Finite Group Databasebeta

← #29662866

Subgroups — #29662866

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456270C211yesmaximal, characteristic
2#21455849C211yesmaximal, characteristic, complemented
3#460×D61C211yesmaximal, characteristic, complemented
4Q8×#11758189C211yesmaximal, characteristic, complemented
5#21445302C211yesmaximal, characteristic, complemented
6#21445300C211yesmaximal, characteristic, complemented
7#21445292C211yesmaximal, characteristic, complemented
8#21445290C211yesmaximal, characteristic, complemented
9#21445177C211yesmaximal, characteristic, complemented
10#21445167C211yesmaximal, characteristic, complemented
11#21445165C211yesmaximal, characteristic, complemented
12#21444570C211yesmaximal, characteristic, complemented
13#21444493C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1668C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C2×#14519801—21no—
18C2×SD16×D61—21no—
19#16840310C2211yescharacteristic
20#127×D61C2211yescharacteristic
21C4×Q8×D61C2211yescharacteristic
22#16840150—21no—
23#16840150—21no—
24#16840138C2211yescharacteristic
25#16840135C2211yescharacteristic
26#16840134C2211yescharacteristic
27#16840079—21no—
28#16840048C2211yescharacteristic, complemented
29#16840047C2211yescharacteristic, complemented
30#16839963C2211yescharacteristic
31#16839962C2211yescharacteristic
32#16839961—21no—
33#16839960—21no—
34#16839959C2211yescharacteristic
35#16839959C2211yescharacteristic, complemented
36#16839899—21no—
37#16839892—21no—
38#16839892C2211yescharacteristic, complemented
39#16839889—21no—
40#16839889C2211yescharacteristic, complemented
41Q8×#92418C2211yescharacteristic, complemented
42#16839853—21no—
43SD16×Dic61—21no—
44#16839814C2211yescharacteristic
45#16839784—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#102×D61—21no—
49C4×#11758189—21no—
50C4×#11758189C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839513C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#493C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#460C2211yescharacteristic, complemented
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, complemented
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61D411yescharacteristic
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519801—44no—
73#14519801—44no—
74C2×#11758190—21no—
75C2×#11758190—21no—
76C2×#11758190—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758200—21no—
80C2×#11758200—21no—
81C2×#11758201—21no—
82SD16×D61—44no—
83C2×#11758199—21no—
84C2×#11758199—21no—
85C2×#11758189—21no—
86C2×#11758189D411yescharacteristic
87C2×#11758189D411yescharacteristic
88C2×C8×D61—21no—
89#14519777C2311yescharacteristic
90#14519774C2311yescharacteristic
91#14519773—21no—
92#14519773—21no—
93#14519773—21no—
94#14519764—21no—
95#14519748—21no—
96#14519748—21no—
97Q8×Dic61D411yescharacteristic
98C4×Dic122—21no—
99C4×Dic122C2311yescharacteristic
100#14519742—21no—
101#14519742D411yescharacteristic
102D4×Dic61—21no—
103D4×Dic61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61D411yescharacteristic
106C42×D61—21no—
107C42×D61C2311yescharacteristic
108#14519709—21no—
109#14519708—21no—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519702—21no—
119#14519701—21no—
120#14519699—21no—
121#14519698—21no—
122#14519698—21no—
123#14519697—21no—
124#14519689—21no—
125#14519687—21no—
126#14519687—21no—
127#14519687D411yescharacteristic
128#14519687D411yescharacteristic
129C4×#92418—21no—
130C4×#92418C2311yescharacteristic
131C8×Dic61—21no—
132C2×SD16×C61—21no—
133C2×SD16×C61—21no—
134C2×D8×C61—21no—
135C2×D8×C61—21no—
136#127×C61C2311yescharacteristic
137#127×C61C2311yescharacteristic
138C4×Q8×C61C2311yescharacteristic
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143#102×C61—21no—
144#102×C61—21no—
145C4×C8×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C23×D61—41no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159C22×Dic61—41no—
160Q8×D61—22no—
161Q8×D61—41no—
162#11758224—42no—
163#11758224—42no—
164D4×D61—42no—
165D4×D61—42no—
166D4×D61—42no—
167D4×D61—42no—
168D4×D61—82no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61#4111yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C2×D411yescharacteristic
180SD16×C61—42no—
181SD16×C61—42no—
182D8×C61—42no—
183D8×C61—42no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C4⋊C4×C61—21no—
190C4⋊C4×C61C2×D411yescharacteristic
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194C22⋊Dic61—41no—
195#11758201—42no—
196#11758200—42no—
197#11758200—42no—
198#11758200—42no—
199#11758200—42no—
200#11758199—42no—
201#11758199—42no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61#4111yescharacteristic
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220#11758190—42no—
221#11758190—42no—
222#11758190—42no—
223#11758189—22no—
224#11758189—42no—
225#11758189SD1614yescomplemented
226C8×D61—42no—
227C23×C61—41no—
228C23×C61—41no—
229C22×D61—41no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61—82no—
233C22×D61#12111yescharacteristic
234Q8×C61—21no—
235Q8×C61AGL(1,ℤ/8)12yescomplemented
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—81no—
245D4×C61—81no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C22⋊D61—82no—
252C22⋊D61—82no—
253C22⋊D61—82no—
254C22⋊D61—82no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—82no—
259C2×Dic61#12111yescharacteristic
260C2×Dic61#12111yescharacteristic
261C2×Dic61#12111yescharacteristic
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—81no—
267C4×D61—21no—
268C4×D61—21no—
269C4×D61—41no—
270C4×D61—42no—
271C4×D61—81no—
272C4×D61C2×SD1612yes—
273C4×D61AGL(1,ℤ/8)12yes—
274Dic122—21no—
275Dic122—21no—
276Dic122—42no—
277Dic122—41no—
278Dic122—41no—
279C8×C61—21no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61—41no—
283C8×C61C2×SD1612yescomplemented
284#92418—21no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418C2×SD1612yescomplemented
289#19170—611nomaximal, Sylow
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61C2×#12111yescharacteristic
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—164no—
303C2×D61#46012yes—
304C4×C61—22no—
305C4×C61—21no—
306C4×C61—41no—
307C4×C61C22×SD1611yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61#57511yescharacteristic
311Dic61—21no—
312Dic61—21no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316Dic61#46012yes—
317#3151—611no—
318#3103—611no—
319C2×#460—611no—
320Q8×M16—611no—
321#1688—611no—
322#1688—611no—
323#1686—611no—
324#1686—611no—
325#1668—611no—
326#1668D6111yescharacteristic, complemented
327#1666—611no—
328#1666—611no—
329#1565—611no—
330#1547—611no—
331#1545—611no—
332C2×C61—161no—
333C2×C61—161no—
334C2×C61C2×#46011yescharacteristic
335C2×C61#305511yescharacteristic
336C2×C61#306811yescharacteristic
337D61—22no—
338D61—161no—
339C2×AGL(1,ℤ/8)—1221no—
340C2×AGL(1,ℤ/8)—1221no—
341C22×SD16—1221no—
342#532—611no—
343C2×#127—611no—
344C2×C4×Q8—611no—
345#496—1221no—
346#496—1221no—
347#493—611no—
348#493C2×D6111yescharacteristic
349#492—611no—
350#492—611no—
351#481—1221no—
352#475—611no—
353#475—611no—
354#460—611no—
355#460—611no—
356#460—611no—
357#460—611no—
358#460—1221no—
359#460—1221no—
360#460C2×D6111yescharacteristic
361#460C2×D6111yescharacteristic, complemented
362#448—1221no—
363#446—611no—
364#446—611no—
365#446—1221no—
366#446—1221no—
367C8×Q8—611no—
368C8×Q8C2×D6111yescharacteristic, complemented
369C4×SD16—1221no—
370C4×SD16—1221no—
371#431—611no—
372#423—611no—
373#423—1221no—
374C2×C4⋊C8—611no—
375C2×#102—1221no—
376C4×M16—611no—
377C4×M16—1221no—
378#335—611no—
379#335C2×D6111yescharacteristic, complemented
380#334—611no—
381#334—611no—
382#332—611no—
383#332—611no—
384#331—611no—
385#331—611no—
386#331—611no—
387#331—611no—
388#331C2×D6111yescharacteristic
389#331C2×D6111yescharacteristic, complemented
390C61#1917011yescharacteristic, complemented, Sylow
391C2×#41—1221no—
392C22×Q8—611no—
393C22×D4—1221no—
394C22×D4—1221no—
395C22×D4—1221no—
396AGL(1,ℤ/8)—2444no—
397AGL(1,ℤ/8)—2444no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×SD16—1221no—
407C2×SD16—1221no—
408C2×SD16—2444no—
409C2×D8—21no—
410C2×D8—21no—
411C2×D8—1221no—
412C2×D8—1221no—
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—1221no—
416C22×C8—1221no—
417C4⋊Q8—611no—
418#127—611no—
419#127—1221no—
420#127—1221no—
421#127—1221no—
422#127C22×D6111yescharacteristic
423#127C22×D6111yescharacteristic
424#124—1221no—
425#121—1221no—
426#121—1221no—
427C4×Q8—611no—
428C4×Q8—611no—
429C4×Q8—611no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8C22×D6111yescharacteristic
433C4×D4—1221no—
434C4×D4—1221no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C42—611no—
438C2×C42—1221no—
439#106—1221no—
440#106—1221no—
441C4⋊C8—21no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—611no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8—1221no—
450C4⋊C8C22×D6111yescharacteristic
451C4⋊C8C22×D6111yescharacteristic
452C4⋊C8C22×D6111yescharacteristic
453#103—1221no—
454#103—1221no—
455#102—21no—
456#102—21no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460#102—1221no—
461C22⋊C8—1221no—
462#97—611no—
463#97—611no—
464#97—1221no—
465#97—1221no—
466C4×C8—21no—
467C4×C8—611no—
468C4×C8—1221no—
469C4×C8—1221no—
470C4×C8C22×D6111yescharacteristic
471C24—2441no—
472#41—2442no—
473#41—2442no—
474C2×Q8—611no—
475C2×Q8—1221no—
476C2×Q8—1222no—
477C2×Q8—1221no—
478C2×Q8—2441no—
479C2×Q8D4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—4882no—
490C2×D4—2442no—
491C2×D4—2442no—
492C2×D4—2441no—
493C2×D4—2441no—
494C2×D4—2441no—
495C2×D4—2442no—
496C2×D4—2442no—
497C2×D4—2441no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—2441no—
504SD16—42no—
505SD16—42no—
506SD16—2442no—
507SD16—2442no—
508SD16—2442no—
509SD16—2442no—
510SD16—2442no—
511SD16—2442no—
512SD16—2442no—
513SD16—2442no—
514D8—42no—
515D8—42no—
516D8—2442no—
517D8—2442no—
518M16—614no—
519M16—1222no—
520M16—2442no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—611no—
525C2×C8—611no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—2442no—
530C2×C8D4×D6111yescharacteristic
531C2×C8D4×D6111yescharacteristic
532C4⋊C4—21no—
533C4⋊C4—611no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4D4×D6111yescharacteristic
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C42—21no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42C23×D6111yescharacteristic
551C23—41no—
552C23—41no—
553C23—611no—
554C23—4882no—
555C23—4882no—
556C23—4882no—
557C23—2441no—
558Q8—21no—
559Q8—1221no—
560Q8—1221no—
561Q8—2442no—
562Q8—2441no—
563Q8—2441no—
564Q8#1451980112yescomplemented
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—81no—
574D4—81no—
575D4—4881no—
576D4—4882no—
577D4—4882no—
578D4—4882no—
579D4—4882no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—611no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—4881no—
596C2×C4—4882no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C8—21no—
604C8—41no—
605C8—41no—
606C8—41no—
607C8—612no—
608C8—1221no—
609C8—2441no—
610C8—2441no—
611C8—2441no—
612C8SD16×D6112yescomplemented
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—612no—
620C22—1221no—
621C22—1221no—
622C22—9764no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22#121×D6111yescharacteristic
627C4—22no—
628C4—21no—
629C4—41no—
630C4—612no—
631C4—1221no—
632C4—1221no—
633C4—4881no—
634C4—2441no—
635C4—2441no—
636C4C2×SD16×D6111yescharacteristic
637C4AGL(1,ℤ/8)×D6111yescharacteristic
638C4C2×#1451980111yescharacteristic
639C4#1684055911yescharacteristic
640C2—161no—
641C2—161no—
642C2—1222no—
643C2—9761no—
644C2#460×D6111yescharacteristic
645C2#2145543511yescharacteristic
646C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1668C2
socleC22×C61C2×#121
prime radical#21456270C2
hypercenter#1668D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. #21455849C2semidirect
3. #460×D61C2semidirect
4. Q8×#11758189C2semidirect
5. #21445302C2semidirect
6. #21445300C2semidirect
7. #21445292C2semidirect
8. #21445290C2semidirect
9. #21445177C2semidirect
10. #21445167C2semidirect
11. #21445165C2semidirect
12. #21444570C2semidirect
13. #21444493C2semidirect
14. #21444485C2semidirect
15. C61×#1668C2semidirect
28. #16840048C22semidirect
29. #16840047C22semidirect
35. #16839959C22semidirect
38. #16839892C22semidirect
40. #16839889C22semidirect
41. Q8×#92418C22semidirect
51. #16839514C22semidirect
52. #16839513C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839507C22semidirect
57. #16839506C22semidirect
62. C61×#460C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#335C22semidirect
66. C61×#331C22semidirect
225. #11758189SD16semidirect
235. Q8×C61AGL(1,ℤ/8)semidirect
283. C8×C61C2×SD16semidirect
288. #92418C2×SD16semidirect
326. #1668D61semidirect
361. #460C2×D61semidirect
368. C8×Q8C2×D61semidirect
379. #335C2×D61semidirect
389. #331C2×D61semidirect
390. C61#19170semidirect
564. Q8#14519801semidirect
612. C8SD16×D61semidirect

Maximal subgroups

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

subgroupindex
#214562702
#214558492
#460×D612
Q8×#117581892
#214453022
#214453002
#214452922
#214452902
#214451772
#214451672
#214451652
#214445702
#214444932
#214444852
C61×#16682
#1917061