Finite Group Databasebeta

← #30620621

Subgroups — #30620621

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462512C211yesmaximal, characteristic, complemented
2#21462409C211yesmaximal, characteristic
3#21461914C212yesmaximal, complemented
4#21461903C212yesmaximal, complemented
5#21461786C212yesmaximal, complemented
6#21461727C211yesmaximal, characteristic, complemented
7#21461699C211yesmaximal, characteristic, complemented
8#21461661C211yesmaximal, characteristic
9#21461598C212yesmaximal, complemented
10#21461555C211yesmaximal, characteristic, complemented
11#416×D61C211yesmaximal, characteristic
12C2×#14519864C2212yes—
13#16840786—22no—
14#16840784—21no—
15#16840780—21no—
16C2×#14519848C2212yescomplemented
17#16840769C2212yescomplemented
18C2×#14519842C2212yes—
19C2×#14519841C2212yes—
20#16840737C2212yes—
21#16840735—22no—
22#16840734—21no—
23#16840733C2211yescharacteristic
24#16840733C2211yescharacteristic
25C2×#14519834—21no—
26D61⋊(C22⋊C8)C2211yescharacteristic
27#16840727C2212yescomplemented
28C2×#14519831C2212yescomplemented
29C4×C22⋊#6923C2211yescharacteristic
30#16840638C2212yescomplemented
31#16840632C2212yescomplemented
32#16840632C2212yescomplemented
33(C2×#6923)⋊C8C2212yescomplemented
34C2×#41×D61C2211yescharacteristic
35C22×C8×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839759C411yescharacteristic, complemented
38#16839758C411yescharacteristic, complemented
39#16839757C2211yescharacteristic
40#16839757C2211yescharacteristic
41#16839756C411yescharacteristic, complemented
42#16839755C2211yescharacteristic
43#16839755C2211yescharacteristic
44#103×D61C411yescharacteristic, complemented
45#103×D61C411yescharacteristic, complemented
46#102×D61C411yescharacteristic, complemented
47#102×D61C411yescharacteristic, complemented
48C61×#416C411yescharacteristic, complemented
49#14519864—24no—
50#14519864—42no—
51C2×#11758213C2312yes—
52C2×#11758213C2312yes—
53C2×#11758217—22no—
54C2×D61⋊C8C2312yes—
55C22×#92420—22no—
56C2×C22⋊#6923—21no—
57C2×C4×#6923C2312yes—
58#14519848—42no—
59#14519844—22no—
60C4⋊#92420—22no—
61#14519842—42no—
62#14519841—24no—
63#14519834—21no—
64#14519834—21no—
65#14519834—22no—
66#14519834—22no—
67#14519834D411yescharacteristic
68#14519834D411yescharacteristic
69C22⋊#92420—21no—
70C22⋊#92420—21no—
71C22⋊#92420D412yes—
72#14519832—22no—
73#14519831—42no—
74C4×#92420—22no—
75(C2×#6923)⋊C4—21no—
76C2×#11758226C2311yescharacteristic
77C2×#11758222—21no—
78C2×#11758222C2×C411yescharacteristic
79C2×#11758224—21no—
80C2×#11758224C2311yescharacteristic
81#41×D61—24no—
82#41×D61—42no—
83C2×Q8×D61C2×C411yescharacteristic
84C2×D4×D61—21no—
85C2×D4×D61C2×C411yescharacteristic
86C22×C4×D61—21no—
87C22×C4×D61C2311yescharacteristic
88C22×#92418C2×C411yescharacteristic
89C2×C8×D61—42no—
90C2×C8×D61C2×C411yescharacteristic
91C2×C8×D61C2×C411yescharacteristic
92C2×C8×D61C2311yescharacteristic
93C2×C8×D61C2311yescharacteristic
94#14519736C2×C411yescharacteristic
95#14519735C2311yescharacteristic
96#14519735C2311yescharacteristic
97#14519734—21no—
98#14519733—21no—
99C4⋊C4×D61C2×C411yescharacteristic
100C4⋊C4×D61C2×C411yescharacteristic
101C22⋊C4×D61—21no—
102C42×D61—21no—
103#14519702C2×C411yescharacteristic
104#14519702C2×C411yescharacteristic
105#14519701C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519700C2×C411yescharacteristic
108#14519700C2×C411yescharacteristic
109#14519699C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114#14519697C2×C411yescharacteristic
115C2×#41×C61C2×C411yescharacteristic
116C22×C8×C61C2×C411yescharacteristic
117#117×C61C2×C411yescharacteristic
118#103×C61C2×C411yescharacteristic
119#103×C61C2×C411yescharacteristic
120#102×C61C2×C411yescharacteristic
121#102×C61C2×C411yescharacteristic
122C23×D61—21no—
123C23×D61#4111yescharacteristic
124C22×#6923—22no—
125#41×C61—22no—
126#41×C61—41no—
127C2×Q8×C61C22×C411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61C22×C411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61C22×C411yescharacteristic
132C2×C22⋊D61—21no—
133C2×C22⋊D61—21no—
134C2×C22⋊D61—21no—
135C22×Dic61—21no—
136C22×Dic61C2×D411yescharacteristic
137#11758226—22no—
138#11758226—22no—
139Q8×D61—22no—
140Q8×D61—41no—
141#11758224—22no—
142#11758224—44no—
143#11758224#4114yes—
144D4×D61—22no—
145D4×D61—41no—
146D4×D61—44no—
147#11758222—22no—
148#11758222—41no—
149#11758222—44no—
150C2×D244—21no—
151C2×D244C22×C411yescharacteristic
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—44no—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61#4111yescharacteristic
163C2×C4×D61C2411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122C22×C411yescharacteristic
166C22⋊#6923—42no—
167#11758217—44no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—44no—
171C2×#92420C2×D412yes—
172C2×#92420C2×D412yes—
173C4×#6923—22no—
174C4×#6923—22no—
175#11758213—22no—
176#11758213—22no—
177#11758213—22no—
178#11758213—22no—
179D61⋊C8—22no—
180D61⋊C8—22no—
181C2×C8×C61—41no—
182C2×C8×C61C22×C411yescharacteristic
183C2×C8×C61C22×C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C4⋊C4×C61C22×C411yescharacteristic
186C22⋊C4×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—21no—
189#11758198—21no—
190#11758198—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C2×#92418—41no—
199C2×#92418C22×C411yescharacteristic
200C2×#92418C22×C411yescharacteristic
201C8×D61—22no—
202C8×D61—22no—
203C8×D61—41no—
204C8×D61—41no—
205C23×C61—21no—
206C23×C61C4×D411yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—21no—
210C22×D61—41no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—81no—
214C22×D61C4×D411yescharacteristic
215C22×D61C2×#4111yescharacteristic
216C2×#6923—42no—
217C2×#6923—42no—
218C2×#6923—42no—
219Q8×C61—21no—
220Q8×C61—21no—
221D4×C61—21no—
222D4×C61—42no—
223D4×C61#11712yes—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—42no—
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—24no—
233C22⋊D61—42no—
234C22⋊D61—42no—
235C2×Dic61—21no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61C4×D411yescharacteristic
241C2×Dic61C22×D411yescharacteristic
242D244—21no—
243D244—21no—
244D244—42no—
245C4×D61—22no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—22no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—81no—
254C4×D61C2×#4112yes—
255C4×D61H(5,𝔽2)12yes—
256Dic122—21no—
257Dic122—42no—
258Dic122#11712yes—
259#92420—22no—
260#92420—22no—
261#92420—42no—
262#92420—42no—
263C8×C61—21no—
264C8×C61—21no—
265#92418—21no—
266#92418—21no—
267#22586—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C2×C4×D411yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—44no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—84no—
284C2×D61#44412yes—
285C2×D61#55112yes—
286#6923—82no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61C2×#11711yescharacteristic
292C4×C61#51811yescharacteristic
293Dic61—22no—
294Dic61—21no—
295Dic61—41no—
296Dic61—41no—
297Dic61#54612yes—
298#2924—611no—
299C2×#416—611no—
300#2880—611no—
301#1938—612no—
302#1929—612no—
303#1866—612no—
304#1826—611no—
305#1810—611no—
306#1789—611no—
307#1765—612no—
308#1752—611no—
309C2×C61—41no—
310C2×C61—42no—
311C2×C61#234211yescharacteristic
312C2×C61C2×#44411yescharacteristic
313C2×C61#295711yescharacteristic
314D61—22no—
315D61—22no—
316D61—81no—
317C22×#41—611no—
318C2×#131—612no—
319C23×C8—611no—
320C2×#117—611no—
321#433—1222no—
322#432—1221no—
323#431—1221no—
324C2×#108—612no—
325#424—612no—
326C2×#104—612no—
327C2×#104—612no—
328#416—611no—
329#416—611no—
330#416—611no—
331#416—611no—
332#416—611no—
333#416—611no—
334#416—611no—
335#416#692311yescharacteristic, complemented
336C2×#103—611no—
337C2×#103—611no—
338C2×#102—611no—
339C2×#102—611no—
340#408—612no—
341#408—1222no—
342#407—611no—
343#407—611no—
344#407—1221no—
345C2×C22⋊C8—611no—
346C2×C22⋊C8—1221no—
347#405—612no—
348C2×#97—612no—
349C4×C22⋊C4—611no—
350#341—612no—
351#339—612no—
352#339—612no—
353#336—612no—
354C61#2258611yescharacteristic, complemented, Sylow
355C2×#41—611no—
356C2×#41—611no—
357C2×#41—611no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1224no—
361C2×#41—2442no—
362C2×#41C2×#692311yescharacteristic
363C22×Q8—611no—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—611no—
367C23×C4—1221no—
368#131—1224no—
369#131—2442no—
370C2×M16—612no—
371C2×M16—612no—
372C2×M16—1222no—
373C22×C8—612no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—1222no—
380C22×C8—2442no—
381C22×C8C2×#692311yescharacteristic
382#117—611no—
383#117—611no—
384#117—611no—
385#117—1221no—
386#117—1221no—
387#117C2×#692311yescharacteristic
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—612no—
393C2×C42—1221no—
394#108—2442no—
395C4⋊C8—1222no—
396C4⋊C8—1222no—
397#104—1224no—
398#104—2442no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#103—611no—
404#103—611no—
405#103C2×#692311yescharacteristic
406#103C2×#692311yescharacteristic
407#102—611no—
408#102—611no—
409#102—611no—
410#102—611no—
411#102—611no—
412#102—611no—
413#102C2×#692311yescharacteristic
414#102C2×#692311yescharacteristic
415C22⋊C8—612no—
416C22⋊C8—611no—
417C22⋊C8—611no—
418C22⋊C8—1222no—
419C22⋊C8—1222no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—1222no—
425#97—2442no—
426C4×C8—1222no—
427#95—1221no—
428C24—611no—
429C24—1221no—
430#41—22no—
431#41—41no—
432#41—614no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—1222no—
437#41—2441no—
438#41—2444no—
439#41—2444no—
440C2×Q8—611no—
441C2×Q8—1221no—
442C2×Q8—1222no—
443C2×Q8—2441no—
444C2×Q8C22×#692311yescharacteristic
445C2×D4—21no—
446C2×D4—611no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1222no—
451C2×D4—1221no—
452C2×D4—2441no—
453C2×D4—2444no—
454C2×D4C22×#692311yescharacteristic
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—2444no—
471C22×C4C22×#692311yescharacteristic
472M16—1222no—
473M16—1222no—
474M16—1222no—
475M16—1222no—
476M16—2444no—
477C2×C8—41no—
478C2×C8—612no—
479C2×C8—612no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—2441no—
489C2×C8—2441no—
490C2×C8—2441no—
491C2×C8—2444no—
492C2×C8C22×#692311yescharacteristic
493C2×C8C22×#692311yescharacteristic
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4C22×#692311yescharacteristic
499C4⋊C4C22×#692311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—2442no—
505C42—21no—
506C42—1221no—
507C42—1221no—
508C42—1222no—
509C42—1222no—
510C42—1221no—
511C23—21no—
512C23—611no—
513C23—611no—
514C23—1222no—
515C23—1222no—
516C23—1221no—
517C23—4881no—
518C23—2441no—
519C23—2442no—
520C23—2442no—
521C23D4×#692311yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—612no—
525Q8—1221no—
526Q8—2442no—
527D4—21no—
528D4—42no—
529D4—1224no—
530D4—1221no—
531D4—1221no—
532D4—2442no—
533D4—2442no—
534D4—2442no—
535D4#1451985612yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—42no—
542C2×C4—611no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—611no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—4881no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4D4×#692311yescharacteristic
564C2×C4C23×#692311yescharacteristic
565C8—21no—
566C8—21no—
567C8—1222no—
568C8—1222no—
569C8—1221no—
570C8—1221no—
571C8—2442no—
572C8—2442no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—21no—
577C22—41no—
578C22—41no—
579C22—612no—
580C22—612no—
581C22—1221no—
582C22—1221no—
583C22—1221no—
584C22—1221no—
585C22—4884no—
586C22—2444no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22C2×D4×#692311yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—612no—
596C4—1221no—
597C4—1222no—
598C4—4882no—
599C4—2441no—
600C4—2441no—
601C4C2×#1451985611yescharacteristic
602C4#1684082911yescharacteristic
603C2—41no—
604C2—42no—
605C2—1222no—
606C2—1222no—
607C2—4881no—
608C2#2146232511yescharacteristic
609C2D61⋊#44411yescharacteristic
610C2#2146259611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#416C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#416#6923
hypocenterC61#22586
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #21462512C2semidirect
3. #21461914C2semidirect
4. #21461903C2semidirect
5. #21461786C2semidirect
6. #21461727C2semidirect
7. #21461699C2semidirect
9. #21461598C2semidirect
10. #21461555C2semidirect
16. C2×#14519848C22semidirect
17. #16840769C22semidirect
27. #16840727C22semidirect
28. C2×#14519831C22semidirect
30. #16840638C22semidirect
31. #16840632C22semidirect
32. #16840632C22semidirect
33. (C2×#6923)⋊C8C22semidirect
37. #16839759C4semidirect
38. #16839758C4semidirect
41. #16839756C4semidirect
44. #103×D61C4semidirect
45. #103×D61C4semidirect
46. #102×D61C4semidirect
47. #102×D61C4semidirect
48. C61×#416C4semidirect
335. #416#6923semidirect
354. C61#22586semidirect

Maximal subgroups

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

subgroupindex
#214625122
#214624092
#214619142
#214619032
#214617862
#214617272
#214616992
#214616612
#214615982
#214615552
#416×D612
#2258661