Finite Group Databasebeta

← #29647049

Subgroups — #29647049

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459441C211yesmaximal, characteristic
2#21454944C211yesmaximal, characteristic, complemented
3#21444293C212yesmaximal, complemented
4#21444275C211yesmaximal, characteristic, complemented
5#21444259C211yesmaximal, characteristic, complemented
6#21444254C212yesmaximal, complemented
7#21444241C211yesmaximal, characteristic, complemented
8#21444223C212yesmaximal, complemented
9#21444207C212yesmaximal, complemented
10#21444202C211yesmaximal, characteristic, complemented
11C61×#2922C211yesmaximal, characteristic, complemented
12#16840491C411yescharacteristic
13#16840482C411yescharacteristic
14#16840482C412yes—
15#16840451—21no—
16#16840451—22no—
17Q8×D244C411yescharacteristic
18#16840419—21no—
19#16840417C412yes—
20#16840417C411yescharacteristic
21#16840319C2212yes—
22#16840319C2211yescharacteristic
23#16840294—21no—
24#16840294—22no—
25C4×#11758222C2211yescharacteristic
26C4×#11758222C2212yes—
27C4×Q8×D61—21no—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2212yes—
32#16839815C2211yescharacteristic
33#16839786C2211yescharacteristic
34C2×C4⋊#92418C2212yes—
35#16839494C2211yescharacteristic
36#16839494C2211yescharacteristic
37#16839494C2212yes—
38#16839494C2212yes—
39#16839494C2212yes—
40#16839494C2212yes—
41#16839494C2212yes—
42#16839491C2211yescharacteristic
43#16839491C2211yescharacteristic
44#16839491C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#431C2212yes—
48C61×#431C2212yes—
49C61×#424—21no—
50C61×#408—21no—
51C4×M16×C61C2211yescharacteristic
52C2×C4×C8×C61C2211yescharacteristic
53C2×#11758222C2×C411yescharacteristic
54C2×#11758222C2×C412yes—
55C2×Q8×D61—21no—
56C2×#11758194—21no—
57C22×#92418—22no—
58#14519777C2×C412yes—
59#14519777C2×C411yescharacteristic
60#14519777C2×C412yes—
61#14519777C2×C411yescharacteristic
62#14519770—22no—
63#14519770—22no—
64#14519770—22no—
65#14519769—21no—
66#14519769—22no—
67#14519768—22no—
68#14519768—21no—
69#14519764—21no—
70#14519764—22no—
71#14519759—22no—
72#14519759—22no—
73#14519759—22no—
74#14519757—22no—
75#14519757—21no—
76#14519756—21no—
77#14519756—22no—
78#14519756C2×C412yes—
79#14519756C2×C412yes—
80#14519756C2×C412yes—
81#14519748—21no—
82#14519748—22no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122—22no—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C412yes—
88C4×Dic122C2311yescharacteristic
89C4×Dic122C2312yes—
90#14519742—22no—
91#14519742—21no—
92C4×D244C2×C411yescharacteristic
93C4×D244C2×C412yes—
94C4×D244C2311yescharacteristic
95C4×C22⋊D61—21no—
96C4×C22⋊D61—22no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—21no—
99#14519736—22no—
100#14519736—21no—
101#14519736C2×C411yescharacteristic
102#14519736C2×C412yes—
103#14519734—21no—
104#14519734—22no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—21no—
107C42×D61—21no—
108C42×D61—22no—
109C4⋊#92418—22no—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2312yes—
113C4⋊#92418C2311yescharacteristic
114C22⋊#92418—21no—
115C22⋊#92418—22no—
116#14519688—22no—
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—
136C4×C8×C61—21no—
137C4×C8×C61C2311yescharacteristic
138C4×C8×C61C2311yescharacteristic
139C4×C8×C61C2312yes—
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×D412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—22no—
152Q8×D61—44no—
153#11758222—22no—
154#11758222—24no—
155#11758222—41no—
156#11758222—42no—
157C2×D244C22×C411yescharacteristic
158C2×C4×D61—22no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—22no—
162C2×C4×D61—21no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122C22×C411yescharacteristic
166C2×Dic122C22×C412yes—
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—22no—
171C2×C8×C61—42no—
172C4⋊C4×C61C22⋊C412yes—
173C4⋊C4×C61C22⋊C411yescharacteristic
174C4⋊C4×C61C22⋊C412yes—
175C4⋊C4×C61C22⋊C411yescharacteristic
176C22⋊C4×C61C22⋊C412yes—
177C22⋊C4×C61C22⋊C412yes—
178C22⋊C4×C61C22⋊C412yes—
179C42×C61C22×C412yes—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D412yes—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D412yes—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—21no—
190C22⋊Dic61—22no—
191#11758198—21no—
192#11758198—22no—
193#11758198—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208#11758196—21no—
209#11758196—22no—
210C4×Dic61—21no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—22no—
216#11758194—81no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—22no—
220C2×#92418—82no—
221C23×C61C2×C22⋊C411yescharacteristic
222C23×C61C2×C22⋊C412yes—
223C22×D61—21no—
224Q8×C61—22no—
225D4×C61—22no—
226D4×C61—42no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C22⋊D61—42no—
247C22⋊D61—44no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253D244—21no—
254D244#13112yes—
255C4×D61—42no—
256C4×D61—44no—
257C4×D61—44no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—42no—
261Dic122—21no—
262Dic122—22no—
263Dic122—21no—
264Dic122—22no—
265Dic122—44no—
266Dic122—42no—
267C8×C61—41no—
268C8×C61—41no—
269C8×C61—42no—
270#92418—41no—
271#92418—42no—
272#92418—41no—
273#17523—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—22no—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—22no—
282C2×D61—41no—
283C2×D61—41no—
284C4×C61—21no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61#42112yes—
291C4×C61C2×#13111yescharacteristic
292C4×C61#56811yescharacteristic
293C4×C61#57711yescharacteristic
294C4×C61#57811yescharacteristic
295Dic61—42no—
296Dic61—41no—
297Dic61—41no—
298Dic61—41no—
299Dic61—42no—
300#3482—611no—
301#2990—611no—
302#2922D6111yescharacteristic, complemented
303#1497—612no—
304#1493—611no—
305#1489—611no—
306#1488—612no—
307#1485—611no—
308#1481—612no—
309#1477—612no—
310#1476—611no—
311C2×C61—41no—
312C2×C61—42no—
313C2×C61#288011yescharacteristic
314C2×C61#289811yescharacteristic
315C2×C61C2×#42111yescharacteristic
316D61—42no—
317#557—611no—
318#556—612no—
319#556—611no—
320#552—1221no—
321#552—1222no—
322D4×Q8—611no—
323#548—1221no—
324#547—611no—
325#547—612no—
326#533—612no—
327#533—611no—
328#529—1221no—
329#529—1222no—
330C4×#41—611no—
331C4×#41—612no—
332C4×#41C2×D6111yescharacteristic
333C2×C4×Q8—1221no—
334#446—1221no—
335#435—1222no—
336C8×D4—1221no—
337#433—22no—
338#432—612no—
339#431—611no—
340#431C2×D6112yes—
341#431C2×D6112yes—
342#424—21no—
343#423—611no—
344C2×C4⋊C8—612no—
345#408—21no—
346C4×M16C2×D6111yescharacteristic
347C2×C4×C8C2×D6111yescharacteristic
348Q8⋊C8—611no—
349Q8⋊C8—611no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—612no—
358C61#1752311yescharacteristic, complemented, Sylow
359C2×#41—611no—
360C2×#41—612no—
361C2×#41C22⋊D6111yescharacteristic
362C22×Q8—1221no—
363C2×M16—21no—
364C2×M16—1221no—
365C22×C8—21no—
366C22×C8—1222no—
367C4⋊Q8—612no—
368C4⋊Q8—611no—
369C4⋊Q8—612no—
370C4⋊Q8—611no—
371#126—1222no—
372#126—1222no—
373#126—1222no—
374#125—1222no—
375#125—1221no—
376#125—1221no—
377#125—1222no—
378#124—1221no—
379#124—1222no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—612no—
386C22⋊Q8—1221no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1221no—
390#121—1221no—
391#121—1222no—
392C4×Q8—611no—
393C4×Q8—612no—
394C4×Q8—611no—
395C4×Q8—612no—
396C4×Q8—1221no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1222no—
401C4×Q8C4×D6111yescharacteristic
402C4×Q8D24411yescharacteristic
403C4×D4—611no—
404C4×D4—612no—
405C4×D4—611no—
406C4×D4—1222no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4—1222no—
410C4×D4C4×D6111yescharacteristic
411C4×D4C4×D6112yes—
412C4×D4D24411yescharacteristic
413C4×D4D24412yes—
414#117—611no—
415#117—612no—
416#117—1222no—
417#117—1221no—
418#117—1221no—
419#117—1222no—
420#117C22⋊D6111yescharacteristic
421#117C22⋊D6112yes—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1221no—
424C2×C42—1221no—
425C2×C42—1222no—
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6112yes—
428C4⋊C8—22no—
429C4⋊C8—22no—
430C4⋊C8—611no—
431C4⋊C8—612no—
432C4⋊C8—611no—
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C22⋊C8—22no—
436C22⋊C8—22no—
437C22⋊C8—1221no—
438C22⋊C8—1222no—
439#97—21no—
440#97—1222no—
441C4×C8—21no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6112yes—
446#41—42no—
447#41—1222no—
448#41—1224no—
449#41—2441no—
450#41—2442no—
451C2×Q8—611no—
452C2×Q8—612no—
453C2×Q8—1222no—
454C2×Q8—1221no—
455C2×Q8—2444no—
456C2×Q8#1175819811yescharacteristic
457C2×D4—611no—
458C2×D4—1221no—
459C2×D4—1222no—
460C2×D4#1175819811yescharacteristic
461C2×D4#1175819812yes—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4C2×C4×D6112yes—
469C22×C4C2×D24412yes—
470C22×C4C2×D24411yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6112yes—
473M16—42no—
474M16—4881no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—22no—
478C2×C8—42no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1222no—
482C2×C8—4882no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—612no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4#1175819812yes—
500C4⋊C4#1175819811yescharacteristic
501C4⋊C4#1175819812yes—
502C4⋊C4#1175819811yescharacteristic
503C22⋊C4—1221no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4#1175819812yes—
512C22⋊C4#1175819812yes—
513C22⋊C4#1175819812yes—
514C42—1221no—
515C42—1222no—
516C42—1221no—
517C42—1222no—
518C42—1222no—
519C42—1221no—
520C42C2×C4×D6112yes—
521C42C2×C4×D6111yescharacteristic
522C42C2×D24411yescharacteristic
523C42C2×D24412yes—
524C42C2×C22⋊D6111yescharacteristic
525C42C2×C22⋊D6112yes—
526C42C23×D6111yescharacteristic
527C23—1221no—
528C23C2×#1175819811yescharacteristic
529C23C2×#1175819812yes—
530Q8—22no—
531Q8—1221no—
532Q8—1222no—
533Q8—1221no—
534Q8—1222no—
535Q8—2444no—
536Q8—2442no—
537D4—22no—
538D4—42no—
539D4—612no—
540D4—1221no—
541D4—2442no—
542D4—2444no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4—2444no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4C2×#1175819812yes—
565C2×C4C2×#1175819811yescharacteristic
566C2×C4C2×#1175819812yes—
567C2×C4C2×#1175819811yescharacteristic
568C2×C4C2×#1175819812yes—
569C2×C4C2×#1175819811yescharacteristic
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C22×D24411yescharacteristic
573C8—41no—
574C8—41no—
575C8—42no—
576C8—2441no—
577C8—2442no—
578C8—2441no—
579C22—22no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—21no—
584C22—22no—
585C22—1222no—
586C22—2441no—
587C22—2441no—
588C22C22×#1175819811yescharacteristic
589C4—22no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—22no—
595C4—2441no—
596C4—2442no—
597C4—2441no—
598C4—2442no—
599C4—2441no—
600C4#1683978012yes—
601C4#131×D6111yescharacteristic
602C4#1684052711yescharacteristic
603C4#1684057411yescharacteristic
604C4#1684057911yescharacteristic
605C2—41no—
606C2—42no—
607C2—2442no—
608C2#2145426611yescharacteristic
609C2#2145435311yescharacteristic
610C2C2×#1683978011yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2922C2
socleC22×C61C22×C22⋊C4
prime radical#21459441C2
cosocleC42×C61C24
hypercenter#2922D61
hypocenterC61#17523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2922

Complements

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

NH ≅ G/Ntype
2. #21454944C2semidirect
3. #21444293C2semidirect
4. #21444275C2semidirect
5. #21444259C2semidirect
6. #21444254C2semidirect
7. #21444241C2semidirect
8. #21444223C2semidirect
9. #21444207C2semidirect
10. #21444202C2semidirect
11. C61×#2922C2semidirect
302. #2922D61semidirect
358. C61#17523semidirect

Maximal subgroups

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

subgroupindex
#214594412
#214549442
#214442932
#214442752
#214442592
#214442542
#214442412
#214442232
#214442072
#214442022
C61×#29222
#1752361