Finite Group Databasebeta

← #29706583

Subgroups — #29706583

Order 15616.

688 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456274C211yesmaximal, characteristic
2#21446315C211yesmaximal, characteristic, complemented
3#21446312C211yesmaximal, characteristic, complemented
4#21446311C211yesmaximal, characteristic, complemented
5#21446310C211yesmaximal, characteristic, complemented
6#21446295C211yesmaximal, characteristic, complemented
7#21446293C211yesmaximal, characteristic, complemented
8#21446292C211yesmaximal, characteristic, complemented
9#21446210C211yesmaximal, characteristic, complemented
10#21446209C211yesmaximal, characteristic, complemented
11#21446202C211yesmaximal, characteristic
12#21446199C211yesmaximal, characteristic
13#21445969C211yesmaximal, characteristic, complemented
14#21445967C211yesmaximal, characteristic, complemented
15C61×#1902C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519808C2211yescharacteristic
18C2×#14519807C2211yescharacteristic
19#16840528C2211yescharacteristic
20#16840308C2211yescharacteristic
21C2×#14519775C2211yescharacteristic
22#16840240C2211yescharacteristic
23#16840239C2211yescharacteristic
24#16840155C412yescomplemented
25#16840153C412yescomplemented
26#16840140C412yescomplemented
27#16840133C412yescomplemented
28#16839779C2211yescharacteristic
29#16839778C2211yescharacteristic
30#16839778C2211yescharacteristic
31#16839772C2211yescharacteristic
32#16839772C2211yescharacteristic
33#16839771C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35#16839736C2211yescharacteristic
36#16839735C2211yescharacteristic
37M16×Dic61C2211yescharacteristic
38#16839534C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839533C2211yescharacteristic
41#16839523C2211yescharacteristic
42#16839523C2211yescharacteristic
43#16839523C2211yescharacteristic
44#16839521C2211yescharacteristic
45#16839520C2211yescharacteristic
46#16839519C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×#137×C61C2211yescharacteristic
49C61×#568C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#419C2211yescharacteristic
52C2×#101×C61C2211yescharacteristic
53C61×#344C2211yescharacteristic
54C61×#338C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57#14519810—41no—
58#14519810—41no—
59#14519808—41no—
60#14519808—41no—
61#14519807—41no—
62#14519807—41no—
63C2×Dic244C2×C412yes—
64C2×#11758202C2×C412yes—
65C2×#11758202C2×C412yes—
66C2×#11758190C2×C412yes—
67C2×#11758200C2×C412yes—
68C2×#11758201C2×C412yes—
69#14519785—41no—
70#14519783—41no—
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73M16×D61—41no—
74#14519775—41no—
75#14519775C2×C412yes—
76#14519775C2×C412yes—
77#14519775C2×C411yescharacteristic
78#14519773C2×C411yescharacteristic
79#14519766C2×C411yescharacteristic
80#14519764C2×C411yescharacteristic
81#14519749—41no—
82Q8×Dic61—41no—
83Q8×Dic61—41no—
84C4×Dic122—41no—
85#14519742—41no—
86C4×C22⋊D61—41no—
87#14519736—41no—
88#14519736C2311yescharacteristic
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91#14519733—41no—
92C2×#11758196—41no—
93C2×C4×Dic61C2311yescharacteristic
94#14519704—41no—
95#14519703—41no—
96#14519703—41no—
97#14519702—41no—
98#14519702—41no—
99#14519701—41no—
100#14519695—41no—
101#14519692—41no—
102#14519691—41no—
103#14519687C2×C412yes—
104C8×Dic61C2×C412yes—
105C2×#41×C61C2311yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107#137×C61—41no—
108#137×C61—41no—
109C2×Q16×C61C2×C412yes—
110C2×SD16×C61C2×C412yes—
111#131×C61—41no—
112#131×C61—41no—
113C2×M16×C61—41no—
114C2×M16×C61C2311yescharacteristic
115C2×M16×C61C2311yescharacteristic
116C2×M16×C61C2311yescharacteristic
117C2×M16×C61C2311yescharacteristic
118#117×C61C2311yescharacteristic
119#104×C61—41no—
120#103×C61—41no—
121#101×C61—41no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C2×Q8×C61C22×C411yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61C22×C411yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—21no—
139C2×C22⋊D61—41no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142#11758222—21no—
143#11758222—21no—
144#11758222—21no—
145#11758222—21no—
146C2×D244C22×C411yescharacteristic
147C2×C4×D61—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122C22×C412yes—
154C2×Dic122C22×C411yescharacteristic
155Q16×C61—42no—
156SD16×C61—42no—
157M16×C61—21no—
158M16×C61—21no—
159M16×C61—21no—
160M16×C61—21no—
161M16×C61—21no—
162M16×C61—21no—
163M16×C61—21no—
164M16×C61—21no—
165M16×C61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—42no—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C42×C61—21no—
174C22⋊Dic61—21no—
175C22⋊Dic61—21no—
176C22⋊Dic61—21no—
177C22⋊Dic61—41no—
178#11758202—42no—
179#11758202—42no—
180#11758201—42no—
181#11758200—42no—
182#11758198—41no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—41no—
190#11758196—42no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61C22×C411yescharacteristic
198C4×Dic61C22×C411yescharacteristic
199#11758194—21no—
200#11758194—21no—
201#11758194—21no—
202#11758194—21no—
203C2×#92418—21no—
204C2×#92418C22×C412yes—
205Dic244—42no—
206#11758190—42no—
207#11758189—42no—
208C8×D61—42no—
209C23×C61—21no—
210C23×C61H(5,𝔽2)11yescharacteristic
211C22×D61—21no—
212Q8×C61—22no—
213Q8×C61—21no—
214Q8×C61—22no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—41no—
221D4×C61—41no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C22⋊D61—41no—
236C22⋊D61—41no—
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248D244—21no—
249D244—21no—
250C4×D61—41no—
251C4×D61—41no—
252Dic122—22no—
253Dic122—21no—
254Dic122—22no—
255Dic122—21no—
256Dic122—41no—
257Dic122—41no—
258C8×C61—22no—
259C8×C61—41no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—22no—
263#92418—41no—
264#23360—611nomaximal, Sylow
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—81no—
268C22×C61#51811yescharacteristic
269C22×C61#56011yescharacteristic
270C22×C61#56011yescharacteristic
271C2×D61—41no—
272C2×D61—81no—
273C4×C61—41no—
274C4×C61—42no—
275C4×C61—81no—
276C4×C61#51811yescharacteristic
277C4×C61#51811yescharacteristic
278C4×C61#56011yescharacteristic
279C4×C61#56011yescharacteristic
280Dic61—22no—
281Dic61—41no—
282Dic61—42no—
283Dic61—41no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287#3152—611no—
288#1905—611no—
289#1905—611no—
290#1905—611no—
291#1905—611no—
292#1902—611no—
293#1902—611no—
294#1902—611no—
295#1902D6111yescharacteristic, complemented
296#1886—611no—
297#1886—611no—
298#1884—611no—
299#1884—611no—
300#1841—611no—
301#1841—611no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—81no—
306C2×C61#238011yescharacteristic
307D61—81no—
308C2×#137—611no—
309C2×#137—611no—
310C2×#137—611no—
311C2×#137C2×D6111yescharacteristic
312#568—611no—
313#568C2×D6111yescharacteristic
314#532—611no—
315C2×C4⋊Q8—611no—
316#519—611no—
317#519—611no—
318#497—612no—
319#497—612no—
320#494—612no—
321#492—612no—
322#421—611no—
323#421—611no—
324#421—611no—
325#421C2×D6111yescharacteristic
326#419—611no—
327#419—611no—
328#419—611no—
329#419C2×D6111yescharacteristic
330C2×#101—611no—
331C2×#101C2×D6111yescharacteristic
332#410—611no—
333#410—611no—
334C4×M16—611no—
335#344—611no—
336#344—611no—
337#344—611no—
338#344C2×D6111yescharacteristic
339#339—611no—
340#339—611no—
341#339—611no—
342#339—611no—
343#338—611no—
344#338C2×D6111yescharacteristic
345H(3,ℤ/4)—611no—
346H(3,ℤ/4)—611no—
347C61#2336011yescharacteristic, complemented, Sylow
348C2×#41—611no—
349C2×#41C22×D6111yescharacteristic
350C22×Q8—611no—
351C22×Q8C22×D6111yescharacteristic
352#137—41no—
353#137—41no—
354#137—2441no—
355#137—2441no—
356#137—2441no—
357#137—2441no—
358#137—2441no—
359#137—2441no—
360C2×Q16—612no—
361C2×Q16—612no—
362C2×Q16—612no—
363C2×Q16C4×D6112yes—
364C2×SD16—612no—
365C2×SD16—612no—
366C2×SD16—612no—
367C2×SD16C4×D6112yes—
368#131—41no—
369#131—41no—
370#131—2441no—
371#131—2441no—
372C2×M16—41no—
373C2×M16—611no—
374C2×M16—611no—
375C2×M16—2441no—
376C2×M16C22×D6111yescharacteristic
377C2×M16C22×D6111yescharacteristic
378C2×M16C22×D6111yescharacteristic
379C2×M16C22×D6111yescharacteristic
380C4⋊Q8—612no—
381C4⋊Q8—612no—
382C4⋊Q8—611no—
383C4⋊Q8—2441no—
384#127—611no—
385#124—611no—
386#124—611no—
387#121—2441no—
388C4×Q8—2441no—
389C4×Q8—2441no—
390C4×Q8—2441no—
391C4×Q8—2441no—
392C4×D4—2441no—
393#117—611no—
394#117—611no—
395#117—611no—
396#117—2441no—
397#117—2441no—
398#117C22×D6111yescharacteristic
399C2×C4⋊C4—2441no—
400C2×C42—611no—
401#104—41no—
402#104—2441no—
403#104—2441no—
404#104—2441no—
405#103—41no—
406#103—2441no—
407#103—2441no—
408#103—2441no—
409#101—41no—
410#101—2441no—
411#99—2441no—
412#99—2441no—
413#97—612no—
414C4×C8—612no—
415#41—21no—
416#41—21no—
417#41—21no—
418#41—21no—
419#41—1221no—
420#41—1221no—
421#41—1221no—
422#41—1221no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—611no—
429C2×Q8—612no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8C2×C4×D6111yescharacteristic
436C2×Q8C2×C4×D6112yes—
437C2×D4—21no—
438C2×D4—611no—
439C2×D4—1221no—
440C2×D4—2441no—
441C2×D4C2×C4×D6111yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4C23×D6111yescharacteristic
448Q16—42no—
449Q16—2442no—
450Q16—2442no—
451Q16—2442no—
452SD16—42no—
453SD16—2442no—
454SD16—2442no—
455SD16—2442no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—21no—
463M16—21no—
464M16—42no—
465M16—1221no—
466M16—1221no—
467M16—1221no—
468M16—1221no—
469M16—2442no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—42no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—2442no—
477C2×C8C2×C4×D6112yes—
478C4⋊C4—21no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C4⋊C4—2441no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C22⋊C4—21no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—21no—
494C42—611no—
495C42—611no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C42—2442no—
501C42—2442no—
502C23—21no—
503C23—1221no—
504C23#1451982311yescharacteristic
505Q8—22no—
506Q8—21no—
507Q8—22no—
508Q8—21no—
509Q8—41no—
510Q8—41no—
511Q8—1222no—
512Q8—1221no—
513Q8—1222no—
514Q8—1221no—
515Q8—2441no—
516Q8—2441no—
517D4—21no—
518D4—21no—
519D4—41no—
520D4—41no—
521D4—1221no—
522D4—1221no—
523D4—4882no—
524D4—2441no—
525D4—2441no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4C22×C4×D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C2×C4#1451982211yescharacteristic
548C2×C4#1451982211yescharacteristic
549C2×C4#1451982211yescharacteristic
550C2×C4#1451982311yescharacteristic
551C8—22no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—1222no—
556C8—2441no—
557C22—41no—
558C22—41no—
559C22—81no—
560C22—4881no—
561C22—2441no—
562C22#1684023411yescharacteristic
563C22#1684050011yescharacteristic
564C22#1684050211yescharacteristic
565C4—41no—
566C4—42no—
567C4—81no—
568C4—1222no—
569C4—4881no—
570C4—4881no—
571C4—4881no—
572C4—2441no—
573C4—2441no—
574C4—2442no—
575C4#1684023411yescharacteristic
576C4#1684023511yescharacteristic
577C4#1684050011yescharacteristic
578C4#1684050111yescharacteristic
579C2—21no—
580C2—21no—
581C2—21no—
582C2—81no—
583C2—4881no—
584C2#2144899411yescharacteristic
subgroupG/H
centerC2#21448994
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1902C2
socleC2×C61#2380
prime radical#16840308C22
cosocleC22×C4×C61C24
hypercenter#1902D61
hypocenterC61#23360
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1902

Complements

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

NH ≅ G/Ntype
2. #21446315C2semidirect
3. #21446312C2semidirect
4. #21446311C2semidirect
5. #21446310C2semidirect
6. #21446295C2semidirect
7. #21446293C2semidirect
8. #21446292C2semidirect
9. #21446210C2semidirect
10. #21446209C2semidirect
13. #21445969C2semidirect
14. #21445967C2semidirect
15. C61×#1902C2semidirect
24. #16840155C4semidirect
25. #16840153C4semidirect
26. #16840140C4semidirect
27. #16840133C4semidirect
295. #1902D61semidirect
347. C61#23360semidirect

Maximal subgroups

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

subgroupindex
#214562742
#214463152
#214463122
#214463112
#214463102
#214462952
#214462932
#214462922
#214462102
#214462092
#214462022
#214461992
#214459692
#214459672
C61×#19022
#2336061