Finite Group Databasebeta

← #29650152

Subgroups — #29650152

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455825C211yesmaximal, characteristic
2#21454910C211yesmaximal, characteristic, complemented
3#21454840C211yesmaximal, characteristic
4#21444677C211yesmaximal, characteristic, complemented
5#21444675C211yesmaximal, characteristic, complemented
6#21444658C211yesmaximal, characteristic, complemented
7#21444652C211yesmaximal, characteristic, complemented
8#21444633C211yesmaximal, characteristic, complemented
9C8×#11758201C211yesmaximal, characteristic, complemented
10C8×#11758200C211yesmaximal, characteristic, complemented
11#21444598C211yesmaximal, characteristic, complemented
12#21444260C211yesmaximal, characteristic
13#21444238C211yesmaximal, characteristic, complemented
14#21444115C211yesmaximal, characteristic, complemented
15C61×SD16⋊C8C211yesmaximal, characteristic, complemented
16#16840316C2211yescharacteristic
17C4×D4×D61C2211yescharacteristic
18#16840094C411yescharacteristic
19#16840091C411yescharacteristic
20#16840051C411yescharacteristic
21#16840050C411yescharacteristic
22#16839966C411yescharacteristic
23#16839965C411yescharacteristic
24#16839914C411yescharacteristic
25#16839911C411yescharacteristic
26#16839893C2211yescharacteristic, complemented
27#16839892—21no—
28#16839891—21no—
29#16839889C2211yescharacteristic, complemented
30C8×Dic122C2211yescharacteristic, complemented
31C4×#11758190C2211yescharacteristic
32C4×#11758201C2211yescharacteristic
33C4×#11758200C2211yescharacteristic
34C8×D244C2211yescharacteristic
35D4×#92418C2211yescharacteristic
36#16839827C2211yescharacteristic
37C8×C22⋊D61—21no—
38#16839825—21no—
39#16839822—21no—
40#16839821—21no—
41#16839818C2211yescharacteristic
42#16839814C2211yescharacteristic
43#16839813C2211yescharacteristic
44#16839813C2211yescharacteristic
45C2×#14519689—21no—
46C22⋊C8×D61—21no—
47C4×#11758189C2211yescharacteristic
48#16839515C2211yescharacteristic, complemented
49#16839512C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53#16839492C2211yescharacteristic, complemented
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C8×Q8×C61C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758201C2×C411yescharacteristic
72C2×#11758189—21no—
73C22×#92418—21no—
74C2×C8×D61—21no—
75C2×C8×D61—21no—
76#14519776C2×C411yescharacteristic
77#14519775C2×C411yescharacteristic
78#14519767—21no—
79#14519757C2×C411yescharacteristic
80#14519755—21no—
81#14519753C2×C411yescharacteristic
82C4×Dic122—21no—
83C4×Dic122C2311yescharacteristic
84C4×D244C2311yescharacteristic
85D4×Dic61C2×C411yescharacteristic
86#14519739C2×C411yescharacteristic
87C4×C22⋊D61—21no—
88#14519737—21no—
89#14519734—21no—
90C4⋊C4×D61C2×C411yescharacteristic
91C22⋊C4×D61—21no—
92C42×D61C2311yescharacteristic
93#14519709C2×C411yescharacteristic
94#14519708C2×C411yescharacteristic
95#14519708C2×C411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519706—21no—
101#14519706C2×C411yescharacteristic
102#14519706C2×C411yescharacteristic
103#14519706D411yescharacteristic
104#14519706D411yescharacteristic
105#14519702C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519700C2×C411yescharacteristic
108#14519699C2×C411yescharacteristic
109#14519698C2×C411yescharacteristic
110#14519697C2×C411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689C2×C411yescharacteristic
117#14519689C2×C411yescharacteristic
118#14519689D411yescharacteristic
119#14519689D411yescharacteristic
120#14519688—21no—
121#14519687—21no—
122#14519687—21no—
123#14519687—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×SD16×C61C2×C411yescharacteristic
130C22×C8×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#106×C61C2×C411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#102×C61C2×C411yescharacteristic
139C22⋊C8×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150D4×D61—22no—
151D4×D61—22no—
152C2×D244C22×C411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—81no—
157C2×C4×D61—81no—
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61#4111yescharacteristic
161C2×Dic122—21no—
162C2×Dic122C22×C411yescharacteristic
163SD16×C61—22no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—81no—
168C2×C8×C61C22×C411yescharacteristic
169C2×C8×C61C22×C411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C22⋊C4×C61—21no—
176C42×C61—21no—
177C42×C61C2411yescharacteristic
178C22⋊Dic61—21no—
179#11758201—22no—
180#11758200—22no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C22×C411yescharacteristic
187C4⋊Dic61C22×C411yescharacteristic
188#11758196—21no—
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194C4×Dic61—21no—
195C4×Dic61C22×C411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C4×Dic61#4111yescharacteristic
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—81no—
202C2×#92418C22×C411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206#11758190—22no—
207#11758189—42no—
208C8×D61—81no—
209C8×D61—81no—
210C23×C61—21no—
211C22×D61—21no—
212C22×D61—24no—
213C22×D61—81no—
214C22×D61—81no—
215C22×D61C4×D411yescharacteristic
216Q8×C61—21no—
217Q8×C61—21no—
218D4×C61—21no—
219D4×C61#13112yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—24no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—81no—
233C2×Dic61C4×D411yescharacteristic
234C2×Dic61C4×D411yescharacteristic
235C2×Dic61C4×D411yescharacteristic
236D244—21no—
237D244#13112yes—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—81no—
246C4×D61—81no—
247Dic122—21no—
248Dic122—21no—
249Dic122—81no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61—21no—
253C8×C61—21no—
254C8×C61—41no—
255C8×C61—41no—
256C8×C61—41no—
257#92418—22no—
258#92418—21no—
259#92418—21no—
260#92418—21no—
261#92418—21no—
262#92418—41no—
263#92418—41no—
264#17924—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×C4×D411yescharacteristic
269C2×D61—21no—
270C2×D61—22no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—84no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61C2×#13111yescharacteristic
281C4×C61C2×#13111yescharacteristic
282C4×C61#57711yescharacteristic
283C4×C61#57711yescharacteristic
284Dic61—21no—
285Dic61—21no—
286Dic61—21no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290#3101—611no—
291#2985—611no—
292#2972—611no—
293#1585—611no—
294#1585—611no—
295#1582—611no—
296#1582—611no—
297SD16⋊C8—611no—
298SD16⋊C8D6111yescharacteristic, complemented
299C8×SD16—611no—
300C8×SD16—611no—
301#1572—611no—
302#1490—611no—
303#1485—611no—
304#1453—611no—
305C2×C61—42no—
306C2×C61#292911yescharacteristic
307C2×C61#294911yescharacteristic
308C2×C61#295611yescharacteristic
309D61—41no—
310D61—42no—
311#533—611no—
312C2×C4×D4—611no—
313#483—611no—
314#483—611no—
315#476—611no—
316#476—611no—
317#461—611no—
318#461—611no—
319#450—611no—
320#450—611no—
321#446—611no—
322#446—611no—
323#446—1221no—
324#446—1221no—
325C8×Q8—611no—
326C8×Q8C2×D6111yescharacteristic, complemented
327C4×SD16—611no—
328C4×SD16—611no—
329C4×SD16—611no—
330C4×SD16C2×D6111yescharacteristic
331C8×D4—611no—
332C8×D4—611no—
333C8×D4—611no—
334C8×D4—1221no—
335C8×D4—1221no—
336C8×D4C2×D6111yescharacteristic
337#433—1221no—
338#433—1221no—
339#432—611no—
340#431—611no—
341#431—611no—
342#431—611no—
343C2×C22⋊C8—1221no—
344C2×C22⋊C8—1221no—
345C4×M16—611no—
346#335—611no—
347#335C2×D6111yescharacteristic, complemented
348#334—611no—
349#334—611no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352Q8⋊C8—611no—
353Q8⋊C8C2×D6111yescharacteristic, complemented
354D4⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8C2×D6111yescharacteristic
358#322—611no—
359#322—611no—
360#322C2×D6111yescharacteristic, complemented
361C82—611no—
362C61#1792411yescharacteristic, complemented, Sylow
363C22×D4—611no—
364C23×C4—1221no—
365C2×SD16—611no—
366C2×SD16—611no—
367C2×SD16—611no—
368C2×SD16C4×D6111yescharacteristic
369C2×M16—1221no—
370C22×C8—21no—
371C22×C8—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376#125—1221no—
377C22⋊Q8—611no—
378C22⋊Q8—611no—
379C22⋊Q8—1221no—
380C4×Q8—611no—
381C4×Q8—1221no—
382C4×Q8C22×D6111yescharacteristic
383C4×D4—611no—
384C4×D4—611no—
385C4×D4—611no—
386C4×D4—1221no—
387C4×D4—1221no—
388C4×D4C22×D6111yescharacteristic
389#117—1221no—
390C2×C4⋊C4—611no—
391C2×C22⋊C4—1221no—
392C2×C42—611no—
393#106—611no—
394#106—611no—
395#106—611no—
396#106C4×D6111yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8C22×D6111yescharacteristic
408C4⋊C8C22×D6111yescharacteristic
409#103—611no—
410#103—611no—
411#103—611no—
412#103C4×D6111yescharacteristic
413#102—611no—
414#102—611no—
415#102—611no—
416#102C4×D6111yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—611no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C22⋊C8—611no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428#97—1221no—
429#97—1221no—
430#97—1221no—
431C4×C8—21no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436C4×C8—1221no—
437C4×C8C22×D6111yescharacteristic
438C4×C8C22×D6111yescharacteristic
439C4×C8C22×D6111yescharacteristic
440C24—1221no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×Q8C2×C4×D6111yescharacteristic
444C2×D4—611no—
445C2×D4—1221no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4C2×C4×D6111yescharacteristic
449C22×C4—21no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—4881no—
458C22×C4—4881no—
459SD16—22no—
460SD16—1222no—
461SD16—1222no—
462SD16—1222no—
463M16—2442no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—81no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—4881no—
476C2×C8—4881no—
477C2×C8—4881no—
478C2×C8C2×C4×D6111yescharacteristic
479C2×C8C2×C4×D6111yescharacteristic
480C2×C8D4×D6111yescharacteristic
481C2×C8D4×D6111yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4C2×C4×D6111yescharacteristic
493C4⋊C4C2×C4×D6111yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1221no—
500C42—21no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—1221no—
505C42C23×D6111yescharacteristic
506C23—21no—
507C23—611no—
508C23—1224no—
509C23—1221no—
510C23—4881no—
511C23—4881no—
512Q8—21no—
513Q8—21no—
514Q8—1221no—
515Q8—1221no—
516Q8—4881no—
517D4—21no—
518D4—612no—
519D4—1224no—
520D4—1221no—
521D4#1451978312yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—4881no—
538C2×C4—4881no—
539C2×C4—4881no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C22×C4×D6111yescharacteristic
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4#41×D6111yescharacteristic
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—1222no—
554C8—1221no—
555C8—1221no—
556C8—1221no—
557C8—1221no—
558C8—2441no—
559C8—2441no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1222no—
567C22—4884no—
568C22—2441no—
569C22—2441no—
570C22C4×D4×D6111yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—1221no—
576C4—1221no—
577C4—1221no—
578C4—2441no—
579C4—2441no—
580C4—2441no—
581C4#131×D6111yescharacteristic
582C4C2×#1451978311yescharacteristic
583C4#1684057111yescharacteristic
584C4#1684057211yescharacteristic
585C2—42no—
586C2—2441no—
587C2—2442no—
588C2#2145450811yescharacteristic
589C2#2145467511yescharacteristic
590C2#2145472711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×SD16⋊C8C2
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C2×C4
cosocleC42×C61C24
hypercenterSD16⋊C8D61
hypocenterC61#17924
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #21454910C2semidirect
4. #21444677C2semidirect
5. #21444675C2semidirect
6. #21444658C2semidirect
7. #21444652C2semidirect
8. #21444633C2semidirect
9. C8×#11758201C2semidirect
10. C8×#11758200C2semidirect
11. #21444598C2semidirect
13. #21444238C2semidirect
14. #21444115C2semidirect
15. C61×SD16⋊C8C2semidirect
26. #16839893C22semidirect
29. #16839889C22semidirect
30. C8×Dic122C22semidirect
48. #16839515C22semidirect
49. #16839512C22semidirect
50. #16839512C22semidirect
51. #16839494C22semidirect
52. Q8⋊#92418C22semidirect
53. #16839492C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. C8×#92418C22semidirect
60. C8×Q8×C61C22semidirect
63. C61×#335C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
298. SD16⋊C8D61semidirect
326. C8×Q8C2×D61semidirect
347. #335C2×D61semidirect
353. Q8⋊C8C2×D61semidirect
360. #322C2×D61semidirect
362. C61#17924semidirect

Maximal subgroups

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

subgroupindex
#214558252
#214549102
#214548402
#214446772
#214446752
#214446582
#214446522
#214446332
C8×#117582012
C8×#117582002
#214445982
#214442602
#214442382
#214441152
C61×SD16⋊C82
#1792461