Finite Group Databasebeta

← #29718918

Subgroups — #29718918

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455759C211yesmaximal, characteristic, complemented
2C2×#16839896C211yesmaximal, characteristic
3#21454477C211yesmaximal, characteristic, complemented
4#21447179C212yesmaximal, complemented
5#21447074C212yesmaximal, complemented
6#21446728C212yesmaximal, complemented
7#21446484C212yesmaximal, complemented
8#21446282C211yesmaximal, characteristic, complemented
9#21446024C211yesmaximal, characteristic, complemented
10#21445740C211yesmaximal, characteristic, complemented
11C2×C61×#418C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519751C2211yescharacteristic
14#16840086—21no—
15#16840086—21no—
16#16840080—21no—
17#16840075—21no—
18#16839896—21no—
19#16839896—21no—
20#16839896—22no—
21#16839823—22no—
22#16839819—21no—
23#16839815—21no—
24C2×#14519712C2212yescomplemented
25#16839805C2211yescharacteristic, complemented
26#16839805C2211yescharacteristic, complemented
27#16839789C2212yescomplemented
28#16839766C2211yescharacteristic, complemented
29#16839766C2211yescharacteristic, complemented
30C2×#14519698C2212yes—
31C2×#14519694C2212yes—
32C2×#14519694C2212yes—
33C2×#14519693C2212yescomplemented
34#16839726C2211yescharacteristic
35#16839726C2211yescharacteristic
36C2×C22⋊#92418C2211yescharacteristic
37#16839718C2212yescomplemented
38#16839530C2212yescomplemented
39#16839526C2212yescomplemented
40C23×D4×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#418—21no—
44C61×#418—21no—
45C61×#418C2212yes—
46C61×#418C2212yes—
47C2×#102×C61—21no—
48C2×#102×C61—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C2×#11758199—24no—
52C2×#11758199—22no—
53C2×#11758199—22no—
54C2×#11758199—42no—
55C2×#11758194C2312yes—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58C22×#92418C2312yes—
59#14519751—21no—
60#14519751—21no—
61#14519751—21no—
62#14519751—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758198—21no—
65#14519712—42no—
66#14519710—21no—
67#14519710—21no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—22no—
71C4⋊#92418—22no—
72#14519698—22no—
73#14519698—22no—
74#14519697—21no—
75#14519697—21no—
76#14519697—21no—
77#14519697—21no—
78#14519694—24no—
79#14519694—42no—
80#14519693—42no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85C22⋊#92418D411yescharacteristic
86C22⋊#92418D411yescharacteristic
87C22⋊#92418D412yes—
88#14519688—22no—
89C4×#92418—22no—
90C25×C61—21no—
91C22×D4×C61—24no—
92C22×D4×C61—21no—
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61—42no—
98C22×D4×C61C2312yes—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—21no—
101C2×M16×C61—42no—
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61D411yescharacteristic
107#117×C61D411yescharacteristic
108#117×C61C2311yescharacteristic
109#117×C61C2311yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#102×C61—22no—
115#102×C61—22no—
116#102×C61—22no—
117#102×C61—22no—
118C24×C61—22no—
119C24×C61—44no—
120C24×C61—41no—
121C24×C61—41no—
122C24×C61—82no—
123C24×C61—82no—
124C24×C61#4111yescharacteristic
125C23×D61—21no—
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61—44no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—41no—
139C2×D4×C61—41no—
140C2×D4×C61#4114yes—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61#4111yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—82no—
154C22×Dic61—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—22no—
158C2×D244—22no—
159M16×C61—22no—
160M16×C61—22no—
161M16×C61—42no—
162C2×C8×C61—22no—
163C2×C8×C61—42no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C2×D411yescharacteristic
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C4⋊C4×C61C2×D411yescharacteristic
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178#11758199—44no—
179#11758199—44no—
180#11758198—41no—
181#11758198—41no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418C2×D412yes—
194C2×#92418C2×D412yes—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—41no—
199C23×C61—44no—
200C23×C61—44no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—82no—
204C23×C61—82no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—82no—
208C23×C61—816no—
209C23×C61—82no—
210C23×C61#12111yescharacteristic
211C23×C61#12111yescharacteristic
212C23×C61C2×#4111yescharacteristic
213C22×D61—41no—
214C22×D61—42no—
215C22×D61—81no—
216C22×D61—81no—
217D4×C61—24no—
218D4×C61—24no—
219D4×C61—44no—
220D4×C61—44no—
221D4×C61—42no—
222D4×C61—42no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—42no—
235C2×C4×C61—81no—
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4112yes—
241C2×C4×C61H(5,𝔽2)12yes—
242C22⋊D61—82no—
243C22⋊D61—82no—
244C2×Dic61—41no—
245C2×Dic61—81no—
246D244—41no—
247D244—41no—
248D244—42no—
249C8×C61—42no—
250#92418—22no—
251#92418—22no—
252#92418—41no—
253#92418—41no—
254#92418—42no—
255#24520—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—44no—
264C22×C61—44no—
265C22×C61—84no—
266C22×C61—84no—
267C22×C61—88no—
268C22×C61—88no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61#47212yes—
272C22×C61C2×#12111yescharacteristic
273C22×C61#55112yes—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—162no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61#52511yescharacteristic
285C4×C61#53511yescharacteristic
286C4×C61#54612yes—
287Dic61—81no—
288#3094—611no—
289C2×#447—611no—
290#2924—611no—
291C2×#418D6111yescharacteristic, complemented
292#2063—612no—
293#2045—612no—
294#1978—612no—
295#1931—612no—
296#1900—611no—
297#1855—611no—
298#1783—611no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—41no—
302C2×C61—84no—
303C2×C61#241111yescharacteristic
304C2×C61C2×#47211yescharacteristic
305C2×C61#306811yescharacteristic
306D61—161no—
307C23×D4C2×D6111yescharacteristic
308C22×D8—612no—
309C22×M16C2×D6111yescharacteristic
310C2×#121—611no—
311C2×#117C2×D6111yescharacteristic
312#482—1221no—
313#482—1221no—
314#481—1221no—
315#480—1221no—
316#447—1221no—
317#447—1221no—
318#447—1222no—
319#433—1222no—
320#432—1221no—
321#431—1221no—
322C2×#108—612no—
323#428—611no—
324#428—611no—
325#424—612no—
326#418—21no—
327#418—21no—
328#418—611no—
329#418—611no—
330#418C2×D6112yes—
331#418C2×D6112yes—
332C2×#102—21no—
333C2×#102—21no—
334C2×#102—612no—
335C2×#100—612no—
336C2×#100—612no—
337C2×#100—612no—
338#407—611no—
339#407—611no—
340C2×C22⋊C8—611no—
341#405—612no—
342#343—612no—
343#341—612no—
344C61#2452011yescharacteristic, complemented, Sylow
345C25—21no—
346C22×D4—24no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4—21no—
351C22×D4—21no—
352C22×D4—42no—
353C22×D4—611no—
354C22×D4—1221no—
355C22×D4C22×D6112yes—
356C23×C4C22×D6111yescharacteristic
357C2×D8—1222no—
358C2×D8—1222no—
359C2×D8—1224no—
360C2×D8—2442no—
361C2×M16—21no—
362C2×M16—42no—
363C2×M16—611no—
364C2×M16—612no—
365C2×M16—611no—
366C2×M16C22×D6112yes—
367C22×C8—21no—
368C22×C8—612no—
369#121—1221no—
370#121—1221no—
371#121—1221no—
372#121—1221no—
373#117—21no—
374#117—21no—
375#117C22⋊D6111yescharacteristic
376#117C22⋊D6111yescharacteristic
377#117C22×D6111yescharacteristic
378#117C22×D6111yescharacteristic
379C2×C4⋊C4—611no—
380C2×C4⋊C4C22⋊D6111yescharacteristic
381C2×C4⋊C4C22⋊D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C42—21no—
385#108—2442no—
386#107—1221no—
387#107—1221no—
388#106—1221no—
389#106—1221no—
390C4⋊C8—1222no—
391C4⋊C8—1222no—
392#102—22no—
393#102—22no—
394#102—22no—
395#102—22no—
396#102—1222no—
397#102—1222no—
398#102—1221no—
399#102—1221no—
400#102—1221no—
401#102—1221no—
402#100—1224no—
403#100—2442no—
404#100—2442no—
405C22⋊C8—611no—
406C22⋊C8—611no—
407C22⋊C8—612no—
408C22⋊C8—1221no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412#97—1222no—
413C4×C8—1222no—
414C24—22no—
415C24—44no—
416C24—41no—
417C24—41no—
418C24—82no—
419C24—82no—
420C24—1221no—
421C24#1175822411yescharacteristic
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—4882no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4#1175822214yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—1221no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4#1175822411yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455D8—2444no—
456D8—2444no—
457M16—22no—
458M16—22no—
459M16—42no—
460M16—1222no—
461M16—1222no—
462M16—1222no—
463M16—1222no—
464C2×C8—22no—
465C2×C8—42no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—41no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—816no—
505C23—82no—
506C23—4881no—
507C23—4881no—
508C23—2442no—
509C23—2441no—
510C23#1451974911yescharacteristic
511C23#1451974911yescharacteristic
512C23C2×#1175822411yescharacteristic
513D4—24no—
514D4—24no—
515D4—44no—
516D4—44no—
517D4—42no—
518D4—42no—
519D4—4882no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523D4—2442no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—42no—
536C2×C4—81no—
537C2×C4—4881no—
538C2×C4—2441no—
539C2×C4#1451974911yescharacteristic
540C2×C4#1451974911yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C2×#1175822212yes—
544C2×C4#1451982212yes—
545C8—42no—
546C8—1222no—
547C8—1222no—
548C8—2441no—
549C8—2441no—
550C8—2442no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—44no—
559C22—44no—
560C22—84no—
561C22—84no—
562C22—88no—
563C22—88no—
564C22—82no—
565C22—82no—
566C22—9762no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22#1684003012yes—
571C22C2×#1451974911yescharacteristic
572C22#1684043812yes—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—4881no—
580C4#1684027311yescharacteristic
581C4#1684032911yescharacteristic
582C4#1684040212yes—
583C2—22no—
584C2—22no—
585C2—41no—
586C2—84no—
587C2—9761no—
588C2#2144920411yescharacteristic
589C2C2×#1684003011yescharacteristic
590C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC22×C61C2×#121
prime radicalC2×#16839896C2
hypercenterC2×#418D61
hypocenterC61#24520
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
1. #21455759C2semidirect
3. #21454477C2semidirect
4. #21447179C2semidirect
5. #21447074C2semidirect
6. #21446728C2semidirect
7. #21446484C2semidirect
8. #21446282C2semidirect
9. #21446024C2semidirect
10. #21445740C2semidirect
11. C2×C61×#418C2semidirect
24. C2×#14519712C22semidirect
25. #16839805C22semidirect
26. #16839805C22semidirect
27. #16839789C22semidirect
28. #16839766C22semidirect
29. #16839766C22semidirect
33. C2×#14519693C22semidirect
37. #16839718C22semidirect
38. #16839530C22semidirect
39. #16839526C22semidirect
291. C2×#418D61semidirect
344. C61#24520semidirect

Maximal subgroups

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

subgroupindex
#214557592
C2×#168398962
#214544772
#214471792
#214470742
#214467282
#214464842
#214462822
#214460242
#214457402
C2×C61×#4182
#2452061