
336C2×#103—612no—
337C2×#103—614no—
338C2×#102—612no—
339C2×#102—614no—
340C2×C22⋊C8—611no—
341C2×C22⋊C8—611no—
342C2×C22⋊C8—614no—
343#386—1221no—
344#386—1222no—
345#380—611no—
346#380—612no—
347C61#3288011yescharacteristic, complemented, Sylow
348C25—21no—
349C25C22⋊D6111yescharacteristic
350C2×#41—264no—
351C2×#41—264no—
352C2×#41—216no—
353C2×#41—216no—
354C2×#41—216no—
355C2×#41—28no—
356C2×#41—28no—
357C2×#41—28no—
358C2×#41—28no—
359C2×#41—28no—
360C2×#41C22⋊D61116yescomplemented
361C22×Q8—28no—
362C22×Q8—21no—
363C22×Q8—22no—
364C22×Q8C22⋊D6114yescomplemented
365C22×Q8C22×D6112yes—
366C22×D4—216no—
367C22×D4—28no—
368C22×D4—24no—
369C22×D4—28no—
370C22×D4—22no—
371C22×D4—21no—
372C22×D4—611no—
373C22×D4—1222no—
374C22×D4—1221no—
375C22×D4C22⋊D6114yescomplemented
376C22×D4C22×D6112yes—
377C23×C4—216no—
378C23×C4—28no—
379C23×C4—21no—
380C23×C4—24no—
381C23×C4—22no—
382C23×C4—22no—
383C23×C4—28no—
384C23×C4C22⋊D6111yescharacteristic
385C23×C4C22⋊D6112yes—
386C23×C4C22×D6111yescharacteristic
387C23×C4C22×D6112yes—
388C2×SD16—1224no—
389C2×SD16—1228no—
390C2×D8—1224no—
391C2×D8—1228no—
392C22×C8—612no—
393C22×C8—612no—
394#121—1222no—
395#121—1222no—
396#121—1228no—
397#120—2444no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—612no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1221no—
404#103—1224no—
405#103—1228no—
406#102—1224no—
407#102—1228no—
408C22⋊C8—1222no—
409C22⋊C8—1222no—
410C22⋊C8—1228no—
411#95—1222no—
412#95—1221no—
413C24—28no—
414C24—22no—
415C24—24no—
416C24—22no—
417C24—24no—
418C24—22no—
419C24—22no—
420C24—42no—
421C24—44no—
422C24—42no—
423C24—42no—
424C24—1221no—
425C24C2×C22⋊D6112yes—
426C24C2×C22⋊D6111yescharacteristic
427#41—2128no—
428#41—232no—
429#41—232no—
430#41—232no—
431#41—232no—
432C2×Q8—216no—
433C2×Q8—28no—
434C2×Q8—216no—
435C2×Q8—24no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8C2×C22⋊D6118yes—
439C2×Q8C2×C22⋊D6114yes—
440C2×D4—264no—
441C2×D4—216no—
442C2×D4—216no—
443C2×D4—28no—
444C2×D4—216no—
445C2×D4—216no—
446C2×D4—216no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—1221no—
451C2×D4—1224no—
452C2×D4—1221no—
453C2×D4—2444no—
454C2×D4—2444no—
455C2×D4—2444no—
456C2×D4—2442no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4C2×C22⋊D6118yes—
460C2×D4C2×C22⋊D6114yes—
461C22×C4—28no—
462C22×C4—264no—
463C22×C4—216no—
464C22×C4—216no—
465C22×C4—216no—
466C22×C4—216no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—216no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4C2×C22⋊D6112yes—
482C22×C4C2×C22⋊D6112yes—
483C22×C4C2×C22⋊D6118yes—
484C22×C4C2×C22⋊D6112yes—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487SD16—2448no—
488D8—2448no—
489C2×C8—1222no—
490C2×C8—1224no—
491C2×C8—1224no—
492C2×C8—1222no—
493C4⋊C4—1221no—
494C4⋊C4—1224no—
495C4⋊C4—1221no—
496C22⋊C4—2444no—
497C22⋊C4—2442no—
498C22⋊C4—2444no—
499C22⋊C4—2442no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—216no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—48no—
513C23—48no—
514C23—416no—
515C23—44no—
516C23—48no—
517C23—416no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—48no—
523C23—2441no—
524C23—2442no—
525C23—2441no—
526C23—2441no—
527C23—2442no—
528C23#1451974614yes—
529C23#1451974612yes—
530C23#1451979712yes—
531C23#1451980212yes—
532C23C22×C22⋊D6111yescharacteristic
533Q8—216no—
534Q8—24no—
535Q8—28no—
536Q8#1451974618yescomplemented
537D4—264no—
538D4—216no—
539D4—28no—
540D4—24no—
541D4—4888no—
542D4—4884no—
543D4—2442no—
544D4—2442no—
545D4#1451974618yescomplemented
546C2×C4—28no—
547C2×C4—216no—
548C2×C4—264no—
549C2×C4—216no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—2441no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4#1451974614yes—
566C2×C4#1451974612yes—
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×C22⋊D6114yes—
569C2×C4C22×C22⋊D6111yescharacteristic
570C8—2442no—
571C8—2442no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—28no—
578C22—24no—
579C22—24no—
580C22—416no—
581C22—432no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—48no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—48no—
590C22—4881no—
591C22—4881no—
592C22—4882no—
593C22—4881no—
594C22—4882no—
595C22#1683990714yescomplemented
596C22C2×#1451974611yescharacteristic
597C22C2×#1451974612yes—
598C22C2×#1451979711yescharacteristic
599C22C2×#1451980211yescharacteristic
600C22#1684056012yes—
601C4—24no—
602C4—28no—
603C4—22no—
604C4—4881no—
605C4—4882no—
606C4C2×#1451974612yes—
607C4C2×#1451974612yes—
608C2—24no—
609C2—44no—
610C2—48no—
611C2—9761no—
612C2#2145409611yescharacteristic
613C2C2×#1683990711yescharacteristic
614C2#2145503712yes—
615C2#2145506111yescharacteristic
616C2#2145507112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC23×#41×C61C2
socleC23×C61C22×D4
hypercenterC23×#41D61
hypocenterC61#32880
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16839907C2semidirect
2. C2×#16839901C2semidirect
3. C2×#16839896C2semidirect
4. #21454101C2semidirect
5. #21445759C2semidirect
6. #21445754C2semidirect
7. #21445750C2semidirect
8. #21445729C2semidirect
9. C23×#41×C61C2semidirect
10. C22×#11758201C22semidirect
11. C22×#11758199C22semidirect
13. C2×#14519751C22semidirect
18. C2×#14519701C22semidirect
19. C2×#14519701C22semidirect
20. C2×#14519698C22semidirect
21. C2×#14519698C22semidirect
24. C2×C22⋊#92418C22semidirect
28. #16839622C22semidirect
79. C2×#41×C61D4semidirect
83. C22×Q8×C61D4semidirect
91. C22×D4×C61D4semidirect
227. Q8×C61#120semidirect
232. D4×C61#120semidirect
282. C22×C61#449semidirect
300. C23×#41D61semidirect
347. C61#32880semidirect
360. C2×#41C22⋊D61semidirect
364. C22×Q8C22⋊D61semidirect
375. C22×D4C22⋊D61semidirect
536. Q8#14519746semidirect
545. D4#14519746semidirect
595. C22#16839907semidirect

Maximal subgroups

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

subgroupindex
C2×#168399072
C2×#168399012
C2×#168398962
#214541012
#214457592
#214457542
#214457502
#214457292
C23×#41×C612
#3288061