Finite Group Databasebeta

← #29648216

Subgroups — #29648216

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459825C211yesmaximal, characteristic
2#21454833C211yesmaximal, characteristic, complemented
3#21444453C212yesmaximal, complemented
4#21444448C212yesmaximal, complemented
5#21444440C212yesmaximal, complemented
6#21444431C212yesmaximal, complemented
7#21444429C211yesmaximal, characteristic, complemented
8#21444424C211yesmaximal, characteristic, complemented
9#21444410C211yesmaximal, characteristic, complemented
10#21444405C211yesmaximal, characteristic, complemented
11C61×#2923C211yesmaximal, characteristic, complemented
12#16840491C411yescharacteristic
13#16840456—21no—
14#16840456—22no—
15#16840434C411yescharacteristic
16#16840433C412yes—
17#16840433C411yescharacteristic
18#16840431—21no—
19#16840417C412yes—
20#16840417C411yescharacteristic
21#16840363—21no—
22#16840363—22no—
23#16840312C2211yescharacteristic
24#16840312C2212yes—
25#16840296C2211yescharacteristic
26#16840296C2212yes—
27#16840244—21no—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2211yescharacteristic
32#16839819C2212yes—
33C4×#11758194C2211yescharacteristic
34C2×#14519688C2212yes—
35#16839503C2211yescharacteristic
36#16839503C2211yescharacteristic
37#16839503C2212yes—
38#16839501C2211yescharacteristic
39#16839501C2211yescharacteristic
40#16839501C2212yes—
41#16839501C2212yes—
42#16839501C2212yes—
43#16839501C2212yes—
44#16839501C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#432C2212yes—
48C61×#432C2212yes—
49C61×#424—21no—
50C61×#408—21no—
51C4×M16×C61C2211yescharacteristic
52C2×#97×C61C2211yescharacteristic
53C2×#11758226—21no—
54C2×#11758222C2×C412yes—
55C2×#11758222C2×C412yes—
56C2×#11758222C2×C412yes—
57C2×#11758194—21no—
58C22×#92418—22no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C412yes—
61#14519774C2×C411yescharacteristic
62#14519774C2×C412yes—
63#14519773—21no—
64#14519773—22no—
65#14519771—21no—
66#14519771—22no—
67#14519769C2311yescharacteristic
68#14519768—22no—
69#14519768—21no—
70#14519765C2311yescharacteristic
71#14519765C2312yes—
72#14519760—22no—
73#14519760—21no—
74#14519759—22no—
75#14519759—22no—
76#14519759—22no—
77#14519756C2×C411yescharacteristic
78#14519756C2×C412yes—
79#14519752—22no—
80#14519752—21no—
81#14519751—21no—
82#14519751—22no—
83#14519751—22no—
84#14519751—21no—
85#14519751C2×C411yescharacteristic
86#14519751C2×C412yes—
87#14519750—22no—
88#14519750—22no—
89#14519750—22no—
90#14519748—21no—
91#14519748—22no—
92Q8×Dic61—21no—
93C4×Dic122C2×C411yescharacteristic
94C4×Dic122C2×C412yes—
95C4×D244—21no—
96C4×D244—22no—
97C4×D244—21no—
98C4×D244—22no—
99C4×D244C2×C411yescharacteristic
100C4×D244C2×C412yes—
101#14519739—22no—
102#14519739—21no—
103C4×C22⋊D61—21no—
104C4×C22⋊D61—22no—
105#14519734—21no—
106#14519734—22no—
107C4⋊C4×D61—22no—
108C4⋊C4×D61—21no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—21no—
113#14519688—22no—
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116#14519688C2311yescharacteristic
117C4×#92418—21no—
118C2×#41×C61D411yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—21no—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127#117×C61D411yescharacteristic
128#117×C61D412yes—
129C2×C42×C61C2312yes—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2312yes—
139C4×C8×C61—21no—
140#41×C61—42no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—22no—
155#11758226—44no—
156#11758222—82no—
157#11758222—82no—
158#11758222—82no—
159C2×D244—22no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—22no—
163C2×D244C22×C412yes—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—22no—
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×Dic122C22×C412yes—
171C2×Dic122C22×C411yescharacteristic
172M16×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C4⋊C4×C61C22⋊C412yes—
178C4⋊C4×C61C22⋊C411yescharacteristic
179C4⋊C4×C61C22⋊C412yes—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C22⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C42×C61C22×C412yes—
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D412yes—
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D412yes—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—21no—
192C22⋊Dic61—22no—
193#11758198—21no—
194#11758198—22no—
195#11758198—21no—
196#11758198—22no—
197#11758198—21no—
198#11758198—22no—
199#11758198—21no—
200#11758198—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C411yescharacteristic
206C4⋊Dic61C22×C411yescharacteristic
207#11758196—21no—
208#11758196—22no—
209#11758196—21no—
210#11758196—22no—
211#11758196—21no—
212#11758196—21no—
213C4×Dic61—22no—
214C4×Dic61—21no—
215#11758194—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—22no—
219C2×#92418—44no—
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61C2×C22⋊C412yes—
222C22×D61—22no—
223C22×D61—21no—
224Q8×C61—22no—
225D4×C61—22no—
226D4×C61—42no—
227C2×C4×C61—22no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C22⋊D61—82no—
243C22⋊D61—82no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C22⋊D61—81no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251D244—42no—
252D244—41no—
253D244—44no—
254D244—42no—
255D244—82no—
256D244—81no—
257C4×D61—44no—
258C4×D61—42no—
259C4×D61—82no—
260C4×D61—82no—
261C4×D61—82no—
262Dic122—42no—
263Dic122—41no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—42no—
267#92418—22no—
268#92418—42no—
269#92418—41no—
270#17684—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—21no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—21no—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—42no—
279C2×D61—41no—
280C2×D61—42no—
281C2×D61—41no—
282C2×D61—42no—
283C2×D61—41no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—22no—
287C4×C61—21no—
288C4×C61—22no—
289C4×C61—21no—
290C4×C61—22no—
291C4×C61—21no—
292C4×C61#42112yes—
293Dic61—42no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297#3517—611no—
298#2970—611no—
299#2923D6111yescharacteristic, complemented
300#1536—612no—
301#1535—612no—
302#1532—612no—
303#1530—612no—
304#1530—611no—
305#1529—611no—
306#1525—611no—
307#1524—611no—
308C2×C61—41no—
309C2×C61—42no—
310C2×C61#288811yescharacteristic
311C2×C61C2×#42111yescharacteristic
312C2×C61#290111yescharacteristic
313D61—82no—
314D61—81no—
315#557—611no—
316#553—1221no—
317#553—1222no—
318#550—611no—
319#550—611no—
320#550—612no—
321#550—1221no—
322#547—611no—
323#547—612no—
324#540—1221no—
325#540—1222no—
326#532—612no—
327#532—611no—
328#529—611no—
329#529—612no—
330#520—1221no—
331C4×#41C2×D6111yescharacteristic
332#446—1221no—
333#435—1222no—
334C8×D4—1221no—
335#433—22no—
336#432—612no—
337#432—611no—
338#432C2×D6112yes—
339#432C2×D6112yes—
340#424—21no—
341#408—21no—
342C4×M16—611no—
343C4×M16C2×D6111yescharacteristic
344C2×#97—612no—
345C2×#97C2×D6111yescharacteristic
346#330—611no—
347#330—611no—
348#330—612no—
349#329—611no—
350#329—612no—
351#329—612no—
352#329—612no—
353#329—611no—
354#329—612no—
355#329—612no—
356C61#1768411yescharacteristic, complemented, Sylow
357C2×#41—612no—
358C2×#41—612no—
359C2×#41—612no—
360C2×#41—1221no—
361C2×#41C22⋊D6111yescharacteristic
362C2×M16—21no—
363C2×M16—1221no—
364C22×C8—21no—
365C22×C8—1222no—
366C4⋊Q8—612no—
367C4⋊Q8—611no—
368#127—612no—
369#127—611no—
370#127—1221no—
371#127—1222no—
372#126—1221no—
373#126—1222no—
374#125—611no—
375#125—1222no—
376#125—1221no—
377#124—611no—
378#124—612no—
379#123—1222no—
380#123—1221no—
381#123—1222no—
382#123—1222no—
383#123—1222no—
384C22⋊Q8—611no—
385C22⋊Q8—612no—
386#121—611no—
387#121—612no—
388#121—1221no—
389#121—1222no—
390#121—1222no—
391#121—1221no—
392#121—1221no—
393#121—1222no—
394#121—1222no—
395#121—1221no—
396#121—1222no—
397#121—1222no—
398#121—1222no—
399C4×Q8—611no—
400C4×Q8—612no—
401C4×Q8—1221no—
402C4×Q8C4×D6111yescharacteristic
403C4×Q8D24411yescharacteristic
404C4×D4—611no—
405C4×D4—612no—
406C4×D4—1221no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4—1221no—
411C4×D4—1222no—
412C4×D4—1221no—
413C4×D4—1222no—
414C4×D4C4×D6111yescharacteristic
415C4×D4C4×D6112yes—
416C4×D4D24411yescharacteristic
417C4×D4D24412yes—
418#117—1221no—
419#117—1222no—
420#117C22⋊D6111yescharacteristic
421#117C22⋊D6112yes—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1221no—
424C2×C42C22×D6112yes—
425C2×C42C22×D6111yescharacteristic
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—1221no—
429C4⋊C8—1222no—
430C22⋊C8—22no—
431C22⋊C8—22no—
432C22⋊C8—1222no—
433C22⋊C8—1221no—
434#97—21no—
435#97—611no—
436#97—612no—
437#97—611no—
438#97—1222no—
439#97C22×D6111yescharacteristic
440#97C22×D6111yescharacteristic
441#97C22×D6112yes—
442C4×C8—21no—
443C4×C8—1221no—
444#41—42no—
445#41—4882no—
446#41—4882no—
447#41—4882no—
448#41—2444no—
449C2×Q8—611no—
450C2×Q8—612no—
451C2×Q8#1175819811yescharacteristic
452C2×D4—611no—
453C2×D4—612no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1221no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1221no—
463C2×D4#1175819811yescharacteristic
464C2×D4#1175819812yes—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4C2×C4×D6112yes—
471C22×C4C2×C4×D6111yescharacteristic
472C22×C4C2×D24412yes—
473C22×C4C2×D24411yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6112yes—
476M16—42no—
477M16—2442no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1222no—
485C2×C8—2444no—
486C4⋊C4—611no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1222no—
498C4⋊C4#1175819812yes—
499C4⋊C4#1175819811yescharacteristic
500C4⋊C4#1175819812yes—
501C4⋊C4#1175819811yescharacteristic
502C22⋊C4—1221no—
503C22⋊C4—1222no—
504C22⋊C4—1221no—
505C22⋊C4—1222no—
506C22⋊C4—1221no—
507C22⋊C4—1222no—
508C22⋊C4—1221no—
509C22⋊C4—1222no—
510C22⋊C4—1221no—
511C22⋊C4—1222no—
512C22⋊C4#1175819812yes—
513C22⋊C4#1175819812yes—
514C22⋊C4#1175819812yes—
515C42—1222no—
516C42—1221no—
517C42C2×C4×D6112yes—
518C42C2×C4×D6111yescharacteristic
519C42C2×D24411yescharacteristic
520C42C2×D24412yes—
521C42C2×C22⋊D6111yescharacteristic
522C42C2×C22⋊D6112yes—
523C42C23×D6111yescharacteristic
524C23—1222no—
525C23—1221no—
526C23C2×#1175819811yescharacteristic
527C23C2×#1175819812yes—
528Q8—22no—
529Q8—2441no—
530Q8—2442no—
531D4—22no—
532D4—42no—
533D4—4882no—
534D4—4881no—
535D4—4882no—
536D4—4881no—
537D4—4882no—
538D4—4882no—
539D4—4882no—
540D4—2441no—
541D4—2442no—
542D4—2444no—
543D4—2442no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—1221no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—4882no—
555C2×C4—4882no—
556C2×C4—4882no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4C2×#1175819812yes—
560C2×C4C2×#1175819811yescharacteristic
561C2×C4C2×#1175819812yes—
562C2×C4C2×#1175819811yescharacteristic
563C2×C4C2×#1175819812yes—
564C2×C4C2×#1175819811yescharacteristic
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C22×D24411yescharacteristic
568C8—41no—
569C8—41no—
570C8—42no—
571C8—1222no—
572C8—2442no—
573C8—2441no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—2442no—
581C22—2441no—
582C22—2442no—
583C22—2441no—
584C22—2442no—
585C22—2441no—
586C22C22×#1175819811yescharacteristic
587C4—21no—
588C4—22no—
589C4—21no—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—22no—
595C4—2441no—
596C4—2441no—
597C4—2442no—
598C4—2441no—
599C4#1683978012yes—
600C2—41no—
601C2—42no—
602C2—4882no—
603C2—4881no—
604C2#2145429711yescharacteristic
605C2C2×#1683978011yescharacteristic
606C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2923C2
socleC22×C61C22×C22⋊C4
prime radical#21459825C2
cosocleC42×C61C24
hypercenter#2923D61
hypocenterC61#17684
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2923

Complements

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

NH ≅ G/Ntype
2. #21454833C2semidirect
3. #21444453C2semidirect
4. #21444448C2semidirect
5. #21444440C2semidirect
6. #21444431C2semidirect
7. #21444429C2semidirect
8. #21444424C2semidirect
9. #21444410C2semidirect
10. #21444405C2semidirect
11. C61×#2923C2semidirect
299. #2923D61semidirect
356. C61#17684semidirect

Maximal subgroups

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

subgroupindex
#214598252
#214548332
#214444532
#214444482
#214444402
#214444312
#214444292
#214444242
#214444102
#214444052
C61×#29232
#1768461