Finite Group Databasebeta

← #29697364

Subgroups — #29697364

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#21454479C211yesmaximal, characteristic, complemented
2C2×#16839758C211yesmaximal, characteristic
3#21454268C211yesmaximal, characteristic
4#21446527C212yesmaximal, complemented
5#21446470C212yesmaximal, complemented
6#21446092C212yesmaximal, complemented
7#21445919C211yesmaximal, characteristic, complemented
8#21445855C211yesmaximal, characteristic, complemented
9#21445664C212yesmaximal, complemented
10#21445597C211yesmaximal, characteristic, complemented
11C61×#1789C211yesmaximal, characteristic
12C2×#14519784C2212yes—
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15#16839820—22no—
16#16839816—21no—
17#16839812—21no—
18C2×#14519712C2212yescomplemented
19#16839790C2212yescomplemented
20#16839758C411yescharacteristic, complemented
21#16839758C411yescharacteristic, complemented
22#16839758C411yescharacteristic, complemented
23#16839758C411yescharacteristic, complemented
24#16839758C2211yescharacteristic
25#16839758C2211yescharacteristic
26#16839758C2211yescharacteristic
27#16839758C2211yescharacteristic
28C2×#14519701C411yescharacteristic, complemented
29C2×#14519701C411yescharacteristic, complemented
30C2×#14519698C411yescharacteristic, complemented
31C2×#14519698C411yescharacteristic, complemented
32#16839729—22no—
33#16839729C2212yes—
34#16839726—21no—
35#16839726C2211yescharacteristic
36C2×C22⋊#92418—21no—
37C2×C22⋊#92418C2211yescharacteristic
38#16839720C2212yescomplemented
39C2×#14519688C2212yescomplemented
40#16839526C2212yescomplemented
41#16839522C2212yescomplemented
42#16839522C2212yescomplemented
43#16839517C2212yescomplemented
44C22×#41×C61C2211yescharacteristic
45C2×#104×C61C2212yes—
46C2×#104×C61C2212yes—
47C61×#407C2211yescharacteristic
48C4×C22⋊C4×C61C2211yescharacteristic
49#14519784—24no—
50#14519784—42no—
51C2×#11758194—22no—
52C2×#11758194C2312yes—
53C22×#92418—22no—
54C22×#92418—42no—
55C22×#92418C2×C411yescharacteristic
56C22×#92418C2×C411yescharacteristic
57C22×#92418C2×C411yescharacteristic
58C22×#92418C2×C411yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2312yes—
62#14519736—21no—
63#14519736—21no—
64#14519736C2×C411yescharacteristic
65#14519736C2×C411yescharacteristic
66#14519736C2311yescharacteristic
67#14519736C2311yescharacteristic
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×C4⋊Dic61C2×C411yescharacteristic
70C2×C22⋊Dic61—21no—
71C2×C4×Dic61—21no—
72#14519712—42no—
73#14519706—22no—
74#14519706—22no—
75#14519701C2×C411yescharacteristic
76#14519701C2×C411yescharacteristic
77#14519701C2×C411yescharacteristic
78#14519701C2×C411yescharacteristic
79#14519701C2×C411yescharacteristic
80#14519701C2×C411yescharacteristic
81#14519701C2×C411yescharacteristic
82#14519701C2×C411yescharacteristic
83#14519698C2×C411yescharacteristic
84#14519698C2×C411yescharacteristic
85#14519698C2×C411yescharacteristic
86#14519698C2×C411yescharacteristic
87#14519698C2×C411yescharacteristic
88#14519698C2×C411yescharacteristic
89#14519698C2×C411yescharacteristic
90#14519698C2×C411yescharacteristic
91C22⋊#92418—21no—
92C22⋊#92418—21no—
93C22⋊#92418—22no—
94C22⋊#92418—22no—
95C22⋊#92418D411yescharacteristic
96C22⋊#92418D411yescharacteristic
97C22⋊#92418D412yes—
98#14519688—42no—
99#14519687—22no—
100C8×Dic61—22no—
101C2×#41×C61—21no—
102C2×#41×C61—21no—
103C2×#41×C61—24no—
104C2×#41×C61—42no—
105C2×#41×C61C2×C411yescharacteristic
106C2×#41×C61C2×C411yescharacteristic
107C2×#41×C61C2311yescharacteristic
108C2×#41×C61C2311yescharacteristic
109C22×Q8×C61C2×C411yescharacteristic
110C22×D4×C61—21no—
111C22×D4×C61C2×C411yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114C2×M16×C61C2312yes—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61C2312yes—
117#104×C61—24no—
118#104×C61—42no—
119C22⋊C8×C61—21no—
120C22⋊C8×C61—21no—
121#95×C61—21no—
122C24×C61—21no—
123C24×C61#4111yescharacteristic
124#41×C61—22no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—22no—
128#41×C61—22no—
129#41×C61—44no—
130#41×C61—41no—
131#41×C61—41no—
132#41×C61—44no—
133#41×C61#4114yes—
134C2×Q8×C61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61—41no—
137C2×Q8×C61C22×C411yescharacteristic
138C2×Q8×C61C22×C411yescharacteristic
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—41no—
146C2×D4×C61—44no—
147C2×D4×C61C22×C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—22no—
157C22×C4×C61—44no—
158C22×C4×C61C22×C411yescharacteristic
159C22×C4×C61C22×C411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—21no—
165C22×Dic61—21no—
166M16×C61—22no—
167M16×C61—22no—
168C2×C8×C61—22no—
169C22⋊C4×C61—42no—
170C42×C61—22no—
171C42×C61—22no—
172C22⋊Dic61—21no—
173C22⋊Dic61—21no—
174C22⋊Dic61—21no—
175C22⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22×C411yescharacteristic
179C4⋊Dic61C22×C411yescharacteristic
180C4⋊Dic61C22×C411yescharacteristic
181C4⋊Dic61C22×C411yescharacteristic
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186#11758194—22no—
187#11758194—22no—
188#11758194—44no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—41no—
195C2×#92418—41no—
196C2×#92418—41no—
197C2×#92418—41no—
198C2×#92418—44no—
199C2×#92418C22×C411yescharacteristic
200C2×#92418C22×C411yescharacteristic
201C2×#92418C22×C411yescharacteristic
202C2×#92418C22×C411yescharacteristic
203C2×#92418C2×D412yes—
204C2×#92418C2×D412yes—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—41no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—81no—
213C23×C61C4×D411yescharacteristic
214C23×C61C4×D411yescharacteristic
215C23×C61C2×#4111yescharacteristic
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—42no—
220Q8×C61#11712yes—
221D4×C61—24no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—42no—
229D4×C61#11712yes—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—24no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61C4×D411yescharacteristic
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4112yes—
252C2×C4×C61H(5,𝔽2)12yes—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—42no—
258C2×Dic61—81no—
259C8×C61—42no—
260#92418—21no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—22no—
265#92418—22no—
266#92418—42no—
267#22586—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—22no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—44no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—84no—
283C22×C61#44412yes—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61#55112yes—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—82no—
291C4×C61C2×#11711yescharacteristic
292C4×C61#51811yescharacteristic
293C4×C61#54612yes—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
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#1789D6111yescharacteristic
307#1765—612no—
308#1752—611no—
309C2×C61—22no—
310C2×C61—22no—
311C2×C61—41no—
312C2×C61—42no—
313C2×C61—81no—
314C2×C61#234211yescharacteristic
315C2×C61C2×#44411yescharacteristic
316C2×C61#295711yescharacteristic
317C22×#41C2×D6111yescharacteristic
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×#104C2×D6112yes—
327C2×#104C2×D6112yes—
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—611no—
338C2×#102—611no—
339C2×#102—611no—
340#408—612no—
341#408—1222no—
342#407—611no—
343#407—1221no—
344#407C2×D6111yescharacteristic
345C2×C22⋊C8—611no—
346C2×C22⋊C8—1221no—
347#405—612no—
348C2×#97—612no—
349C4×C22⋊C4C2×D6111yescharacteristic
350#341—612no—
351#339—612no—
352#339—612no—
353#336—612no—
354C61#2258611yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41—21no—
357C2×#41—24no—
358C2×#41—42no—
359C2×#41C4×D6111yescharacteristic
360C2×#41C4×D6111yescharacteristic
361C2×#41C22×D6111yescharacteristic
362C2×#41C22×D6111yescharacteristic
363C22×Q8C4×D6111yescharacteristic
364C22×D4—21no—
365C22×D4C4×D6111yescharacteristic
366C23×C4—21no—
367C23×C4C22×D6111yescharacteristic
368#131—1224no—
369#131—2442no—
370C2×M16—612no—
371C2×M16—1222no—
372C2×M16C22×D6112yes—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—612no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—1222no—
381C22×C8—2442no—
382#117—611no—
383#117—611no—
384#117—611no—
385#117—611no—
386#117—1221no—
387#117—1221no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1221no—
392C2×C42—1221no—
393C2×C42C22×D6112yes—
394#108—2442no—
395C4⋊C8—1222no—
396C4⋊C8—1222no—
397#104—24no—
398#104—42no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#103—611no—
404#103—611no—
405#103—611no—
406#103—611no—
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—21no—
416C22⋊C8—21no—
417C22⋊C8—611no—
418C22⋊C8—611no—
419C22⋊C8—612no—
420C22⋊C8—1222no—
421C22⋊C8—1222no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—1222no—
425#97—2442no—
426C4×C8—1222no—
427#95—21no—
428C24—21no—
429C24#1175822411yescharacteristic
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—44no—
436#41—41no—
437#41—41no—
438#41—44no—
439#41#1175822414yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8C2×C4×D6111yescharacteristic
444C2×Q8C2×C4×D6111yescharacteristic
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—44no—
453C2×D4C2×C4×D6111yescharacteristic
454C2×D4C2×C4×D6111yescharacteristic
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4C2×C4×D6111yescharacteristic
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4#1175822411yescharacteristic
471C22×C4C23×D6111yescharacteristic
472M16—22no—
473M16—22no—
474M16—1222no—
475M16—1222no—
476M16—2444no—
477C2×C8—22no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—612no—
483C2×C8—612no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—2441no—
490C2×C8—2441no—
491C2×C8—2441no—
492C2×C8—2441no—
493C2×C8—2444no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C22⋊C4—42no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C42—22no—
506C42—22no—
507C42—1221no—
508C42—1221no—
509C42—1221no—
510C42—1221no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—81no—
519C23#1451973711yescharacteristic
520C23#1451973711yescharacteristic
521C23C2×#1175822411yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—21no—
525Q8—42no—
526Q8#1451973312yes—
527D4—24no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4#1451973312yes—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4881no—
558C2×C4—2442no—
559C2×C4#1451973711yescharacteristic
560C2×C4#1451973711yescharacteristic
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×#1175822412yes—
564C2×C4#1451982212yes—
565C8—42no—
566C8—1221no—
567C8—1221no—
568C8—1221no—
569C8—1221no—
570C8—1222no—
571C8—1222no—
572C8—2442no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—44no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—84no—
588C22#1683988112yes—
589C22C2×#1451973711yescharacteristic
590C22#1684044312yes—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—41no—
595C4—82no—
596C4—2441no—
597C4—2441no—
598C4—2441no—
599C4—2441no—
600C4C2×#1451973311yescharacteristic
601C4#1684023511yescharacteristic
602C4#1684039912yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—42no—
607C2—81no—
608C2#2144867011yescharacteristic
609C2C2×#1683988111yescharacteristic
610C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1789C2
socleC22×C61C2×C4×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#1789D61
hypocenterC61#22586
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
1. #21454479C2semidirect
4. #21446527C2semidirect
5. #21446470C2semidirect
6. #21446092C2semidirect
7. #21445919C2semidirect
8. #21445855C2semidirect
9. #21445664C2semidirect
10. #21445597C2semidirect
18. C2×#14519712C22semidirect
19. #16839790C22semidirect
20. #16839758C4semidirect
21. #16839758C4semidirect
22. #16839758C4semidirect
23. #16839758C4semidirect
28. C2×#14519701C4semidirect
29. C2×#14519701C4semidirect
30. C2×#14519698C4semidirect
31. C2×#14519698C4semidirect
38. #16839720C22semidirect
39. C2×#14519688C22semidirect
40. #16839526C22semidirect
41. #16839522C22semidirect
42. #16839522C22semidirect
43. #16839517C22semidirect
354. C61#22586semidirect

Maximal subgroups

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

subgroupindex
#214544792
C2×#168397582
#214542682
#214465272
#214464702
#214460922
#214459192
#214458552
#214456642
#214455972
C61×#17892
#2258661