Finite Group Databasebeta

← #30613722

Subgroups — #30613722

Order 15616.

612 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#21462769C211yesmaximal, characteristic, complemented
2#21462700C211yesmaximal, characteristic, complemented
3#21461469C211yesmaximal, characteristic, complemented
4#21461455C211yesmaximal, characteristic, complemented
5#21461440C211yesmaximal, characteristic, complemented
6#21461436C211yesmaximal, characteristic, complemented
7#21461428C211yesmaximal, characteristic, complemented
8#21461420C211yesmaximal, characteristic, complemented
9SD16⋊#92420C211yesmaximal, characteristic, complemented
10D8×#92420C211yesmaximal, characteristic, complemented
11#21461371C211yesmaximal, characteristic, complemented
12#21461131C211yesmaximal, characteristic, complemented
13#21461125C211yesmaximal, characteristic, complemented
14#21461039C211yesmaximal, characteristic, complemented
15#21456411C211yesmaximal, characteristic
16#16840813C2211yescharacteristic
17Q8×#92420C2211yescharacteristic
18#16840797—21no—
19#16840797C2211yescharacteristic
20#16840797C2211yescharacteristic
21#16840797C2211yescharacteristic
22#16840794—21no—
23#16840789—21no—
24D4×#92420—21no—
25D4×#92420C2211yescharacteristic
26D4×#92420C2211yescharacteristic
27D4×#92420C2211yescharacteristic
28#16840786—21no—
29#16840781C2211yescharacteristic
30#16840781C2211yescharacteristic
31#16840781C2211yescharacteristic
32#16840768—21no—
33D61⋊(C4⋊C8)C2211yescharacteristic
34#16840735—21no—
35#16840735—21no—
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42D4⋊#92420C2211yescharacteristic
43D4⋊#92420C2211yescharacteristic
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840310C2211yescharacteristic
53#16840291C2211yescharacteristic
54#16840171C411yescharacteristic
55#16840126C411yescharacteristic
56#16840121C411yescharacteristic
57#16840012C411yescharacteristic
58#16840012C411yescharacteristic
59#16840005C411yescharacteristic
60#16840005C411yescharacteristic
61SD16×Dic61C2211yescharacteristic
62SD16×Dic61C2211yescharacteristic
63D8×Dic61C2211yescharacteristic
64D8×Dic61C2211yescharacteristic
65#16839714C2211yescharacteristic
66C61×#496C411yescharacteristic
67C2×#11758217—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70C2×D61⋊C8—21no—
71C22×#92420—21no—
72C22×#92420—21no—
73C22×#92420—21no—
74#14519844—21no—
75#14519844—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420D411yescharacteristic
79C4⋊#92420D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86#14519834D411yescharacteristic
87#14519834D411yescharacteristic
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91C22⋊#92420—21no—
92C22⋊#92420—21no—
93C22⋊#92420—21no—
94#14519832—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758224—21no—
102C2×#11758189C2×C411yescharacteristic
103C2×#11758189C2×C411yescharacteristic
104#14519777C2×C411yescharacteristic
105#14519770—21no—
106#14519769C2×C411yescharacteristic
107#14519764—21no—
108#14519759—21no—
109#14519753C2×C411yescharacteristic
110#14519748—21no—
111#14519748C2×C411yescharacteristic
112#14519748C2×C411yescharacteristic
113#14519748C2×C411yescharacteristic
114Q8×Dic61C2311yescharacteristic
115D4×Dic61—21no—
116D4×Dic61C2311yescharacteristic
117D4×Dic61C2311yescharacteristic
118D4×Dic61C2311yescharacteristic
119#14519737—21no—
120#14519733—21no—
121C42×D61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519709C2×C411yescharacteristic
126#14519701C2×C411yescharacteristic
127#14519701C2×C411yescharacteristic
128#14519698C2×C411yescharacteristic
129#14519698C2×C411yescharacteristic
130#14519698C2×C411yescharacteristic
131#14519698C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134#14519688C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×SD16×C61C2×C411yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139C2×D8×C61C2×C411yescharacteristic
140C2×D8×C61C2×C411yescharacteristic
141#127×C61C2×C411yescharacteristic
142#124×C61C2×C411yescharacteristic
143#97×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C2×D4×C61C22×C411yescharacteristic
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156#11758224—81no—
157#11758224—81no—
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61#4111yescharacteristic
161C2×Dic122—21no—
162#11758217—81no—
163#11758217—81no—
164#11758217—81no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—81no—
172C2×#92420—81no—
173C2×#92420—81no—
174C2×#92420C2×D411yescharacteristic
175C2×#92420C2×D411yescharacteristic
176C2×#92420C2×D411yescharacteristic
177C2×#92420C2×D411yescharacteristic
178D61⋊C8—81no—
179SD16×C61—22no—
180SD16×C61—41no—
181D8×C61—22no—
182D8×C61—41no—
183C2×C8×C61C22×C411yescharacteristic
184C2×C8×C61C22×C411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C42×C61C22×C411yescharacteristic
188C22⋊Dic61—21no—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192C22⋊Dic61—21no—
193C22⋊Dic61—21no—
194#11758198—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C22×C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204C4×Dic61—21no—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61#4111yescharacteristic
207C4×Dic61C2411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210#11758189—22no—
211#11758189—41no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C22×D61C4×D411yescharacteristic
216Q8×C61—21no—
217Q8×C61—21no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—42no—
225D4×C61—81no—
226C2×C4×C61—21no—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61C23×C411yescharacteristic
230C22⋊D61—42no—
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×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61C22×D411yescharacteristic
245C2×Dic61C2×#4111yescharacteristic
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—41no—
251Dic122—81no—
252#92420—21no—
253#92420—21no—
254#92420—21no—
255#92420—21no—
256#92420—41no—
257#92420—41no—
258#92420—41no—
259#92420—41no—
260#92420—41no—
261#92420—41no—
262C8×C61—21no—
263C8×C61#13112yes—
264#92418—21no—
265#92418—21no—
266#18080—611nomaximal, Sylow
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C2×C4×D411yescharacteristic
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61C2×#13111yescharacteristic
284C4×C61#56811yescharacteristic
285Dic61—21no—
286Dic61—21no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61#57611yescharacteristic
292Dic61#57711yescharacteristic
293#3171—611no—
294#2994—611no—
295#2979—611no—
296#1594—611no—
297#1590—611no—
298#1586—611no—
299#1585—611no—
300#1583—611no—
301#1581—611no—
302SD16⋊C8—611no—
303C8×D8—611no—
304#1565—611no—
305#1482—611no—
306#1480—611no—
307#1453—611no—
308C2×C61—81no—
309C2×C61—81no—
310C2×C61—81no—
311C2×C61#293511yescharacteristic
312C2×C61#294611yescharacteristic
313C2×C61#295511yescharacteristic
314D61—41no—
315#532—611no—
316#529—611no—
317#501—611no—
318#496#692311yescharacteristic
319#490—611no—
320#489—611no—
321#469—611no—
322#469—611no—
323#468—611no—
324#468—611no—
325#446—611no—
326C8×Q8—611no—
327C4×SD16—611no—
328C4×SD16—611no—
329C4×D8—611no—
330C4×D8—611no—
331#436—611no—
332#436—611no—
333#436—611no—
334#436—1221no—
335#435—1221no—
336C8×D4—611no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—1221no—
340C8×D4—1221no—
341#433—1221no—
342#431—611no—
343#431—611no—
344#431—611no—
345#424—1221no—
346C2×C4⋊C8—611no—
347#408—1221no—
348#408—1221no—
349C4×M16—611no—
350#335—611no—
351#335—611no—
352#334—611no—
353#334—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362#322—611no—
363#322—611no—
364#322—611no—
365C82—611no—
366C61#1808011yescharacteristic, complemented, Sylow
367C2×#41—1221no—
368C2×SD16C2×#692311yescharacteristic
369C2×SD16C2×#692311yescharacteristic
370C2×D8C2×#692311yescharacteristic
371C2×D8C2×#692311yescharacteristic
372C2×M16—611no—
373C2×M16—611no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C4⋊Q8—611no—
382#127C2×#692311yescharacteristic
383#126—1221no—
384#125—611no—
385#124—1221no—
386#124C2×#692311yescharacteristic
387#123—1221no—
388C22⋊Q8—611no—
389#121—611no—
390#121—611no—
391#121—611no—
392#121—1221no—
393C4×Q8—611no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—611no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—1221no—
400C2×C42—611no—
401#107—611no—
402#107—611no—
403#106—611no—
404#106—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415#103—611no—
416#103—611no—
417#102—611no—
418#102—611no—
419#102—611no—
420#102—611no—
421#102—611no—
422#102—611no—
423C22⋊C8—611no—
424C22⋊C8—611no—
425C22⋊C8—1221no—
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—1221no—
435#97C2×#692311yescharacteristic
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444#41—4881no—
445#41—4881no—
446C2×Q8—1221no—
447C2×Q8C22×#692311yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4C22×#692311yescharacteristic
454C2×D4C22×#692311yescharacteristic
455C2×D4C22×#692311yescharacteristic
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462SD16—22no—
463SD16—41no—
464D8—22no—
465D8—41no—
466M16—1222no—
467M16—4881no—
468M16—4881no—
469M16—4881no—
470M16—2441no—
471C2×C8—611no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—4881no—
484C2×C8—4881no—
485C2×C8—4881no—
486C2×C8—4881no—
487C2×C8C22×#692311yescharacteristic
488C2×C8C22×#692311yescharacteristic
489C4⋊C4—611no—
490C4⋊C4—611no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—1221no—
511C42C22×#692311yescharacteristic
512C23—21no—
513C23—21no—
514C23—21no—
515C23—611no—
516Q8—21no—
517Q8—21no—
518Q8—4881no—
519D4—21no—
520D4—21no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—42no—
526D4—81no—
527D4—4881no—
528D4—4881no—
529D4—2442no—
530C2×C4—21no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4D4×#692311yescharacteristic
550C2×C4D4×#692311yescharacteristic
551C2×C4C23×#692311yescharacteristic
552C8—21no—
553C8—1221no—
554C8—1221no—
555C8—1221no—
556C8—1221no—
557C8—1221no—
558C8—1221no—
559C8—2441no—
560C8—2441no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C8#1451986412yes—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—1221no—
576C22—1221no—
577C22—1221no—
578C22C2×D4×#692311yescharacteristic
579C4—21no—
580C4—21no—
581C4—41no—
582C4—611no—
583C4—611no—
584C4—1221no—
585C4—1221no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4—2441no—
590C4C2×#1451986411yescharacteristic
591C4#1684084011yescharacteristic
592C2—81no—
593C2—81no—
594C2—81no—
595C2—2441no—
596C2#2146256211yescharacteristic
597C2#2146257611yescharacteristic
598C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#496C4
socleC22×C61C2×C4×D4
prime radical#21456411C2
cosocleC4×Dic61C24
hypercenter#496#6923
hypocenterC61#18080
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #21462769C2semidirect
2. #21462700C2semidirect
3. #21461469C2semidirect
4. #21461455C2semidirect
5. #21461440C2semidirect
6. #21461436C2semidirect
7. #21461428C2semidirect
8. #21461420C2semidirect
9. SD16⋊#92420C2semidirect
10. D8×#92420C2semidirect
11. #21461371C2semidirect
12. #21461131C2semidirect
13. #21461125C2semidirect
14. #21461039C2semidirect
366. C61#18080semidirect

Maximal subgroups

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

subgroupindex
#214627692
#214627002
#214614692
#214614552
#214614402
#214614362
#214614282
#214614202
SD16⋊#924202
D8×#924202
#214613712
#214611312
#214611252
#214610392
#214564112
#1808061