Finite Group Databasebeta

← #29732171

Subgroups — #29732171

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456368C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840059C211yesmaximal, characteristic
4#21454766C211yesmaximal, characteristic, complemented
5#21447333C211yesmaximal, characteristic, complemented
6#21447328C211yesmaximal, characteristic, complemented
7#21447180C211yesmaximal, characteristic
8#21447180C211yesmaximal, characteristic
9#21446603C211yesmaximal, characteristic, complemented
10#21446598C211yesmaximal, characteristic, complemented
11#21446592C211yesmaximal, characteristic, complemented
12#21446479C211yesmaximal, characteristic
13#21446479C211yesmaximal, characteristic
14C4×#14519693C211yesmaximal, characteristic
15C61×#1998C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840172—21no—
19#16840172—21no—
20#16840161—21no—
21#16840161—21no—
22#16840063—21no—
23#16840063—21no—
24#16840059—21no—
25#16840059—21no—
26#16840059—21no—
27#16840059—21no—
28#16840053—21no—
29#16840043—21no—
30#16839892—21no—
31#16839890—21no—
32#16839890—21no—
33C8×Dic122—21no—
34C2×#14519712C2211yescharacteristic
35C2×#14519712C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839802C2211yescharacteristic
41#16839802C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519708C2211yescharacteristic
44#16839790C2211yescharacteristic
45#16839790C2211yescharacteristic
46C2×C4⋊#92418C2211yescharacteristic
47#16839768C2211yescharacteristic
48#16839768C2211yescharacteristic
49C2×#14519697C2211yescharacteristic
50C2×#14519697C2211yescharacteristic
51C2×#14519693C2211yescharacteristic
52C2×#14519693C2211yescharacteristic
53#16839720C2211yescharacteristic
54#16839720C2211yescharacteristic
55C4×#11758194C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839526C2211yescharacteristic
59#16839526C2211yescharacteristic
60C2×#125×C61C2211yescharacteristic
61C61×#423C2211yescharacteristic
62C4×M16×C61C2211yescharacteristic
63C61×#343C2211yescharacteristic
64C61×#343C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519777D411yescharacteristic
74#14519777D411yescharacteristic
75#14519769D411yescharacteristic
76#14519769D411yescharacteristic
77#14519756—21no—
78#14519756—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—21no—
85#14519736C2311yescharacteristic
86#14519736C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758198—21no—
89#14519712—22no—
90#14519712—22no—
91#14519711—21no—
92#14519711—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519709—21no—
100#14519709—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708—21no—
106#14519708—21no—
107C4⋊#92418D411yescharacteristic
108C4⋊#92418D411yescharacteristic
109C4⋊#92418D411yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118#14519699—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122#14519697—21no—
123#14519693—22no—
124#14519693Q814yescomplemented
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128C4×#92418—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C2×M16×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135#125×C61—21no—
136#125×C61—21no—
137#125×C61—21no—
138#125×C61—21no—
139C2×C4⋊C4×C61—21no—
140C2×C4⋊C4×C61—21no—
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C42×C61C2311yescharacteristic
144C4⋊C8×C61—21no—
145C4⋊C8×C61—21no—
146#97×C61—21no—
147C4×C8×C61—21no—
148C23×D61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61#4111yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C2×Q812yes—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—82no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166M16×C61—21no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—21no—
172M16×C61—21no—
173M16×C61C2×Q812yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199C4⋊Dic61C2×D411yescharacteristic
200C4⋊Dic61C2×D411yescharacteristic
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208#11758194—21no—
209#11758194—21no—
210#11758194C2×Q812yes—
211#11758194#4112yes—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C23×C61C2×#4111yescharacteristic
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—81no—
227C22×D61—81no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—81no—
233C2×C4×C61—81no—
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—81no—
250D244—22no—
251D244—22no—
252D244—41no—
253D244—41no—
254C4×D61—82no—
255C4×D61—82no—
256Dic122—42no—
257Dic122—81no—
258C8×C61—22no—
259C8×C61—41no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—21no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—22no—
267#92418—22no—
268#25493—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C2×#12111yescharacteristic
272C22×C61#53311yescharacteristic
273C22×C61#54111yescharacteristic
274C2×D61—42no—
275C2×D61—82no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—82no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61#54711yescharacteristic
286C4×C61#54811yescharacteristic
287C4×C61D4×Q811yescharacteristic
288C4×C61D4×Q811yescharacteristic
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294#3166—611no—
295#3077—611no—
296C2×#478—611no—
297#2962—611no—
298#2090—611no—
299#2089—611no—
300#2063—611no—
301#2063—611no—
302#1998D6111yescharacteristic, complemented
303#1952—611no—
304#1951—611no—
305#1950—611no—
306#1931—611no—
307#1931—611no—
308C4×#100—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61#242911yescharacteristic
312C2×C61#306711yescharacteristic
313C2×C61#306811yescharacteristic
314D61—82no—
315#533—611no—
316C2×#125C2×D6111yescharacteristic
317C2×C4×D4—611no—
318#501—1221no—
319#501—1221no—
320#499—1221no—
321#499—1221no—
322#478—1221no—
323#478—1221no—
324#478—1221no—
325#478—1221no—
326#478—1221no—
327#478—1221no—
328#476—1221no—
329#474—1221no—
330#446—1221no—
331#446—1221no—
332#446—1221no—
333C8×Q8—1221no—
334C2×#108—611no—
335C2×#108—611no—
336#428—611no—
337#428—611no—
338#428—611no—
339#428—611no—
340#428—611no—
341#428—611no—
342C2×#107—611no—
343C2×#106—611no—
344#424—611no—
345#424—611no—
346#423C2×D6111yescharacteristic
347C2×C4⋊C8—611no—
348#418—611no—
349#418—611no—
350C2×#102—611no—
351C2×#102—611no—
352C2×#100—611no—
353C2×#100—611no—
354#405—611no—
355#405—611no—
356C4×M16—611no—
357C4×M16C2×D6111yescharacteristic
358#343—611no—
359#343—611no—
360#343C2×D6111yescharacteristic
361#343C2×D6111yescharacteristic
362#343C2×D6111yescharacteristic
363#343C2×D6111yescharacteristic
364#341—611no—
365#341—611no—
366C61#2549311yescharacteristic, complemented, Sylow
367C22×D4—611no—
368C23×C4—1221no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16C22×D6111yescharacteristic
372C2×M16C22×D6111yescharacteristic
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C22×C8—611no—
376C22×C8—611no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379#125—21no—
380#125—21no—
381#125—21no—
382#125—21no—
383#125—611no—
384#125—611no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393#117—611no—
394#117—611no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C42C22×D6111yescharacteristic
402#108—1222no—
403#108—1222no—
404#107—1221no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#100—614no—
439#100—1222no—
440#97—21no—
441#97—1221no—
442#97—1221no—
443#97—1221no—
444C4×C8—21no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C24—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×D4—612no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—1221no—
460C22×C4—4882no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4#1175822211yescharacteristic
464C22×C4#1175822211yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—21no—
471M16—21no—
472M16—21no—
473M16—612no—
474M16—612no—
475M16—1221no—
476M16—1221no—
477M16Q8×D6112yes—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—1221no—
517C42—1221no—
518C42D4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42D4×D6111yescharacteristic
521C42D4×D6111yescharacteristic
522C23—1222no—
523C23—4881no—
524C23—4881no—
525C23—2441no—
526C23—2441no—
527C23C2×#1175822211yescharacteristic
528Q8—4881no—
529Q8—2442no—
530D4—1222no—
531D4—1222no—
532D4—2441no—
533D4—2441no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—4882no—
545C2×C4—4882no—
546C2×C4—4881no—
547C2×C4—2441no—
548C2×C4#1451975011yescharacteristic
549C2×C4#1451975011yescharacteristic
550C2×C4#1451975011yescharacteristic
551C2×C4#1451975011yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×Q8×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—22no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—1221no—
563C8—1221no—
564C8—1221no—
565C8—1221no—
566C8—1222no—
567C8—1222no—
568C22—21no—
569C22—21no—
570C22—4882no—
571C22—4882no—
572C22—4881no—
573C22—4881no—
574C22—2442no—
575C22C2×#1451975011yescharacteristic
576C22#1684031611yescharacteristic
577C22#1684037011yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—81no—
583C4—81no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684041411yescharacteristic
590C4#1684042011yescharacteristic
591C4#1684042511yescharacteristic
592C4#1684042711yescharacteristic
593C2—21no—
594C2—21no—
595C2—4882no—
596C2#2144941911yescharacteristic
597C2#2145553811yescharacteristic
598C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1998C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#1998D61
hypocenterC61#25493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1998

Complements

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

NH ≅ G/Ntype
1. #21456368C2semidirect
4. #21454766C2semidirect
5. #21447333C2semidirect
6. #21447328C2semidirect
9. #21446603C2semidirect
10. #21446598C2semidirect
11. #21446592C2semidirect
15. C61×#1998C2semidirect
124. #14519693Q8semidirect
302. #1998D61semidirect
366. C61#25493semidirect

Maximal subgroups

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

subgroupindex
#214563682
#214556182
C2×#168400592
#214547662
#214473332
#214473282
#214471802
#214466032
#214465982
#214465922
#214464792
C4×#145196932
C61×#19982
#2549361