Finite Group Databasebeta

← #30613521

Subgroups — #30613521

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462745C211yesmaximal, characteristic, complemented
2#21462675C211yesmaximal, characteristic, complemented
3#21461462C211yesmaximal, characteristic, complemented
4#21461455C211yesmaximal, characteristic, complemented
5#21461440C211yesmaximal, characteristic, complemented
6#21461436C211yesmaximal, characteristic, complemented
7#21461424C211yesmaximal, characteristic, complemented
8#21461420C211yesmaximal, characteristic, complemented
9#21461415C211yesmaximal, characteristic, complemented
10D8×#92420C211yesmaximal, characteristic, complemented
11#21461383C211yesmaximal, characteristic, complemented
12#21461150C211yesmaximal, characteristic, complemented
13#21461144C211yesmaximal, characteristic, complemented
14#21461038C211yesmaximal, characteristic, complemented
15AGL(1,ℤ/8)×Dic61C211yesmaximal, characteristic
16#16840813C2211yescharacteristic, complemented
17#16840813C2211yescharacteristic, complemented
18#16840797C2211yescharacteristic
19#16840797C2211yescharacteristic
20#16840793—21no—
21#16840793—21no—
22#16840793—21no—
23#16840793—21no—
24D4×#92420C2211yescharacteristic
25D4×#92420C2211yescharacteristic
26D4×#92420C2211yescharacteristic, complemented
27D4×#92420C2211yescharacteristic, complemented
28#16840783C2211yescharacteristic
29#16840782—21no—
30#16840779C2211yescharacteristic
31#16840779C2211yescharacteristic
32#16840765C2211yescharacteristic
33C2×#14519844—21no—
34#16840732—21no—
35C2×C22⋊#92420—21no—
36#16840626C2211yescharacteristic, complemented
37#16840625C2211yescharacteristic, complemented
38#16840624C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40Q8⋊#92420C2211yescharacteristic, complemented
41Q8⋊#92420C2211yescharacteristic, complemented
42D4⋊#92420C2211yescharacteristic
43D4⋊#92420C2211yescharacteristic
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic, complemented
47D4⋊#92420C2211yescharacteristic, complemented
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50#16840603C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#41×Dic61C2211yescharacteristic
53C2×D4×Dic61C2211yescharacteristic
54#16839874C411yescharacteristic
55#16839874C411yescharacteristic
56#16839863C411yescharacteristic
57#16839863C411yescharacteristic
58SD16×Dic61C2211yescharacteristic
59SD16×Dic61C2211yescharacteristic
60D8×Dic61C2211yescharacteristic
61D8×Dic61C2211yescharacteristic
62#16839804C411yescharacteristic
63#16839767C411yescharacteristic
64#16839763C411yescharacteristic
65M16×Dic61C2211yescharacteristic
66C2×AGL(1,ℤ/8)×C61C411yescharacteristic
67C2×#11758217—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70C22×#92420—21no—
71C22×#92420—21no—
72C22×#92420—21no—
73C22×#92420—21no—
74#14519844—21no—
75#14519844—21no—
76#14519844—21no—
77#14519844—21no—
78#14519844D411yescharacteristic
79#14519844D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84C22⋊#92420—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91C22⋊#92420—21no—
92C22⋊#92420D411yescharacteristic
93C22⋊#92420D411yescharacteristic
94#14519832—21no—
95#14519832—21no—
96#14519832—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C23×Dic61—21no—
102C2×#11758194C2×C411yescharacteristic
103Q8×Dic61C2×C411yescharacteristic
104Q8×Dic61C2311yescharacteristic
105D4×Dic61—21no—
106D4×Dic61—21no—
107D4×Dic61—21no—
108D4×Dic61—21no—
109D4×Dic61C2×C411yescharacteristic
110D4×Dic61C2×C411yescharacteristic
111D4×Dic61C2×C411yescharacteristic
112D4×Dic61C2311yescharacteristic
113D4×Dic61C2311yescharacteristic
114D4×Dic61C2311yescharacteristic
115#14519736—21no—
116#14519736C2×C411yescharacteristic
117C2×C4⋊Dic61C2×C411yescharacteristic
118C2×C22⋊Dic61—21no—
119C2×C4×Dic61—21no—
120C2×C4×Dic61C2311yescharacteristic
121#14519711C2×C411yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519709C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519701C2×C411yescharacteristic
126#14519701C2×C411yescharacteristic
127#14519698C2×C411yescharacteristic
128#14519698C2×C411yescharacteristic
129#14519698C2×C411yescharacteristic
130#14519698C2×C411yescharacteristic
131#14519698C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519687C2×C411yescharacteristic
134#14519687C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×#41×C61C2×C411yescharacteristic
138C22×D4×C61C2×C411yescharacteristic
139AGL(1,ℤ/8)×C61—24no—
140C2×SD16×C61C2×C411yescharacteristic
141C2×SD16×C61C2×C411yescharacteristic
142C2×D8×C61C2×C411yescharacteristic
143C2×D8×C61C2×C411yescharacteristic
144C2×M16×C61C2×C411yescharacteristic
145C24×C61—21no—
146#41×C61—22no—
147#41×C61—41no—
148C2×Q8×C61C22×C411yescharacteristic
149C2×D4×C61—22no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61C22×C411yescharacteristic
154C2×D4×C61C22×C411yescharacteristic
155C2×D4×C61C22×C411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61C22×C411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61#4111yescharacteristic
169#11758217—81no—
170#11758217—81no—
171#11758217—81no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—81no—
179C2×#92420—81no—
180C2×#92420—81no—
181C2×#92420—81no—
182C2×#92420C2×D411yescharacteristic
183C2×#92420C2×D411yescharacteristic
184C2×#92420C2×D411yescharacteristic
185C2×#92420C2×D411yescharacteristic
186SD16×C61—22no—
187SD16×C61—22no—
188D8×C61—22no—
189D8×C61M1614yescomplemented
190M16×C61—22no—
191C2×C8×C61C22×C411yescharacteristic
192C2×C8×C61C22×C411yescharacteristic
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—21no—
199C22⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C22×C411yescharacteristic
204C4⋊Dic61C22×C411yescharacteristic
205C4⋊Dic61C22×C411yescharacteristic
206C4⋊Dic61C22×C411yescharacteristic
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61C22×C411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61#4111yescharacteristic
213C4×Dic61C2411yescharacteristic
214#11758194—41no—
215C2×#92418C22×C411yescharacteristic
216C2×#92418C22×C411yescharacteristic
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—81no—
223C23×C61C4×D411yescharacteristic
224Q8×C61—21no—
225Q8×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61C2×M1612yes—
233D4×C61C2×M1612yes—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—42no—
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—81no—
255C2×Dic61—81no—
256C2×Dic61—81no—
257C2×Dic61C4×D411yescharacteristic
258C2×Dic61C4×D411yescharacteristic
259C2×Dic61C22×D411yescharacteristic
260C2×Dic61C2×#4111yescharacteristic
261#92420—21no—
262#92420—21no—
263#92420—21no—
264#92420—21no—
265#92420—41no—
266#92420—41no—
267#92420—41no—
268#92420—41no—
269#92420—41no—
270#92420—41no—
271C8×C61—21no—
272C8×C61C2×M1612yes—
273#92418—21no—
274#92418—21no—
275#18050—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—84no—
289C22×C61C2×C4×D411yescharacteristic
290C4×C61—21no—
291C4×C61—41no—
292C4×C61C22×M1611yescharacteristic
293C4×C61#56811yescharacteristic
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61#57611yescharacteristic
302Dic61#57711yescharacteristic
303#2990—611no—
304#2974—611no—
305C4×AGL(1,ℤ/8)—611no—
306#1592—611no—
307#1590—611no—
308#1586—611no—
309#1585—611no—
310#1582—611no—
311#1581—611no—
312#1580—611no—
313C8×D8—611no—
314#1569—611no—
315#1488—611no—
316#1486—611no—
317#1453—611no—
318C2×C61—41no—
319C2×C61—42no—
320C2×C61—42no—
321C2×C61—81no—
322C2×C61D4×M1611yescharacteristic
323C2×C61#294611yescharacteristic
324C2×C61#295511yescharacteristic
325C2×AGL(1,ℤ/8)#692311yescharacteristic
326C4×#41—611no—
327C2×C4×D4—611no—
328#446—611no—
329#446—611no—
330#442—611no—
331#442—611no—
332#440—611no—
333#440—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×D8—611no—
337C4×D8—611no—
338#436—611no—
339#436—611no—
340#435—1221no—
341#435—1221no—
342#435—1221no—
343#435—1221no—
344C8×D4—611no—
345C8×D4—611no—
346C8×D4—611no—
347C8×D4—611no—
348#432—611no—
349#432—1221no—
350#431—611no—
351#431—611no—
352#428—611no—
353#423—611no—
354C2×C4⋊C8—1221no—
355#418—611no—
356#417—611no—
357#407—1221no—
358C2×C22⋊C8—1221no—
359C4×M16—611no—
360#335—611no—
361#335—611no—
362#334—611no—
363#334—611no—
364Q8⋊C8—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372#322—611no—
373#322—611no—
374#322—611no—
375C82—611no—
376C61#1805011yescharacteristic, complemented, Sylow
377C2×#41C2×#692311yescharacteristic
378C22×D4C2×#692311yescharacteristic
379C23×C4—1221no—
380AGL(1,ℤ/8)—24no—
381C2×SD16C2×#692311yescharacteristic
382C2×SD16C2×#692311yescharacteristic
383C2×D8C2×#692311yescharacteristic
384C2×D8C2×#692311yescharacteristic
385C2×M16—611no—
386C2×M16—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16C2×#692311yescharacteristic
390C22×C8—1221no—
391C22×C8—1221no—
392C22×C8—1221no—
393C22×C8—1221no—
394C4×Q8—611no—
395C4×Q8—611no—
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4—1221no—
406#117—611no—
407#117—1221no—
408C2×C4⋊C4—611no—
409C2×C22⋊C4—1221no—
410C2×C42—611no—
411C2×C42—1221no—
412#107—611no—
413#107—611no—
414#106—611no—
415#106—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426#103—611no—
427#103—611no—
428#102—611no—
429#102—611no—
430#102—611no—
431#102—611no—
432#102—611no—
433#102—611no—
434C22⋊C8—611no—
435C22⋊C8—611no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444#97—611no—
445#97—611no—
446#97—1221no—
447#97—1221no—
448#97—1221no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—611no—
455C24—21no—
456#41—22no—
457#41—41no—
458C2×Q8C22×#692311yescharacteristic
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4C22×#692311yescharacteristic
464C2×D4C22×#692311yescharacteristic
465C2×D4C22×#692311yescharacteristic
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—4881no—
477C22×C4—4881no—
478C22×C4C22×#692311yescharacteristic
479SD16—22no—
480SD16—22no—
481D8—22no—
482D8#1175821714yescomplemented
483M16—22no—
484M16—4881no—
485M16—4881no—
486M16—4881no—
487M16—2441no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—4881no—
501C2×C8—4881no—
502C2×C8—4881no—
503C2×C8—4881no—
504C2×C8C22×#692311yescharacteristic
505C2×C8C22×#692311yescharacteristic
506C4⋊C4—611no—
507C4⋊C4—611no—
508C4⋊C4—611no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—42no—
531C23—42no—
532C23—81no—
533C23D4×#692311yescharacteristic
534Q8—21no—
535Q8—21no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—21no—
540D4—42no—
541D4—42no—
542D4C2×#1175821712yes—
543D4C2×#1175821712yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—4881no—
559C2×C4—4881no—
560C2×C4—4881no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2442no—
569C2×C4D4×#692311yescharacteristic
570C2×C4C23×#692311yescharacteristic
571C8—21no—
572C8—1221no—
573C8—1221no—
574C8—1221no—
575C8—1221no—
576C8—1221no—
577C8—1221no—
578C8—2441no—
579C8—2441no—
580C8—2441no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8C2×#1175821712yes—
585C22—21no—
586C22—22no—
587C22—22no—
588C22—21no—
589C22—21no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—84no—
598C22C2×D4×#692311yescharacteristic
599C4—21no—
600C4—41no—
601C4—611no—
602C4—611no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—2441no—
607C4—2441no—
608C4—2441no—
609C4—2441no—
610C4C22×#1175821711yescharacteristic
611C4#1684084011yescharacteristic
612C2—41no—
613C2—42no—
614C2—42no—
615C2—81no—
616C2D4×#1175821711yescharacteristic
617C2#2146257611yescharacteristic
618C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC2×AGL(1,ℤ/8)×C61C4
socleC22×C61C2×C4×D4
cosocleC4×Dic61C24
hypercenterC2×AGL(1,ℤ/8)#6923
hypocenterC61#18050
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21462745C2semidirect
2. #21462675C2semidirect
3. #21461462C2semidirect
4. #21461455C2semidirect
5. #21461440C2semidirect
6. #21461436C2semidirect
7. #21461424C2semidirect
8. #21461420C2semidirect
9. #21461415C2semidirect
10. D8×#92420C2semidirect
11. #21461383C2semidirect
12. #21461150C2semidirect
13. #21461144C2semidirect
14. #21461038C2semidirect
16. #16840813C22semidirect
17. #16840813C22semidirect
26. D4×#92420C22semidirect
27. D4×#92420C22semidirect
36. #16840626C22semidirect
37. #16840625C22semidirect
38. #16840624C22semidirect
39. #16840623C22semidirect
40. Q8⋊#92420C22semidirect
41. Q8⋊#92420C22semidirect
46. D4⋊#92420C22semidirect
47. D4⋊#92420C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. #16840603C22semidirect
51. C8×#92420C22semidirect
189. D8×C61M16semidirect
376. C61#18050semidirect
482. D8#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627452
#214626752
#214614622
#214614552
#214614402
#214614362
#214614242
#214614202
#214614152
D8×#924202
#214613832
#214611502
#214611442
#214610382
AGL(1,ℤ/8)×Dic612
#1805061