Finite Group Databasebeta

← #29718925

Subgroups — #29718925

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455763C211yesmaximal, characteristic, complemented
2#21454473C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic
4#21447177C212yesmaximal, complemented
5#21447077C212yesmaximal, complemented
6#21446724C212yesmaximal, complemented
7#21446478C212yesmaximal, complemented
8#21446288C211yesmaximal, characteristic, complemented
9#21446026C211yesmaximal, characteristic, complemented
10#21445740C211yesmaximal, characteristic
11C2×C61×#450C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16840100—21no—
15#16840100—21no—
16#16840095—21no—
17#16840090—21no—
18#16839820—22no—
19#16839816—21no—
20#16839812—21no—
21C2×#14519713C2212yescomplemented
22#16839802C2211yescharacteristic, complemented
23#16839802C2211yescharacteristic, complemented
24#16839788C2212yescomplemented
25#16839771C2211yescharacteristic, complemented
26#16839771C2211yescharacteristic, complemented
27#16839767—21no—
28#16839767—21no—
29#16839767C2212yes—
30#16839767C2212yes—
31C2×#14519698—21no—
32C2×#14519698—21no—
33C2×#14519696C2212yescomplemented
34C2×#14519694C2212yes—
35C2×#14519694C2212yes—
36#16839726C2211yescharacteristic
37#16839726C2211yescharacteristic
38#16839717C2212yescomplemented
39#16839531C2212yescomplemented
40#16839527C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C2×C22⋊Q8×C61C2211yescharacteristic
44C61×#450—21no—
45C61×#450—21no—
46C61×#450—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#14519701—21no—
74#14519701—21no—
75#14519701—21no—
76#14519701—21no—
77#14519698—22no—
78#14519698—22no—
79#14519698—22no—
80#14519698—22no—
81#14519696—42no—
82#14519694—24no—
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×Q8×C61C2311yescharacteristic
92C22×D4×C61—24no—
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×C61C2312yes—
100C23×C4×C61C2311yescharacteristic
101C2×SD16×C61—24no—
102C2×SD16×C61—22no—
103C2×SD16×C61—22no—
104C2×SD16×C61—42no—
105C22×C8×C61C2312yes—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61—21no—
110C2×C4⋊C4×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—44no—
120C24×C61—41no—
121C24×C61—41no—
122C24×C61—82no—
123C24×C61—82no—
124C24×C61#4111yescharacteristic
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×D4×C61—24no—
130C2×D4×C61—24no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—44no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61—42no—
140C2×D4×C61—42no—
141C2×D4×C61—41no—
142C2×D4×C61—41no—
143C2×D4×C61#4114yes—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156SD16×C61—44no—
157SD16×C61—44no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61C2×D412yes—
161C2×C8×C61C2×D412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—41no—
165C4⋊C4×C61—41no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C22⋊Dic61—21no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D411yescharacteristic
175C4⋊Dic61C2×D411yescharacteristic
176C4⋊Dic61C2×D411yescharacteristic
177C4⋊Dic61C2×D411yescharacteristic
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182#11758194—22no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—22no—
188#11758194—42no—
189C2×#92418—22no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—42no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—24no—
197C23×C61—41no—
198C23×C61—44no—
199C23×C61—44no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—82no—
203C23×C61—82no—
204C23×C61—82no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—816no—
208C23×C61—82no—
209C23×C61#12111yescharacteristic
210C23×C61#12111yescharacteristic
211C23×C61C2×#4111yescharacteristic
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—42no—
215D4×C61—24no—
216D4×C61—24no—
217D4×C61—44no—
218D4×C61—44no—
219D4×C61—42no—
220D4×C61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—81no—
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4112yes—
237C2×C4×C61H(5,𝔽2)12yes—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—81no—
244C8×C61—22no—
245C8×C61—22no—
246#92418—41no—
247#92418—42no—
248#92418—41no—
249#92418—42no—
250#24521—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—84no—
261C22×C61—84no—
262C22×C61—88no—
263C22×C61—88no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61#47112yes—
267C22×C61C2×#12111yescharacteristic
268C22×C61#55112yes—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61#52511yescharacteristic
274C4×C61#53511yescharacteristic
275C4×C61#54612yes—
276Dic61—41no—
277Dic61—41no—
278Dic61—41no—
279Dic61—41no—
280#3095—611no—
281C2×#450D6111yescharacteristic
282#2924—611no—
283C2×#418—611no—
284#2063—612no—
285#2046—612no—
286#1977—612no—
287#1931—612no—
288#1901—611no—
289#1855—611no—
290#1783—611no—
291C2×C61—22no—
292C2×C61—22no—
293C2×C61—41no—
294C2×C61—84no—
295C2×C61#241111yescharacteristic
296C2×C61C2×#47111yescharacteristic
297C2×C61#306811yescharacteristic
298C23×D4C2×D6111yescharacteristic
299C22×SD16C2×D6112yes—
300C22×M16—611no—
301C2×C22⋊Q8C2×D6111yescharacteristic
302C2×#117—611no—
303#485—1221no—
304#485—1221no—
305#484—1221no—
306#483—1221no—
307#450—21no—
308#450—21no—
309#450—22no—
310#433—1222no—
311#432—1221no—
312#431—1221no—
313C2×#108—612no—
314#428—611no—
315#428—611no—
316#424—612no—
317#419—611no—
318#419—611no—
319#418—612no—
320#418—612no—
321#418—1221no—
322#418—1221no—
323C2×#102—1221no—
324C2×#102—1221no—
325C2×#102C2×D6112yes—
326C2×#101—612no—
327C2×#100—612no—
328C2×#100—612no—
329#407—611no—
330#407—611no—
331C2×C22⋊C8C2×D6111yescharacteristic
332#405—612no—
333#343—612no—
334#341—612no—
335C61#2452111yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8C22×D6111yescharacteristic
338C22×D4—24no—
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—21no—
344C22×D4—42no—
345C22×D4C22×D6112yes—
346C23×C4C22×D6111yescharacteristic
347C2×SD16—24no—
348C2×SD16—22no—
349C2×SD16—22no—
350C2×SD16—42no—
351C2×M16—612no—
352C2×M16—611no—
353C2×M16—612no—
354C2×M16—611no—
355C2×M16—1221no—
356C2×M16—2442no—
357C22×C8—1221no—
358C22×C8C22×D6112yes—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363#117—611no—
364#117—611no—
365#117—611no—
366#117—611no—
367#117—1221no—
368#117—1221no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1221no—
375C2×C42—1221no—
376#108—2442no—
377#107—1221no—
378#107—1221no—
379#106—1221no—
380#106—1221no—
381C4⋊C8—1222no—
382C4⋊C8—1222no—
383#103—1221no—
384#103—1221no—
385#103—1221no—
386#103—1221no—
387#102—22no—
388#102—22no—
389#102—1222no—
390#102—1222no—
391#102—1222no—
392#102—1222no—
393#101—2442no—
394#100—1224no—
395#100—2442no—
396C22⋊C8—1221no—
397C22⋊C8—1221no—
398C22⋊C8—1221no—
399C22⋊C8—1221no—
400C22⋊C8C22⋊D6111yescharacteristic
401C22⋊C8C22⋊D6111yescharacteristic
402C22⋊C8C22⋊D6112yes—
403#97—1222no—
404C4×C8—1222no—
405C24—22no—
406C24—44no—
407C24—41no—
408C24—41no—
409C24—82no—
410C24—82no—
411C24#1175822411yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4#1175822414yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4#1175822411yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443SD16—44no—
444SD16—44no—
445M16—1222no—
446M16—1222no—
447M16—1222no—
448M16—1222no—
449M16—1222no—
450M16—1222no—
451M16—2442no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—2442no—
459C2×C8C2×C22⋊D6112yes—
460C2×C8C2×C22⋊D6112yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—611no—
466C4⋊C4—611no—
467C4⋊C4—611no—
468C4⋊C4—611no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—1221no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C42—1221no—
478C42—1221no—
479C42—1221no—
480C42—1221no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—41no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—816no—
495C23—82no—
496C23#1451974811yescharacteristic
497C23#1451974811yescharacteristic
498C23C2×#1175822411yescharacteristic
499Q8—41no—
500Q8—41no—
501Q8—42no—
502D4—24no—
503D4—24no—
504D4—44no—
505D4—44no—
506D4—42no—
507D4—42no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—4881no—
524C2×C4—2442no—
525C2×C4#1451974811yescharacteristic
526C2×C4#1451974811yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822412yes—
530C2×C4#1451982212yes—
531C8—22no—
532C8—22no—
533C8—2441no—
534C8—2442no—
535C8—2441no—
536C8—2442no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—44no—
546C22—84no—
547C22—88no—
548C22—84no—
549C22—88no—
550C22—82no—
551C22—82no—
552C22#1684002212yes—
553C22C2×#1451974811yescharacteristic
554C22#1684044312yes—
555C4—21no—
556C4—21no—
557C4—81no—
558C4—81no—
559C4—2441no—
560C4—2441no—
561C4—2441no—
562C4—2441no—
563C4#1684026811yescharacteristic
564C4#1684032711yescharacteristic
565C4#1684040612yes—
566C2—22no—
567C2—22no—
568C2—41no—
569C2—84no—
570C2#2144920011yescharacteristic
571C2C2×#1684002211yescharacteristic
572C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#450C2
socleC22×C61C2×#121
prime radicalC23×D4×C61C22
hypercenterC2×#450D61
hypocenterC61#24521
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
2. #21454473C2semidirect
4. #21447177C2semidirect
5. #21447077C2semidirect
6. #21446724C2semidirect
7. #21446478C2semidirect
8. #21446288C2semidirect
9. #21446026C2semidirect
21. C2×#14519713C22semidirect
22. #16839802C22semidirect
23. #16839802C22semidirect
24. #16839788C22semidirect
25. #16839771C22semidirect
26. #16839771C22semidirect
33. C2×#14519696C22semidirect
38. #16839717C22semidirect
39. #16839531C22semidirect
40. #16839527C22semidirect
335. C61#24521semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214544732
C2×#168397672
#214471772
#214470772
#214467242
#214464782
#214462882
#214460262
#214457402
C2×C61×#4502
#2452161