Finite Group Databasebeta

← #30570333

Subgroups — #30570333

Order 15616.

1664 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459629C211yesmaximal, characteristic, complemented
2#21458751C213yesmaximal, complemented
3#21458729C213yesmaximal, complemented
4Q8×#11758202C211yesmaximal, characteristic, complemented
5Q8×#11758200C211yesmaximal, characteristic, complemented
6#21458622C213yesmaximal, complemented
7#21458600C213yesmaximal, complemented
8#21458497C211yesmaximal, characteristic, complemented
9#21458461C211yesmaximal, characteristic, complemented
10#21457033C213yesmaximal, complemented
11#21456978C213yesmaximal, complemented
12#21455981C213yesmaximal
13#21455890C213yesmaximal
14#21454908C211yesmaximal, characteristic, complemented
15Q8×#41×C61C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17Q8×Dic122C2213yes—
18#16840491C2211yescharacteristic
19#16840491C2213yes—
20#16840488—23no—
21#16840488—21no—
22#16840393—23no—
23#16840319C2213yes—
24#16840319C2213yes—
25#16840240—21no—
26#16840172C2216yes—
27#16840165C2213yes—
28#16840158C2213yes—
29#16840132C2216yes—
30#16840114C2213yes—
31#16840114C2213yes—
32#16840103C2213yes—
33#16840103C2213yes—
34#16840103C2216yes—
35#16840066C2216yes—
36#16840066C2216yes—
37#16840057C2213yes—
38#16840054C2213yes—
39#16840054C2213yes—
40#16840052C2213yes—
41#16840047C2213yes—
42#16840044C2213yes—
43#16840044C2213yes—
44#16840042C2213yes—
45#16839972C2213yes—
46#16839972C2213yes—
47#16839967C2213yes—
48#16839967C2213yes—
49#16839924C2211yescharacteristic
50#16839924C2213yes—
51#16839893C2213yes—
52#16839893C2213yes—
53Q8×#92418C2211yescharacteristic
54Q8×#92418C2211yescharacteristic
55#16839867C2216yes—
56#16839862C2216yes—
57C4×#11758202C2213yes—
58C4×#11758202C2213yes—
59C4×#11758200C2213yes—
60C4×#11758200C2213yes—
61#16839819C2213yes—
62#16839815C2213yes—
63Q82×C61C2211yescharacteristic
64C61×#557C2213yes—
65C61×#554—23no—
66C61×#554C2213yes—
67D4×Q8×C61—21no—
68D4×Q8×C61C2211yescharacteristic
69C61×#533—23no—
70C61×#533C2213yes—
71C4×#41×C61C2213yes—
72C2×C4×Q8×C61—21no—
73C2×C4×Q8×C61C2211yescharacteristic
74C2×#11758202C2311yescharacteristic
75C2×#11758202C2313yes—
76C2×#11758200C2311yescharacteristic
77C2×#11758200C2313yes—
78#14519777C2313yes—
79#14519777C2313yes—
80#14519777C2313yes—
81#14519777C2313yes—
82#14519776—23no—
83#14519776—26no—
84#14519776—23no—
85#14519769—23no—
86#14519769—23no—
87#14519768—23no—
88#14519768—26no—
89#14519768—23no—
90#14519756—23no—
91#14519756—23no—
92#14519756C2311yescharacteristic
93#14519756C2313yes—
94Q8×Dic61—23no—
95Q8×Dic61—21no—
96Q8×Dic61—23no—
97Q8×Dic61—21no—
98C4×Dic122—23no—
99C4×Dic122—23no—
100C4×Dic122C2313yes—
101C4×Dic122C2313yes—
102C4×Dic122C2316yes—
103#14519736—23no—
104#14519736—23no—
105#14519710C2313yes—
106#14519710C2313yes—
107#14519710C2316yes—
108#14519708C2313yes—
109#14519708C2316yes—
110#14519708C2313yes—
111C4⋊#92418C2313yes—
112C4⋊#92418C2313yes—
113C4⋊#92418C2313yes—
114C4⋊#92418C2313yes—
115#14519701C2311yescharacteristic
116#14519701C2313yes—
117#14519700C2316yes—
118#14519700C2313yes—
119#14519700C2313yes—
120#14519700C2313yes—
121#14519700C2316yes—
122#14519700C2313yes—
123#14519698C2311yescharacteristic
124#14519698C2313yes—
125C22⋊#92418C2311yescharacteristic
126C22⋊#92418C2313yes—
127#14519688C2316yes—
128C4×#92418C2313yes—
129C4×#92418C2313yes—
130C2×#41×C61C2311yescharacteristic
131C22×Q8×C61—21no—
132C22×Q8×C61D411yescharacteristic
133C4⋊Q8×C61—23no—
134C4⋊Q8×C61—23no—
135C4⋊Q8×C61C2313yes—
136C4⋊Q8×C61C2313yes—
137#125×C61—26no—
138#125×C61C2316yes—
139C22⋊Q8×C61—23no—
140C22⋊Q8×C61—23no—
141C22⋊Q8×C61—23no—
142C22⋊Q8×C61—23no—
143C22⋊Q8×C61—26no—
144C4×Q8×C61—21no—
145C4×Q8×C61—23no—
146C4×Q8×C61—23no—
147C4×Q8×C61—23no—
148C4×Q8×C61D413yes—
149C4×Q8×C61D411yescharacteristic
150C4×Q8×C61C2313yes—
151C4×Q8×C61C2313yes—
152C4×Q8×C61C2311yescharacteristic
153C4×Q8×C61C2313yes—
154C4×D4×C61—23no—
155C4×D4×C61—23no—
156C4×D4×C61C2313yes—
157C4×D4×C61C2313yes—
158#117×C61—23no—
159#117×C61C2313yes—
160C2×C4⋊C4×C61—23no—
161C2×C4⋊C4×C61D413yes—
162C2×C42×C61—23no—
163C2×C42×C61C2313yes—
164#41×C61—22no—
165#41×C61—22no—
166C2×Q8×C61—24no—
167C2×Q8×C61—23no—
168C2×Q8×C61—23no—
169C2×Q8×C61—44no—
170C2×Q8×C61C2×D413yes—
171C2×Q8×C61C2×D411yescharacteristic
172C2×Q8×C61C2411yescharacteristic
173C2×D4×C61—21no—
174C2×D4×C61C2411yescharacteristic
175C22×C4×C61—21no—
176C22×C4×C61—23no—
177C22×C4×C61—23no—
178C22×C4×C61C2×D413yes—
179C22×C4×C61C2×D413yes—
180C22×C4×C61C2411yescharacteristic
181C2×Dic122—23no—
182C2×Dic122—23no—
183C2×Dic122C2413yes—
184C2×Dic122C2411yescharacteristic
185C4⋊C4×C61—26no—
186C4⋊C4×C61—23no—
187C4⋊C4×C61—23no—
188C4⋊C4×C61—23no—
189C4⋊C4×C61—23no—
190C4⋊C4×C61—26no—
191C4⋊C4×C61C2×D413yes—
192C4⋊C4×C61C2×D416yes—
193C4⋊C4×C61C2×D413yes—
194C4⋊C4×C61C2413yes—
195C4⋊C4×C61C2413yes—
196C22⋊C4×C61—23no—
197C22⋊C4×C61—23no—
198C22⋊C4×C61—23no—
199C42×C61—23no—
200C42×C61—23no—
201C42×C61—23no—
202C42×C61C2×D413yes—
203C42×C61C2×D413yes—
204C42×C61C2413yes—
205C42×C61C2413yes—
206C22⋊Dic61—21no—
207C22⋊Dic61—23no—
208#11758202—22no—
209#11758202—26no—
210#11758200—22no—
211#11758200—26no—
212C4⋊Dic61—23no—
213C4⋊Dic61—23no—
214C4⋊Dic61—23no—
215C4⋊Dic61—23no—
216C4⋊Dic61—23no—
217C4⋊Dic61—23no—
218C4⋊Dic61C2411yescharacteristic
219C4⋊Dic61C2413yes—
220#11758196—21no—
221#11758196—23no—
222#11758196—26no—
223#11758196—23no—
224#11758196—23no—
225C4×Dic61—26no—
226C4×Dic61—23no—
227C4×Dic61—23no—
228C2×#92418C2411yescharacteristic
229C2×#92418C2411yescharacteristic
230C2×#92418C2413yes—
231C2×#92418C2413yes—
232C23×C61—21no—
233C23×C61C22×D411yescharacteristic
234Q8×C61—26no—
235Q8×C61—21no—
236Q8×C61—46no—
237Q8×C61—83no—
238Q8×C61#13714yes—
239Q8×C612−1+412yes—
240D4×C61—24no—
241D4×C61—21no—
242D4×C612−1+412yes—
243C2×C4×C61—24no—
244C2×C4×C61—21no—
245C2×C4×C61—26no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—23no—
249C2×C4×C61—23no—
250C2×C4×C61—43no—
251C2×C4×C61—46no—
252C2×C4×C61—83no—
253C2×C4×C61C22×D413yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D413yes—
256C2×C4×C61C22×D413yes—
257C2×C4×C61C22×D413yes—
258C2×C4×C61C2511yescharacteristic
259C2×Dic61—23no—
260C2×Dic61—21no—
261C2×Dic61—23no—
262C2×Dic61—21no—
263Dic122—21no—
264Dic122—23no—
265Dic122—21no—
266Dic122—23no—
267Dic122—83no—
268Dic122—83no—
269#92418—21no—
270#92418—23no—
271#92418—21no—
272#92418—23no—
273#61846—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—22no—
277C22×C61—21no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C23×D411yescharacteristic
281C4×C61—23no—
282C4×C61—23no—
283C4×C61—21no—
284C4×C61—22no—
285C4×C61—23no—
286C4×C61—43no—
287C4×C61—43no—
288C4×C61C2×#13716yes—
289C4×C61C2×2−1+411yescharacteristic
290C4×C61C2×2−1+411yescharacteristic
291Dic61—41no—
292Dic61—43no—
293Dic61—41no—
294Dic61—43no—
295#3502—611no—
296Q8×#41D6111yescharacteristic
297#3387—613no—
298#3385—613no—
299Q8×Q16—611no—
300Q8×SD16—611no—
301#3374—613no—
302#3372—613no—
303#3363—611no—
304#3359—611no—
305#3237—613no—
306#3231—613no—
307#3116—613no—
308#3107—613no—
309#2984—611no—
310C2×C61—41no—
311C2×C61—42no—
312C2×C61#348611yescharacteristic
313C2×C61C22×#13711yescharacteristic
314C2×C61#358411yescharacteristic
315Q82—611no—
316Q82—613no—
317Q82C2×D6111yescharacteristic
318#557—611no—
319#557—613no—
320#557—1221no—
321#557—1223no—
322#557C2×D6113yes—
323#554—23no—
324#554C2×D6113yes—
325D4×Q8—21no—
326D4×Q8C2×D6111yescharacteristic
327#544—1223no—
328#533—23no—
329#533—613no—
330#533—613no—
331#533C2×D6113yes—
332#519—1221no—
333C4×#41C2×D6113yes—
334C2×C4×Q8—21no—
335C2×C4×Q8C2×D6111yescharacteristic
336#501—616no—
337#500—613no—
338#498—613no—
339#491—616no—
340#487—613no—
341#487—613no—
342#485—613no—
343#485—613no—
344#485—616no—
345#479—616no—
346#479—616no—
347#477—613no—
348#477—613no—
349#477—613no—
350#476—613no—
351#475—613no—
352#475—613no—
353#475—613no—
354#474—613no—
355#462—613no—
356#462—613no—
357#461—613no—
358#461—613no—
359#452—611no—
360#452—613no—
361#446—613no—
362#446—613no—
363C8×Q8—611no—
364C8×Q8—611no—
365#441—616no—
366#440—616no—
367C4×Q16—613no—
368C4×Q16—613no—
369C4×SD16—613no—
370C4×SD16—613no—
371#432—613no—
372#431—613no—
373C61#6184611yescharacteristic, complemented, Sylow
374C2×#41C22×D6111yescharacteristic
375C22×Q8—21no—
376C22×Q8C22⋊D6111yescharacteristic
377C2×Q16—611no—
378C2×Q16—613no—
379C2×SD16—611no—
380C2×SD16—613no—
381C4⋊Q8—23no—
382C4⋊Q8—23no—
383C4⋊Q8—613no—
384C4⋊Q8—613no—
385C4⋊Q8—613no—
386C4⋊Q8—613no—
387C4⋊Q8—1223no—
388C4⋊Q8—1226no—
389C4⋊Q8—1223no—
390C4⋊Q8C22×D6113yes—
391C4⋊Q8C22×D6113yes—
392#125—26no—
393#125—1223no—
394#125—1226no—
395#125—1223no—
396#125—1223no—
397#125—1223no—
398#125C22×D6116yes—
399C22⋊Q8—23no—
400C22⋊Q8—23no—
401C22⋊Q8—23no—
402C22⋊Q8—23no—
403C22⋊Q8—26no—
404C22⋊Q8—611no—
405C22⋊Q8—613no—
406C22⋊Q8—1223no—
407C22⋊Q8—1223no—
408C4×Q8—21no—
409C4×Q8—23no—
410C4×Q8—23no—
411C4×Q8—23no—
412C4×Q8—613no—
413C4×Q8—613no—
414C4×Q8—616no—
415C4×Q8—1223no—
416C4×Q8—1221no—
417C4×Q8—1223no—
418C4×Q8—1221no—
419C4×Q8—1223no—
420C4×Q8—1223no—
421C4×Q8C22⋊D6111yescharacteristic
422C4×Q8C22⋊D6113yes—
423C4×Q8C22×D6113yes—
424C4×Q8C22×D6113yes—
425C4×Q8C22×D6111yescharacteristic
426C4×Q8C22×D6113yes—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4C22×D6113yes—
430C4×D4C22×D6113yes—
431#117—23no—
432#117—1223no—
433#117—1223no—
434#117C22×D6113yes—
435C2×C4⋊C4—23no—
436C2×C4⋊C4C22⋊D6113yes—
437C2×C42—23no—
438C2×C42C22×D6113yes—
439#107—613no—
440#107—613no—
441#107—616no—
442#106—613no—
443#106—613no—
444#106—616no—
445C4⋊C8—613no—
446C4⋊C8—613no—
447C4⋊C8—613no—
448C4⋊C8—613no—
449#103—611no—
450#103—613no—
451#103—613no—
452#103—613no—
453#103—613no—
454#103—613no—
455#103—616no—
456#103—616no—
457#102—611no—
458#102—613no—
459C22⋊C8—611no—
460C22⋊C8—613no—
461#97—616no—
462C4×C8—613no—
463C4×C8—613no—
464#41—22no—
465#41—22no—
466C2×Q8—24no—
467C2×Q8—23no—
468C2×Q8—23no—
469C2×Q8—44no—
470C2×Q8—611no—
471C2×Q8—613no—
472C2×Q8—1223no—
473C2×Q8—1223no—
474C2×Q8C2×C22⋊D6113yes—
475C2×Q8C2×C22⋊D6111yescharacteristic
476C2×Q8C23×D6111yescharacteristic
477C2×D4—21no—
478C2×D4C23×D6111yescharacteristic
479C22×C4—21no—
480C22×C4—23no—
481C22×C4—23no—
482C22×C4C2×C22⋊D6113yes—
483C22×C4C2×C22⋊D6113yes—
484C22×C4C23×D6111yescharacteristic
485Q16—1222no—
486Q16—1226no—
487SD16—1222no—
488SD16—1226no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—613no—
492C2×C8—613no—
493C4⋊C4—26no—
494C4⋊C4—23no—
495C4⋊C4—23no—
496C4⋊C4—23no—
497C4⋊C4—23no—
498C4⋊C4—26no—
499C4⋊C4—611no—
500C4⋊C4—613no—
501C4⋊C4—1226no—
502C4⋊C4—1223no—
503C4⋊C4—1223no—
504C4⋊C4—1221no—
505C4⋊C4—1223no—
506C4⋊C4—1223no—
507C4⋊C4—1223no—
508C4⋊C4—1223no—
509C4⋊C4—1223no—
510C4⋊C4—1223no—
511C4⋊C4—1223no—
512C4⋊C4C2×C22⋊D6113yes—
513C4⋊C4C2×C22⋊D6116yes—
514C4⋊C4C2×C22⋊D6113yes—
515C4⋊C4C23×D6113yes—
516C4⋊C4C23×D6113yes—
517C22⋊C4—23no—
518C22⋊C4—23no—
519C22⋊C4—23no—
520C22⋊C4—1221no—
521C22⋊C4—1223no—
522C42—23no—
523C42—23no—
524C42—23no—
525C42—1226no—
526C42—1223no—
527C42—1223no—
528C42C2×C22⋊D6113yes—
529C42C2×C22⋊D6113yes—
530C42C23×D6113yes—
531C42C23×D6113yes—
532C23—21no—
533C23C22×C22⋊D6111yescharacteristic
534Q8—26no—
535Q8—21no—
536Q8—46no—
537Q8—83no—
538Q8—1221no—
539Q8—1223no—
540Q8—1221no—
541Q8—1223no—
542Q8—4883no—
543Q8—4883no—
544Q8#1451980714yes—
545Q8#1451982512yes—
546D4—24no—
547D4—21no—
548D4#1451982512yes—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—26no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—23no—
555C2×C4—23no—
556C2×C4—43no—
557C2×C4—46no—
558C2×C4—83no—
559C2×C4—1223no—
560C2×C4—1221no—
561C2×C4—1223no—
562C2×C4—1221no—
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C22×C22⋊D6113yes—
565C2×C4C22×C22⋊D6113yes—
566C2×C4C22×C22⋊D6113yes—
567C2×C4C22×C22⋊D6113yes—
568C2×C4C24×D6111yescharacteristic
569C8—1221no—
570C8—1223no—
571C8—1221no—
572C8—1223no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—21no—
577C22—41no—
578C22—41no—
579C22C23×C22⋊D6111yescharacteristic
580C4—21no—
581C4—22no—
582C4—23no—
583C4—23no—
584C4—23no—
585C4—43no—
586C4—43no—
587C4—2443no—
588C4—2441no—
589C4—2441no—
590C4—2443no—
591C4C2×#1451980716yes—
592C4C2×#1451982511yescharacteristic
593C4C2×#1451982511yescharacteristic
594C2—41no—
595C2—42no—
596C2#2145947011yescharacteristic
597C2C22×#1451980711yescharacteristic
598C2#2146059011yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingQ8×#41×C61C2
socleC22×C61C23×D4
prime radicalC2×#41×C61C23
hypercenterQ8×#41D61
hypocenterC61#61846
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. #21459629C2semidirect
2. #21458751C2semidirect
3. #21458729C2semidirect
4. Q8×#11758202C2semidirect
5. Q8×#11758200C2semidirect
6. #21458622C2semidirect
7. #21458600C2semidirect
8. #21458497C2semidirect
9. #21458461C2semidirect
10. #21457033C2semidirect
11. #21456978C2semidirect
14. #21454908C2semidirect
373. C61#61846semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214587512
#214587292
Q8×#117582022
Q8×#117582002
#214586222
#214586002
#214584972
#214584612
#214570332
#214569782
#214559812
#214558902
#214549082
Q8×#41×C612
#6184661