Finite Group Databasebeta

← #29648202

Subgroups — #29648202

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459792C211yesmaximal, characteristic
2#21454837C211yesmaximal, characteristic, complemented
3#21454471C211yesmaximal, characteristic, complemented
4#21444449C212yesmaximal, complemented
5#21444444C212yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444425C212yesmaximal, complemented
8#21444424C211yesmaximal, characteristic, complemented
9#21444419C212yesmaximal, complemented
10#21444408C211yesmaximal, characteristic, complemented
11C61×#1526C211yesmaximal, characteristic, complemented
12#16840487C411yescharacteristic
13#16840473C411yescharacteristic
14#16840473C411yescharacteristic
15#16840456C412yes—
16#16840453—22no—
17#16840451C412yes—
18#16840446—21no—
19#16840415C411yescharacteristic
20#16840414—21no—
21#16840370—21no—
22#16840358—22no—
23#16840297C2212yes—
24#16840296C2211yescharacteristic
25#125×D61C2212yes—
26#16840244—21no—
27C4×#11758222C2211yescharacteristic
28C8×Dic122—21no—
29#16839841—21no—
30#16839838—22no—
31#16839821—22no—
32#16839819C2211yescharacteristic
33#16839817C2212yes—
34#16839817C2212yes—
35#16839817C2212yes—
36#16839790—21no—
37#16839730—21no—
38#16839714C2212yes—
39C2×#14519688C2211yescharacteristic
40#16839504C2212yes—
41#16839504C2212yes—
42#16839503C2211yescharacteristic
43#16839503C2211yescharacteristic
44#16839503C2212yes—
45#16839501C2211yescharacteristic
46#16839501C2211yescharacteristic
47#16839501C2212yes—
48C61×#533C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×#330C2212yes—
52C61×#330C2212yes—
53C2×#11758226—21no—
54C2×#11758222D411yescharacteristic
55C2×#11758189—22no—
56C22×#92418—21no—
57#14519775—21no—
58#14519772—22no—
59#14519771—21no—
60#14519771—22no—
61#14519770—22no—
62#14519769—22no—
63#14519769C2311yescharacteristic
64#14519768—21no—
65#14519768C2×C412yes—
66#14519768C2×C412yes—
67#14519767—22no—
68#14519767—22no—
69#14519767—22no—
70#14519765C2311yescharacteristic
71#14519762—22no—
72#14519762—22no—
73#14519760—21no—
74#14519759—22no—
75#14519759—22no—
76#14519757—21no—
77#14519757C2×C412yes—
78#14519756C2×C411yescharacteristic
79#14519753—22no—
80#14519753—21no—
81#14519753—22no—
82#14519752C2×C412yes—
83#14519751C2×C411yescharacteristic
84#14519750—22no—
85Q8×Dic61—21no—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122D411yescharacteristic
89#14519742C2×C412yes—
90C4×D244—22no—
91C4×D244C2×C411yescharacteristic
92C4×D244C2×C411yescharacteristic
93C4×D244D411yescharacteristic
94#14519739—21no—
95#14519739—22no—
96#14519739C2×C412yes—
97C4×C22⋊D61—21no—
98C4×C22⋊D61C2×C412yes—
99C4×C22⋊D61D412yes—
100#14519736D411yescharacteristic
101#14519735—22no—
102#14519734—21no—
103#14519734—22no—
104#14519734D412yes—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61C2×C412yes—
109C4⋊C4×D61C2×C412yes—
110C42×D61C2312yes—
111C4⋊#92418—21no—
112#14519706—22no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418—21no—
116#14519689—22no—
117#14519689—22no—
118#14519689—22no—
119#14519688—21no—
120#14519688C2311yescharacteristic
121#14519688C2311yescharacteristic
122#14519688C2312yes—
123C8×Dic61—22no—
124C2×M16×C61—21no—
125C2×M16×C61—21no—
126C4⋊Q8×C61C2×C411yescharacteristic
127C4⋊Q8×C61C2×C411yescharacteristic
128#125×C61C2312yes—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—21no—
131C4×Q8×C61—21no—
132C4×Q8×C61—21no—
133#117×C61C2×C412yes—
134C2×C42×C61C2311yescharacteristic
135#97×C61C2311yescharacteristic
136#97×C61C2312yes—
137#97×C61C2311yescharacteristic
138C4×C8×C61—21no—
139C4×C8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—21no—
146C2×C22⋊D61C22⋊C412yes—
147#11758226—44no—
148#11758222—42no—
149C2×D244—22no—
150C2×D244C22⋊C411yescharacteristic
151C2×D244C22×C411yescharacteristic
152C2×C4×D61—22no—
153C2×C4×D61—21no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×Dic122C22⋊C411yescharacteristic
160C2×Dic122C22×C411yescharacteristic
161M16×C61—42no—
162M16×C61—42no—
163C2×C8×C61—21no—
164C2×C8×C61—22no—
165C2×C8×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C412yes—
171C4⋊C4×C61C22×C412yes—
172C22⋊C4×C61—22no—
173C42×C61—22no—
174C42×C61C22×C411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2411yescharacteristic
178C22⋊Dic61—21no—
179C22⋊Dic61C22⋊C412yes—
180#11758198—22no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185#11758198—22no—
186#11758198—22no—
187#11758198C22⋊C412yes—
188#11758198C22⋊C412yes—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C22⋊C411yescharacteristic
193C4⋊Dic61C22⋊C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207C4×Dic61—21no—
208C4×Dic61—22no—
209C4×Dic61C22×C412yes—
210C4×Dic61C2×D412yes—
211C4×Dic61C2×D412yes—
212C2×#92418—22no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—42no—
216#11758189—44no—
217C23×C61C2×C22⋊C411yescharacteristic
218C22×D61—21no—
219C22×D61C2×C22⋊C412yes—
220Q8×C61—42no—
221Q8×C61—81no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—42no—
234C22⋊D61—81no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241D244—22no—
242D244—41no—
243D244—44no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61—44no—
248C4×D61—42no—
249C4×D61—82no—
250C4×D61—82no—
251Dic122—22no—
252Dic122—41no—
253C8×C61—22no—
254C8×C61—42no—
255C8×C61—41no—
256#92418—42no—
257#92418—41no—
258#92418—41no—
259#17682—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61C22×C22⋊C411yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61#42112yes—
278Dic61—22no—
279Dic61—22no—
280Dic61—22no—
281Dic61—41no—
282Dic61—41no—
283Dic61—41no—
284#3515—611no—
285#2971—611no—
286#2923—611no—
287#1535—612no—
288#1534—612no—
289#1531—611no—
290#1529—611no—
291#1529—612no—
292#1528—612no—
293#1526D6111yescharacteristic, complemented
294#1524—611no—
295C2×C61—41no—
296C2×C61#288911yescharacteristic
297C2×C61C2×#42111yescharacteristic
298C2×C61#290111yescharacteristic
299D61—42no—
300D61—81no—
301#557—611no—
302#554—611no—
303#554—611no—
304#553—612no—
305#552—612no—
306#552—1222no—
307#552—1221no—
308#547—611no—
309#547—1221no—
310#541—1221no—
311#539—1222no—
312#533C2×D6111yescharacteristic
313#529—612no—
314#529—611no—
315C2×#125—612no—
316#520—1221no—
317C4×#41—611no—
318C8×Q8—1221no—
319#436—1221no—
320#436—1222no—
321#433—1222no—
322#432—611no—
323#432—612no—
324#432—612no—
325#432—612no—
326#424—1221no—
327#408—1221no—
328C4×M16—612no—
329C4×M16C2×D6111yescharacteristic
330C4×M16C2×D6111yescharacteristic
331C2×#97—611no—
332#330—612no—
333#330—612no—
334#330—611no—
335#330—611no—
336#330—612no—
337#330C2×D6112yes—
338#330C2×D6112yes—
339#329—611no—
340#329—611no—
341#329—612no—
342C61#1768211yescharacteristic, complemented, Sylow
343C2×#41—611no—
344C2×#41—1221no—
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—1222no—
348C22×C8—1221no—
349C4⋊Q8—1221no—
350C4⋊Q8C4×D6111yescharacteristic
351C4⋊Q8C4×D6111yescharacteristic
352#126—1222no—
353#126—1222no—
354#126—1222no—
355#126—1221no—
356#125—612no—
357#125—612no—
358#125—611no—
359#125—1222no—
360#125—1222no—
361#125—1222no—
362#125—1222no—
363#125—1221no—
364#125C22×D6112yes—
365#124—611no—
366#123—1222no—
367#123—1222no—
368#123—1222no—
369#123—1222no—
370#123—1221no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—611no—
374C22⋊Q8—612no—
375C22⋊Q8—1222no—
376C22⋊Q8—1221no—
377C22⋊Q8—1222no—
378C22⋊Q8—1221no—
379#121—611no—
380#121—612no—
381#121—1222no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—611no—
385C4×Q8—611no—
386C4×Q8—612no—
387C4×Q8—611no—
388C4×Q8—1221no—
389C4×D4—612no—
390C4×D4—612no—
391C4×D4—612no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—611no—
400#117—612no—
401#117—1222no—
402#117—1222no—
403#117—1221no—
404#117C4×D6112yes—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1222no—
410C2×C42—612no—
411C2×C42C22×D6111yescharacteristic
412C4⋊C8—1222no—
413C4⋊C8—1222no—
414C4⋊C8—1222no—
415C4⋊C8—1221no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1222no—
419C22⋊C8—1221no—
420#97—611no—
421#97—611no—
422#97—612no—
423#97—1221no—
424#97C22×D6111yescharacteristic
425#97C22×D6111yescharacteristic
426#97C22×D6112yes—
427C4×C8—21no—
428C4×C8—21no—
429C4×C8—1222no—
430#41—2442no—
431#41—2444no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—611no—
435C2×Q8—611no—
436C2×D4—612no—
437C2×D4—611no—
438C2×D4—611no—
439C2×D4—1222no—
440C2×D4—1221no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1221no—
448C22×C4C2×C4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451M16—42no—
452M16—42no—
453M16—2444no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—1222no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—2442no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—611no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—611no—
469C4⋊C4—611no—
470C4⋊C4—611no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4C2×C4×D6112yes—
484C4⋊C4C2×C4×D6112yes—
485C22⋊C4—22no—
486C22⋊C4—612no—
487C22⋊C4—612no—
488C22⋊C4—612no—
489C22⋊C4—1222no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1221no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C42—22no—
498C42—612no—
499C42—612no—
500C42—612no—
501C42—1222no—
502C42—1221no—
503C42C2×C4×D6111yescharacteristic
504C42D4×D6111yescharacteristic
505C42D4×D6111yescharacteristic
506C42C23×D6111yescharacteristic
507C23—612no—
508C23—1221no—
509C23C22⋊C4×D6111yescharacteristic
510Q8—42no—
511Q8—81no—
512Q8—1222no—
513Q8—2441no—
514D4—1222no—
515D4—4881no—
516D4—2441no—
517D4—2444no—
518D4—2442no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—4882no—
531C2×C4—4882no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4—2442no—
537C2×C4C22⋊C4×D6111yescharacteristic
538C2×C4C22⋊C4×D6111yescharacteristic
539C2×C4C22⋊C4×D6111yescharacteristic
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C8—22no—
544C8—42no—
545C8—41no—
546C8—2441no—
547C8—2441no—
548C8—2442no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—2441no—
556C22—2441no—
557C22—2441no—
558C22C2×C22⋊C4×D6111yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—1222no—
567C4—1222no—
568C4—1222no—
569C4—2441no—
570C4—2441no—
571C4—2441no—
572C4#1683977712yes—
573C2—41no—
574C2—4881no—
575C2—2442no—
576C2#2145429911yescharacteristic
577C2C2×#1683977711yescharacteristic
578C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1526C2
socleC22×C61C22×C22⋊C4
prime radical#21459792C2
cosocleC42×C61C24
hypercenter#1526D61
hypocenterC61#17682
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #21454837C2semidirect
3. #21454471C2semidirect
4. #21444449C2semidirect
5. #21444444C2semidirect
6. #21444434C2semidirect
7. #21444425C2semidirect
8. #21444424C2semidirect
9. #21444419C2semidirect
10. #21444408C2semidirect
11. C61×#1526C2semidirect
293. #1526D61semidirect
342. C61#17682semidirect

Maximal subgroups

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

subgroupindex
#214597922
#214548372
#214544712
#214444492
#214444442
#214444342
#214444252
#214444242
#214444192
#214444082
C61×#15262
#1768261