Finite Group Databasebeta

← #29743474

Subgroups — #29743474

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#21455670C211yesmaximal, characteristic, complemented
2#21455653C211yesmaximal, characteristic, complemented
3#21454647C211yesmaximal, characteristic
4#21447341C211yesmaximal, characteristic, complemented
5#21447330C211yesmaximal, characteristic, complemented
6#21447176C211yesmaximal, characteristic, complemented
7#21447167C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446420C211yesmaximal, characteristic, complemented
10#21446327C211yesmaximal, characteristic
11#21446195C211yesmaximal, characteristic
12#21446094C211yesmaximal, characteristic, complemented
13#21446052C211yesmaximal, characteristic, complemented
14#21446027C211yesmaximal, characteristic, complemented
15C61×#2912C211yesmaximal, 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#16839627C2211yescharacteristic
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×#106×C61—21no—
66C2×#106×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×#11758198—41no—
88C22⋊C4×D61—41no—
89C2×C4×Dic61C2311yescharacteristic
90#14519712—22no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519708—21no—
97#14519708—21no—
98#14519708—21no—
99C4⋊#92418—21no—
100C4⋊#92418—21no—
101#14519706—21no—
102#14519706—21no—
103#14519704—22no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519701D411yescharacteristic
107#14519699—21no—
108#14519699—21no—
109#14519698—21no—
110#14519698D411yescharacteristic
111#14519698D411yescharacteristic
112#14519697—21no—
113#14519697—21no—
114#14519697—21no—
115#14519697—21no—
116#14519693—22no—
117#14519688—21no—
118C4×#92418—21no—
119#14519684—41no—
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×C61D411yescharacteristic
144#107×C61D411yescharacteristic
145#106×C61—21no—
146#106×C61—21no—
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—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194C2×Q812yes—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C23×C61—21no—
233C23×C61H(5,𝔽2)11yescharacteristic
234C22×D61—22no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—81no—
238C22×D61—81no—
239Q8×C61—21no—
240Q8×C61C22⋊Q812yes—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—21no—
244D4×C61C22⋊Q812yes—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—81no—
255C2×C4×C61—81no—
256C2×C4×C61—161no—
257C2×C4×C61—161no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×Q811yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—81no—
271D244—22no—
272D244—22no—
273D244—41no—
274D244—41no—
275C4×D61—161no—
276C4×D61—161no—
277Dic122—161no—
278C8×C61—21no—
279C8×C61—22no—
280C8×C61—21no—
281#92418—22no—
282#92418—21no—
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#26402—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61#53911yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61#55411yescharacteristic
295C2×D61—42no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—82no—
299C2×D61—161no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61C2×C22⋊Q811yescharacteristic
308C4×C61#52511yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61D4×Q811yescharacteristic
311Dic61—41no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315#3083—611no—
316#3081—611no—
317#2946—611no—
318#2912D6111yescharacteristic, complemented
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—42no—
333C2×C61#244911yescharacteristic
334C2×C61#328811yescharacteristic
335C2×C61#329011yescharacteristic
336D61—82no—
337C2×AGL(1,ℤ/8)—611no—
338C2×#131C2×D6111yescharacteristic
339#544—611no—
340#520C2×D6111yescharacteristic
341#520C2×D6111yescharacteristic
342#518—611no—
343#479—1221no—
344#478—1221no—
345#478—1221no—
346#478—1221no—
347#477—1221no—
348#476—1221no—
349#474—1221no—
350#474—1221no—
351#442—1221no—
352#440—1221no—
353C4×SD16—1221no—
354C4×D8—1221no—
355C2×#108—611no—
356#428—21no—
357#428—21no—
358#428—611no—
359#428—611no—
360#428—611no—
361#428C2×D6111yescharacteristic
362#428C2×D6111yescharacteristic
363#427—21no—
364#427C2×D6111yescharacteristic
365C2×#107—611no—
366C2×#106—21no—
367C2×#106C2×D6111yescharacteristic
368#424—611no—
369#424—611no—
370C2×#104—611no—
371#418—611no—
372#417—611no—
373#416—611no—
374C2×#102—611no—
375C2×#102—611no—
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×#41C22×D6111yescharacteristic
390C22×D4—611no—
391AGL(1,ℤ/8)—614no—
392AGL(1,ℤ/8)—1222no—
393C2×SD16—1221no—
394C2×D8—1221no—
395#131—22no—
396#131—22no—
397C2×M16—21no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—611no—
401C2×M16C22×D6111yescharacteristic
402C22×C8—21no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—1221no—
406C4⋊Q8—1221no—
407#125—1221no—
408#125—1221no—
409C22⋊Q8—2441no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—41no—
415C4×D4—41no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420#117—21no—
421#117—21no—
422#117—611no—
423#117C22×D6111yescharacteristic
424#117C22×D6111yescharacteristic
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—2441no—
431C2×C22⋊C4—2441no—
432C2×C42—611no—
433#108—1222no—
434#107—21no—
435#107—21no—
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—21no—
445#106—21no—
446#106—21no—
447#106—1221no—
448#106—1221no—
449#106—1221no—
450#106C22⋊D6111yescharacteristic
451#106C22⋊D6111yescharacteristic
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456#104—1222no—
457#103—611no—
458#103—611no—
459#103—1221no—
460#102—611no—
461#102—611no—
462#102—1221no—
463#102—1221no—
464#102—1221no—
465#102—1221no—
466#102—1221no—
467#102—1221no—
468#102—1221no—
469#100—1222no—
470#97—1221no—
471C4×C8—1221no—
472#95—2441no—
473C24—1221no—
474#41—21no—
475#41—21no—
476#41#1175822212yes—
477#41Q8×D6112yes—
478C2×Q8—2441no—
479C2×Q8D4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4—612no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4D4×D6111yescharacteristic
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497C22×C4C23×D6111yescharacteristic
498SD16—1224no—
499D8—1224no—
500M16—24no—
501M16—21no—
502M16—612no—
503M16—1221no—
504M16—1221no—
505M16—1221no—
506M16—1221no—
507M16—1221no—
508M16#1175822212yes—
509C2×C8—24no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—611no—
514C2×C8—611no—
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—21no—
525C4⋊C4—21no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—611no—
529C4⋊C4—611no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2441no—
533C4⋊C4—2441no—
534C4⋊C4—2441no—
535C4⋊C4C2×C22⋊D6111yescharacteristic
536C4⋊C4C2×C22⋊D6111yescharacteristic
537C4⋊C4C2×C22⋊D6111yescharacteristic
538C4⋊C4C2×C22⋊D6111yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—1221no—
544C22⋊C4—4882no—
545C22⋊C4—2441no—
546C22⋊C4—2441no—
547C42—21no—
548C42—21no—
549C42—41no—
550C42—41no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C23—21no—
555C23—1222no—
556C23—4881no—
557C23—4881no—
558C23—2441no—
559C23—2441no—
560C23#1451982211yescharacteristic
561Q8—21no—
562Q8—9761no—
563Q8#1451975512yes—
564D4—22no—
565D4—22no—
566D4—21no—
567D4—1222no—
568D4—1222no—
569D4—2441no—
570D4—2441no—
571D4#1451975512yes—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—161no—
584C2×C4—161no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—9761no—
588C2×C4—9761no—
589C2×C4—4881no—
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×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4#1451982511yescharacteristic
600C8—21no—
601C8—22no—
602C8—21no—
603C8—1222no—
604C8—1221no—
605C8—1221no—
606C8—2441no—
607C8—2441no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—9761no—
614C22—4882no—
615C22—4881no—
616C22—4881no—
617C22—2442no—
618C22#1684035911yescharacteristic
619C22D4×C22⋊D6111yescharacteristic
620C22#1684046711yescharacteristic
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—4881no—
629C4—4881no—
630C4—2441no—
631C4—2441no—
632C4C2×#1451975511yescharacteristic
633C4#1684027311yescharacteristic
634C4#1684041311yescharacteristic
635C4Q8×C22⋊D6111yescharacteristic
636C2—21no—
637C2—21no—
638C2—42no—
639C2—4882no—
640C2#2144962811yescharacteristic
641C2#2145758111yescharacteristic
642C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2912C2
socleC22×C61D42
prime radicalC2×#14519802C22
hypercenter#2912D61
hypocenterC61#26402
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
1. #21455670C2semidirect
2. #21455653C2semidirect
4. #21447341C2semidirect
5. #21447330C2semidirect
6. #21447176C2semidirect
7. #21447167C2semidirect
8. #21446601C2semidirect
9. #21446420C2semidirect
12. #21446094C2semidirect
13. #21446052C2semidirect
14. #21446027C2semidirect
15. C61×#2912C2semidirect
69. #14519802Q8semidirect
318. #2912D61semidirect
388. C61#26402semidirect

Maximal subgroups

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

subgroupindex
#214556702
#214556532
#214546472
#214473412
#214473302
#214471762
#214471672
#214466012
#214464202
#214463272
#214461952
#214460942
#214460522
#214460272
C61×#29122
#2640261