Finite Group Databasebeta

← #29721518

Subgroups — #29721518

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455642C211yesmaximal, characteristic, complemented
2C2×#16840062C211yesmaximal, characteristic, complemented
3C4×#14519803C211yesmaximal, characteristic
4#21454487C211yesmaximal, characteristic, complemented
5#21447268C211yesmaximal, characteristic, complemented
6#21447263C211yesmaximal, characteristic, complemented
7#21447111C211yesmaximal, characteristic
8#21447111C211yesmaximal, characteristic
9#21446811C211yesmaximal, characteristic, complemented
10#21446603C211yesmaximal, characteristic, complemented
11#21446598C211yesmaximal, characteristic, complemented
12#21446592C211yesmaximal, characteristic, complemented
13#21446483C211yesmaximal, characteristic, complemented
14#21446483C211yesmaximal, characteristic, complemented
15C61×#1800C211yesmaximal, characteristic, complemented
16C2×#14519803C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C4×#11758222C2211yescharacteristic
19#16840070—21no—
20#16840070—21no—
21#16840063—21no—
22#16840063—21no—
23#16840062—21no—
24#16840062—21no—
25#16840062—21no—
26#16840062—21no—
27#16839872—21no—
28#16839862—21no—
29C4×#11758200—21no—
30C4×#11758199—21no—
31#16839822—21no—
32#16839822—21no—
33#16839818—21no—
34#16839814—21no—
35C2×#14519712C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839802C2211yescharacteristic
40C2×#14519710C2211yescharacteristic
41C2×#14519708C2211yescharacteristic
42C2×C4⋊#92418C2211yescharacteristic
43#16839767C2211yescharacteristic
44#16839764C2211yescharacteristic
45#16839764C2211yescharacteristic
46#16839761C2211yescharacteristic
47C2×#14519698C2211yescharacteristic
48C2×#14519698C2211yescharacteristic
49#16839730C2211yescharacteristic
50#16839730C2211yescharacteristic
51#16839720C2211yescharacteristic
52#16839720C2211yescharacteristic
53C4×#11758194C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839526C2211yescharacteristic
59#16839526C2211yescharacteristic
60C2×C4×D4×C61C2211yescharacteristic
61C61×#423C2211yescharacteristic
62C61×#423C2211yescharacteristic
63C2×#100×C61C2211yescharacteristic
64C2×#100×C61C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519803—22no—
69#14519803—22no—
70C2×#11758200—21no—
71C2×#11758199—21no—
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519769—21no—
77#14519769—21no—
78#14519769—21no—
79#14519769—21no—
80C4×Dic122D411yescharacteristic
81C4×Dic122D411yescharacteristic
82C4×D244D411yescharacteristic
83C4×D244D411yescharacteristic
84C4×C22⋊D61—21no—
85C4×C22⋊D61—21no—
86#14519736C2311yescharacteristic
87#14519734—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C42×D61—21no—
93#14519712—22no—
94#14519712—22no—
95#14519711—21no—
96#14519711—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710—21no—
100#14519709—21no—
101#14519709—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105C4⋊#92418D411yescharacteristic
106C4⋊#92418D411yescharacteristic
107C4⋊#92418D411yescharacteristic
108C4⋊#92418D411yescharacteristic
109#14519702—21no—
110#14519702—21no—
111#14519700—21no—
112#14519699—21no—
113#14519699—21no—
114#14519698—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698—21no—
120#14519697—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128C4×#92418—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C22×D4×C61C2311yescharacteristic
132C23×C4×C61—21no—
133C2×M16×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C2×M16×C61C2311yescharacteristic
137C4×D4×C61—21no—
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140C4×D4×C61—21no—
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C22⋊C4×C61—21no—
143C2×C42×C61C2311yescharacteristic
144C4⋊C8×C61—21no—
145C4⋊C8×C61—21no—
146C4⋊C8×C61—21no—
147C4⋊C8×C61—21no—
148#100×C61—22no—
149#100×C61—22no—
150C24×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61#4112yes—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×C4×C61—82no—
161C22×C4×C61#4111yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C2×C22⋊D61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167#11758222—21no—
168#11758222—21no—
169#11758222—21no—
170#11758222—21no—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×Dic122C2×D411yescharacteristic
176M16×C61—21no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181M16×C61—21no—
182M16×C61—21no—
183M16×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—41no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C22⋊Dic61—21no—
197#11758200—24no—
198#11758199—24no—
199#11758198—21no—
200#11758198—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—21no—
208#11758196—21no—
209#11758196—41no—
210#11758196—41no—
211#11758196—41no—
212#11758196—41no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194—21no—
219#11758194#4112yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C2×D411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229C2×#92418C2×D411yescharacteristic
230C23×C61—22no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—81no—
234C23×C61—81no—
235C23×C61C2×#4111yescharacteristic
236C22×D61—21no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—41no—
240D4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—82no—
245C2×C4×C61—82no—
246C2×C4×C61—81no—
247C2×C4×C61#12111yescharacteristic
248C2×C4×C61#12111yescharacteristic
249C2×C4×C61#12111yescharacteristic
250C2×C4×C61#12111yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C22⋊D61—41no—
258C22⋊D61—41no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—81no—
265C2×Dic61—81no—
266D244—21no—
267D244—21no—
268C4×D61—42no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—81no—
272Dic122—21no—
273Dic122—21no—
274C8×C61—41no—
275C8×C61—41no—
276C8×C61—41no—
277C8×C61—41no—
278#92418—22no—
279#92418—21no—
280#92418—21no—
281#92418—21no—
282#92418—21no—
283#92418—41no—
284#24724—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—42no—
288C22×C61—82no—
289C22×C61—82no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C22×C61#55211yescharacteristic
294C22×C61#55611yescharacteristic
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—81no—
303C4×C61#52511yescharacteristic
304C4×C61#53611yescharacteristic
305C4×C61#54711yescharacteristic
306C4×C61#54811yescharacteristic
307Dic61—41no—
308Dic61—41no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312#3080—611no—
313C2×#478—611no—
314C4×AGL(1,ℤ/8)—611no—
315#2925—611no—
316#2079—611no—
317#2078—611no—
318#2051—611no—
319#2051—611no—
320#1998—611no—
321#1952—611no—
322#1951—611no—
323#1950—611no—
324#1931—611no—
325#1931—611no—
326#1800D6111yescharacteristic, complemented
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—82no—
330C2×C61#241511yescharacteristic
331C2×C61#306711yescharacteristic
332C2×C61#306811yescharacteristic
333D61—81no—
334C2×AGL(1,ℤ/8)—611no—
335C2×#125—611no—
336C4×#41—611no—
337C2×C4×D4C2×D6111yescharacteristic
338#479—1221no—
339#479—1221no—
340#478—1221no—
341#478—1221no—
342#478—1221no—
343#478—1221no—
344#478—1221no—
345#478—1221no—
346#442—1221no—
347#440—1221no—
348C4×SD16—1221no—
349C4×D8—1221no—
350#433—1221no—
351#433—1221no—
352#432—1221no—
353#431—1221no—
354C2×#108—611no—
355C2×#108—611no—
356#428—611no—
357#428—611no—
358#428—611no—
359C2×#107—611no—
360C2×#106—611no—
361#423C2×D6111yescharacteristic
362#423C2×D6111yescharacteristic
363C2×C4⋊C8—611no—
364#418—611no—
365#417—611no—
366#417—611no—
367#417—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#100C2×D6111yescharacteristic
371C2×#100C2×D6111yescharacteristic
372#408—611no—
373#408—611no—
374#405—611no—
375#405—611no—
376C4×M16—611no—
377#343—611no—
378#343—611no—
379#343—611no—
380#343—611no—
381#343C2×D6111yescharacteristic
382#343C2×D6111yescharacteristic
383#341—611no—
384#341—611no—
385C61#2472411yescharacteristic, complemented, Sylow
386C2×#41—611no—
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389AGL(1,ℤ/8)—1222no—
390AGL(1,ℤ/8)—1222no—
391C2×SD16—1221no—
392C2×D8—1221no—
393C2×M16—611no—
394C2×M16—611no—
395C2×M16C22×D6111yescharacteristic
396C2×M16C22×D6111yescharacteristic
397C2×M16C22×D6111yescharacteristic
398C2×M16C22×D6111yescharacteristic
399C22×C8—611no—
400C22×C8—611no—
401#125—1221no—
402#125—1221no—
403#125—1221no—
404#125—1221no—
405C4×Q8—611no—
406C4×Q8—611no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—1221no—
414C4×D4—1221no—
415#117—611no—
416#117—1221no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C42—1221no—
424C2×C42C22×D6111yescharacteristic
425#108—1222no—
426#108—1222no—
427#107—1221no—
428#107—1221no—
429#107—1221no—
430#107—1221no—
431#107—1221no—
432#106—1221no—
433#106—1221no—
434#106—1221no—
435#106—1221no—
436#106—1221no—
437C4⋊C8—21no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8—611no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—611no—
445#103—1221no—
446#103—1221no—
447#103—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#100—22no—
458#100—22no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463#97—1221no—
464#97—1221no—
465#97—1221no—
466C4×C8—1221no—
467C4×C8—1221no—
468C4×C8—1221no—
469C24—21no—
470#41—1221no—
471#41—1221no—
472#41—1221no—
473#41—1221no—
474C2×Q8—611no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—611no—
481C2×D4—1221no—
482C2×D4#1175822412yes—
483C22×C4—21no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—82no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4#1175822211yescharacteristic
493C22×C4#1175822211yescharacteristic
494C22×C4C23×D6111yescharacteristic
495SD16—1224no—
496D8—1224no—
497M16—21no—
498M16—21no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—21no—
503M16—21no—
504M16—21no—
505M16—612no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—2441no—
534C4⋊C4—2441no—
535C4⋊C4—2441no—
536C4⋊C4—2441no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42D4×D6111yescharacteristic
548C42D4×D6111yescharacteristic
549C23—22no—
550C23—41no—
551C23—41no—
552C23—81no—
553C23—81no—
554C23—1221no—
555C23C2×#1175822211yescharacteristic
556Q8—1221no—
557Q8—1221no—
558D4—22no—
559D4—22no—
560D4—41no—
561D4—41no—
562D4—1221no—
563D4—1221no—
564D4—2441no—
565D4—2441no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—4881no—
576C2×C4—4881no—
577C2×C4—4881no—
578C2×C4—2442no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4#1451975011yescharacteristic
584C2×C4#1451975011yescharacteristic
585C2×C4#1451975011yescharacteristic
586C2×C4#1451975011yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C2×C4C2×#1175822411yescharacteristic
591C2×C4#1451982411yescharacteristic
592C2×C4#1451982511yescharacteristic
593C8—41no—
594C8—41no—
595C8—41no—
596C8—41no—
597C8—1222no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—1221no—
602C8—2441no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22—81no—
609C22—81no—
610C22—2441no—
611C22—2441no—
612C22—2441no—
613C22C2×#1451975011yescharacteristic
614C22#1684044711yescharacteristic
615C22#1684048011yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—81no—
621C4—4881no—
622C4—4881no—
623C4—2441no—
624C4—2441no—
625C4—2441no—
626C4#1684027211yescharacteristic
627C4#1684033511yescharacteristic
628C4#1684041411yescharacteristic
629C4#1684041811yescharacteristic
630C2—21no—
631C2—21no—
632C2—82no—
633C2—4881no—
634C2#2144925011yescharacteristic
635C2#2145553811yescharacteristic
636C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1800C2
socleC22×C61C2×#121
prime radicalC2×#14519803C22
hypercenter#1800D61
hypocenterC61#24724
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #21455642C2semidirect
2. C2×#16840062C2semidirect
4. #21454487C2semidirect
5. #21447268C2semidirect
6. #21447263C2semidirect
9. #21446811C2semidirect
10. #21446603C2semidirect
11. #21446598C2semidirect
12. #21446592C2semidirect
13. #21446483C2semidirect
14. #21446483C2semidirect
15. C61×#1800C2semidirect
326. #1800D61semidirect
385. C61#24724semidirect

Maximal subgroups

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

subgroupindex
#214556422
C2×#168400622
C4×#145198032
#214544872
#214472682
#214472632
#214471112
#214468112
#214466032
#214465982
#214465922
#214464832
C61×#18002
#2472461