Finite Group Databasebeta

← #30617099

Subgroups — #30617099

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#137×#6923C212yesmaximal
2C2×#16840802C211yesmaximal, characteristic, complemented
3#21462389C211yesmaximal, characteristic
4#21461981C211yesmaximal, characteristic, complemented
5#21461936C211yesmaximal, characteristic, complemented
6#21461904C212yesmaximal, complemented
7#21461859C212yesmaximal
8#21461707C211yesmaximal, characteristic
9#21461640C211yesmaximal, characteristic, complemented
10#101×#6923C212yesmaximal, complemented
11#473×D61C211yesmaximal, characteristic
12D61⋊#131C2211yescharacteristic
13#41×#6923C2211yescharacteristic
14C2×Q8×#6923C2212yes—
15#16840802—22no—
16#16840802—21no—
17#16840802—21no—
18#16840802—21no—
19#16840802—21no—
20#16840801—22no—
21Q16×#6923—22no—
22SD16×#6923—22no—
23D61⋊#108C2211yescharacteristic
24#16840773C2212yes—
25C2×#14519845C2211yescharacteristic
26#16840755C2212yes—
27#16840753C2212yes—
28C2×#14519840C2212yes—
29D61⋊#101C2212yes—
30#16840728—21no—
31#16840728C2211yescharacteristic
32#16840725—21no—
33#16840725C2211yescharacteristic
34M16×#6923—21no—
35M16×#6923C2212yes—
36M16×#6923C2211yescharacteristic
37C2×#14519831—21no—
38C2×#14519831C2211yescharacteristic
39#16840641C2212yes—
40#16840641C2212yes—
41#16840637C2212yes—
42#137×D61C2212yes—
43C2×Q16×D61C2211yescharacteristic
44#131×D61C2211yescharacteristic
45#16840038C411yescharacteristic, complemented
46#16840037C411yescharacteristic, complemented
47#16840036C412yescomplemented
48#16840035C412yescomplemented
49#16840034C411yescharacteristic, complemented
50#108×D61C2211yescharacteristic
51#101×D61C2212yes—
52C61×#473C411yescharacteristic, complemented
53#14519865—21no—
54#14519864—21no—
55#14519864—22no—
56D61⋊M16—21no—
57D61⋊M16C2312yes—
58D61⋊M16C2311yescharacteristic
59C2×D61⋊C8—21no—
60C2×D61⋊C8C2311yescharacteristic
61Q8×#6923—22no—
62Q8×#6923—22no—
63Q8×#6923—22no—
64Q8×#6923—22no—
65Q8×#6923D412yes—
66D4×#6923—22no—
67D4×#6923D412yes—
68#14519856—21no—
69#14519856C2311yescharacteristic
70C2×C4⋊#6923—22no—
71C2×C4⋊#6923C2312yes—
72C2×C4×#6923—21no—
73C2×C4×#6923—22no—
74C2×C4×#6923C2311yescharacteristic
75#14519848—21no—
76#14519847—21no—
77#14519846—22no—
78#14519845—22no—
79#14519845—21no—
80#14519845—21no—
81#14519840—22no—
82#14519840—22no—
83#14519840—22no—
84#14519840—22no—
85#14519840—22no—
86#14519839—22no—
87#14519838—24no—
88#14519831—21no—
89#14519831—22no—
90#14519831—21no—
91#14519831—21no—
92#14519831D411yescharacteristic
93#14519831D411yescharacteristic
94#14519831D411yescharacteristic
95#14519831D411yescharacteristic
96C8×#6923—21no—
97C8×#6923—22no—
98C8×#6923—21no—
99C8×#6923—21no—
100#41×D61C2311yescharacteristic
101C2×Q8×D61C2312yes—
102#14519810C2×C412yes—
103#14519809—22no—
104#14519808C2×C412yes—
105#14519807C2×C412yes—
106#14519806—22no—
107C2×Dic244C2×C411yescharacteristic, complemented
108C2×#11758202C2×C412yescomplemented
109Q16×D61—22no—
110Q16×D61—21no—
111Q16×D61—21no—
112Q16×D61—21no—
113Q16×D61—21no—
114SD16×D61—22no—
115#14519785—21no—
116#14519785C2×C411yescharacteristic
117#14519784C2×C411yescharacteristic
118#14519783—21no—
119#14519783C2×C411yescharacteristic
120M16×D61—21no—
121M16×D61C2311yescharacteristic
122M16×D61C2312yes—
123C2×C8×D61—21no—
124C2×C8×D61C2311yescharacteristic
125#14519713C2×C411yescharacteristic, complemented
126#14519712C2×C412yescomplemented
127#14519696C2×C412yescomplemented
128#14519695C2×C412yescomplemented
129#14519695C2×C412yescomplemented
130#137×C61C2×C412yes—
131C2×Q16×C61C2×C411yescharacteristic, complemented
132#131×C61C2×C411yescharacteristic
133#108×C61C2×C411yescharacteristic, complemented
134#101×C61C2×C412yescomplemented
135C22×#6923—22no—
136C22×#6923—22no—
137C22×#6923#4112yes—
138#41×C61C22×C411yescharacteristic
139C2×Q8×C61C22×C412yes—
140#11758226#4111yescharacteristic
141Q8×D61—22no—
142Q8×D61—22no—
143Q8×D61—22no—
144Q8×D61—22no—
145Q8×D61C2×D411yescharacteristic
146#11758224—21no—
147#11758224#4111yescharacteristic
148D4×D61—21no—
149D4×D61C2×D411yescharacteristic
150#11758222—21no—
151#11758222C22×C411yescharacteristic
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61C2411yescharacteristic
155C2×Dic122—22no—
156C2×Dic122C22×C412yes—
157C22⋊#6923—21no—
158C22⋊#6923—22no—
159#11758217—22no—
160#11758217—22no—
161#11758217#4112yes—
162C2×#92420—22no—
163C2×#92420—21no—
164C4⋊#6923—22no—
165C4⋊#6923—22no—
166C4⋊#6923—22no—
167C4⋊#6923—42no—
168C4⋊#6923—42no—
169C4⋊#6923C2×D412yes—
170C4×#6923—22no—
171C4×#6923—21no—
172C4×#6923—42no—
173C4×#6923—42no—
174C4×#6923C2×D412yes—
175C4×#6923C2×D411yescharacteristic
176C4×#6923C2×D411yescharacteristic
177#11758213—22no—
178#11758213—21no—
179#11758213—22no—
180D61⋊C8—22no—
181D61⋊C8—21no—
182D61⋊C8—21no—
183D61⋊C8—22no—
184D61⋊C8C2×D411yescharacteristic
185D61⋊C8C2×D411yescharacteristic
186Q16×C61—22no—
187Q16×C61—21no—
188Q16×C61—21no—
189SD16×C61—22no—
190M16×C61—21no—
191M16×C61C22×C412yes—
192M16×C61C22×C411yescharacteristic
193C2×C8×C61—21no—
194C2×C8×C61C22×C411yescharacteristic
195#11758202—22no—
196#11758202—22no—
197#11758202—22no—
198#11758202—22no—
199#11758201—22no—
200#11758200—22no—
201#11758194—21no—
202#11758194C22×C412yes—
203#11758194C22×C411yescharacteristic
204C2×#92418—21no—
205C2×#92418C22×C411yescharacteristic
206Dic244—22no—
207Dic244—21no—
208Dic244—21no—
209#11758190—22no—
210#11758189—22no—
211#11758189—21no—
212#11758189—21no—
213#11758189—21no—
214C8×D61—21no—
215C8×D61—22no—
216C8×D61—21no—
217C8×D61—21no—
218C8×D61C2×D411yescharacteristic
219C8×D61C2×D411yescharacteristic
220C8×D61#4111yescharacteristic
221C8×D61#4111yescharacteristic
222C22×D61—21no—
223C22×D61C2×#4111yescharacteristic
224C2×#6923—22no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—82no—
228C2×#6923—82no—
229C2×#6923#12112yes—
230C2×#6923#12112yes—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61C4×D411yescharacteristic
234D4×C61—21no—
235D4×C61C4×D411yescharacteristic
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61C23×C411yescharacteristic
239C22⋊D61—21no—
240C22⋊D61—21no—
241C22⋊D61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61C2×#4111yescharacteristic
245D244—21no—
246D244C4×D411yescharacteristic
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61C22×D411yescharacteristic
253C4×D61C22×D411yescharacteristic
254C4×D61C2×#4111yescharacteristic
255C4×D61C2×#4111yescharacteristic
256Dic122—21no—
257Dic122—22no—
258Dic122—22no—
259Dic122—42no—
260Dic122—42no—
261Dic122C4×D411yescharacteristic
262#92420—21no—
263#92420—22no—
264#92420—21no—
265#92420—42no—
266C8×C61—21no—
267C8×C61—22no—
268C8×C61C4×D411yescharacteristic
269C8×C61C4×D411yescharacteristic
270#92418—21no—
271#92418—22no—
272#92418C4×D411yescharacteristic
273#92418C4×D411yescharacteristic
274#21556—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61C4×#4111yescharacteristic
277C2×D61—22no—
278C2×D61—21no—
279C2×D61—42no—
280C2×D61C2×#12111yescharacteristic
281C2×D61#52911yescharacteristic
282#6923—22no—
283#6923—22no—
284#6923—42no—
285#6923—82no—
286C4×C61—21no—
287C4×C61—42no—
288C4×C61C2×C4×D411yescharacteristic
289C4×C61C2×C4×D411yescharacteristic
290Dic61—21no—
291Dic61—42no—
292Dic61#52511yescharacteristic
293Dic61#53211yescharacteristic
294C2×#473—611no—
295C4×#137—612no—
296C2×#441—611no—
297#2875—611no—
298#1979—611no—
299#1951—611no—
300#1930—612no—
301#1908—612no—
302#1815—611no—
303#1778—611no—
304C4×#101—612no—
305C2×C61—21no—
306C2×C61—41no—
307C2×C61C4×#12111yescharacteristic
308D61—21no—
309D61—41no—
310D61#306912yes—
311C2×#137—612no—
312C22×Q16—611no—
313C2×#131—611no—
314C2×#131—611no—
315C4×#41—611no—
316C2×C4×Q8—612no—
317#473—612no—
318#473—611no—
319#473—611no—
320#473—611no—
321#473—612no—
322#473#692311yescharacteristic, complemented
323#441—1221no—
324#441—1221no—
325#441—1221no—
326#441—1221no—
327#441—1222no—
328#440—1222no—
329C4×Q16—1222no—
330C4×SD16—1222no—
331C2×#108—611no—
332C2×#108—611no—
333#428—612no—
334C2×#106—611no—
335#419—612no—
336#417—612no—
337C2×#103—612no—
338C2×#101—612no—
339C2×#101—612no—
340#405—611no—
341#405—611no—
342#405—1221no—
343#405—1221no—
344C4×M16—611no—
345C4×M16—612no—
346C4×M16—1221no—
347C2×#97—611no—
348C2×#97—1221no—
349#343—612no—
350#343—612no—
351#341—612no—
352C61#2155611yescharacteristic, complemented, Sylow
353C2×#41—611no—
354C22×Q8—612no—
355#137—612no—
356#137—612no—
357#137—612no—
358#137—1222no—
359#137—1222no—
360#137C2×#692312yes—
361C2×Q16—612no—
362C2×Q16—611no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×Q16—1221no—
367C2×Q16—1222no—
368C2×Q16C2×#692311yescharacteristic, complemented
369C2×SD16—1222no—
370#131—611no—
371#131—611no—
372#131—611no—
373#131—1221no—
374#131—1221no—
375#131—1222no—
376#131—1221no—
377#131—1221no—
378#131C2×#692311yescharacteristic
379C2×M16—611no—
380C2×M16—611no—
381C2×M16—612no—
382C2×M16—612no—
383C2×M16—1221no—
384C2×M16—1221no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—1221no—
388C22×C8—1221no—
389C4×Q8—612no—
390C4×Q8—1222no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×D4—612no—
395C4×D4—1222no—
396#117—611no—
397#117—1221no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1222no—
400C2×C42—611no—
401C2×C42—1221no—
402C2×C42—1222no—
403#108—612no—
404#108—611no—
405#108—1221no—
406#108—1221no—
407#108C2×#692311yescharacteristic, complemented
408#107—1222no—
409#106—1222no—
410#106—1221no—
411#106—1221no—
412#103—1222no—
413#103—1222no—
414#103—1222no—
415#103—1222no—
416#103—1222no—
417#102—1222no—
418#101—612no—
419#101—612no—
420#101—612no—
421#101—1224no—
422#101C2×#692312yescomplemented
423#97—611no—
424#97—611no—
425#97—611no—
426#97—611no—
427#97—1221no—
428#97—1222no—
429#97—1221no—
430#97—1221no—
431C4×C8—1221no—
432C4×C8—1222no—
433C4×C8—1221no—
434C4×C8—1221no—
435#41—611no—
436#41—611no—
437#41—611no—
438#41—1221no—
439#41—1221no—
440#41C22×#692311yescharacteristic
441C2×Q8—611no—
442C2×Q8—612no—
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×Q8—1222no—
446C2×Q8—1222no—
447C2×Q8—1222no—
448C2×Q8C22×#692312yes—
449C2×D4—611no—
450C2×D4—1221no—
451C22×C4—611no—
452C22×C4—612no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457Q16—22no—
458Q16—21no—
459Q16—21no—
460Q16—1222no—
461Q16—1222no—
462Q16—1222no—
463Q16—1222no—
464Q16—1222no—
465Q16—1221no—
466Q16—1221no—
467SD16—22no—
468SD16—1222no—
469SD16—1222no—
470SD16—1222no—
471M16—21no—
472M16—612no—
473M16—611no—
474M16—612no—
475M16—1221no—
476M16—1221no—
477M16—1222no—
478M16—1221no—
479M16—1221no—
480M16—1222no—
481M16—1222no—
482M16—1221no—
483M16—1222no—
484M16—1222no—
485M16C22×#692311yescharacteristic
486M16C22×#692312yes—
487C2×C8—21no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1222no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1221no—
505C2×C8—1222no—
506C2×C8C22×#692311yescharacteristic
507C4⋊C4—612no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—2442no—
512C4⋊C4—2442no—
513C22⋊C4—1222no—
514C22⋊C4—1221no—
515C42—611no—
516C42—612no—
517C42—611no—
518C42—1221no—
519C42—1222no—
520C42—2442no—
521C42—2442no—
522C23—611no—
523C23—1221no—
524Q8—22no—
525Q8—22no—
526Q8—611no—
527Q8—1221no—
528Q8—1222no—
529Q8—1222no—
530Q8—2442no—
531Q8—2442no—
532Q8D4×#692311yescharacteristic
533D4—21no—
534D4—611no—
535D4—1221no—
536D4—1221no—
537D4—1221no—
538D4—1221no—
539D4D4×#692311yescharacteristic
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—4882no—
556C2×C4—4882no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4C23×#692311yescharacteristic
562C8—21no—
563C8—22no—
564C8—611no—
565C8—611no—
566C8—1221no—
567C8—1221no—
568C8—1222no—
569C8—1221no—
570C8—1222no—
571C8—2442no—
572C8D4×#692311yescharacteristic
573C8D4×#692311yescharacteristic
574C22—21no—
575C22—611no—
576C22—611no—
577C22—1222no—
578C22—1221no—
579C22—2442no—
580C22#41×#692311yescharacteristic
581C4—21no—
582C4—42no—
583C4—611no—
584C4—611no—
585C4—1222no—
586C4—1222no—
587C4—1221no—
588C4—4882no—
589C4—2442no—
590C4—2442no—
591C4C2×D4×#692311yescharacteristic
592C4C2×D4×#692311yescharacteristic
593C2—21no—
594C2—41no—
595C2—612no—
596C2—1221no—
597C2—2441no—
598C2#121×#692311yescharacteristic
subgroupG/H
centerC2#121×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#473C4
socleC2×C61C4×#121
prime radical#41×D61C23
cosocleC2×C4×D61C24
hypercenter#473#6923
hypocenterC61#21556
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
2. C2×#16840802C2semidirect
4. #21461981C2semidirect
5. #21461936C2semidirect
6. #21461904C2semidirect
9. #21461640C2semidirect
10. #101×#6923C2semidirect
45. #16840038C4semidirect
46. #16840037C4semidirect
47. #16840036C4semidirect
48. #16840035C4semidirect
49. #16840034C4semidirect
52. C61×#473C4semidirect
107. C2×Dic244C2×C4semidirect
108. C2×#11758202C2×C4semidirect
125. #14519713C2×C4semidirect
126. #14519712C2×C4semidirect
127. #14519696C2×C4semidirect
128. #14519695C2×C4semidirect
129. #14519695C2×C4semidirect
131. C2×Q16×C61C2×C4semidirect
133. #108×C61C2×C4semidirect
134. #101×C61C2×C4semidirect
322. #473#6923semidirect
352. C61#21556semidirect
368. C2×Q16C2×#6923semidirect
407. #108C2×#6923semidirect
422. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#137×#69232
C2×#168408022
#214623892
#214619812
#214619362
#214619042
#214618592
#214617072
#214616402
#101×#69232
#473×D612
#2155661