Finite Group Databasebeta

← #29693939

Subgroups — #29693939

Order 15616.

820 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456304C211yesmaximal, characteristic
2#21454734C211yesmaximal, characteristic, complemented
3#21454667C211yesmaximal, characteristic, complemented
4#21446726C211yesmaximal, characteristic, complemented
5#21446716C211yesmaximal, characteristic, complemented
6#21446349C211yesmaximal, characteristic, complemented
7#21446348C211yesmaximal, characteristic, complemented
8#21446346C211yesmaximal, characteristic, complemented
9#21446343C211yesmaximal, characteristic, complemented
10#21446308C211yesmaximal, characteristic, complemented
11#21446305C211yesmaximal, characteristic, complemented
12#21446300C211yesmaximal, characteristic, complemented
13#21445940C211yesmaximal, characteristic, complemented
14#21445575C211yesmaximal, characteristic, complemented
15C61×#1903C211yesmaximal, characteristic, complemented
16C2×#14519805C2211yescharacteristic
17C2×#14519797C2211yescharacteristic
18C22×#11758199C2211yescharacteristic
19#16840528C2211yescharacteristic
20#16840329C2211yescharacteristic
21#16840326C2211yescharacteristic
22#16840237C2211yescharacteristic
23#16840236C2211yescharacteristic
24#16840150C412yescomplemented
25#16840148C412yescomplemented
26#16840143C412yescomplemented
27#16840137C412yescomplemented
28#16839882—22no—
29#16839877—22no—
30#16839871—22no—
31#16839842—22no—
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839779C2211yescharacteristic
35#16839778C2211yescharacteristic
36#16839778C2211yescharacteristic
37#16839768C2211yescharacteristic
38#16839768C2211yescharacteristic
39#16839767C2211yescharacteristic
40C2×#14519693C2211yescharacteristic
41#16839736C2211yescharacteristic
42#16839735C2211yescharacteristic
43#16839720C2211yescharacteristic
44#16839720C2211yescharacteristic
45#16839720C2211yescharacteristic
46#16839533C2211yescharacteristic
47#16839519C2211yescharacteristic
48#16839519C2211yescharacteristic
49#16839519C2211yescharacteristic
50#16839519C2211yescharacteristic
51#16839518C2211yescharacteristic
52C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
53C61×#568C2211yescharacteristic
54C61×#421C2211yescharacteristic
55C61×#418C2211yescharacteristic
56C2×#100×C61C2211yescharacteristic
57C61×#344C2211yescharacteristic
58C61×#338C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60C22×D244C2311yescharacteristic
61#14519805—41no—
62#14519805—41no—
63#14519797—21no—
64#14519797—21no—
65#14519797—21no—
66#14519797—21no—
67C2×#11758202C2×C411yescharacteristic, complemented
68C2×#11758190C2×C412yescomplemented
69C2×#11758200C2×C411yescharacteristic, complemented
70C2×#11758201C2×C411yescharacteristic, complemented
71C2×D488C2×C412yescomplemented
72C2×#11758199—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199C2×C412yes—
77C2×#11758199C2×C412yes—
78C2×#11758199C2×C411yescharacteristic, complemented
79#14519785—41no—
80#14519783—22no—
81C2×#11758194C2311yescharacteristic
82M16×D61—41no—
83C22×#92418C2311yescharacteristic
84#14519773C2×C412yescomplemented
85#14519773C2×C412yescomplemented
86#14519766C2×C412yescomplemented
87#14519764C2×C412yescomplemented
88#14519751—41no—
89#14519748—41no—
90#14519746—41no—
91#14519745—41no—
92C4×D244—41no—
93D4×Dic61—22no—
94D4×Dic61—41no—
95#14519739—22no—
96#14519736C2311yescharacteristic
97#14519736C2311yescharacteristic
98#14519736C2311yescharacteristic
99#14519736C2311yescharacteristic
100C2×C22⋊Dic61—41no—
101C22⋊C4×D61—41no—
102#14519712—21no—
103#14519712—21no—
104#14519710—21no—
105#14519708—21no—
106#14519704—22no—
107#14519703—22no—
108#14519703—41no—
109#14519699—22no—
110#14519699—41no—
111#14519698—41no—
112#14519693—41no—
113#14519692—41no—
114#14519691—41no—
115#14519687—21no—
116#14519687C2×C412yescomplemented
117#14519687D412yescomplemented
118C8×Dic61—21no—
119C8×Dic61C2×C412yescomplemented
120C8×Dic61D412yescomplemented
121C2×#41×C61C2311yescharacteristic
122C22×D4×C61C2311yescharacteristic
123AGL(1,ℤ/8)×C61—41no—
124AGL(1,ℤ/8)×C61—41no—
125C2×SD16×C61C2×C412yescomplemented
126C2×D8×C61C2×C412yescomplemented
127#131×C61—22no—
128#131×C61—41no—
129C2×M16×C61—41no—
130C2×M16×C61C2311yescharacteristic
131C2×M16×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134#117×C61C2311yescharacteristic
135#104×C61—41no—
136#102×C61—22no—
137#100×C61—41no—
138C24×C61—21no—
139C23×D61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143#41×C61—21no—
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C22×C411yescharacteristic
152C2×D4×C61C22×C412yes—
153C22×C4×C61—21no—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—21no—
156C2×C22⋊D61—42no—
157C2×C22⋊D61—42no—
158C22×Dic61—42no—
159#11758222—21no—
160#11758222—21no—
161#11758222—21no—
162#11758222—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C22×C412yes—
169C2×D244C22×C411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—42no—
172C2×Dic122C22×C411yescharacteristic
173SD16×C61—42no—
174D8×C61—42no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181M16×C61—21no—
182M16×C61—21no—
183M16×C61—42no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—42no—
187C2×C8×C61C22×C412yes—
188C2×C8×C61C2×D412yes—
189C4⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C42×C61—21no—
192C22⋊Dic61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196C22⋊Dic61—42no—
197C22⋊Dic61—42no—
198#11758202—21no—
199#11758202—21no—
200#11758201—21no—
201#11758201—21no—
202#11758200—21no—
203#11758200—21no—
204#11758199—22no—
205#11758199—22no—
206#11758199—22no—
207#11758199—22no—
208#11758199—21no—
209#11758199—21no—
210#11758198—42no—
211#11758198—42no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61C22×C412yes—
219C4×Dic61C2×D412yes—
220#11758194—21no—
221#11758194—21no—
222C2×#92418—21no—
223C2×#92418C22×C411yescharacteristic
224C2×#92418C22×C411yescharacteristic
225C2×#92418C2×D411yescharacteristic
226C2×#92418C2×D411yescharacteristic
227C2×#92418#4111yescharacteristic
228C2×#92418#4111yescharacteristic
229D488—42no—
230#11758190—42no—
231#11758189—42no—
232C8×D61—42no—
233C23×C61—22no—
234C23×C61—21no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—161no—
238C23×C61H(5,𝔽2)11yescharacteristic
239C22×D61—22no—
240C22×D61—21no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—161no—
244Q8×C61—21no—
245Q8×C61—21no—
246D4×C61—22no—
247D4×C61—21no—
248D4×C61—21no—
249D4×C61—22no—
250D4×C61—41no—
251D4×C61—41no—
252D4×C61—41no—
253D4×C61—41no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C22⋊D61—41no—
265C22⋊D61—41no—
266C22⋊D61—162no—
267C22⋊D61—162no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—162no—
274D244—22no—
275D244—21no—
276D244—22no—
277D244—21no—
278D244—41no—
279D244—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—162no—
283Dic122—21no—
284Dic122—21no—
285C8×C61—22no—
286C8×C61—22no—
287C8×C61—41no—
288C8×C61—41no—
289#92418—41no—
290#92418C4×D411yescharacteristic
291#92418C4×D411yescharacteristic
292#92418C4×D411yescharacteristic
293#92418C4×D411yescharacteristic
294#22286—611nomaximal, Sylow
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—42no—
298C22×C61—81no—
299C22×C61—82no—
300C22×C61—82no—
301C22×C61—161no—
302C22×C61#51811yescharacteristic
303C22×C61#53511yescharacteristic
304C22×C61#54011yescharacteristic
305C2×D61—41no—
306C2×D61—42no—
307C2×D61—82no—
308C2×D61—81no—
309C2×D61—82no—
310C2×D61—161no—
311C4×C61—41no—
312C4×C61—81no—
313C4×C61C2×C4×D411yescharacteristic
314C4×C61#51811yescharacteristic
315C4×C61#53511yescharacteristic
316C4×C61#54011yescharacteristic
317Dic61—42no—
318Dic61—42no—
319Dic61—41no—
320Dic61—81no—
321Dic61—81no—
322#3155—611no—
323#2957—611no—
324#2948—611no—
325#1978—611no—
326#1976—611no—
327#1911—611no—
328#1911—611no—
329#1910—611no—
330#1910—611no—
331#1904—611no—
332#1904—611no—
333#1903—611no—
334#1903D6111yescharacteristic, complemented
335#1833—611no—
336#1747—611no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—81no—
341C2×C61—82no—
342C2×C61#232811yescharacteristic
343D61—81no—
344D61—82no—
345C2×AGL(1,ℤ/8)—611no—
346C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
347C2×#135—611no—
348C22×D8—611no—
349#568—611no—
350#568C2×D6111yescharacteristic
351#535—611no—
352#535—611no—
353#518—611no—
354#518—611no—
355#496—612no—
356#496—612no—
357#495—612no—
358#493—612no—
359#444—1222no—
360#443—1222no—
361#442—1222no—
362C4×D8—1222no—
363C2×#108—611no—
364#427—611no—
365#421—611no—
366#421—611no—
367#421—611no—
368#421C2×D6111yescharacteristic
369#418—611no—
370#418—611no—
371#418—611no—
372#418C2×D6111yescharacteristic
373C2×#100—611no—
374C2×#100C2×D6111yescharacteristic
375#410—611no—
376#410—611no—
377#405—611no—
378#405—611no—
379#405—611no—
380#344—611no—
381#344C2×D6111yescharacteristic
382#338C2×D6111yescharacteristic
383H(3,ℤ/4)—611no—
384H(3,ℤ/4)—611no—
385H(3,ℤ/4)—611no—
386H(3,ℤ/4)—611no—
387H(3,ℤ/4)—611no—
388C61#2228611yescharacteristic, complemented, Sylow
389C2×#41—611no—
390C2×#41C22×D6111yescharacteristic
391C22×D4—611no—
392C22×D4C22×D6111yescharacteristic
393AGL(1,ℤ/8)—41no—
394AGL(1,ℤ/8)—41no—
395AGL(1,ℤ/8)—2441no—
396AGL(1,ℤ/8)—2441no—
397#135—1221no—
398#135—1221no—
399#135—1221no—
400#135—1221no—
401C2×Q16—611no—
402C2×SD16—611no—
403C2×SD16—611no—
404C2×SD16—612no—
405C2×SD16C4×D6112yescomplemented
406C2×D8—612no—
407C2×D8—612no—
408C2×D8—611no—
409C2×D8—612no—
410C2×D8—1221no—
411C2×D8—1221no—
412C2×D8—1221no—
413C2×D8—1221no—
414C2×D8C4×D6112yescomplemented
415#131—22no—
416#131—41no—
417#131—1222no—
418#131—2441no—
419C2×M16—41no—
420C2×M16—611no—
421C2×M16—2441no—
422C2×M16C22×D6111yescharacteristic
423C2×M16C22×D6111yescharacteristic
424C2×M16C22×D6111yescharacteristic
425C2×M16C22×D6111yescharacteristic
426C22×C8—611no—
427#127—612no—
428#127—612no—
429#124—612no—
430#124—612no—
431#121—2441no—
432#121—2441no—
433#120—2441no—
434#120—2441no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—2441no—
438C4×D4—2441no—
439#117—611no—
440#117—611no—
441#117—611no—
442#117—611no—
443#117C22×D6111yescharacteristic
444C2×C22⋊C4—2441no—
445C2×C22⋊C4—2441no—
446#108—1221no—
447#108—1221no—
448#107—1221no—
449#106—1221no—
450#104—41no—
451#104—1222no—
452#104—1222no—
453#104—2441no—
454#102—22no—
455#102—1222no—
456#102—2441no—
457#102—2441no—
458#100—41no—
459#100—2441no—
460#99—2441no—
461#99—2441no—
462#97—612no—
463#97—612no—
464#97—1221no—
465C4×C8—612no—
466C4×C8—612no—
467C4×C8—1221no—
468C24—21no—
469C24—1221no—
470#41—21no—
471#41—21no—
472#41—21no—
473#41—21no—
474#41—1221no—
475#41—1221no—
476#41—1221no—
477#41—1221no—
478C2×Q8—611no—
479C2×Q8C2×C4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—612no—
487C2×D4—611no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—2442no—
495C2×D4—2442no—
496C2×D4C2×C4×D6112yes—
497C2×D4C2×C4×D6111yescharacteristic
498C22×C4—21no—
499C22×C4—1221no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4C23×D6111yescharacteristic
503Q16—1221no—
504Q16—1221no—
505SD16—42no—
506SD16—1221no—
507SD16—1221no—
508SD16—1221no—
509SD16—1221no—
510SD16—2442no—
511D8—42no—
512D8—1222no—
513D8—1222no—
514D8—1222no—
515D8—1222no—
516D8—1221no—
517D8—1221no—
518D8—2442no—
519M16—21no—
520M16—21no—
521M16—21no—
522M16—21no—
523M16—21no—
524M16—21no—
525M16—21no—
526M16—21no—
527M16—42no—
528M16—1221no—
529M16—1221no—
530M16—2442no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—42no—
534C2×C8—611no—
535C2×C8—611no—
536C2×C8—611no—
537C2×C8—611no—
538C2×C8—611no—
539C2×C8—611no—
540C2×C8—1221no—
541C2×C8—2442no—
542C2×C8C2×C4×D6112yes—
543C2×C8D4×D6112yes—
544C4⋊C4—21no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C22⋊C4—21no—
550C22⋊C4—1221no—
551C22⋊C4—1221no—
552C22⋊C4—1221no—
553C22⋊C4—1221no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C42—21no—
559C42—612no—
560C42—612no—
561C42—1221no—
562C42—1221no—
563C23—22no—
564C23—21no—
565C23—41no—
566C23—41no—
567C23—161no—
568C23—1222no—
569C23—1221no—
570C23—9761no—
571C23—2441no—
572C23—2441no—
573C23#1451982311yescharacteristic
574Q8—21no—
575Q8—21no—
576Q8—1221no—
577Q8—1221no—
578D4—22no—
579D4—21no—
580D4—21no—
581D4—22no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—1222no—
587D4—1221no—
588D4—1221no—
589D4—1222no—
590D4—9762no—
591D4—9762no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595D4—2441no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—9762no—
604C2×C4—9762no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4C22×C4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4#41×D6111yescharacteristic
612C2×C4#1451982211yescharacteristic
613C2×C4#1451982211yescharacteristic
614C2×C4#1451982211yescharacteristic
615C8—22no—
616C8—22no—
617C8—41no—
618C8—41no—
619C8—611no—
620C8—611no—
621C8—611no—
622C8—611no—
623C8—2441no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—81no—
628C22—82no—
629C22—82no—
630C22—161no—
631C22—9761no—
632C22—4881no—
633C22—4882no—
634C22—4882no—
635C22—2441no—
636C22—2442no—
637C22#1684023411yescharacteristic
638C22#1684032811yescharacteristic
639C22#1684036411yescharacteristic
640C4—41no—
641C4—81no—
642C4—4881no—
643C4—4881no—
644C4—2442no—
645C4—2442no—
646C4—2441no—
647C4C4×D4×D6111yescharacteristic
648C4#1684023511yescharacteristic
649C4#1684032711yescharacteristic
650C4#1684036211yescharacteristic
651C2—21no—
652C2—21no—
653C2—21no—
654C2—82no—
655C2—81no—
656C2—4882no—
657C2—4881no—
658C2#2144854011yescharacteristic
subgroupG/H
centerC2#21448540
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1903C2
socleC2×C61#2328
prime radical#21456304C2
cosocleC22×C4×C61C24
hypercenter#1903D61
hypocenterC61#22286
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1903

Complements

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

NH ≅ G/Ntype
2. #21454734C2semidirect
3. #21454667C2semidirect
4. #21446726C2semidirect
5. #21446716C2semidirect
6. #21446349C2semidirect
7. #21446348C2semidirect
8. #21446346C2semidirect
9. #21446343C2semidirect
10. #21446308C2semidirect
11. #21446305C2semidirect
12. #21446300C2semidirect
13. #21445940C2semidirect
14. #21445575C2semidirect
15. C61×#1903C2semidirect
24. #16840150C4semidirect
25. #16840148C4semidirect
26. #16840143C4semidirect
27. #16840137C4semidirect
67. C2×#11758202C2×C4semidirect
68. C2×#11758190C2×C4semidirect
69. C2×#11758200C2×C4semidirect
70. C2×#11758201C2×C4semidirect
71. C2×D488C2×C4semidirect
78. C2×#11758199C2×C4semidirect
84. #14519773C2×C4semidirect
85. #14519773C2×C4semidirect
86. #14519766C2×C4semidirect
87. #14519764C2×C4semidirect
116. #14519687C2×C4semidirect
117. #14519687D4semidirect
119. C8×Dic61C2×C4semidirect
120. C8×Dic61D4semidirect
125. C2×SD16×C61C2×C4semidirect
126. C2×D8×C61C2×C4semidirect
334. #1903D61semidirect
388. C61#22286semidirect
405. C2×SD16C4×D61semidirect
414. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214563042
#214547342
#214546672
#214467262
#214467162
#214463492
#214463482
#214463462
#214463432
#214463082
#214463052
#214463002
#214459402
#214455752
C61×#19032
#2228661