Finite Group Databasebeta

← #30612719

Subgroups — #30612719

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462711C211yesmaximal, characteristic, complemented
2#21462667C211yesmaximal, characteristic, complemented
3#21461439C211yesmaximal, characteristic, complemented
4#21461438C211yesmaximal, characteristic, complemented
5#21461419C211yesmaximal, characteristic, complemented
6#21461418C211yesmaximal, characteristic, complemented
7#21461417C211yesmaximal, characteristic, complemented
8#21461416C211yesmaximal, characteristic, complemented
9#21461400C211yesmaximal, characteristic, complemented
10#21461399C211yesmaximal, characteristic, complemented
11#21461391C211yesmaximal, characteristic, complemented
12#21461159C211yesmaximal, characteristic, complemented
13#21461140C211yesmaximal, characteristic, complemented
14#21461037C211yesmaximal, characteristic, complemented
15#21455821C211yesmaximal, characteristic
16#16840796C2211yescharacteristic, complemented
17#16840796C2211yescharacteristic, complemented
18#16840792—21no—
19#16840792—21no—
20#16840790C2211yescharacteristic
21#16840790C2211yescharacteristic
22#16840790C2211yescharacteristic, complemented
23#16840790C2211yescharacteristic, complemented
24#16840789—21no—
25#16840789—21no—
26D4×#92420C2211yescharacteristic
27D4×#92420C2211yescharacteristic
28#16840784C2211yescharacteristic
29#16840781C2211yescharacteristic
30#16840780C2211yescharacteristic
31#16840737—21no—
32#16840737—21no—
33C2×#14519834—21no—
34D61⋊(C22⋊C8)—21no—
35C4×#11758213C2211yescharacteristic
36#16840626C2211yescharacteristic, complemented
37#16840625C2211yescharacteristic, complemented
38#16840624C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40#16840610C2211yescharacteristic
41#16840610C2211yescharacteristic
42#16840610C2211yescharacteristic, complemented
43#16840610C2211yescharacteristic, complemented
44#16840610C2211yescharacteristic, complemented
45#16840610C2211yescharacteristic, complemented
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50#16840603C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#16840313C2211yescharacteristic
53C4×D4×D61C2211yescharacteristic
54#16840077C411yescharacteristic
55#16840077C411yescharacteristic
56#16840039C411yescharacteristic
57#16839958C411yescharacteristic
58#16839957C411yescharacteristic
59#16839897C411yescharacteristic
60#16839897C411yescharacteristic
61#16839844C2211yescharacteristic
62#16839844C2211yescharacteristic
63#16839842C2211yescharacteristic
64#16839842C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#474C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758213—21no—
69C2×#11758213—21no—
70C2×D61⋊C8—21no—
71C2×D61⋊C8—21no—
72C2×D61⋊C8—21no—
73C2×D61⋊C8—21no—
74C22×#92420—21no—
75C22×#92420—21no—
76#14519844D411yescharacteristic
77#14519844D411yescharacteristic
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82#14519834—21no—
83#14519834—21no—
84#14519834—21no—
85#14519834—21no—
86#14519834—21no—
87#14519834—21no—
88#14519834—21no—
89#14519834—21no—
90#14519834D411yescharacteristic
91#14519834D411yescharacteristic
92C22⋊#92420—21no—
93C22⋊#92420—21no—
94#14519832—21no—
95#14519832—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758226—21no—
102C2×D4×D61C2×C411yescharacteristic
103C22×C4×D61—21no—
104C2×D488C2×C411yescharacteristic
105C2×D488C2×C411yescharacteristic
106C2×#11758199C2×C411yescharacteristic
107C2×#11758199C2×C411yescharacteristic
108#14519774C2×C411yescharacteristic
109#14519766—21no—
110#14519752—21no—
111#14519752C2×C411yescharacteristic
112#14519752C2×C411yescharacteristic
113C4×D244C2×C411yescharacteristic
114D4×Dic61C2311yescharacteristic
115#14519739—21no—
116#14519739C2311yescharacteristic
117#14519739C2311yescharacteristic
118#14519739C2311yescharacteristic
119C4×C22⋊D61—21no—
120#14519737—21no—
121C4⋊C4×D61C2×C411yescharacteristic
122C22⋊C4×D61—21no—
123C42×D61C2311yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519710C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519699C2×C411yescharacteristic
128#14519699C2×C411yescharacteristic
129#14519697C2×C411yescharacteristic
130#14519697C2×C411yescharacteristic
131#14519697C2×C411yescharacteristic
132#14519697C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C4⋊Q8×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic
139#107×C61C2×C411yescharacteristic
140#107×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C23×D61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150#11758226—81no—
151#11758226—81no—
152D4×D61—22no—
153D4×D61—22no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244C22×C411yescharacteristic
159C2×D244C22×C411yescharacteristic
160C2×D244C22×C411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—81no—
167C2×C4×D61—81no—
168C2×C4×D61C22×C411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61#4111yescharacteristic
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—81no—
178C2×#92420—81no—
179C2×#92420C2×D411yescharacteristic
180C2×#92420C2×D411yescharacteristic
181C2×#92420C2×D411yescharacteristic
182C2×#92420C2×D411yescharacteristic
183#11758213—42no—
184#11758213—81no—
185#11758213—81no—
186D61⋊C8—81no—
187D61⋊C8—81no—
188D61⋊C8—81no—
189D61⋊C8—81no—
190C2×C8×C61C22×C411yescharacteristic
191C2×C8×C61C22×C411yescharacteristic
192C4⋊C4×C61—21no—
193C4⋊C4×C61C22×C411yescharacteristic
194C4⋊C4×C61C22×C411yescharacteristic
195C4⋊C4×C61C22×C411yescharacteristic
196C22⋊C4×C61—21no—
197C42×C61C22×C411yescharacteristic
198C22⋊Dic61—21no—
199#11758199—22no—
200#11758199—22no—
201#11758198—21no—
202#11758198—21no—
203#11758198—21no—
204#11758198—21no—
205#11758198—21no—
206#11758198—21no—
207C4⋊Dic61C22×C411yescharacteristic
208#11758196—21no—
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61#4111yescharacteristic
211C4×Dic61C2411yescharacteristic
212C2×#92418C22×C411yescharacteristic
213C2×#92418C22×C411yescharacteristic
214D488—22no—
215D488—22no—
216C23×C61—21no—
217C22×D61—21no—
218C22×D61—24no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—81no—
222C22×D61—81no—
223C22×D61C4×D411yescharacteristic
224Q8×C61—81no—
225D4×C61—21no—
226D4×C61#13112yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—81no—
231C2×C4×C61C4×D411yescharacteristic
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C22⋊D61—24no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C22×D411yescharacteristic
240C2×Dic61C2×#4111yescharacteristic
241D244—21no—
242D244—21no—
243D244—21no—
244D244—21no—
245D244—21no—
246D244—81no—
247D244—81no—
248D244—81no—
249D244—81no—
250D244#13112yes—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—81no—
263C4×D61—81no—
264#92420—22no—
265#92420—21no—
266#92420—21no—
267#92420—21no—
268#92420—21no—
269#92420—41no—
270#92420—41no—
271#92420—41no—
272#92420—41no—
273#92420—41no—
274C8×C61—21no—
275C8×C61—21no—
276#92418—21no—
277#92418—21no—
278#17922—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61C2×C4×D411yescharacteristic
283C2×D61—21no—
284C2×D61—22no—
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—84no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61C2×#13111yescharacteristic
302C4×C61C2×#13111yescharacteristic
303Dic61—21no—
304Dic61—21no—
305Dic61—41no—
306Dic61#57611yescharacteristic
307Dic61#57611yescharacteristic
308#3100—611no—
309#2981—611no—
310#2972—611no—
311#1586—611no—
312#1586—611no—
313#1581—611no—
314#1581—611no—
315#1580—611no—
316#1580—611no—
317C8×D8—611no—
318C8×D8—611no—
319#1572—611no—
320#1490—611no—
321#1484—611no—
322#1453—611no—
323C2×C61—42no—
324C2×C61#292911yescharacteristic
325C2×C61#294811yescharacteristic
326C2×C61#295611yescharacteristic
327D61—41no—
328D61—42no—
329D61—81no—
330D61—81no—
331#532—611no—
332C2×C4×D4—611no—
333#480—611no—
334#480—611no—
335#474—611no—
336#474#692311yescharacteristic
337#459—611no—
338#459—611no—
339#447—611no—
340#447—611no—
341C4×D8—611no—
342C4×D8—611no—
343C4×D8—611no—
344C4×D8—611no—
345#436—611no—
346#436—611no—
347#435—1221no—
348#435—1221no—
349C8×D4—611no—
350C8×D4—611no—
351C8×D4—611no—
352C8×D4—611no—
353C8×D4—611no—
354C8×D4—611no—
355C8×D4—1221no—
356C8×D4—1221no—
357#432—611no—
358#431—611no—
359#431—611no—
360#431—611no—
361#408—1221no—
362#408—1221no—
363C2×C22⋊C8—1221no—
364C2×C22⋊C8—1221no—
365C4×M16—611no—
366#335—611no—
367#335—611no—
368#334—611no—
369#334—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378#322—611no—
379#322—611no—
380#322—611no—
381C82—611no—
382C61#1792211yescharacteristic, complemented, Sylow
383C2×#41—1221no—
384C22×D4—611no—
385C23×C4—1221no—
386C2×D8—611no—
387C2×D8—611no—
388C2×D8—611no—
389C2×D8—611no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C22×C8—1221no—
397C22×C8—1221no—
398C22×C8—1221no—
399C4⋊Q8C2×#692311yescharacteristic
400#127—611no—
401#124—1221no—
402#121—611no—
403#121—611no—
404#121—1221no—
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—611no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—1221no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4C2×#692311yescharacteristic
414C2×C4⋊C4—611no—
415C2×C22⋊C4—1221no—
416C2×C42—611no—
417#107—611no—
418#107—611no—
419#107C2×#692311yescharacteristic
420#107C2×#692311yescharacteristic
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8C2×#692311yescharacteristic
429#102—611no—
430#102—611no—
431#102—611no—
432#102—611no—
433#102—611no—
434#102—611no—
435#102C2×#692311yescharacteristic
436#102C2×#692311yescharacteristic
437C22⋊C8—611no—
438C22⋊C8—611no—
439C22⋊C8—611no—
440C22⋊C8—611no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451#97—1221no—
452#97—1221no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—611no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—1221no—
460C24—1221no—
461#41—4881no—
462#41—4881no—
463C2×Q8—21no—
464C2×D4—611no—
465C2×D4—611no—
466C2×D4—611no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4C22×#692311yescharacteristic
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—4881no—
486C22×C4—4881no—
487D8—1222no—
488D8—1222no—
489D8—1222no—
490D8—1222no—
491M16—4881no—
492M16—4881no—
493M16—2442no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—4881no—
507C2×C8—4881no—
508C2×C8—4881no—
509C2×C8—4881no—
510C2×C8—4881no—
511C2×C8—4881no—
512C2×C8C22×#692311yescharacteristic
513C2×C8C22×#692311yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—611no—
516C4⋊C4—1221no—
517C4⋊C4C22×#692311yescharacteristic
518C4⋊C4C22×#692311yescharacteristic
519C4⋊C4C22×#692311yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C22⋊C4—1221no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42C22×#692311yescharacteristic
532C23—21no—
533C23—611no—
534C23—1224no—
535C23—1221no—
536C23—1221no—
537C23—1221no—
538C23—4881no—
539C23—4881no—
540Q8—81no—
541D4—21no—
542D4—612no—
543D4—1224no—
544D4—1221no—
545D4—1221no—
546D4—1221no—
547D4—1221no—
548D4—1221no—
549D4—4881no—
550D4—4881no—
551D4—4881no—
552D4—4881no—
553D4#1451986512yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—81no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—4881no—
567C2×C4—4881no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4D4×#692311yescharacteristic
578C2×C4D4×#692311yescharacteristic
579C2×C4C23×#692311yescharacteristic
580C8—21no—
581C8—21no—
582C8—1221no—
583C8—1221no—
584C8—1221no—
585C8—1221no—
586C8—1222no—
587C8—1221no—
588C8—1221no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C8—2441no—
593C8—2441no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—1221no—
598C22—1222no—
599C22—1221no—
600C22—1221no—
601C22—4884no—
602C22—2441no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22—2441no—
607C22—2441no—
608C22—2441no—
609C22—2441no—
610C22C2×D4×#692311yescharacteristic
611C4—21no—
612C4—21no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—611no—
617C4—611no—
618C4—1221no—
619C4—1221no—
620C4—2441no—
621C4D61⋊#13111yescharacteristic
622C4C2×#1451986511yescharacteristic
623C2—42no—
624C2—4881no—
625C2—4881no—
626C2—2441no—
627C2—2442no—
628C2#2146253511yescharacteristic
629C2#2146257811yescharacteristic
630C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#474C4
socleC22×C61C2×C4×D4
prime radical#21455821C2
cosocleC4×Dic61C24
hypercenter#474#6923
hypocenterC61#17922
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #21462711C2semidirect
2. #21462667C2semidirect
3. #21461439C2semidirect
4. #21461438C2semidirect
5. #21461419C2semidirect
6. #21461418C2semidirect
7. #21461417C2semidirect
8. #21461416C2semidirect
9. #21461400C2semidirect
10. #21461399C2semidirect
11. #21461391C2semidirect
12. #21461159C2semidirect
13. #21461140C2semidirect
14. #21461037C2semidirect
16. #16840796C22semidirect
17. #16840796C22semidirect
22. #16840790C22semidirect
23. #16840790C22semidirect
36. #16840626C22semidirect
37. #16840625C22semidirect
38. #16840624C22semidirect
39. #16840623C22semidirect
42. #16840610C22semidirect
43. #16840610C22semidirect
44. #16840610C22semidirect
45. #16840610C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. #16840603C22semidirect
51. C8×#92420C22semidirect
382. C61#17922semidirect

Maximal subgroups

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

subgroupindex
#214627112
#214626672
#214614392
#214614382
#214614192
#214614182
#214614172
#214614162
#214614002
#214613992
#214613912
#214611592
#214611402
#214610372
#214558212
#1792261