Finite Group Databasebeta

← #29718914

Subgroups — #29718914

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#21455755C211yesmaximal, characteristic, complemented
2#21454473C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4#21447177C212yesmaximal, complemented
5#21447072C212yesmaximal, complemented
6#21446730C212yesmaximal, complemented
7#21446478C212yesmaximal, complemented
8#21446283C211yesmaximal, characteristic, complemented
9#21446026C211yesmaximal, characteristic, complemented
10#21445740C211yesmaximal, characteristic, complemented
11C2×C61×#447C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16840083—21no—
15#16840083—21no—
16#16840078—21no—
17#16840073—21no—
18#16839820—22no—
19#16839816—21no—
20#16839812—21no—
21C2×#14519713C2212yescomplemented
22#16839802C2211yescharacteristic, complemented
23#16839802C2211yescharacteristic, complemented
24#16839788C2212yescomplemented
25#16839767—21no—
26#16839767—21no—
27#16839767C2212yes—
28#16839767C2212yes—
29#16839767C2211yescharacteristic, complemented
30#16839767C2211yescharacteristic, complemented
31C2×#14519698—21no—
32C2×#14519698—21no—
33C2×#14519694C2212yes—
34C2×#14519694C2212yes—
35C2×#14519694C2212yescomplemented
36#16839726C2211yescharacteristic
37#16839726C2211yescharacteristic
38#16839717C2212yescomplemented
39#16839531C2212yescomplemented
40#16839527C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×D8×C61C2212yes—
43C2×#121×C61C2211yescharacteristic
44C61×#447—21no—
45C61×#447—21no—
46C61×#447—22no—
47C2×#102×C61C2212yes—
48C2×C22⋊C8×C61C2211yescharacteristic
49C2×#11758194—21no—
50C2×#11758194—42no—
51C2×#11758194C2312yes—
52C2×#11758194C2311yescharacteristic
53C2×#11758194C2311yescharacteristic
54C2×#11758194C2312yes—
55C22×#92418—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736D411yescharacteristic
59#14519736D411yescharacteristic
60#14519736C2311yescharacteristic
61#14519736C2311yescharacteristic
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×C22⋊Dic61—21no—
65C2×C4×Dic61—21no—
66#14519713—42no—
67#14519710—21no—
68#14519710—21no—
69#14519708—21no—
70#14519708—21no—
71#14519706—22no—
72#14519706—22no—
73#14519698—22no—
74#14519698—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—21no—
78#14519698—21no—
79#14519698—21no—
80#14519698—21no—
81#14519694—24no—
82#14519694—42no—
83#14519694—42no—
84C22⋊#92418—21no—
85C22⋊#92418—21no—
86C22⋊#92418—21no—
87C22⋊#92418—21no—
88#14519687—22no—
89C8×Dic61—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—21no—
98C22×D4×C61—42no—
99C22×D4×C61C2311yescharacteristic
100C22×D4×C61C2312yes—
101C23×C4×C61C2311yescharacteristic
102C2×D8×C61—24no—
103C2×D8×C61—22no—
104C2×D8×C61—22no—
105C2×D8×C61—42no—
106C22×C8×C61C2312yes—
107#121×C61—21no—
108#121×C61—21no—
109#121×C61—21no—
110#121×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113#102×C61—22no—
114#102×C61—22no—
115C22⋊C8×C61D411yescharacteristic
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D412yes—
118C24×C61—22no—
119C24×C61—21no—
120C24×C61—44no—
121C24×C61—41no—
122C24×C61—41no—
123C24×C61—82no—
124C24×C61—82no—
125C24×C61#4111yescharacteristic
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61—44no—
140C2×D4×C61—42no—
141C2×D4×C61—42no—
142C2×D4×C61—41no—
143C2×D4×C61—41no—
144C2×D4×C61—41no—
145C2×D4×C61—41no—
146C2×D4×C61—82no—
147C2×D4×C61#4114yes—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159D8×C61—44no—
160D8×C61—44no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61C2×D412yes—
164C2×C8×C61C2×D412yes—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172C22⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61C2×D411yescharacteristic
176C4⋊Dic61C2×D411yescharacteristic
177C4⋊Dic61C2×D411yescharacteristic
178C4⋊Dic61C2×D411yescharacteristic
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—42no—
190C2×#92418—22no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—22no—
194C2×#92418—42no—
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—41no—
204C23×C61—42no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—82no—
208C23×C61—82no—
209C23×C61—82no—
210C23×C61—81no—
211C23×C61—81no—
212C23×C61—816no—
213C23×C61—82no—
214C23×C61#12111yescharacteristic
215C23×C61#12111yescharacteristic
216C23×C61C2×#4111yescharacteristic
217D4×C61—24no—
218D4×C61—24no—
219D4×C61—44no—
220D4×C61—44no—
221D4×C61—41no—
222D4×C61—42no—
223D4×C61—41no—
224D4×C61—42no—
225D4×C61—42no—
226D4×C61—82no—
227D4×C61—82no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—81no—
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4112yes—
242C2×C4×C61H(5,𝔽2)12yes—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—42no—
248C2×Dic61—81no—
249C8×C61—22no—
250C8×C61—22no—
251#92418—41no—
252#92418—42no—
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—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—162no—
275C22×C61#47212yes—
276C22×C61C2×#12111yescharacteristic
277C22×C61#55112yes—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—81no—
281C4×C61#52511yescharacteristic
282C4×C61#53511yescharacteristic
283C4×C61#54612yes—
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288#3094—611no—
289C2×#447D6111yescharacteristic
290#2924—611no—
291C2×#418—611no—
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—161no—
304C2×C61#241111yescharacteristic
305C2×C61C2×#47211yescharacteristic
306C2×C61#306811yescharacteristic
307C23×D4C2×D6111yescharacteristic
308C22×D8C2×D6112yes—
309C22×M16—611no—
310C2×#121C2×D6111yescharacteristic
311C2×#117—611no—
312#482—1221no—
313#482—1221no—
314#481—1221no—
315#480—1221no—
316#447—21no—
317#447—21no—
318#447—22no—
319#433—1222no—
320#432—1221no—
321#431—1221no—
322C2×#108—612no—
323#428—611no—
324#428—611no—
325#424—612no—
326#418—612no—
327#418—611no—
328#418—611no—
329#418—612no—
330#418—1221no—
331#418—1221no—
332C2×#102—1221no—
333C2×#102—1221no—
334C2×#102C2×D6112yes—
335C2×#100—612no—
336C2×#100—612no—
337C2×#100—612no—
338#407—611no—
339#407—611no—
340C2×C22⋊C8C2×D6111yescharacteristic
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—21no—
353C22×D4—42no—
354C22×D4C22×D6111yescharacteristic
355C22×D4C22×D6112yes—
356C23×C4C22×D6111yescharacteristic
357C2×D8—24no—
358C2×D8—22no—
359C2×D8—22no—
360C2×D8—42no—
361C2×M16—612no—
362C2×M16—611no—
363C2×M16—612no—
364C2×M16—611no—
365C2×M16—1221no—
366C2×M16—2442no—
367C22×C8—1221no—
368C22×C8C22×D6112yes—
369#121—21no—
370#121—21no—
371#121—21no—
372#121—21no—
373#117—611no—
374#117—611no—
375#117—611no—
376#117—611no—
377#117—1221no—
378#117—1221no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C42—1221no—
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—1222no—
395#102—1222no—
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—1221no—
406C22⋊C8—1221no—
407C22⋊C8—1221no—
408C22⋊C8—1221no—
409C22⋊C8C22⋊D6111yescharacteristic
410C22⋊C8C22⋊D6111yescharacteristic
411C22⋊C8C22⋊D6112yes—
412#97—1222no—
413C4×C8—1222no—
414C24—22no—
415C24—21no—
416C24—44no—
417C24—41no—
418C24—41no—
419C24—82no—
420C24—82no—
421C24#1175822411yescharacteristic
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—82no—
443C2×D4#1175822414yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4#1175822411yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455D8—44no—
456D8—44no—
457M16—1222no—
458M16—1222no—
459M16—1222no—
460M16—1222no—
461M16—1222no—
462M16—1222no—
463M16—2442no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—2442no—
471C2×C8C2×C22⋊D6112yes—
472C2×C8C2×C22⋊D6112yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—611no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—1221no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C42—1221no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—41no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—81no—
507C23—81no—
508C23—816no—
509C23—82no—
510C23#1451974811yescharacteristic
511C23#1451974811yescharacteristic
512C23C2×#1175822411yescharacteristic
513D4—24no—
514D4—24no—
515D4—44no—
516D4—44no—
517D4—41no—
518D4—42no—
519D4—41no—
520D4—42no—
521D4—42no—
522D4—82no—
523D4—82no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—81no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—4881no—
538C2×C4—2442no—
539C2×C4#1451974811yescharacteristic
540C2×C4#1451974811yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C2×#1175822412yes—
544C2×C4#1451982212yes—
545C8—22no—
546C8—22no—
547C8—2441no—
548C8—2442no—
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—81no—
567C22—81no—
568C22—81no—
569C22—162no—
570C22#1684002912yes—
571C22C2×#1451974811yescharacteristic
572C22#1684044312yes—
573C4—21no—
574C4—21no—
575C4—81no—
576C4—2441no—
577C4—2441no—
578C4—2441no—
579C4—2441no—
580C4#1684026811yescharacteristic
581C4#1684032711yescharacteristic
582C4#1684040612yes—
583C2—22no—
584C2—22no—
585C2—41no—
586C2—84no—
587C2—161no—
588C2#2144920011yescharacteristic
589C2C2×#1684002911yescharacteristic
590C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#447C2
socleC22×C61C2×#121
hypercenterC2×#447D61
hypocenterC61#24520
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. #21455755C2semidirect
2. #21454473C2semidirect
3. C2×#16839767C2semidirect
4. #21447177C2semidirect
5. #21447072C2semidirect
6. #21446730C2semidirect
7. #21446478C2semidirect
8. #21446283C2semidirect
9. #21446026C2semidirect
10. #21445740C2semidirect
21. C2×#14519713C22semidirect
22. #16839802C22semidirect
23. #16839802C22semidirect
24. #16839788C22semidirect
29. #16839767C22semidirect
30. #16839767C22semidirect
35. C2×#14519694C22semidirect
38. #16839717C22semidirect
39. #16839531C22semidirect
40. #16839527C22semidirect
344. C61#24520semidirect

Maximal subgroups

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

subgroupindex
#214557552
#214544732
C2×#168397672
#214471772
#214470722
#214467302
#214464782
#214462832
#214460262
#214457402
C2×C61×#4472
#2452061