Finite Group Databasebeta

← #30620620

Subgroups — #30620620

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#21462509C211yesmaximal, characteristic, complemented
2#21462408C211yesmaximal, characteristic
3#21461914C212yesmaximal, complemented
4#21461903C212yesmaximal, complemented
5#21461787C212yesmaximal, complemented
6#21461725C211yesmaximal, characteristic, complemented
7#21461699C211yesmaximal, characteristic, complemented
8#21461662C211yesmaximal, characteristic
9#21461600C212yesmaximal, complemented
10#21461555C211yesmaximal, characteristic, complemented
11C2×#16839757C211yesmaximal, characteristic
12C2×#14519865C2212yes—
13#16840787—22no—
14#16840782—21no—
15#16840778—21no—
16C2×#14519848C2212yescomplemented
17#16840769C2212yescomplemented
18C2×#14519842C2212yes—
19C2×#14519842C2212yes—
20#16840737—22no—
21#16840737C2212yes—
22#16840733—21no—
23#16840733C2211yescharacteristic
24#16840733C2211yescharacteristic
25D61⋊(C22⋊C8)—21no—
26D61⋊(C22⋊C8)C2211yescharacteristic
27#16840727C2212yescomplemented
28C2×#14519831C2212yescomplemented
29C4×C22⋊#6923C2211yescharacteristic
30#16840638C2212yescomplemented
31#16840632C2212yescomplemented
32#16840632C2212yescomplemented
33(C2×#6923)⋊C8C2212yescomplemented
34C22×#11758226C2211yescharacteristic
35C22×C8×D61C2211yescharacteristic
36C2×#14519735C2211yescharacteristic
37#16839757C411yescharacteristic, complemented
38#16839757C411yescharacteristic, complemented
39#16839757C411yescharacteristic, complemented
40#16839757C411yescharacteristic, complemented
41#16839757C2211yescharacteristic
42#16839757C2211yescharacteristic
43#16839757C2211yescharacteristic
44#16839757C2211yescharacteristic
45C2×#14519701C411yescharacteristic, complemented
46C2×#14519699C411yescharacteristic, complemented
47C2×#14519697C411yescharacteristic, complemented
48C2×#103×C61C411yescharacteristic, complemented
49#14519865—24no—
50#14519865—42no—
51C2×#11758213—22no—
52C2×#11758213C2312yes—
53C2×#11758213C2312yes—
54C2×D61⋊C8—22no—
55C2×D61⋊C8C2312yes—
56C2×C22⋊#6923—21no—
57C2×C4×#6923C2312yes—
58#14519848—42no—
59#14519844—22no—
60C4⋊#92420—22no—
61#14519842—24no—
62#14519842—42no—
63#14519834—21no—
64#14519834—21no—
65#14519834—22no—
66#14519834—22no—
67#14519834D412yes—
68C22⋊#92420—21no—
69C22⋊#92420—21no—
70C22⋊#92420D411yescharacteristic
71C22⋊#92420D411yescharacteristic
72#14519832—22no—
73#14519831—42no—
74C4×#92420—22no—
75(C2×#6923)⋊C4—21no—
76C2×#11758226—21no—
77C2×#11758226—21no—
78C2×#11758226—24no—
79C2×#11758226—42no—
80C2×#11758226C2×C411yescharacteristic
81C2×#11758226C2×C411yescharacteristic
82C2×#11758226C2311yescharacteristic
83C2×#11758226C2311yescharacteristic
84C22×D244—21no—
85C22×D244C2×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#14519735—21no—
95#14519735—21no—
96#14519735C2×C411yescharacteristic
97#14519735C2×C411yescharacteristic
98#14519735C2311yescharacteristic
99#14519735C2311yescharacteristic
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103#14519701C2×C411yescharacteristic
104#14519701C2×C411yescharacteristic
105#14519701C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519699C2×C411yescharacteristic
108#14519699C2×C411yescharacteristic
109#14519699C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519697C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114#14519697C2×C411yescharacteristic
115C22×Q8×C61C2×C411yescharacteristic
116C22×C8×C61C2×C411yescharacteristic
117C2×C4⋊C4×C61C2×C411yescharacteristic
118#103×C61C2×C411yescharacteristic
119#103×C61C2×C411yescharacteristic
120#103×C61C2×C411yescharacteristic
121#103×C61C2×C411yescharacteristic
122C23×D61—21no—
123C23×D61#4111yescharacteristic
124C22×#6923—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—41no—
128C2×Q8×C61C22×C411yescharacteristic
129C2×Q8×C61C22×C411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C22×C411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61C2×D411yescharacteristic
135#11758226—22no—
136#11758226—22no—
137#11758226—22no—
138#11758226—22no—
139#11758226—22no—
140#11758226—44no—
141#11758226—41no—
142#11758226—41no—
143#11758226—44no—
144#11758226#4114yes—
145C2×D244—21no—
146C2×D244—22no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—41no—
152C2×D244—44no—
153C2×D244C22×C411yescharacteristic
154C2×D244C22×C411yescharacteristic
155C2×C4×D61—21no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—44no—
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165C2×C4×D61C2411yescharacteristic
166C22⋊#6923—42no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420C2×D412yes—
170C2×#92420C2×D412yes—
171C4×#6923—22no—
172C4×#6923—22no—
173#11758213—22no—
174#11758213—22no—
175#11758213—22no—
176#11758213—22no—
177#11758213—44no—
178D61⋊C8—22no—
179D61⋊C8—22no—
180D61⋊C8—44no—
181C2×C8×C61—41no—
182C2×C8×C61C22×C411yescharacteristic
183C2×C8×C61C22×C411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22×C411yescharacteristic
186C4⋊C4×C61C22×C411yescharacteristic
187#11758198—21no—
188#11758198—21no—
189#11758198—21no—
190#11758198—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194C4×Dic61—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×C61C4×D411yescharacteristic
206C22×D61—21no—
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—
221Q8×C61—21no—
222Q8×C61—42no—
223Q8×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—21no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—81no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C22×D411yescharacteristic
240D244—24no—
241D244—21no—
242D244—21no—
243D244—21no—
244D244—42no—
245D244—42no—
246D244—42no—
247D244—42no—
248D244#11712yes—
249C4×D61—22no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—24no—
253C4×D61—22no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61C2×#4112yes—
258C4×D61H(5,𝔽2)12yes—
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—21no—
269C22×C61—21no—
270C22×C61#44412yes—
271C22×C61C2×C4×D411yescharacteristic
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—22no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—44no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—84no—
285C2×D61#55112yes—
286#6923—82no—
287C4×C61—21no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61C2×#11711yescharacteristic
293C4×C61#51811yescharacteristic
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—22no—
310C2×C61#234211yescharacteristic
311C2×C61C2×#44411yescharacteristic
312C2×C61#295711yescharacteristic
313D61—22no—
314D61—41no—
315D61—42no—
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—611no—
336C2×#103—611no—
337C2×#103#692311yescharacteristic, complemented
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—611no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1224no—
362C2×#41—2442no—
363C22×Q8C2×#692311yescharacteristic
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—611no—
386#117—1221no—
387#117—1221no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4C2×#692311yescharacteristic
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#103C2×#692311yescharacteristic
404#103C2×#692311yescharacteristic
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#102—611no—
414#102—611no—
415C22⋊C8—611no—
416C22⋊C8—611no—
417C22⋊C8—612no—
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—614no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—2444no—
437#41—2441no—
438#41—2441no—
439#41—2444no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8C22×#692311yescharacteristic
444C2×Q8C22×#692311yescharacteristic
445C2×D4—611no—
446C2×D4—611no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—2441no—
454C2×D4—2444no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1222no—
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—21no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—1221no—
498C4⋊C4C22×#692311yescharacteristic
499C4⋊C4C22×#692311yescharacteristic
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—2442no—
505C42—1221no—
506C42—1221no—
507C42—1221no—
508C42—1221no—
509C42—1222no—
510C42—1222no—
511C23—611no—
512C23—611no—
513C23—1222no—
514C23—1222no—
515C23—1221no—
516C23—1221no—
517C23—4881no—
518C23—2441no—
519C23—2442no—
520C23—2442no—
521C23D4×#692311yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—21no—
525Q8—42no—
526Q8#1451985612yes—
527D4—612no—
528D4—1224no—
529D4—1221no—
530D4—1221no—
531D4—1221no—
532D4—2442no—
533D4—2442no—
534D4—2442no—
535D4—2442no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—611no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—611no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—4881no—
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—21no—
575C22—612no—
576C22—1221no—
577C22—1221no—
578C22—1221no—
579C22—1222no—
580C22—1221no—
581C22—1221no—
582C22—4884no—
583C22—2444no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22—2441no—
589C22#1684080712yes—
590C22C2×D4×#692311yescharacteristic
591C4—21no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—612no—
597C4—1221no—
598C4—4882no—
599C4—2441no—
600C4—2441no—
601C4C2×#1451985611yescharacteristic
602C4#1684082911yescharacteristic
603C2—22no—
604C2—1222no—
605C2—4881no—
606C2—2441no—
607C2—2442no—
608C2#2146232511yescharacteristic
609C2C2×#1684080711yescharacteristic
610C2#2146259911yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#103×C61C4
socleC22×C61C2×C4×D4
prime radicalC22×#11758226C22
cosocleC2×C4×D61C24
hypercenterC2×#103#6923
hypocenterC61#22586
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #21462509C2semidirect
3. #21461914C2semidirect
4. #21461903C2semidirect
5. #21461787C2semidirect
6. #21461725C2semidirect
7. #21461699C2semidirect
9. #21461600C2semidirect
10. #21461555C2semidirect
16. C2×#14519848C22semidirect
17. #16840769C22semidirect
27. #16840727C22semidirect
28. C2×#14519831C22semidirect
30. #16840638C22semidirect
31. #16840632C22semidirect
32. #16840632C22semidirect
33. (C2×#6923)⋊C8C22semidirect
37. #16839757C4semidirect
38. #16839757C4semidirect
39. #16839757C4semidirect
40. #16839757C4semidirect
45. C2×#14519701C4semidirect
46. C2×#14519699C4semidirect
47. C2×#14519697C4semidirect
48. C2×#103×C61C4semidirect
337. C2×#103#6923semidirect
354. C61#22586semidirect

Maximal subgroups

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

subgroupindex
#214625092
#214624082
#214619142
#214619032
#214617872
#214617252
#214616992
#214616622
#214616002
#214615552
C2×#168397572
#2258661