Finite Group Databasebeta

← #29828675

Subgroups — #29828675

Order 15616.

3926 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839924C214yesmaximal, complemented
2C2×#16839918C211yesmaximal, characteristic, complemented
3C2×#16839913C211yesmaximal, characteristic, complemented
4#21454106C211yesmaximal, characteristic, complemented
5#21445759C214yesmaximal, complemented
6#21445754C211yesmaximal, characteristic, complemented
7#21445750C211yesmaximal, characteristic, complemented
8#21445729C211yesmaximal, characteristic, complemented
9C23×#41×C61C211yesmaximal, characteristic
10C22×#11758202C2212yescomplemented
11C22×#11758200C2212yescomplemented
12C2×#14519756C2211yescharacteristic
13C2×#14519756C2212yescomplemented
14#16839924—216no—
15#16839918—24no—
16#16839913—24no—
17C2×#14519701C2214yescomplemented
18C2×#14519701C2212yescomplemented
19C2×#14519698C2214yescomplemented
20C2×#14519698C2212yescomplemented
21C2×C22⋊#92418C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2214yescomplemented
24#16839635—21no—
25#16839635—22no—
26#16839622C2211yescharacteristic
27#16839622C2212yescomplemented
28#16839618—21no—
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×Dic122C2311yescharacteristic
40C2×#11758202—24no—
41C2×#11758202—28no—
42C2×#11758200—24no—
43C2×#11758200—28no—
44C22×#92418C2312yes—
45C22×#92418C2312yes—
46#14519756—22no—
47#14519756—22no—
48#14519756—28no—
49C2×C4⋊Dic61C2311yescharacteristic
50C2×C4⋊Dic61C2312yes—
51C2×#11758196—21no—
52C2×#11758196—22no—
53C2×C22⋊Dic61—21no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—22no—
57#14519701—24no—
58#14519701—28no—
59#14519698—24no—
60#14519698—28no—
61C22⋊#92418—22no—
62C22⋊#92418—22no—
63C22⋊#92418—28no—
64#14519684—21no—
65#14519684—22no—
66C25×C61—21no—
67C25×C61D411yescharacteristic
68C2×#41×C61—264no—
69C2×#41×C61—264no—
70C2×#41×C61—216no—
71C2×#41×C61—216no—
72C2×#41×C61—216no—
73C2×#41×C61—28no—
74C2×#41×C61—28no—
75C2×#41×C61—28no—
76C2×#41×C61—28no—
77C2×#41×C61—28no—
78C2×#41×C61D4116yes—
79C22×Q8×C61—28no—
80C22×Q8×C61—22no—
81C22×Q8×C61—21no—
82C22×Q8×C61D414yes—
83C22×Q8×C61C2312yes—
84C22×D4×C61—216no—
85C22×D4×C61—28no—
86C22×D4×C61—24no—
87C22×D4×C61—28no—
88C22×D4×C61—21no—
89C22×D4×C61—22no—
90C22×D4×C61D414yes—
91C22×D4×C61C2312yes—
92C23×C4×C61—216no—
93C23×C4×C61—28no—
94C23×C4×C61—21no—
95C23×C4×C61—24no—
96C23×C4×C61—22no—
97C23×C4×C61—22no—
98C23×C4×C61—28no—
99C23×C4×C61D411yescharacteristic
100C23×C4×C61D412yes—
101C23×C4×C61C2311yescharacteristic
102C23×C4×C61C2312yes—
103C24×C61—28no—
104C24×C61—24no—
105C24×C61—22no—
106C24×C61—24no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—42no—
111C24×C61—44no—
112C24×C61—42no—
113C24×C61—42no—
114C24×C61C2×D412yes—
115C24×C61C2×D411yescharacteristic
116#41×C61—2128no—
117#41×C61—232no—
118#41×C61—232no—
119#41×C61—232no—
120#41×C61—232no—
121C2×Q8×C61—216no—
122C2×Q8×C61—28no—
123C2×Q8×C61—216no—
124C2×Q8×C61—24no—
125C2×Q8×C61—24no—
126C2×Q8×C61—22no—
127C2×Q8×C61C2×D418yes—
128C2×Q8×C61C2×D414yes—
129C2×D4×C61—264no—
130C2×D4×C61—216no—
131C2×D4×C61—216no—
132C2×D4×C61—28no—
133C2×D4×C61—216no—
134C2×D4×C61—216no—
135C2×D4×C61—216no—
136C2×D4×C61—24no—
137C2×D4×C61—22no—
138C2×D4×C61—24no—
139C2×D4×C61C2×D418yes—
140C2×D4×C61C2×D414yes—
141C22×C4×C61—28no—
142C22×C4×C61—264no—
143C22×C4×C61—216no—
144C22×C4×C61—216no—
145C22×C4×C61—216no—
146C22×C4×C61—216no—
147C22×C4×C61—22no—
148C22×C4×C61—24no—
149C22×C4×C61—24no—
150C22×C4×C61—22no—
151C22×C4×C61—24no—
152C22×C4×C61—216no—
153C22×C4×C61—24no—
154C22×C4×C61—24no—
155C22×C4×C61—24no—
156C22×C4×C61—22no—
157C22×C4×C61—24no—
158C22×C4×C61—22no—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D418yes—
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—22no—
167C22×Dic61—21no—
168C2×Dic122—21no—
169C2×Dic122—24no—
170C2×Dic122—21no—
171C22⋊Dic61—44no—
172C22⋊Dic61—44no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175#11758202—48no—
176#11758200—48no—
177C4⋊Dic61—21no—
178C4⋊Dic61—24no—
179C4⋊Dic61—21no—
180#11758196—44no—
181#11758196—42no—
182C2×#92418—22no—
183C2×#92418—24no—
184C2×#92418—24no—
185C2×#92418—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—28no—
189C23×C61—216no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—24no—
193C23×C61—28no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—48no—
199C23×C61—48no—
200C23×C61—416no—
201C23×C61—44no—
202C23×C61—48no—
203C23×C61—416no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—44no—
207C23×C61—44no—
208C23×C61—48no—
209C23×C61#12014yes—
210C23×C61#12012yes—
211C23×C61#13512yes—
212C23×C61#13712yes—
213C23×C61C22×D411yescharacteristic
214Q8×C61—216no—
215Q8×C61—24no—
216Q8×C61—28no—
217Q8×C61#12018yes—
218D4×C61—264no—
219D4×C61—216no—
220D4×C61—28no—
221D4×C61—24no—
222D4×C61#12018yes—
223C2×C4×C61—28no—
224C2×C4×C61—216no—
225C2×C4×C61—264no—
226C2×C4×C61—216no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—28no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—28no—
235C2×C4×C61—28no—
236C2×C4×C61#12014yes—
237C2×C4×C61#12012yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D414yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×Dic61—42no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248Dic122—42no—
249Dic122—42no—
250#92418—42no—
251#92418—42no—
252#32883—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—24no—
255C22×C61—22no—
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—28no—
259C22×C61—24no—
260C22×C61—24no—
261C22×C61—416no—
262C22×C61—432no—
263C22×C61—42no—
264C22×C61—44no—
265C22×C61—42no—
266C22×C61—48no—
267C22×C61—42no—
268C22×C61—44no—
269C22×C61—42no—
270C22×C61—48no—
271C22×C61#45214yescomplemented
272C22×C61C2×#12011yescharacteristic
273C22×C61C2×#12012yes—
274C22×C61C2×#13511yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57512yes—
277C4×C61—24no—
278C4×C61—28no—
279C4×C61—22no—
280C4×C61C2×#12012yes—
281C4×C61C2×#12012yes—
282Dic61—81no—
283Dic61—82no—
284Dic61—81no—
285C23×#41D6111yescharacteristic
286C2×#452—614no—
287C2×#451—611no—
288C2×#450—611no—
289#2848—611no—
290#1788—614no—
291#1787—611no—
292#1786—611no—
293#1779—611no—
294C2×C61—24no—
295C2×C61—44no—
296C2×C61—48no—
297C2×C61#284611yescharacteristic
298C2×C61C2×#45211yescharacteristic
299C2×C61#300212yes—
300C2×C61#300511yescharacteristic
301C2×C61#300712yes—
302C22×#41—216no—
303C22×#41—24no—
304C22×#41—24no—
305C22×#41—22no—
306C22×#41C2×D6114yes—
307C23×Q8C2×D6111yescharacteristic
308C23×D4—21no—
309C23×D4C2×D6111yescharacteristic
310C24×C4—21no—
311C24×C4C2×D6111yescharacteristic
312C22×Q16—612no—
313C22×SD16—612no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—612no—
316#452—12216no—
317#451—1224no—
318#450—1224no—
319C2×#103—612no—
320C2×#103—614no—
321C2×#102—612no—
322C2×#102—614no—
323C2×C22⋊C8—611no—
324C2×C22⋊C8—611no—
325C2×C22⋊C8—614no—
326#385—1221no—
327#385—1222no—
328#380—611no—
329#380—612no—
330#379—1221no—
331C61#3288311yescharacteristic, complemented, Sylow
332C25—21no—
333C25C22⋊D6111yescharacteristic
334C2×#41—264no—
335C2×#41—264no—
336C2×#41—216no—
337C2×#41—216no—
338C2×#41—216no—
339C2×#41—28no—
340C2×#41—28no—
341C2×#41—28no—
342C2×#41—28no—
343C2×#41—28no—
344C2×#41C22⋊D61116yes—
345C22×Q8—28no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8—611no—
349C22×Q8C22⋊D6114yes—
350C22×Q8C22×D6112yes—
351C22×D4—216no—
352C22×D4—28no—
353C22×D4—24no—
354C22×D4—28no—
355C22×D4—22no—
356C22×D4—21no—
357C22×D4C22⋊D6114yes—
358C22×D4C22×D6112yes—
359C23×C4—216no—
360C23×C4—28no—
361C23×C4—21no—
362C23×C4—24no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—28no—
366C23×C4C22⋊D6111yescharacteristic
367C23×C4C22⋊D6112yes—
368C23×C4C22×D6111yescharacteristic
369C23×C4C22×D6112yes—
370C2×Q16—1224no—
371C2×Q16—1228no—
372C2×SD16—1224no—
373C2×SD16—1228no—
374C22×C8—612no—
375C22×C8—612no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378C22⋊Q8—1228no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1221no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1221no—
387#103—1224no—
388#103—1228no—
389#102—1224no—
390#102—1228no—
391C22⋊C8—1222no—
392C22⋊C8—1222no—
393C22⋊C8—1228no—
394#95—1222no—
395#95—1221no—
396C24—28no—
397C24—22no—
398C24—24no—
399C24—22no—
400C24—24no—
401C24—22no—
402C24—22no—
403C24—42no—
404C24—44no—
405C24—42no—
406C24—42no—
407C24C2×C22⋊D6112yes—
408C24C2×C22⋊D6111yescharacteristic
409#41—2128no—
410#41—232no—
411#41—232no—
412#41—232no—
413#41—232no—
414C2×Q8—216no—
415C2×Q8—28no—
416C2×Q8—216no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—1221no—
421C2×Q8—1224no—
422C2×Q8—1221no—
423C2×Q8C2×C22⋊D6118yes—
424C2×Q8C2×C22⋊D6114yes—
425C2×D4—264no—
426C2×D4—216no—
427C2×D4—216no—
428C2×D4—28no—
429C2×D4—216no—
430C2×D4—216no—
431C2×D4—216no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4C2×C22⋊D6118yes—
436C2×D4C2×C22⋊D6114yes—
437C22×C4—28no—
438C22×C4—264no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—1222no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4C2×C22⋊D6112yes—
459C22×C4C2×C22⋊D6112yes—
460C22×C4C2×C22⋊D6118yes—
461C22×C4C2×C22⋊D6112yes—
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464Q16—2448no—
465SD16—2448no—
466C2×C8—1222no—
467C2×C8—1224no—
468C2×C8—1224no—
469C2×C8—1222no—
470C4⋊C4—1221no—
471C4⋊C4—1224no—
472C4⋊C4—1221no—
473C4⋊C4—2444no—
474C4⋊C4—2442no—
475C22⋊C4—2444no—
476C22⋊C4—2444no—
477C22⋊C4—2442no—
478C22⋊C4—2442no—
479C23—22no—
480C23—22no—
481C23—28no—
482C23—216no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—48no—
492C23—48no—
493C23—416no—
494C23—44no—
495C23—48no—
496C23—416no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—48no—
502C23#1451974614yes—
503C23#1451974612yes—
504C23#1451979712yes—
505C23#1451980712yes—
506C23C22×C22⋊D6111yescharacteristic
507Q8—216no—
508Q8—24no—
509Q8—28no—
510Q8—2442no—
511Q8—2442no—
512Q8#1451974618yes—
513D4—264no—
514D4—216no—
515D4—28no—
516D4—24no—
517D4#1451974618yes—
518C2×C4—28no—
519C2×C4—216no—
520C2×C4—264no—
521C2×C4—216no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—2442no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4#1451974614yes—
539C2×C4#1451974612yes—
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4C22×C22⋊D6114yes—
542C2×C4C22×C22⋊D6111yescharacteristic
543C8—2442no—
544C8—2442no—
545C22—22no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—28no—
551C22—24no—
552C22—24no—
553C22—416no—
554C22—432no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—48no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—48no—
563C22#1683992414yescomplemented
564C22C2×#1451974611yescharacteristic
565C22C2×#1451974612yes—
566C22C2×#1451979711yescharacteristic
567C22C2×#1451980711yescharacteristic
568C22#1684056012yes—
569C4—24no—
570C4—28no—
571C4—22no—
572C4—4881no—
573C4—4882no—
574C4—4881no—
575C4C2×#1451974612yes—
576C4C2×#1451974612yes—
577C2—24no—
578C2—44no—
579C2—48no—
580C2#2145409611yescharacteristic
581C2C2×#1683992411yescharacteristic
582C2#2145503712yes—
583C2#2145506111yescharacteristic
584C2#2145507912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC23×#41×C61C2
socleC23×C61C22×D4
hypercenterC23×#41D61
hypocenterC61#32883
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×#16839924C2semidirect
2. C2×#16839918C2semidirect
3. C2×#16839913C2semidirect
4. #21454106C2semidirect
5. #21445759C2semidirect
6. #21445754C2semidirect
7. #21445750C2semidirect
8. #21445729C2semidirect
10. C22×#11758202C22semidirect
11. C22×#11758200C22semidirect
13. C2×#14519756C22semidirect
17. C2×#14519701C22semidirect
18. C2×#14519701C22semidirect
19. C2×#14519698C22semidirect
20. C2×#14519698C22semidirect
23. C2×C22⋊#92418C22semidirect
27. #16839622C22semidirect
271. C22×C61#452semidirect
331. C61#32883semidirect
563. C22#16839924semidirect

Maximal subgroups

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

subgroupindex
C2×#168399242
C2×#168399182
C2×#168399132
#214541062
#214457592
#214457542
#214457502
#214457292
C23×#41×C612
#3288361