Finite Group Databasebeta

← #30616062

Subgroups — #30616062

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462278C211yesmaximal, characteristic, complemented
2C4×D4×#6923C211yesmaximal, characteristic
3#21461898C212yesmaximal, complemented
4#21461882C212yesmaximal, complemented
5#21461812C212yesmaximal
6#21461784C212yesmaximal
7C4×#14519845C211yesmaximal, characteristic, complemented
8C4×#14519840C211yesmaximal, characteristic, complemented
9C4×#14519839C211yesmaximal, characteristic
10#21461532C211yesmaximal, characteristic, complemented
11C4×SD16×D61C211yesmaximal, characteristic
12C2×D4×#6923C2212yes—
13C22×C4×#6923—21no—
14C2×#14519845C2212yes—
15C2×#14519840C2212yes—
16C2×#14519839C2212yes—
17C4×D61⋊C8C2211yescharacteristic
18(C2×#6923)⋊Q8C2212yes—
19#16840713C2212yes—
20#16840711—22no—
21C4⋊C4×#6923C2212yes—
22C4×C4⋊#6923—21no—
23C4×C4⋊#6923C2211yescharacteristic
24C4×C4⋊#6923C2211yescharacteristic
25C22⋊C4×#6923—22no—
26C4×C22⋊#6923—21no—
27#16840698—21no—
28C42×#6923C2211yescharacteristic
29#16840636C2212yes—
30#16840635C2212yes—
31#16840635C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C2×SD16×D61C2211yescharacteristic
35C4×Q8×D61C2211yescharacteristic
36C4×D4×D61C2211yescharacteristic
37#16839853C411yescharacteristic, complemented
38C4×#11758190C411yescharacteristic, complemented
39SD16×Dic61C411yescharacteristic, complemented
40C4×#11758201C411yescharacteristic, complemented
41#16839849C411yescharacteristic, complemented
42C4×#11758200C411yescharacteristic, complemented
43#16839847C411yescharacteristic, complemented
44#106×D61C2211yescharacteristic
45#103×D61C2211yescharacteristic
46#102×D61C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48C4×SD16×C61C411yescharacteristic, complemented
49C23×#6923—22no—
50C2×D61⋊C8C2312yes—
51D4×#6923—24no—
52D4×#6923—24no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923C2312yes—
55C2×C4⋊#6923C2312yes—
56C2×C22⋊#6923—22no—
57C2×C4×#6923—22no—
58C2×C4×#6923—21no—
59C2×C4×#6923—21no—
60C2×C4×#6923—44no—
61C2×C4×#6923D411yescharacteristic
62C2×C4×#6923D412yes—
63C2×C4×#6923D411yescharacteristic
64C2×C4×#6923C2312yes—
65#14519845—24no—
66#14519840—24no—
67#14519839—24no—
68C4×#92420—21no—
69C4×#92420—21no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4—22no—
74(C2×#6923)⋊C4—21no—
75(C2×#6923)⋊C4—21no—
76C2×Q8×D61C2311yescharacteristic
77C2×D4×D61C2311yescharacteristic
78C22×C4×D61—21no—
79C2×#11758190C2×C411yescharacteristic
80C2×#11758200C2×C411yescharacteristic
81C2×#11758201C2×C411yescharacteristic
82SD16×D61—24no—
83C2×C8×D61C2311yescharacteristic
84C2×C8×D61C2311yescharacteristic
85Q8×Dic61C2×C411yescharacteristic, complemented
86C4×Dic122—21no—
87C4×Dic122C2×C411yescharacteristic, complemented
88#14519742—21no—
89#14519742C2×C411yescharacteristic, complemented
90C4×D244C2×C411yescharacteristic
91D4×Dic61C2×C411yescharacteristic
92#14519739C2×C411yescharacteristic
93C4×C22⋊D61—21no—
94#14519737—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61C2311yescharacteristic
97C4⋊C4×D61C2311yescharacteristic
98C22⋊C4×D61—21no—
99C42×D61—21no—
100C42×D61C2311yescharacteristic
101#14519709C2×C411yescharacteristic, complemented
102#14519708C2×C411yescharacteristic, complemented
103#14519708C2×C411yescharacteristic, complemented
104#14519702C2×C411yescharacteristic, complemented
105#14519701C2×C411yescharacteristic, complemented
106#14519700C2×C411yescharacteristic, complemented
107#14519699C2×C411yescharacteristic
108#14519698C2×C411yescharacteristic
109#14519697C2×C411yescharacteristic
110C4×#92418C2×C411yescharacteristic, complemented
111C8×Dic61C2×C411yescharacteristic, complemented
112C8×Dic61C2×C411yescharacteristic, complemented
113C2×SD16×C61C2×C411yescharacteristic
114C4×Q8×C61C2×C411yescharacteristic, complemented
115C4×D4×C61C2×C411yescharacteristic
116#106×C61C2×C411yescharacteristic, complemented
117#103×C61C2×C411yescharacteristic, complemented
118#102×C61C2×C411yescharacteristic
119C4×C8×C61C2×C411yescharacteristic, complemented
120C23×D61—21no—
121C22×#6923—28no—
122C22×#6923—22no—
123C22×#6923—22no—
124C22×#6923—44no—
125C22×#6923—44no—
126C22×#6923C2×D412yes—
127C22×#6923C2×D412yes—
128C2×Q8×C61C22×C411yescharacteristic
129C2×D4×C61C22×C411yescharacteristic
130C22×C4×C61—21no—
131C2×C22⋊D61—21no—
132C22×Dic61—21no—
133Q8×D61—22no—
134Q8×D61—22no—
135D4×D61—22no—
136D4×D61#4114yes—
137C2×D244C22×C411yescharacteristic
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122C22×C411yescharacteristic
150C22⋊#6923—28no—
151C2×#92420—22no—
152C4⋊#6923—22no—
153C4⋊#6923—22no—
154C4⋊#6923—22no—
155C4⋊#6923—44no—
156C4⋊#6923#4114yes—
157C4×#6923—22no—
158C4×#6923—22no—
159C4×#6923—22no—
160C4×#6923—42no—
161C4×#6923—44no—
162C4×#6923—42no—
163C4×#6923—42no—
164C4×#6923SD1614yescomplemented
165D61⋊C8—22no—
166D61⋊C8—22no—
167SD16×C61—22no—
168C2×C8×C61C22×C411yescharacteristic
169C2×C8×C61C22×C411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C411yescharacteristic
172C4⋊C4×C61C22×C411yescharacteristic
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61C22×C411yescharacteristic
176C22⋊Dic61—21no—
177#11758201—22no—
178#11758200—22no—
179#11758198—21no—
180#11758198—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C411yescharacteristic
183C4⋊Dic61C22×C411yescharacteristic
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61C22×C411yescharacteristic
192C4×Dic61#4111yescharacteristic
193C4×Dic61#4111yescharacteristic
194C2×#92418C22×C411yescharacteristic
195C2×#92418C22×C411yescharacteristic
196#11758190—22no—
197C8×D61—22no—
198C8×D61—22no—
199C23×C61—21no—
200C22×D61—21no—
201C22×D61—24no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—42no—
205C22×D61C22×D411yescharacteristic
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—24no—
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—416no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—42no—
216C2×#6923—42no—
217C2×#6923—42no—
218C2×#6923C2×SD1614yes—
219Q8×C61—21no—
220Q8×C61—21no—
221D4×C61—21no—
222D4×C61#11712yes—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—42no—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61C23×C411yescharacteristic
230C22⋊D61—24no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—42no—
235C2×Dic61C4×D411yescharacteristic
236C2×Dic61C4×D411yescharacteristic
237C2×Dic61C2×#4111yescharacteristic
238D244—21no—
239D244#11712yes—
240C4×D61—22no—
241C4×D61—42no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61C2×SD1612yes—
248C4×D61#13512yes—
249C4×D61C2×#4112yes—
250C4×D61C2×#4112yes—
251Dic122—21no—
252Dic122—21no—
253Dic122—42no—
254#92420—42no—
255C8×C61—21no—
256C8×C61—21no—
257#92418—21no—
258#92418—21no—
259#21117—611nomaximal, Sylow
260C22×C61—22no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61C2×C4×D411yescharacteristic
264C2×D61—22no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—44no—
268C2×D61—44no—
269C2×D61#52512yes—
270C2×D61C22×SD1611yescharacteristic
271C2×D61C2×#13511yescharacteristic
272C2×D61#57512yes—
273#6923—24no—
274#6923—44no—
275#6923—42no—
276#6923—82no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61C4×SD1612yes—
282C4×C61C2×#11711yescharacteristic
283C4×C61C4×#4111yescharacteristic
284Dic61—21no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61C4×SD1612yes—
289Dic61#52912yes—
290C2×C4×SD16—611no—
291#2295—611no—
292C42×D4—611no—
293#1927—612no—
294#1920—612no—
295#1877—612no—
296#1864—612no—
297C4×#106—611no—
298C4×#103—611no—
299C4×#102—611no—
300#1744—611no—
301C2×C61—42no—
302C2×C61#229011yescharacteristic
303C2×C61C2×C4×SD1611yescharacteristic
304C2×C61#294311yescharacteristic
305D61—42no—
306D61#310112yes—
307D61#310212yes—
308C22×SD16—611no—
309C2×C4×Q8—611no—
310C2×C4×D4—611no—
311C2×C4×D4—612no—
312C22×C42—1221no—
313C4×SD16—611no—
314C4×SD16—611no—
315C4×SD16—611no—
316C4×SD16—611no—
317C4×SD16—611no—
318C4×SD16—611no—
319C4×SD16—611no—
320C4×SD16#692311yescharacteristic, complemented
321C2×#106—611no—
322C2×#106—612no—
323C2×#103—611no—
324C2×#103—612no—
325C2×#102—611no—
326C2×#102—612no—
327C2×C4×C8—611no—
328C2×C4×C8—611no—
329#391—612no—
330#389—612no—
331#387—1222no—
332C4×C4⋊C4—612no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4—611no—
335C4×C4⋊C4—1221no—
336C4×C22⋊C4—1221no—
337C4×C22⋊C4—1222no—
338#376—1221no—
339C43—611no—
340#340—612no—
341#340—612no—
342#340—612no—
343#336—612no—
344#336—612no—
345C61#2111711yescharacteristic, complemented, Sylow
346C22×Q8—611no—
347C22×D4—611no—
348C23×C4—1221no—
349C23×C4—1222no—
350C2×SD16—611no—
351C2×SD16—611no—
352C2×SD16—611no—
353C2×SD16—1224no—
354C2×SD16C2×#692311yescharacteristic
355C22×C8—611no—
356C22×C8—611no—
357C22×C8—612no—
358C4×Q8—611no—
359C4×Q8—611no—
360C4×Q8—611no—
361C4×Q8—1221no—
362C4×Q8—1221no—
363C4×Q8C2×#692311yescharacteristic, complemented
364C4×D4—611no—
365C4×D4—611no—
366C4×D4—611no—
367C4×D4—1221no—
368C4×D4—1221no—
369C4×D4—1224no—
370C4×D4—1224no—
371C4×D4C2×#692311yescharacteristic
372C2×C4⋊C4—612no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1222no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—612no—
383C2×C42—611no—
384C2×C42—612no—
385C2×C42—1221no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1222no—
389C2×C42—2444no—
390#106—611no—
391#106—611no—
392#106—611no—
393#106—1224no—
394#106C2×#692311yescharacteristic, complemented
395#103—611no—
396#103—611no—
397#103—611no—
398#103—1224no—
399#103C2×#692311yescharacteristic, complemented
400#102—611no—
401#102—611no—
402#102—611no—
403#102—1224no—
404#102C2×#692311yescharacteristic
405C4×C8—611no—
406C4×C8—611no—
407C4×C8—611no—
408C4×C8—1221no—
409C4×C8—1221no—
410C4×C8C2×#692311yescharacteristic, complemented
411#95—1222no—
412#95—1222no—
413#95—1221no—
414#95—1222no—
415#95—1221no—
416#95—1222no—
417C24—1221no—
418C2×Q8—611no—
419C2×Q8—1221no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8C22×#692311yescharacteristic
423C2×D4—614no—
424C2×D4—611no—
425C2×D4—1222no—
426C2×D4—1221no—
427C2×D4C22×#692311yescharacteristic
428C22×C4—21no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—1228no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2444no—
446C22×C4—2444no—
447SD16—22no—
448SD16—1222no—
449SD16—1222no—
450SD16—1222no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8C22×#692311yescharacteristic
459C2×C8C22×#692311yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—614no—
462C4⋊C4—611no—
463C4⋊C4—611no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1221no—
472C4⋊C4—2444no—
473C4⋊C4C22×#692311yescharacteristic
474C4⋊C4C22×#692311yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1228no—
479C22⋊C4—1221no—
480C42—21no—
481C42—614no—
482C42—611no—
483C42—611no—
484C42—611no—
485C42—1221no—
486C42—1221no—
487C42—1222no—
488C42—1222no—
489C42—1222no—
490C42—1221no—
491C42—2442no—
492C42—2444no—
493C42—2442no—
494C42—2442no—
495C42C22×#692311yescharacteristic
496C23—21no—
497C23—611no—
498C23—1224no—
499C23—1221no—
500C23—2441no—
501C23—2441no—
502C23—2442no—
503Q8—21no—
504Q8—21no—
505Q8—1221no—
506Q8—1221no—
507Q8—2442no—
508D4—21no—
509D4—612no—
510D4—1224no—
511D4—1221no—
512D4#1451985612yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—42no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—614no—
524C2×C4—611no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1224no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—2442no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—24416no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4D4×#692311yescharacteristic
549C2×C4D4×#692311yescharacteristic
550C2×C4C23×#692311yescharacteristic
551C8—21no—
552C8—21no—
553C8—1221no—
554C8—1221no—
555C8—2442no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—612no—
560C22—611no—
561C22—611no—
562C22—612no—
563C22—1222no—
564C22—2444no—
565C22—2444no—
566C22—2441no—
567C22—2441no—
568C22C2×D4×#692311yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—612no—
574C4—612no—
575C4—1224no—
576C4—1221no—
577C4—4882no—
578C4—2442no—
579C4—2444no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4SD16×#692312yes—
584C4C2×#1451985611yescharacteristic
585C4#41×#692311yescharacteristic
586C2—42no—
587C2—612no—
588C2—612no—
589C2—2442no—
590C2#2146227311yescharacteristic
591C2C2×SD16×#692311yescharacteristic
592C2#2146257311yescharacteristic
subgroupG/H
centerC2×C4D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC4×SD16×C61C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenterC4×SD16#6923
hypocenterC61#21117
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #21462278C2semidirect
3. #21461898C2semidirect
4. #21461882C2semidirect
7. C4×#14519845C2semidirect
8. C4×#14519840C2semidirect
10. #21461532C2semidirect
37. #16839853C4semidirect
38. C4×#11758190C4semidirect
39. SD16×Dic61C4semidirect
40. C4×#11758201C4semidirect
41. #16839849C4semidirect
42. C4×#11758200C4semidirect
43. #16839847C4semidirect
48. C4×SD16×C61C4semidirect
85. Q8×Dic61C2×C4semidirect
87. C4×Dic122C2×C4semidirect
89. #14519742C2×C4semidirect
101. #14519709C2×C4semidirect
102. #14519708C2×C4semidirect
103. #14519708C2×C4semidirect
104. #14519702C2×C4semidirect
105. #14519701C2×C4semidirect
106. #14519700C2×C4semidirect
110. C4×#92418C2×C4semidirect
111. C8×Dic61C2×C4semidirect
112. C8×Dic61C2×C4semidirect
114. C4×Q8×C61C2×C4semidirect
116. #106×C61C2×C4semidirect
117. #103×C61C2×C4semidirect
119. C4×C8×C61C2×C4semidirect
164. C4×#6923SD16semidirect
320. C4×SD16#6923semidirect
345. C61#21117semidirect
363. C4×Q8C2×#6923semidirect
394. #106C2×#6923semidirect
399. #103C2×#6923semidirect
410. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214622782
C4×D4×#69232
#214618982
#214618822
#214618122
#214617842
C4×#145198452
C4×#145198402
C4×#145198392
#214615322
C4×SD16×D612
#2111761