Finite Group Databasebeta

← #29646706

Subgroups — #29646706

Order 15616.

920 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#21459246C211yesmaximal, characteristic
2#21454932C211yesmaximal, characteristic, complemented
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21444290C212yesmaximal, complemented
5#21444281C212yesmaximal, complemented
6#21444276C212yesmaximal, complemented
7#21444272C212yesmaximal, complemented
8#21444213C211yesmaximal, characteristic, complemented
9#21444206C211yesmaximal, characteristic, complemented
10C2×#16839494C211yesmaximal, characteristic, complemented
11C61×#1476C211yesmaximal, characteristic, complemented
12#16840390C411yescharacteristic
13#16840383—21no—
14#16840377C412yes—
15#16840374—21no—
16#16840370C412yes—
17#16840354—22no—
18#16840340C411yescharacteristic
19#16840338C411yescharacteristic
20#16840333C411yescharacteristic
21#16840319—21no—
22#16840316C2212yes—
23#16840296—21no—
24#16840294C2212yes—
25#16840292—22no—
26C4×#11758222C2211yescharacteristic
27C2×C4×Dic122C2211yescharacteristic
28#16839892—21no—
29C8×Dic122—21no—
30#16839841—21no—
31#16839830—22no—
32C8×D244—21no—
33C8×C22⋊D61—22no—
34#16839818C2212yes—
35#16839815C2211yescharacteristic
36#16839813C2212yes—
37#16839784C2212yes—
38C2×C4⋊#92418C2211yescharacteristic
39C4×C8×D61C2212yes—
40#16839494C2212yes—
41#16839494C2212yes—
42Q8⋊#92418C2212yes—
43#16839492C2212yes—
44#16839492C2212yes—
45D4⋊#92418C2212yes—
46C4×#41×C61C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C2×C4×C8×C61C2211yescharacteristic
49C61×Q8⋊C8C2211yescharacteristic
50C61×Q8⋊C8C2211yescharacteristic
51C61×D4⋊C8C2211yescharacteristic
52C61×D4⋊C8C2211yescharacteristic
53C2×#11758222C2×C411yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C2×#11758189—22no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58#14519777—21no—
59#14519777—21no—
60#14519776C2×C412yes—
61#14519775C2×C412yes—
62#14519772—22no—
63#14519771—22no—
64#14519770—22no—
65#14519770—22no—
66#14519769—21no—
67#14519769—22no—
68#14519768C2×C412yes—
69#14519767—22no—
70#14519767—22no—
71#14519765—21no—
72#14519764C2×C412yes—
73#14519763—22no—
74#14519761—22no—
75#14519760—22no—
76#14519759—22no—
77#14519758—22no—
78#14519757C2×C412yes—
79#14519757C2×C412yes—
80#14519756—21no—
81#14519756—21no—
82#14519756—21no—
83#14519756—22no—
84#14519755—22no—
85#14519754—22no—
86#14519753C2×C412yes—
87#14519751—21no—
88#14519749—22no—
89#14519748C2×C412yes—
90C4×Dic122—21no—
91C4×Dic122—21no—
92C4×Dic122—22no—
93C4×Dic122C2×C412yes—
94C4×Dic122D411yescharacteristic
95C4×Dic122D411yescharacteristic
96C4×Dic122C2312yes—
97C4×D244D411yescharacteristic
98C4×D244D411yescharacteristic
99C4×C22⋊D61—22no—
100C4×C22⋊D61D412yes—
101C4×C22⋊D61D412yes—
102#14519736—22no—
103#14519736C2×C411yescharacteristic
104#14519734—22no—
105#14519734C2×C412yes—
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×#11758196—21no—
108C2×C4×Dic61—21no—
109C42×D61C2312yes—
110C4⋊#92418—21no—
111C4⋊#92418—21no—
112C4⋊#92418C2312yes—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418—21no—
116#14519689—22no—
117#14519689—22no—
118#14519687—22no—
119C4×#92418C2312yes—
120C8×Dic61—22no—
121C2×#41×C61C2×C411yescharacteristic
122C22×C8×C61—21no—
123C4×Q8×C61C2×C411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61C2×C411yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61C2×C411yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61—21no—
136C4×C8×C61—21no—
137C4×C8×C61—21no—
138C4×C8×C61C2311yescharacteristic
139C4×C8×C61C2311yescharacteristic
140#41×C61—22no—
141#41×C61—22no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61C22⋊C412yes—
153C22×Dic61—21no—
154C22×Dic61—21no—
155#11758222—22no—
156#11758222—22no—
157C2×D244C22⋊C411yescharacteristic
158C2×C4×D61C22×C412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—21no—
164C2×Dic122—22no—
165C2×Dic122C22⋊C411yescharacteristic
166C2×Dic122C22×C412yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—42no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61C22⋊C412yes—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198C22⋊C412yes—
194#11758198C22⋊C412yes—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196C22⋊C412yes—
210#11758196C22⋊C412yes—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—22no—
217C2×#92418—22no—
218C2×#92418—81no—
219#11758189—82no—
220C8×D61—44no—
221C23×C61—21no—
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61C2×C22⋊C412yes—
224Q8×C61—21no—
225Q8×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—42no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—81no—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61#13112yes—
243C2×C4×C61#13112yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C22⋊D61—24no—
248C22⋊D61—82no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—42no—
252C2×Dic61—81no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256D244—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—42no—
262Dic122—22no—
263Dic122—22no—
264Dic122—22no—
265Dic122—42no—
266Dic122—82no—
267C8×C61—22no—
268C8×C61—41no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—42no—
272#92418—42no—
273#17484—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61#40812yes—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61#40812yes—
291C4×C61C2×#13111yescharacteristic
292C4×C61C2×#13111yescharacteristic
293C4×C61#57711yescharacteristic
294C4×C61#57811yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—22no—
298Dic61—42no—
299Dic61—42no—
300#3461—611no—
301#2989—611no—
302C8×#41—611no—
303#1497—612no—
304#1495—612no—
305#1494—612no—
306#1493—612no—
307#1479—611no—
308#1477—611no—
309#1476D6111yescharacteristic, complemented
310C2×Q8⋊C8—611no—
311C2×C61—22no—
312C2×C61—81no—
313C2×C61C2×#40811yescharacteristic
314C2×C61#289811yescharacteristic
315C2×C61#290111yescharacteristic
316D61—42no—
317#544—611no—
318#542—612no—
319#542—1221no—
320#541—612no—
321#541—1221no—
322#539—1222no—
323#537—611no—
324#536—611no—
325#536—611no—
326#533—612no—
327#533—1221no—
328#529—612no—
329#529—1221no—
330#529—1222no—
331C4×#41—611no—
332C4×#41C2×D6111yescharacteristic
333C2×C4×Q8—611no—
334#446—1221no—
335C8×Q8—1221no—
336#436—1221no—
337#435—1222no—
338C8×D4—1221no—
339C8×D4—1222no—
340#432—612no—
341#431—611no—
342#431—612no—
343#431C2×D6111yescharacteristic
344#423—612no—
345C2×C4⋊C8—611no—
346C2×C4×C8—612no—
347C2×C4×C8C2×D6111yescharacteristic
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353Q8⋊C8C2×D6111yescharacteristic
354Q8⋊C8C2×D6111yescharacteristic
355D4⋊C8—612no—
356D4⋊C8C2×D6111yescharacteristic
357D4⋊C8C2×D6111yescharacteristic
358C61#1748411yescharacteristic, complemented, Sylow
359C2×#41—611no—
360C2×#41C4×D6111yescharacteristic
361C22×Q8—611no—
362C2×M16—1222no—
363C22×C8—21no—
364C22×C8—1222no—
365C22×C8—1221no—
366C4⋊Q8—612no—
367C4⋊Q8—612no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370#126—1222no—
371#126—1222no—
372#126—1222no—
373#126—1222no—
374#125—612no—
375#125—1221no—
376#125—1222no—
377#125—1222no—
378#125—1222no—
379#124—612no—
380#124—1221no—
381#124—1222no—
382#123—1222no—
383#123—1222no—
384#123—1222no—
385#123—1222no—
386C22⋊Q8—612no—
387C22⋊Q8—612no—
388C22⋊Q8—612no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1222no—
394C22⋊Q8—1221no—
395#121—612no—
396#121—1221no—
397#121—1222no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—612no—
401C4×Q8—612no—
402C4×Q8—1222no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8C4×D6111yescharacteristic
406C4×Q8C22×D6111yescharacteristic
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—1222no—
414C4×D4C4×D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416#117—21no—
417#117—611no—
418#117—612no—
419#117—1222no—
420#117—1222no—
421#117C4×D6111yescharacteristic
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C42—21no—
425C2×C42—612no—
426C2×C42—1221no—
427C2×C42C22×D6111yescharacteristic
428C4⋊C8—612no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—1222no—
437C22⋊C8—1222no—
438C22⋊C8—1221no—
439#97—1222no—
440C4×C8—21no—
441C4×C8—21no—
442C4×C8—612no—
443C4×C8—1222no—
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6111yescharacteristic
446#41—22no—
447#41—22no—
448#41—1222no—
449#41—1222no—
450C2×Q8—611no—
451C2×Q8—612no—
452C2×Q8—1222no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×Q8C2×C4×D6111yescharacteristic
457C2×D4—21no—
458C2×D4—612no—
459C2×D4—611no—
460C2×D4—1222no—
461C2×D4C2×C4×D6111yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473M16—4882no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—42no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—4881no—
482C2×C8—2444no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—611no—
488C4⋊C4—611no—
489C4⋊C4—612no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1222no—
501C4⋊C4C2×C4×D6111yescharacteristic
502C4⋊C4C2×C4×D6111yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—612no—
507C22⋊C4—612no—
508C22⋊C4—612no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—612no—
519C42—612no—
520C42—612no—
521C42—1222no—
522C42—1222no—
523C42C2×C4×D6111yescharacteristic
524C42D4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42C23×D6111yescharacteristic
527C23—21no—
528C23—612no—
529C23C22⋊C4×D6111yescharacteristic
530Q8—21no—
531Q8—21no—
532Q8—1222no—
533Q8—1222no—
534Q8—1222no—
535Q8—4882no—
536Q8—2442no—
537D4—21no—
538D4—21no—
539D4—42no—
540D4—1224no—
541D4—1222no—
542D4—4882no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—81no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—4881no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C22⋊C4×D6111yescharacteristic
567C2×C4C22⋊C4×D6111yescharacteristic
568C2×C4#1451978312yes—
569C2×C4#1451978512yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C8—22no—
574C8—41no—
575C8—41no—
576C8—41no—
577C8—2442no—
578C8—2442no—
579C22—21no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—1222no—
585C22—1222no—
586C22—1222no—
587C22#1683972712yes—
588C22C2×C22⋊C4×D6111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—1222no—
596C4—1222no—
597C4—1222no—
598C4—2442no—
599C4—2442no—
600C4#1683972712yes—
601C4C2×#1451978311yescharacteristic
602C4C2×#1451978511yescharacteristic
603C4#1684056911yescharacteristic
604C4#1684057611yescharacteristic
605C2—22no—
606C2—81no—
607C2—2442no—
608C2C2×#1683972711yescharacteristic
609C2#2145434811yescharacteristic
610C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1476C2
socleC22×C61C22×C22⋊C4
prime radical#16840340C4
cosocleC42×C61C24
hypercenter#1476D61
hypocenterC61#17484
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #21454932C2semidirect
3. C8×#11758222C2semidirect
4. #21444290C2semidirect
5. #21444281C2semidirect
6. #21444276C2semidirect
7. #21444272C2semidirect
8. #21444213C2semidirect
9. #21444206C2semidirect
10. C2×#16839494C2semidirect
11. C61×#1476C2semidirect
309. #1476D61semidirect
358. C61#17484semidirect

Maximal subgroups

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

subgroupindex
#214592462
#214549322
C8×#117582222
#214442902
#214442812
#214442762
#214442722
#214442132
#214442062
C2×#168394942
C61×#14762
#1748461