Finite Group Databasebeta

← #29743471

Subgroups — #29743471

Order 15616.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455666C211yesmaximal, characteristic, complemented
2#21454651C211yesmaximal, characteristic
3#21454411C211yesmaximal, characteristic, complemented
4#21447342C211yesmaximal, characteristic, complemented
5#21447329C211yesmaximal, characteristic, complemented
6#21447179C211yesmaximal, characteristic, complemented
7#21447166C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446420C211yesmaximal, characteristic, complemented
10#21446330C211yesmaximal, characteristic
11#21446192C211yesmaximal, characteristic
12#21446093C211yesmaximal, characteristic, complemented
13#21446049C211yesmaximal, characteristic, complemented
14#21446029C211yesmaximal, characteristic, complemented
15C61×#3081C211yesmaximal, characteristic, complemented
16C2×#14519803C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840242C2211yescharacteristic
19#16840242C2211yescharacteristic
20#16840066—21no—
21#16840059—21no—
22#16840054—21no—
23#16840039—21no—
24#16839872—21no—
25#16839862—21no—
26C4×#11758200—21no—
27C4×#11758199—21no—
28C2×#14519712C2211yescharacteristic
29#16839805C2211yescharacteristic
30#16839805C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839805C2211yescharacteristic
33#16839803—21no—
34#16839803—21no—
35#16839797—21no—
36C2×#14519710C2211yescharacteristic
37C2×#14519709C2211yescharacteristic
38#106×D61—21no—
39#16839789C2211yescharacteristic
40C2×#14519703C2211yescharacteristic
41#16839761C2211yescharacteristic
42#16839756C2211yescharacteristic
43C2×#14519698C2211yescharacteristic
44C2×#14519694C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839665C2211yescharacteristic
47#16839627C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839530C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839527C2211yescharacteristic
52#16839521C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C61×#544C2211yescharacteristic
57C61×#518C2211yescharacteristic
58C61×#478—21no—
59C61×#478—21no—
60C61×#476—21no—
61C61×#474—21no—
62C61×#428C2211yescharacteristic
63C61×#427C2211yescharacteristic
64C61×#424C2211yescharacteristic
65C61×#418C2211yescharacteristic
66C2×#102×C61C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519803—22no—
69#14519803Q814yescomplemented
70C2×#11758200—21no—
71C2×#11758199—21no—
72#14519783—22no—
73#14519783—22no—
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758189—21no—
78C22×#92418C2311yescharacteristic
79C2×C8×D61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×D244—21no—
83C4×D244—21no—
84#14519737—41no—
85#14519737—41no—
86#14519735—21no—
87#14519735—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C4⋊C4×D61—21no—
91C4⋊C4×D61—21no—
92C2×C22⋊Dic61—41no—
93C2×C4×Dic61C2311yescharacteristic
94#14519712—22no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519709—21no—
102#14519709—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708—21no—
106#14519708—21no—
107C4⋊#92418—21no—
108C4⋊#92418—21no—
109#14519703—22no—
110#14519700—21no—
111#14519700D411yescharacteristic
112#14519700D411yescharacteristic
113#14519698—21no—
114#14519698—21no—
115#14519697—21no—
116#14519697D411yescharacteristic
117#14519697D411yescharacteristic
118#14519694—22no—
119#14519688—21no—
120C4×#92418—21no—
121#14519684—41no—
122C22×D4×C61C2311yescharacteristic
123C2×M16×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61—21no—
127#125×C61—21no—
128#125×C61—21no—
129C22⋊Q8×C61—41no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134#117×C61C2311yescharacteristic
135#117×C61C2311yescharacteristic
136#117×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C22⋊C4×C61—41no—
139#107×C61—21no—
140#107×C61D411yescharacteristic
141#107×C61D411yescharacteristic
142#106×C61—21no—
143#106×C61D411yescharacteristic
144#106×C61D411yescharacteristic
145C4⋊C8×C61—21no—
146C4⋊C8×C61—21no—
147#102×C61—21no—
148#102×C61—21no—
149#102×C61—21no—
150#102×C61—21no—
151C24×C61—21no—
152C2×Q8×C61—41no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61C2×Q812yes—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C22×C4×C61C2411yescharacteristic
163C2×C22⋊D61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167#11758222—21no—
168#11758222—21no—
169#11758222C2×Q812yes—
170#11758222#4112yes—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×Dic122C2×D411yescharacteristic
176M16×C61—21no—
177M16×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—41no—
198C22⋊C4×C61—41no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C22⋊Dic61—82no—
203#11758200—24no—
204#11758199—24no—
205#11758198—41no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211#11758196—41no—
212#11758196—41no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—41no—
216C4×Dic61—41no—
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194C2×Q812yes—
222#11758194#4112yes—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418C2×D411yescharacteristic
229C2×#92418C2×D411yescharacteristic
230#11758189—24no—
231C8×D61—24no—
232C23×C61—22no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—81no—
236C23×C61—81no—
237C23×C61H(5,𝔽2)11yescharacteristic
238C22×D61—21no—
239Q8×C61—161no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—81no—
252C2×C4×C61—161no—
253C2×C4×C61—161no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C22⋊D61—22no—
261C22⋊D61—22no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—81no—
269C2×Dic61—81no—
270D244—21no—
271D244C22⋊Q812yes—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—161no—
275C4×D61—161no—
276Dic122—21no—
277Dic122C22⋊Q812yes—
278C8×C61—21no—
279C8×C61—21no—
280C8×C61—41no—
281#92418—22no—
282#92418—22no—
283#92418—21no—
284#92418—21no—
285#92418—41no—
286#26402—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—42no—
290C22×C61—82no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—161no—
294C22×C61#53911yescharacteristic
295C22×C61D4211yescharacteristic
296C22×C61#55411yescharacteristic
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61C2×C22⋊Q811yescharacteristic
308C4×C61#52511yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61D4×Q811yescharacteristic
311Dic61—22no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315#3083—611no—
316#3081D6111yescharacteristic, complemented
317#2946—611no—
318#2912—611no—
319#2092—611no—
320#2090—611no—
321#2063—611no—
322#2061—611no—
323#1952—611no—
324#1922—611no—
325#1907—611no—
326#1883—611no—
327#1866—611no—
328#1860—611no—
329#1856—611no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—82no—
333C2×C61#244911yescharacteristic
334C2×C61#328811yescharacteristic
335C2×C61#329011yescharacteristic
336D61—42no—
337C2×AGL(1,ℤ/8)—611no—
338C2×#131—611no—
339#544C2×D6111yescharacteristic
340#520—611no—
341#520—611no—
342#518C2×D6111yescharacteristic
343#479—1221no—
344#478—21no—
345#478—21no—
346#478—1221no—
347#477—1221no—
348#476—21no—
349#474—21no—
350#474—1221no—
351#442—1221no—
352#440—1221no—
353C4×SD16—1221no—
354C4×D8—1221no—
355C2×#108—611no—
356#428—611no—
357#428—611no—
358#428—611no—
359#428—611no—
360#428—1221no—
361#428—1221no—
362#428C2×D6111yescharacteristic
363#427—1221no—
364#427C2×D6111yescharacteristic
365C2×#107—611no—
366C2×#106—611no—
367C2×#106—1221no—
368#424—611no—
369#424C2×D6111yescharacteristic
370C2×#104—611no—
371#418C2×D6111yescharacteristic
372#417—611no—
373#416—611no—
374C2×#102—611no—
375C2×#102C2×D6111yescharacteristic
376C2×#100—611no—
377#405—611no—
378#390—611no—
379#382—611no—
380#343—611no—
381#343—611no—
382#343—611no—
383#341—611no—
384#339—611no—
385#339—611no—
386#339—611no—
387#336—611no—
388C61#2640211yescharacteristic, complemented, Sylow
389C2×#41—611no—
390C22×D4C22×D6111yescharacteristic
391AGL(1,ℤ/8)—614no—
392AGL(1,ℤ/8)—1222no—
393C2×SD16—1221no—
394C2×D8—1221no—
395#131—1222no—
396#131—1222no—
397C2×M16—611no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—1221no—
401C2×M16C22×D6111yescharacteristic
402C22×C8—611no—
403C22×C8—1221no—
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—21no—
406C4⋊Q8—21no—
407#125—21no—
408#125—21no—
409C22⋊Q8—41no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—2441no—
419C4×D4—2441no—
420#117—1221no—
421#117—1221no—
422#117C22×D6111yescharacteristic
423#117C22×D6111yescharacteristic
424#117C22×D6111yescharacteristic
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—41no—
431C2×C22⋊C4—2441no—
432C2×C42—611no—
433#108—1222no—
434#107—21no—
435#107—1221no—
436#107—1221no—
437#107—1221no—
438#107—1221no—
439#107—1221no—
440#107—1221no—
441#107C22⋊D6111yescharacteristic
442#107C22⋊D6111yescharacteristic
443#106—21no—
444#106—1221no—
445#106—1221no—
446#106—1221no—
447#106—1221no—
448#106—1221no—
449#106—1221no—
450#106C22⋊D6111yescharacteristic
451#106C22⋊D6111yescharacteristic
452C4⋊C8—21no—
453C4⋊C8—21no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456#104—1222no—
457#103—611no—
458#103—611no—
459#103—1221no—
460#102—21no—
461#102—21no—
462#102—21no—
463#102—21no—
464#102—611no—
465#102—611no—
466#102—1221no—
467#102—1221no—
468#102—1221no—
469#100—1222no—
470#97—1221no—
471C4×C8—1221no—
472#95—2441no—
473C24—21no—
474#41—612no—
475#41—612no—
476#41—1221no—
477#41—1221no—
478C2×Q8—41no—
479C2×Q8—611no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—611no—
486C2×D4—1221no—
487C2×D4Q8×D6112yes—
488C22×C4—21no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497C22×C4C23×D6111yescharacteristic
498SD16—1224no—
499D8—1224no—
500M16—21no—
501M16—21no—
502M16—612no—
503M16—612no—
504M16—1224no—
505M16—1221no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—1224no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8C2×C22⋊D6111yescharacteristic
521C2×C8C2×C22⋊D6111yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2441no—
532C4⋊C4—2441no—
533C4⋊C4D4×D6111yescharacteristic
534C4⋊C4D4×D6111yescharacteristic
535C4⋊C4C2×C22⋊D6111yescharacteristic
536C4⋊C4C2×C22⋊D6111yescharacteristic
537C4⋊C4C2×C22⋊D6111yescharacteristic
538C4⋊C4C2×C22⋊D6111yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—4882no—
545C22⋊C4—2441no—
546C22⋊C4—2441no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—1221no—
551C42—1221no—
552C42—2441no—
553C42—2441no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—81no—
558C23—81no—
559C23—1221no—
560C23#1451982211yescharacteristic
561Q8—161no—
562Q8—612no—
563Q8—1221no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
568D4—612no—
569D4—1222no—
570D4—1222no—
571D4—1221no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—81no—
580C2×C4—161no—
581C2×C4—161no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—9761no—
586C2×C4—9761no—
587C2×C4—4881no—
588C2×C4—4881no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C2×Q8×D6111yescharacteristic
597C2×C4#41×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C2×C4#1451982511yescharacteristic
600C8—21no—
601C8—21no—
602C8—41no—
603C8—1222no—
604C8—1222no—
605C8—1221no—
606C8—1221no—
607C8—2441no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—82no—
612C22—81no—
613C22—81no—
614C22—161no—
615C22—1222no—
616C22—2441no—
617C22—2441no—
618C22#1684035811yescharacteristic
619C22D4×C22⋊D6111yescharacteristic
620C22#1684046611yescharacteristic
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—1222no—
629C4—4881no—
630C4—4881no—
631C4—4881no—
632C4C22⋊Q8×D6111yescharacteristic
633C4#1684026911yescharacteristic
634C4#1684041411yescharacteristic
635C4Q8×C22⋊D6111yescharacteristic
636C2—21no—
637C2—21no—
638C2—82no—
639C2—2442no—
640C2#2144962511yescharacteristic
641C2#2145758311yescharacteristic
642C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3081C2
socleC22×C61D42
prime radicalC2×#14519803C22
hypercenter#3081D61
hypocenterC61#26402
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3081

Complements

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

NH ≅ G/Ntype
1. #21455666C2semidirect
3. #21454411C2semidirect
4. #21447342C2semidirect
5. #21447329C2semidirect
6. #21447179C2semidirect
7. #21447166C2semidirect
8. #21446601C2semidirect
9. #21446420C2semidirect
12. #21446093C2semidirect
13. #21446049C2semidirect
14. #21446029C2semidirect
15. C61×#3081C2semidirect
69. #14519803Q8semidirect
316. #3081D61semidirect
388. C61#26402semidirect

Maximal subgroups

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

subgroupindex
#214556662
#214546512
#214544112
#214473422
#214473292
#214471792
#214471662
#214466012
#214464202
#214463302
#214461922
#214460932
#214460492
#214460292
C61×#30812
#2640261