Finite Group Databasebeta

← #29743444

Subgroups — #29743444

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455670C211yesmaximal, characteristic, complemented
2#21455653C211yesmaximal, characteristic, complemented
3#21454647C211yesmaximal, characteristic
4#21447344C211yesmaximal, characteristic, complemented
5#21447325C211yesmaximal, characteristic, complemented
6#21447176C211yesmaximal, characteristic, complemented
7#21447162C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446420C211yesmaximal, characteristic, complemented
10#21446327C211yesmaximal, characteristic
11#21446195C211yesmaximal, characteristic
12#21446094C211yesmaximal, characteristic, complemented
13#21446045C211yesmaximal, characteristic, complemented
14#21446027C211yesmaximal, characteristic, complemented
15C61×#2913C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840236C2211yescharacteristic
19#16840069—21no—
20#16840065—21no—
21#16840062—21no—
22#16840061—21no—
23#16840057—21no—
24#16840051—21no—
25#16840042—21no—
26#16840041—21no—
27#16839872—21no—
28#16839860—21no—
29C4×#11758201—21no—
30C4×#11758199—21no—
31C2×#14519712C2211yescharacteristic
32#16839805C2211yescharacteristic
33#16839802C2211yescharacteristic
34#16839802C2211yescharacteristic
35C2×#14519710C2211yescharacteristic
36#16839789C2211yescharacteristic
37#16839788C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39#16839766C2211yescharacteristic
40#16839763C2211yescharacteristic
41#16839758C2211yescharacteristic
42C2×#14519699C2211yescharacteristic
43C2×#14519697C2211yescharacteristic
44C2×#14519693C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839663C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839527C2211yescharacteristic
52#16839523C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#131×C61C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#520C2211yescharacteristic
59C61×#428—21no—
60C61×#428—21no—
61C61×#428C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C61×#427—21no—
64C61×#427C2211yescharacteristic
65C2×#107×C61—21no—
66C2×#107×C61C2211yescharacteristic
67C22×D244C2311yescharacteristic
68#14519802—22no—
69#14519802Q814yescomplemented
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519777—21no—
77#14519776—21no—
78#14519769—21no—
79#14519768—21no—
80#14519754—41no—
81C4×D244—21no—
82C4×D244—21no—
83#14519739—21no—
84#14519739—21no—
85#14519736C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—41no—
88C2×#11758198—41no—
89C22⋊C4×D61—41no—
90C2×C4×Dic61C2311yescharacteristic
91#14519712—22no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519708—21no—
98#14519708—21no—
99#14519708—21no—
100C4⋊#92418—21no—
101C4⋊#92418—21no—
102#14519706—21no—
103#14519706—21no—
104#14519704—22no—
105#14519701—21no—
106#14519701D411yescharacteristic
107#14519701D411yescharacteristic
108#14519699—21no—
109#14519699—21no—
110#14519698—21no—
111#14519698D411yescharacteristic
112#14519698D411yescharacteristic
113#14519697—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117#14519693—22no—
118#14519688—21no—
119C4×#92418—21no—
120C2×#41×C61C2311yescharacteristic
121#131×C61—22no—
122#131×C61—22no—
123C2×M16×C61—21no—
124C2×M16×C61C2311yescharacteristic
125C22×C8×C61—21no—
126C22×C8×C61C2311yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—41no—
132C4×D4×C61—41no—
133#117×C61—21no—
134#117×C61—21no—
135#117×C61C2311yescharacteristic
136#117×C61C2311yescharacteristic
137C2×C4⋊C4×C61—21no—
138C2×C4⋊C4×C61—21no—
139C2×C4⋊C4×C61C2311yescharacteristic
140C2×C4⋊C4×C61C2311yescharacteristic
141#107×C61—21no—
142#107×C61—21no—
143#107×C61—21no—
144#107×C61—21no—
145#107×C61D411yescharacteristic
146#107×C61D411yescharacteristic
147#106×C61—21no—
148#106×C61—21no—
149#106×C61D411yescharacteristic
150#106×C61D411yescharacteristic
151C23×D61—21no—
152#41×C61—21no—
153#41×C61—21no—
154#41×C61C2×Q812yes—
155#41×C61#4112yes—
156C2×Q8×C61C2×D411yescharacteristic
157C2×D4×C61—21no—
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—41no—
163C22×C4×C61—41no—
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244C2×Q812yes—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×Dic122—41no—
176M16×C61—24no—
177M16×C61—21no—
178M16×C61#4112yes—
179C2×C8×C61—24no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61—41no—
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—41no—
198C22⋊C4×C61—41no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—41no—
202C42×C61—41no—
203C22⋊Dic61—21no—
204#11758201—24no—
205#11758199—24no—
206#11758198—41no—
207#11758198—41no—
208#11758198—82no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—41no—
212C4⋊Dic61C2×D411yescharacteristic
213C4⋊Dic61C2×D411yescharacteristic
214#11758196—41no—
215#11758196—41no—
216#11758196—82no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194—21no—
225#11758194C2×Q812yes—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233C23×C61—21no—
234C23×C61H(5,𝔽2)11yescharacteristic
235C22×D61—22no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—81no—
239C22×D61—81no—
240Q8×C61—21no—
241Q8×C61C22⋊Q812yes—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—21no—
245D4×C61C22⋊Q812yes—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—81no—
256C2×C4×C61—81no—
257C2×C4×C61—161no—
258C2×C4×C61—161no—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×Q811yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—81no—
272D244—22no—
273D244—22no—
274D244—41no—
275D244—41no—
276C4×D61—161no—
277C4×D61—161no—
278Dic122—161no—
279C8×C61—21no—
280C8×C61—22no—
281C8×C61—21no—
282#92418—22no—
283#92418—21no—
284#92418—21no—
285#92418—41no—
286#92418—41no—
287#26400—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61#53911yescharacteristic
294C22×C61D4211yescharacteristic
295C22×C61#55411yescharacteristic
296C2×D61—42no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—82no—
300C2×D61—161no—
301C4×C61—22no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C22⋊Q811yescharacteristic
309C4×C61#52511yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61D4×Q811yescharacteristic
312Dic61—42no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316#3083—611no—
317#3081—611no—
318#2946—611no—
319#2913D6111yescharacteristic, complemented
320#2093—611no—
321#2089—611no—
322#2063—611no—
323#2060—611no—
324#1952—611no—
325#1922—611no—
326#1907—611no—
327#1883—611no—
328#1866—611no—
329#1859—611no—
330#1856—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—42no—
334C2×C61#244911yescharacteristic
335C2×C61#328911yescharacteristic
336C2×C61#329211yescharacteristic
337D61—82no—
338C2×AGL(1,ℤ/8)—611no—
339C2×#131C2×D6111yescharacteristic
340#544—611no—
341#520C2×D6111yescharacteristic
342#520C2×D6111yescharacteristic
343#518—611no—
344#479—1221no—
345#478—1221no—
346#478—1221no—
347#478—1221no—
348#477—1221no—
349#476—1221no—
350#474—1221no—
351#474—1221no—
352#442—1221no—
353#440—1221no—
354C4×SD16—1221no—
355C4×D8—1221no—
356C2×#108—611no—
357#428—21no—
358#428—21no—
359#428—611no—
360#428—611no—
361#428—611no—
362#428C2×D6111yescharacteristic
363#428C2×D6111yescharacteristic
364#427—21no—
365#427C2×D6111yescharacteristic
366C2×#107—21no—
367C2×#107—611no—
368C2×#107C2×D6111yescharacteristic
369#424—611no—
370#424—611no—
371C2×#104—611no—
372#418—611no—
373#417—611no—
374#416—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#100—611no—
378#405—611no—
379#390—611no—
380#384—611no—
381#343—611no—
382#343—611no—
383#343—611no—
384#341—611no—
385#339—611no—
386#339—611no—
387#339—611no—
388#336—611no—
389C61#2640011yescharacteristic, complemented, Sylow
390C2×#41C22×D6111yescharacteristic
391C22×D4—611no—
392AGL(1,ℤ/8)—614no—
393AGL(1,ℤ/8)—1222no—
394C2×SD16—1221no—
395C2×D8—1221no—
396#131—22no—
397#131—22no—
398C2×M16—21no—
399C2×M16—611no—
400C2×M16—611no—
401C2×M16—611no—
402C2×M16C22×D6111yescharacteristic
403C22×C8—21no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—1221no—
407C4⋊Q8—1221no—
408#125—1221no—
409#125—1221no—
410C22⋊Q8—2441no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—41no—
416C4×D4—41no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4—1221no—
421#117—21no—
422#117—21no—
423#117—611no—
424#117C22×D6111yescharacteristic
425#117C22×D6111yescharacteristic
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—2441no—
433C2×C22⋊C4—2441no—
434C2×C42—611no—
435#108—1222no—
436#107—21no—
437#107—21no—
438#107—21no—
439#107—21no—
440#107—1221no—
441#107—1221no—
442#107—1221no—
443#107—1221no—
444#107—1221no—
445#107C22⋊D6111yescharacteristic
446#107C22⋊D6111yescharacteristic
447#106—21no—
448#106—21no—
449#106—1221no—
450#106—1221no—
451#106—1221no—
452#106C22⋊D6111yescharacteristic
453#106C22⋊D6111yescharacteristic
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456C4⋊C8—1221no—
457C4⋊C8—1221no—
458#104—1222no—
459#103—611no—
460#103—611no—
461#103—1221no—
462#102—611no—
463#102—611no—
464#102—1221no—
465#102—1221no—
466#102—1221no—
467#102—1221no—
468#102—1221no—
469#102—1221no—
470#102—1221no—
471#100—1222no—
472#97—1221no—
473C4×C8—1221no—
474C24—1221no—
475#41—21no—
476#41—21no—
477#41#1175822212yes—
478#41Q8×D6112yes—
479C2×Q8—2441no—
480C2×Q8D4×D6111yescharacteristic
481C2×D4—21no—
482C2×D4—612no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4D4×D6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4C23×D6111yescharacteristic
499SD16—1224no—
500D8—1224no—
501M16—24no—
502M16—21no—
503M16—612no—
504M16—1221no—
505M16—1221no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509M16#1175822212yes—
510C2×C8—24no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8C2×C22⋊D6111yescharacteristic
522C2×C8C2×C22⋊D6111yescharacteristic
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—611no—
530C4⋊C4—611no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—4882no—
534C4⋊C4—2441no—
535C4⋊C4—2441no—
536C4⋊C4—2441no—
537C4⋊C4C2×C22⋊D6111yescharacteristic
538C4⋊C4C2×C22⋊D6111yescharacteristic
539C4⋊C4C2×C22⋊D6111yescharacteristic
540C4⋊C4C2×C22⋊D6111yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—1221no—
546C22⋊C4—4882no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—21no—
550C42—21no—
551C42—41no—
552C42—41no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C23—21no—
557C23—1222no—
558C23—4881no—
559C23—4881no—
560C23—2441no—
561C23—2441no—
562C23#1451982211yescharacteristic
563Q8—21no—
564Q8—9761no—
565Q8#1451975512yes—
566D4—22no—
567D4—22no—
568D4—21no—
569D4—1222no—
570D4—1222no—
571D4—2441no—
572D4—2441no—
573D4#1451975512yes—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—161no—
586C2×C4—161no—
587C2×C4—1222no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—9761no—
591C2×C4—9761no—
592C2×C4—4881no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C2×Q8×D6111yescharacteristic
599C2×C4C2×#1175822211yescharacteristic
600C2×C4C2×#1175822211yescharacteristic
601C2×C4#1451982511yescharacteristic
602C8—21no—
603C8—22no—
604C8—21no—
605C8—1222no—
606C8—1221no—
607C8—1221no—
608C8—2441no—
609C8—2441no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—9761no—
616C22—4882no—
617C22—4881no—
618C22—4881no—
619C22—2442no—
620C22#1684035911yescharacteristic
621C22D4×C22⋊D6111yescharacteristic
622C22#1684046711yescharacteristic
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—4881no—
631C4—2442no—
632C4—2441no—
633C4—2441no—
634C4C2×#1451975511yescharacteristic
635C4#1684027311yescharacteristic
636C4#1684041311yescharacteristic
637C4Q8×C22⋊D6111yescharacteristic
638C2—21no—
639C2—21no—
640C2—42no—
641C2—4882no—
642C2#2144962811yescharacteristic
643C2#2145759011yescharacteristic
644C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2913C2
socleC22×C61D42
prime radicalC2×#14519802C22
hypercenter#2913D61
hypocenterC61#26400
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. #21455670C2semidirect
2. #21455653C2semidirect
4. #21447344C2semidirect
5. #21447325C2semidirect
6. #21447176C2semidirect
7. #21447162C2semidirect
8. #21446601C2semidirect
9. #21446420C2semidirect
12. #21446094C2semidirect
13. #21446045C2semidirect
14. #21446027C2semidirect
15. C61×#2913C2semidirect
69. #14519802Q8semidirect
319. #2913D61semidirect
389. C61#26400semidirect

Maximal subgroups

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

subgroupindex
#214556702
#214556532
#214546472
#214473442
#214473252
#214471762
#214471622
#214466012
#214464202
#214463272
#214461952
#214460942
#214460452
#214460272
C61×#29132
#2640061