Finite Group Databasebeta

← #30617105

Subgroups — #30617105

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462592C212yesmaximal, complemented
2C2×#16840808C211yesmaximal, characteristic, complemented
3#21462389C211yesmaximal, characteristic, complemented
4#21461978C211yesmaximal, characteristic, complemented
5#21461938C211yesmaximal, characteristic, complemented
6#21461906C212yesmaximal, complemented
7#21461857C212yesmaximal, complemented
8#21461705C211yesmaximal, characteristic, complemented
9#21461628C211yesmaximal, characteristic, complemented
10#21461578C212yesmaximal, complemented
11#21455541C211yesmaximal, characteristic
12C2×#14519865C2212yes—
13D61⋊#131C2211yescharacteristic
14#41×#6923C2211yescharacteristic
15#16840808—22no—
16#16840808—21no—
17#16840808—21no—
18#16840808—21no—
19#16840808—21no—
20#16840807—22no—
21#16840806—22no—
22#16840805—22no—
23#16840777C2212yes—
24C2×#14519847C2211yescharacteristic
25#16840772C2211yescharacteristic
26#16840761C2212yes—
27#16840759C2212yes—
28C2×#14519842C2212yes—
29#16840743C2212yes—
30#16840728—21no—
31#16840728C2212yes—
32#16840728C2211yescharacteristic
33#16840725—21no—
34#16840725C2211yescharacteristic
35M16×#6923—21no—
36M16×#6923C2211yescharacteristic
37C2×#14519831—21no—
38C2×#14519831C2211yescharacteristic
39#16840645C2212yes—
40#16840645C2212yes—
41#16840634C2212yes—
42#16840561C2212yes—
43C2×#14519800C2211yescharacteristic
44#131×D61C2211yescharacteristic
45#16840034C411yescharacteristic, complemented
46#16840033C411yescharacteristic, complemented
47#16840032C411yescharacteristic, complemented
48#16840028C412yescomplemented
49#16840027C412yescomplemented
50#108×D61C2211yescharacteristic
51#16839745C2212yes—
52C61×#473C411yescharacteristic, complemented
53#14519865—22no—
54#14519865—22no—
55#14519865—22no—
56#14519865—22no—
57#14519865D412yes—
58#14519864—22no—
59#14519864D412yes—
60C2×#11758213—22no—
61C2×#11758213C2312yes—
62D61⋊M16—21no—
63D61⋊M16C2311yescharacteristic
64C2×D61⋊C8—21no—
65C2×D61⋊C8—22no—
66C2×D61⋊C8C2311yescharacteristic
67Q8×#6923—21no—
68D4×#6923—21no—
69D4×#6923—22no—
70#14519856—21no—
71#14519856C2312yes—
72#14519856C2311yescharacteristic
73C2×C4×#6923—21no—
74C2×C4×#6923C2311yescharacteristic
75#14519848—22no—
76#14519847—22no—
77#14519847—21no—
78#14519847—21no—
79#14519846—21no—
80#14519845—21no—
81#14519842—22no—
82#14519842—22no—
83#14519842—22no—
84#14519842—22no—
85#14519842—22no—
86#14519841—22no—
87#14519836—24no—
88#14519831—21no—
89#14519831—21no—
90#14519831—22no—
91#14519831—21no—
92#14519831D411yescharacteristic
93#14519831D411yescharacteristic
94#14519831D411yescharacteristic
95#14519831D411yescharacteristic
96C8×#6923—21no—
97C8×#6923—21no—
98C8×#6923—22no—
99C8×#6923—21no—
100C2×#11758226C2312yes—
101#41×D61C2311yescharacteristic
102#14519809—22no—
103#14519808C2×C412yes—
104#14519805C2×C412yes—
105#14519804—22no—
106#14519802C2×C412yes—
107#14519800—22no—
108#14519800—21no—
109#14519800—21no—
110#14519800—21no—
111#14519800—21no—
112#14519798—22no—
113C2×#11758201C2×C412yescomplemented
114C2×D488C2×C411yescharacteristic, complemented
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#14519693C2×C412yescomplemented
129#14519693C2×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—21no—
137#41×C61C22×C411yescharacteristic
138C2×Q8×C61C22×C412yes—
139#11758226—22no—
140#11758226—22no—
141#11758226—22no—
142#11758226—22no—
143#11758226C2×D411yescharacteristic
144Q8×D61#4111yescharacteristic
145#11758224—21no—
146#11758224C2×D411yescharacteristic
147D4×D61—21no—
148D4×D61#4111yescharacteristic
149#11758222—21no—
150#11758222C22×C411yescharacteristic
151C2×D244—22no—
152C2×D244C22×C412yes—
153C2×C4×D61—21no—
154C2×C4×D61—22no—
155C2×C4×D61C2411yescharacteristic
156C22⋊#6923—22no—
157C22⋊#6923—22no—
158C22⋊#6923#4112yes—
159#11758217—21no—
160#11758217—22no—
161C2×#92420—22no—
162C2×#92420—22no—
163C2×#92420#4112yes—
164C4⋊#6923—22no—
165C4⋊#6923—21no—
166C4⋊#6923—22no—
167C4×#6923—22no—
168C4×#6923—21no—
169C4×#6923—22no—
170C4×#6923—21no—
171C4×#6923C2×D411yescharacteristic
172C4×#6923C2×D411yescharacteristic
173#11758213—22no—
174#11758213—22no—
175#11758213—22no—
176#11758213—42no—
177#11758213—42no—
178#11758213C2×D412yes—
179D61⋊C8—21no—
180D61⋊C8—22no—
181D61⋊C8—42no—
182D61⋊C8—42no—
183D61⋊C8C2×D412yes—
184D61⋊C8C2×D411yescharacteristic
185D61⋊C8C2×D411yescharacteristic
186Q16×C61—22no—
187Q16×C61—21no—
188Q16×C61—21no—
189SD16×C61—22no—
190M16×C61—21no—
191M16×C61C22×C411yescharacteristic
192M16×C61C22×C412yes—
193C2×C8×C61—21no—
194C2×C8×C61C22×C411yescharacteristic
195#11758202—22no—
196#11758201—22no—
197#11758201—22no—
198#11758201—22no—
199#11758201—22no—
200#11758199—22no—
201#11758194—21no—
202#11758194C22×C412yes—
203#11758194C22×C411yescharacteristic
204C2×#92418—21no—
205C2×#92418C22×C411yescharacteristic
206D488—22no—
207D488—21no—
208D488—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×D61—22no—
224C22×D61C2×#4111yescharacteristic
225C2×#6923—21no—
226C2×#6923—22no—
227C2×#6923—21no—
228C2×#6923—42no—
229C2×#6923—81no—
230C2×#6923—82no—
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×Dic61C2×#4111yescharacteristic
244D244—21no—
245D244—22no—
246D244—22no—
247D244—42no—
248D244—42no—
249D244C4×D411yescharacteristic
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61C22×D411yescharacteristic
256C4×D61C22×D411yescharacteristic
257C4×D61C2×#4111yescharacteristic
258C4×D61C2×#4111yescharacteristic
259Dic122—21no—
260Dic122C4×D411yescharacteristic
261#92420—22no—
262#92420—42no—
263#92420#12112yes—
264#92420#12112yes—
265C8×C61—21no—
266C8×C61—22no—
267C8×C61C4×D411yescharacteristic
268C8×C61C4×D411yescharacteristic
269#92418—21no—
270#92418—22no—
271#92418C4×D411yescharacteristic
272#92418C4×D411yescharacteristic
273#21559—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61C4×#4111yescharacteristic
276C2×D61—21no—
277C2×D61—41no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—82no—
281C2×D61#52511yescharacteristic
282C2×D61#53211yescharacteristic
283#6923—42no—
284#6923—41no—
285#6923—41no—
286#6923—82no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61C2×C4×D411yescharacteristic
290C4×C61C2×C4×D411yescharacteristic
291Dic61—21no—
292Dic61C2×#12111yescharacteristic
293Dic61#52911yescharacteristic
294#3067—611no—
295#2955—612no—
296C2×#444—611no—
297#2875—611no—
298#1976—611no—
299#1953—611no—
300#1932—612no—
301#1906—612no—
302#1814—611no—
303#1773—611no—
304#1757—612no—
305C2×C61—21no—
306C2×C61—41no—
307C2×C61C4×#12111yescharacteristic
308D61—21no—
309D61—21no—
310D61—41no—
311D61—82no—
312#575—612no—
313C2×#135—611no—
314C2×#131—611no—
315C2×#131—611no—
316C2×#131—612no—
317C4×#41—611no—
318#473—611no—
319#473#692311yescharacteristic, complemented
320#472—611no—
321#472—611no—
322#471—612no—
323#471—612no—
324#444—1221no—
325#444—1221no—
326#444—1221no—
327#444—1221no—
328#444—1222no—
329#444—1222no—
330#443—1222no—
331#443—1222no—
332#430—612no—
333C2×#108—611no—
334C2×#108—611no—
335#427—611no—
336#421—612no—
337#421—612no—
338C2×#104—612no—
339#413—612no—
340#410—612no—
341#405—611no—
342#405—611no—
343#405—612no—
344#405—1221no—
345#405—1221no—
346C4×M16—611no—
347C4×M16—1221no—
348C2×#97—611no—
349C2×#97—1221no—
350#344—612no—
351#344—612no—
352#339—612no—
353C61#2155911yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C2×#41—612no—
356#137—612no—
357#137—1222no—
358#137C2×#692312yes—
359AGL(1,ℤ/8)—612no—
360AGL(1,ℤ/8)—612no—
361AGL(1,ℤ/8)—1222no—
362#135—1221no—
363#135—1221no—
364#135—1221no—
365#135—1221no—
366#135—1222no—
367#135—1222no—
368C2×Q16C2×#692311yescharacteristic, complemented
369C2×SD16—612no—
370C2×D8—611no—
371#131—612no—
372#131—612no—
373#131—611no—
374#131—611no—
375#131—611no—
376#131—1221no—
377#131—1221no—
378#131—1222no—
379#131—1222no—
380#131—1222no—
381#131—1222no—
382#131—1222no—
383#131C2×#692311yescharacteristic
384C2×M16—611no—
385C2×M16—611no—
386C2×M16—612no—
387C2×M16—612no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1222no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1222no—
396C4×Q8—1221no—
397C4×D4—1222no—
398C4×D4—1221no—
399#117—611no—
400#117—612no—
401#117—1221no—
402C2×C42—611no—
403C2×C42—1221no—
404#108—612no—
405#108—611no—
406#108—1221no—
407#108—1221no—
408#108—1222no—
409#108—1222no—
410#108C2×#692311yescharacteristic, complemented
411#107—1221no—
412#106—1221no—
413#104—1222no—
414#104—1222no—
415#104—1222no—
416#104—1222no—
417#104—1222no—
418#104—1222no—
419#101—612no—
420#101C2×#692312yescomplemented
421#100—612no—
422#100—612no—
423#99—1224no—
424#97—611no—
425#97—611no—
426#97—611no—
427#97—611no—
428#97—1221no—
429#97—1221no—
430#97—1221no—
431#97—1222no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8—1221no—
435C4×C8—1222no—
436#41—611no—
437#41—611no—
438#41—611no—
439#41—1221no—
440#41—1221no—
441#41—1222no—
442#41—1222no—
443#41—1222no—
444#41—1222no—
445#41C22×#692311yescharacteristic
446C2×Q8—611no—
447C2×Q8C22×#692312yes—
448C2×D4—611no—
449C2×D4—612no—
450C2×D4—1222no—
451C2×D4—1221no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—1222no—
457Q16—22no—
458Q16—21no—
459Q16—21no—
460Q16—1222no—
461SD16—22no—
462SD16—1222no—
463SD16—1222no—
464SD16—1222no—
465SD16—1222no—
466SD16—1222no—
467D8—1222no—
468D8—1222no—
469D8—1221no—
470D8—1221no—
471M16—21no—
472M16—612no—
473M16—611no—
474M16—612no—
475M16—1221no—
476M16—1221no—
477M16—1222no—
478M16—1221no—
479M16—1222no—
480M16—1222no—
481M16—1221no—
482M16—1221no—
483M16—1222no—
484M16—1222no—
485M16—2442no—
486M16—2442no—
487M16C22×#692311yescharacteristic
488M16C22×#692312yes—
489C2×C8—21no—
490C2×C8—611no—
491C2×C8—612no—
492C2×C8—611no—
493C2×C8—612no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1222no—
502C2×C8—1221no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1222no—
508C2×C8—2442no—
509C2×C8—2442no—
510C2×C8C22×#692311yescharacteristic
511C4⋊C4—1221no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C22⋊C4—612no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C42—611no—
518C42—611no—
519C42—1221no—
520C42—1221no—
521C42—1222no—
522C42—1222no—
523C23—611no—
524C23—1221no—
525C23—1222no—
526Q8—22no—
527Q8—22no—
528Q8—611no—
529Q8—1221no—
530Q8D4×#692311yescharacteristic
531D4—21no—
532D4—611no—
533D4—1221no—
534D4—1221no—
535D4—1221no—
536D4—1221no—
537D4—1222no—
538D4—1222no—
539D4—2442no—
540D4—2442no—
541D4D4×#692311yescharacteristic
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—4881no—
557C2×C4—4882no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4C23×#692311yescharacteristic
562C8—21no—
563C8—22no—
564C8—612no—
565C8—612no—
566C8—611no—
567C8—611no—
568C8—1222no—
569C8—1221no—
570C8—1222no—
571C8—2442no—
572C8D4×#692311yescharacteristic
573C8D4×#692311yescharacteristic
574C22—21no—
575C22—611no—
576C22—611no—
577C22—1221no—
578C22—4882no—
579C22—2441no—
580C22—2442no—
581C22—2442no—
582C22#41×#692311yescharacteristic
583C4—21no—
584C4—42no—
585C4—611no—
586C4—611no—
587C4—1221no—
588C4—4882no—
589C4—2441no—
590C4—2441no—
591C4—2442no—
592C4C2×D4×#692311yescharacteristic
593C4C2×D4×#692311yescharacteristic
594C2—21no—
595C2—41no—
596C2—1221no—
597C2—1221no—
598C2—4882no—
599C2—2441no—
600C2#121×#692311yescharacteristic
subgroupG/H
centerC2#121×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#473C4
socleC2×C61C4×#121
prime radical#21455541C2
cosocleC2×C4×D61C24
hypercenter#473#6923
hypocenterC61#21559
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
1. #21462592C2semidirect
2. C2×#16840808C2semidirect
3. #21462389C2semidirect
4. #21461978C2semidirect
5. #21461938C2semidirect
6. #21461906C2semidirect
7. #21461857C2semidirect
8. #21461705C2semidirect
9. #21461628C2semidirect
10. #21461578C2semidirect
45. #16840034C4semidirect
46. #16840033C4semidirect
47. #16840032C4semidirect
48. #16840028C4semidirect
49. #16840027C4semidirect
52. C61×#473C4semidirect
113. C2×#11758201C2×C4semidirect
114. C2×D488C2×C4semidirect
125. #14519713C2×C4semidirect
126. #14519712C2×C4semidirect
127. #14519696C2×C4semidirect
128. #14519693C2×C4semidirect
129. #14519693C2×C4semidirect
131. C2×Q16×C61C2×C4semidirect
133. #108×C61C2×C4semidirect
134. #101×C61C2×C4semidirect
319. #473#6923semidirect
353. C61#21559semidirect
368. C2×Q16C2×#6923semidirect
410. #108C2×#6923semidirect
420. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625922
C2×#168408082
#214623892
#214619782
#214619382
#214619062
#214618572
#214617052
#214616282
#214615782
#214555412
#2155961