Finite Group Databasebeta

← #30612977

Subgroups — #30612977

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462764C211yesmaximal, characteristic, complemented
2#21462709C211yesmaximal, characteristic, complemented
3#21461468C211yesmaximal, characteristic, complemented
4#21461462C211yesmaximal, characteristic, complemented
5#21461453C211yesmaximal, characteristic, complemented
6#21461450C211yesmaximal, characteristic, complemented
7#21461416C211yesmaximal, characteristic, complemented
8#21461409C211yesmaximal, characteristic, complemented
9SD16×#92420C211yesmaximal, characteristic, complemented
10#21461400C211yesmaximal, characteristic, complemented
11#21461383C211yesmaximal, characteristic, complemented
12#21461172C211yesmaximal, characteristic, complemented
13#21461164C211yesmaximal, characteristic, complemented
14M16×#92420C211yesmaximal, characteristic, complemented
15#21456407C211yesmaximal, characteristic
16#16840813C2211yescharacteristic
17Q8×#92420C2211yescharacteristic
18#16840797C2211yescharacteristic
19#16840796C2211yescharacteristic
20#16840796C2211yescharacteristic
21#16840795—21no—
22#16840794—21no—
23#16840794—21no—
24#16840791—21no—
25#16840790C2211yescharacteristic
26#16840790C2211yescharacteristic
27D4×#92420C2211yescharacteristic
28#16840787—21no—
29#16840783C2211yescharacteristic
30#16840778C2211yescharacteristic
31#16840769—21no—
32#16840765C2211yescharacteristic
33#16840736—21no—
34#16840735—21no—
35C2×C4×#92420C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42#16840610C2211yescharacteristic
43#16840610C2211yescharacteristic
44#16840610C2211yescharacteristic
45#16840610C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#16840603C2211yescharacteristic
49#16840603C2211yescharacteristic
50C8×#92420C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840307C2211yescharacteristic
53#16840298C2211yescharacteristic
54#16840170C411yescharacteristic
55#16840148C411yescharacteristic
56#16840125C411yescharacteristic
57#16840123C411yescharacteristic
58#16840015C411yescharacteristic
59#16840013C411yescharacteristic
60#16840010C411yescharacteristic
61SD16×Dic61C2211yescharacteristic
62#16839847C2211yescharacteristic
63#16839844C2211yescharacteristic
64#16839842C2211yescharacteristic
65M16×Dic61C2211yescharacteristic
66C61×#468C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758213—21no—
69C2×#11758217—21no—
70C2×#11758217—21no—
71C2×D61⋊C8—21no—
72C2×D61⋊C8—21no—
73C22×#92420—21no—
74C22×#92420—21no—
75#14519844—21no—
76#14519844—21no—
77C4⋊#92420—21no—
78C4⋊#92420D411yescharacteristic
79C4⋊#92420D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86#14519834—21no—
87#14519834—21no—
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91C22⋊#92420D411yescharacteristic
92C22⋊#92420D411yescharacteristic
93#14519832—21no—
94#14519832—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758224—21no—
102C2×#11758201C2×C411yescharacteristic
103C2×D488C2×C411yescharacteristic
104C2×#11758199C2×C411yescharacteristic
105C2×#11758194C2×C411yescharacteristic
106#14519776C2×C411yescharacteristic
107#14519773C2×C411yescharacteristic
108#14519772—21no—
109#14519768C2×C411yescharacteristic
110#14519766C2×C411yescharacteristic
111#14519763—21no—
112#14519761—21no—
113#14519751C2×C411yescharacteristic
114#14519751C2×C411yescharacteristic
115#14519749—21no—
116Q8×Dic61C2311yescharacteristic
117D4×Dic61C2311yescharacteristic
118#14519739C2311yescharacteristic
119#14519739C2311yescharacteristic
120#14519737—21no—
121#14519737—21no—
122#14519735—21no—
123C2×C4×Dic61C2311yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519709C2×C411yescharacteristic
126#14519708C2×C411yescharacteristic
127#14519701C2×C411yescharacteristic
128#14519699C2×C411yescharacteristic
129#14519699C2×C411yescharacteristic
130#14519698C2×C411yescharacteristic
131#14519697C2×C411yescharacteristic
132#14519697C2×C411yescharacteristic
133#14519687C2×C411yescharacteristic
134#14519687C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×SD16×C61C2×C411yescharacteristic
138C2×M16×C61C2×C411yescharacteristic
139C22⋊Q8×C61C2×C411yescharacteristic
140#121×C61C2×C411yescharacteristic
141#107×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61#4111yescharacteristic
155#11758224—81no—
156#11758224—81no—
157C2×D244C22×C411yescharacteristic
158C2×D244C22×C411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×Dic122—21no—
162#11758217—81no—
163#11758217—81no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—42no—
171C2×#92420—81no—
172C2×#92420C2×D411yescharacteristic
173C2×#92420C2×D411yescharacteristic
174C2×#92420C2×D411yescharacteristic
175C2×#92420C2×D411yescharacteristic
176#11758213—81no—
177#11758213—81no—
178D61⋊C8—81no—
179D61⋊C8—81no—
180SD16×C61—22no—
181M16×C61—22no—
182C2×C8×C61C22×C411yescharacteristic
183C2×C8×C61C22×C411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22×C411yescharacteristic
186C4⋊C4×C61C22×C411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊Dic61—21no—
190#11758201—22no—
191#11758199—22no—
192#11758198—21no—
193#11758198—21no—
194#11758198—21no—
195#11758198—21no—
196#11758198—21no—
197#11758198—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—21no—
203#11758196—21no—
204C4×Dic61—21no—
205C4×Dic61C22×C411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61#4111yescharacteristic
208C4×Dic61C2411yescharacteristic
209#11758194—22no—
210C2×#92418C22×C411yescharacteristic
211C2×#92418C22×C411yescharacteristic
212D488—22no—
213C23×C61—21no—
214C23×C61C4×D411yescharacteristic
215C22×D61—21no—
216C22×D61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—81no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C22⋊D61—81no—
230C22⋊D61—81no—
231C22⋊D61—81no—
232C22⋊D61—81no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C4×D411yescharacteristic
241C2×Dic61C4×D411yescharacteristic
242C2×Dic61C22×D411yescharacteristic
243C2×Dic61C2×#4111yescharacteristic
244D244—21no—
245D244—21no—
246D244—21no—
247D244—21no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252Dic122—81no—
253#92420—22no—
254#92420—21no—
255#92420—21no—
256#92420—21no—
257#92420—21no—
258#92420—41no—
259#92420—41no—
260#92420—41no—
261#92420—41no—
262#92420—41no—
263C8×C61—21no—
264C8×C61#13112yes—
265#92418—21no—
266#92418#13112yes—
267#17968—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C2×C4×D411yescharacteristic
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C2×#13111yescharacteristic
286C4×C61C2×#13111yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—41no—
291Dic61—41no—
292Dic61#57611yescharacteristic
293Dic61#57711yescharacteristic
294#3171—611no—
295#2993—611no—
296#2981—611no—
297#1594—611no—
298#1592—611no—
299#1590—611no—
300#1589—611no—
301#1580—611no—
302SD16⋊C8—611no—
303C8×SD16—611no—
304C8×D8—611no—
305#1569—611no—
306#1494—611no—
307#1492—611no—
308C8×M16—611no—
309C2×C61—41no—
310C2×C61—81no—
311C2×C61#293111yescharacteristic
312C2×C61#294611yescharacteristic
313C2×C61#295611yescharacteristic
314D61—81no—
315D61—81no—
316#532—611no—
317#529—611no—
318#501—611no—
319#496—611no—
320#490—611no—
321#489—611no—
322#469—611no—
323#469—611no—
324#468—611no—
325#468#692311yescharacteristic
326#446—611no—
327C8×Q8—611no—
328C4×SD16—611no—
329C4×SD16—611no—
330C4×D8—611no—
331C4×D8—611no—
332#436—611no—
333#436—611no—
334#436—611no—
335#436—1221no—
336#435—1221no—
337#435—1221no—
338#435—1221no—
339C8×D4—611no—
340C8×D4—611no—
341C8×D4—611no—
342#433—1221no—
343#432—611no—
344#431—611no—
345#424—1221no—
346#423—611no—
347#408—1221no—
348#408—1221no—
349C4×M16—611no—
350C2×C4×C8—611no—
351#335—611no—
352#335—611no—
353#334—611no—
354#334—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357D4⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363#322—611no—
364#322—611no—
365C82—611no—
366C82—611no—
367C61#1796811yescharacteristic, complemented, Sylow
368C2×#41—1221no—
369C2×SD16—611no—
370C2×SD16C2×#692311yescharacteristic
371C2×D8—611no—
372C2×D8—611no—
373C2×M16—611no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16C2×#692311yescharacteristic
379C22×C8—1221no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C4⋊Q8—611no—
384#127—611no—
385#126—1221no—
386#125—611no—
387#124—611no—
388#124—1221no—
389#123—1221no—
390C22⋊Q8C2×#692311yescharacteristic
391#121—611no—
392#121—611no—
393#121—1221no—
394#121C2×#692311yescharacteristic
395C4×Q8—611no—
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—1221no—
400C4×D4—1221no—
401#117—1221no—
402C2×C42—611no—
403#107—611no—
404#107C2×#692311yescharacteristic
405#106—611no—
406#106—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416#103—611no—
417#103C2×#692311yescharacteristic
418#102—611no—
419#102—611no—
420#102—611no—
421#102—611no—
422#102—611no—
423#102C2×#692311yescharacteristic
424C22⋊C8—611no—
425C22⋊C8—611no—
426C22⋊C8—1221no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433#97—611no—
434#97—611no—
435#97—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8—1221no—
445#41—4881no—
446#41—4881no—
447C2×Q8—1221no—
448C2×Q8C22×#692311yescharacteristic
449C2×D4—21no—
450C2×D4—611no—
451C2×D4—611no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4C22×#692311yescharacteristic
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4C22×#692311yescharacteristic
463SD16—22no—
464SD16—1222no—
465D8—1222no—
466D8—1222no—
467M16—22no—
468M16—1222no—
469M16—4881no—
470M16—4881no—
471M16—4881no—
472M16—4881no—
473C2×C8—611no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—4881no—
486C2×C8—4881no—
487C2×C8—4881no—
488C2×C8—2442no—
489C2×C8C22×#692311yescharacteristic
490C2×C8C22×#692311yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4C22×#692311yescharacteristic
499C4⋊C4C22×#692311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—1221no—
514C23—21no—
515C23—1221no—
516C23—1221no—
517C23D4×#692311yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—4881no—
521D4—21no—
522D4—21no—
523D4—81no—
524D4—1221no—
525D4—1221no—
526D4—1221no—
527D4—1221no—
528D4—4881no—
529D4—4881no—
530D4—4881no—
531D4—4881no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4D4×#692311yescharacteristic
553C2×C4C23×#692311yescharacteristic
554C8—21no—
555C8—612no—
556C8—1221no—
557C8—1221no—
558C8—1221no—
559C8—1221no—
560C8—1222no—
561C8—1221no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C8#1451986412yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22C2×D4×#692311yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—611no—
586C4—611no—
587C4—1221no—
588C4—1221no—
589C4—1221no—
590C4—2441no—
591C4—2441no—
592C4C2×#1451986411yescharacteristic
593C4D61⋊#13111yescharacteristic
594C2—41no—
595C2—81no—
596C2—4881no—
597C2—4881no—
598C2#2146254211yescharacteristic
599C2#2146257611yescharacteristic
600C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#468C4
socleC22×C61C2×C4×D4
prime radical#21456407C2
cosocleC4×Dic61C24
hypercenter#468#6923
hypocenterC61#17968
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #21462764C2semidirect
2. #21462709C2semidirect
3. #21461468C2semidirect
4. #21461462C2semidirect
5. #21461453C2semidirect
6. #21461450C2semidirect
7. #21461416C2semidirect
8. #21461409C2semidirect
9. SD16×#92420C2semidirect
10. #21461400C2semidirect
11. #21461383C2semidirect
12. #21461172C2semidirect
13. #21461164C2semidirect
14. M16×#92420C2semidirect
367. C61#17968semidirect

Maximal subgroups

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

subgroupindex
#214627642
#214627092
#214614682
#214614622
#214614532
#214614502
#214614162
#214614092
SD16×#924202
#214614002
#214613832
#214611722
#214611642
M16×#924202
#214564072
#1796861