Finite Group Databasebeta

← #30546283

Subgroups — #30546283

Order 15616.

1744 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#21459436C211yesmaximal, characteristic
2#21454971C213yesmaximal, complemented
3#21454964C213yesmaximal, complemented
4#21454950C211yesmaximal, characteristic, complemented
5#21454949C213yesmaximal, complemented
6#21454917C213yesmaximal, complemented
7#21454900C211yesmaximal, characteristic, complemented
8#21454898C211yesmaximal, characteristic, complemented
9#21454825C213yesmaximal, complemented
10#21454823C213yesmaximal, complemented
11#21454781C213yesmaximal, complemented
12C8×#11758226C211yesmaximal, characteristic, complemented
13Q8×#11758189C211yesmaximal, characteristic, complemented
14#21454750C213yesmaximal, complemented
15Q8×M16×C61C211yesmaximal, characteristic, complemented
16#16840488C411yescharacteristic
17#16840484C413yes—
18#16840453C413yes—
19#16840447C413yes—
20Q8×C22⋊D61C411yescharacteristic
21#16840424C411yescharacteristic
22#16840423C411yescharacteristic
23#16840413C413yes—
24#16840316C2213yes—
25#16840299C2213yes—
26#16840296C2213yes—
27C4×#11758226C2211yescharacteristic
28C4×#11758222C2213yes—
29C4×Q8×D61C2211yescharacteristic
30#16839892C2213yes—
31#16839890C2213yes—
32#16839890C2213yes—
33#16839889C2213yes—
34Q8×#92418C2211yescharacteristic
35Q8×#92418C2213yes—
36Q8×#92418C2211yescharacteristic
37Q8×#92418C2213yes—
38C8×Dic122C2213yes—
39#16839886C2213yes—
40#16839886C2216yes—
41#16839841C2213yes—
42#16839839C2216yes—
43#16839839C2213yes—
44#16839838C2213yes—
45#16839838C2216yes—
46#16839838C2211yescharacteristic
47#16839835C2213yes—
48#16839835C2213yes—
49#16839831C2213yes—
50#16839831C2213yes—
51C8×D244C2213yes—
52#16839827C2213yes—
53#16839824C2216yes—
54#16839822C2216yes—
55#16839815C2213yes—
56#16839815C2213yes—
57#16839814C2213yes—
58#16839814C2213yes—
59#16839813C2213yes—
60#16839813C2213yes—
61#16839787C2216yes—
62#16839784C2213yes—
63#16839784C2213yes—
64#16839719C2216yes—
65C4×#11758189C2213yes—
66C4×C8×D61C2213yes—
67C2×C4×Q8×C61C2211yescharacteristic
68C61×#446C2213yes—
69C61×#446C2213yes—
70C8×Q8×C61C2211yescharacteristic
71C8×Q8×C61C2211yescharacteristic
72C61×#423C2213yes—
73C4×M16×C61C2213yes—
74C2×#11758226C2×C411yescharacteristic
75C2×#11758222C2×C413yes—
76C2×Q8×D61C2×C411yescharacteristic
77C2×#11758189C2311yescharacteristic
78C2×#11758189C2313yes—
79C2×#11758189C2316yes—
80C2×C8×D61C2313yes—
81C2×C8×D61C2313yes—
82#14519776C2×C413yes—
83#14519775C2×C413yes—
84#14519772C2×C416yes—
85#14519771C2×C416yes—
86#14519769C2×C413yes—
87#14519768C2×C413yes—
88#14519767C2×C416yes—
89#14519766C2×C413yes—
90#14519765C2×C413yes—
91#14519762C2×C416yes—
92#14519760C2×C416yes—
93#14519757C2×C413yes—
94#14519756C2×C413yes—
95#14519755C2×C416yes—
96#14519753C2×C413yes—
97#14519753C2×C413yes—
98#14519752C2×C413yes—
99#14519751C2×C413yes—
100Q8×Dic61C2×C411yescharacteristic
101Q8×Dic61C2×C411yescharacteristic
102C4×Dic122C2×C413yes—
103C4×Dic122C2313yes—
104C4×Dic122C2313yes—
105#14519742C2×C413yes—
106C4×D244C2×C413yes—
107C4×D244C2313yes—
108C4×D244C2313yes—
109#14519739C2×C413yes—
110C4×C22⋊D61C2×C413yes—
111C4×C22⋊D61C2×C413yes—
112C4×C22⋊D61C2311yescharacteristic
113C4×C22⋊D61C2313yes—
114#14519736C2×C413yes—
115#14519735C2×C413yes—
116#14519734C2×C413yes—
117#14519734C2×C413yes—
118#14519734C2316yes—
119C4⋊C4×D61C2×C413yes—
120C42×D61C2313yes—
121C42×D61C2313yes—
122C4⋊#92418C2313yes—
123C4⋊#92418C2313yes—
124C4⋊#92418C2313yes—
125C4⋊#92418C2313yes—
126#14519706C2313yes—
127#14519706C2316yes—
128#14519706C2313yes—
129#14519706C2311yescharacteristic
130#14519706C2313yes—
131C22⋊#92418C2311yescharacteristic
132C22⋊#92418C2313yes—
133#14519689C2311yescharacteristic
134#14519689C2313yes—
135#14519689C2313yes—
136#14519689C2313yes—
137#14519689C2316yes—
138#14519687C2313yes—
139#14519687C2313yes—
140C4×#92418C2313yes—
141C4×#92418C2313yes—
142C4×#92418C2316yes—
143C8×Dic61C2311yescharacteristic
144C8×Dic61C2313yes—
145C8×Dic61C2316yes—
146C22×Q8×C61C2×C411yescharacteristic
147C2×M16×C61C2311yescharacteristic
148C2×M16×C61C2313yes—
149C4×Q8×C61C2×C413yes—
150C4×Q8×C61C2×C411yescharacteristic
151C4×Q8×C61C2311yescharacteristic
152C4×Q8×C61C2313yes—
153C2×C4⋊C4×C61C2×C413yes—
154C2×C42×C61C2313yes—
155C4⋊C8×C61C2313yes—
156C4⋊C8×C61C2313yes—
157C4⋊C8×C61C2316yes—
158#97×C61C2313yes—
159#97×C61C2313yes—
160C4×C8×C61C2313yes—
161C4×C8×C61C2313yes—
162C2×Q8×C61—24no—
163C2×Q8×C61C22×C413yes—
164C2×Q8×C61C22×C411yescharacteristic
165C22×C4×C61C22×C413yes—
166C22×C4×C61C22×C413yes—
167C22×C4×C61C2411yescharacteristic
168C2×C22⋊D61C22×C411yescharacteristic
169C2×C22⋊D61C22×C413yes—
170#11758226—24no—
171Q8×D61—24no—
172#11758222—46no—
173C2×D244C22×C413yes—
174C2×D244C22×C413yes—
175C2×C4×D61C22×C413yes—
176C2×C4×D61C22×C413yes—
177C2×C4×D61C22×C413yes—
178C2×C4×D61C22×C413yes—
179C2×C4×D61C2411yescharacteristic
180C2×C4×D61C2413yes—
181C2×Dic122C22×C413yes—
182C2×Dic122C22×C413yes—
183M16×C61—22no—
184M16×C61—43no—
185C2×C8×C61C2411yescharacteristic
186C2×C8×C61C2413yes—
187C2×C8×C61C2413yes—
188C2×C8×C61C2411yescharacteristic
189C4⋊C4×C61C22×C413yes—
190C4⋊C4×C61C22×C413yes—
191C4⋊C4×C61C22×C416yes—
192C42×C61C22×C413yes—
193C42×C61C22×C413yes—
194C42×C61C2413yes—
195C42×C61C2413yes—
196C22⋊Dic61C22×C411yescharacteristic
197C22⋊Dic61C22×C413yes—
198#11758198C22×C411yescharacteristic
199#11758198C22×C413yes—
200#11758198C22×C413yes—
201#11758198C22×C413yes—
202#11758198C22×C416yes—
203#11758198C22×C416yes—
204#11758198C22×C416yes—
205C4⋊Dic61C22×C413yes—
206C4⋊Dic61C22×C413yes—
207C4⋊Dic61C22×C413yes—
208C4⋊Dic61C22×C413yes—
209#11758196C22×C416yes—
210#11758196C22×C411yescharacteristic
211#11758196C22×C413yes—
212#11758196C22×C413yes—
213#11758196C22×C413yes—
214#11758196C22×C416yes—
215#11758196C22×C416yes—
216C4×Dic61C22×C413yes—
217C4×Dic61C22×C413yes—
218C4×Dic61C22×C413yes—
219C4×Dic61C22×C413yes—
220C4×Dic61C2411yescharacteristic
221C4×Dic61C2413yes—
222C2×#92418C2413yes—
223C2×#92418C2411yescharacteristic
224C2×#92418C2411yescharacteristic
225C2×#92418C2413yes—
226#11758189—22no—
227#11758189—43no—
228#11758189—46no—
229C8×D61—26no—
230C8×D61—43no—
231C23×C61C23×C411yescharacteristic
232C22×D61C23×C413yes—
233C22×D61C23×C411yescharacteristic
234Q8×C61—26no—
235Q8×C61#13114yes—
236C2×C4×C61—26no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—43no—
240C2×C4×C61C23×C413yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C23×C413yes—
243C2×C4×C61C23×C413yes—
244C2×C4×C61C23×C413yes—
245C2×C4×C61C2511yescharacteristic
246C22⋊D61—22no—
247C22⋊D61—43no—
248C2×Dic61C23×C411yescharacteristic
249C2×Dic61C23×C411yescharacteristic
250C2×Dic61C23×C413yes—
251C2×Dic61C23×C413yes—
252C2×Dic61C23×C413yes—
253C2×Dic61C23×C411yescharacteristic
254D244—26no—
255D244—43no—
256C4×D61—26no—
257C4×D61—26no—
258C4×D61—23no—
259C4×D61—21no—
260C4×D61—23no—
261C4×D61—21no—
262C4×D61—26no—
263C4×D61—43no—
264Dic122—26no—
265Dic122—26no—
266C8×C61—21no—
267C8×C61—23no—
268C8×C61—23no—
269C8×C612−1+412yes—
270#92418—23no—
271#92418—21no—
272#92418—21no—
273#92418—23no—
274#60345—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61C24×C411yescharacteristic
279C2×D61—23no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—23no—
283C2×D61—23no—
284C2×D61—21no—
285C4×C61—21no—
286C4×C61—23no—
287C4×C61—23no—
288C4×C61—23no—
289C4×C61C2×#13116yes—
290C4×C61C2×2−1+411yescharacteristic
291C4×C61#58511yescharacteristic
292Dic61—23no—
293Dic61—21no—
294Dic61—23no—
295Dic61—21no—
296Dic61—23no—
297Dic61#56812yes—
298#3482—611no—
299#2993—613no—
300#2993—613no—
301#2991—611no—
302#2991—613no—
303#2986—613no—
304#2982—611no—
305#2982—611no—
306#2969—613no—
307#2969—613no—
308#2965—613no—
309C8×#41—611no—
310Q8×M16—611no—
311Q8×M16D6111yescharacteristic, complemented
312#2960—613no—
313C2×C61—41no—
314C2×C61C4×2−1+411yescharacteristic
315C2×C61C22×#13111yescharacteristic
316C2×C61C2×#56811yescharacteristic
317D61—41no—
318D61—43no—
319#557—611no—
320#556—613no—
321#552—613no—
322#552—613no—
323D4×Q8—611no—
324#548—611no—
325#548—611no—
326#547—613no—
327#533—613no—
328#529—613no—
329#529—613no—
330C4×#41—611no—
331C4×#41—613no—
332C2×C4×Q8—611no—
333C2×C4×Q8C2×D6111yescharacteristic
334#446—613no—
335#446—613no—
336#446—613no—
337#446—613no—
338#446C2×D6113yes—
339#446C2×D6113yes—
340C8×Q8—613no—
341C8×Q8—611no—
342C8×Q8—611no—
343C8×Q8—613no—
344C8×Q8—613no—
345C8×Q8—616no—
346C8×Q8—613no—
347C8×Q8C2×D6111yescharacteristic
348C8×Q8C2×D6111yescharacteristic
349#436—611no—
350#436—613no—
351#436—616no—
352#436—613no—
353#436—616no—
354#436—613no—
355#435—613no—
356#435—613no—
357#435—613no—
358#435—613no—
359C8×D4—613no—
360C8×D4—613no—
361#433—616no—
362#433—616no—
363#431—613no—
364#431—613no—
365#431—613no—
366#431—613no—
367#431—613no—
368#431—613no—
369#424—616no—
370#423—613no—
371#423—613no—
372#423C2×D6113yes—
373#405—616no—
374C4×M16—613no—
375C4×M16C2×D6113yes—
376C2×C4×C8—613no—
377C61#6034511yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C2×#41—613no—
380C22×Q8—611no—
381C22×Q8C4×D6111yescharacteristic
382C2×M16—611no—
383C2×M16—613no—
384C2×M16—616no—
385C2×M16C22×D6111yescharacteristic
386C2×M16C22×D6113yes—
387C22×C8—613no—
388C22×C8—613no—
389C4⋊Q8—613no—
390C4⋊Q8—613no—
391#126—616no—
392#126—616no—
393#125—613no—
394#125—616no—
395#125—613no—
396#124—613no—
397#124—613no—
398#123—616no—
399#123—616no—
400C22⋊Q8—613no—
401C22⋊Q8—613no—
402C22⋊Q8—613no—
403C22⋊Q8—616no—
404C22⋊Q8—613no—
405#121—613no—
406#121—613no—
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—613no—
410C4×Q8—613no—
411C4×Q8—613no—
412C4×Q8—613no—
413C4×Q8C4×D6111yescharacteristic
414C4×Q8C4×D6113yes—
415C4×Q8C22×D6111yescharacteristic
416C4×Q8C22×D6113yes—
417C4×D4—613no—
418C4×D4—613no—
419C4×D4—611no—
420C4×D4—613no—
421C4×D4—613no—
422C4×D4—613no—
423C4×D4—613no—
424C4×D4—613no—
425#117—613no—
426#117—613no—
427#117—613no—
428#117—613no—
429#117—616no—
430C2×C4⋊C4—613no—
431C2×C4⋊C4C4×D6113yes—
432C2×C42—613no—
433C2×C42—613no—
434C2×C42C22×D6113yes—
435C4⋊C8—613no—
436C4⋊C8—616no—
437C4⋊C8—613no—
438C4⋊C8—611no—
439C4⋊C8—613no—
440C4⋊C8—613no—
441C4⋊C8—613no—
442C4⋊C8—613no—
443C4⋊C8—613no—
444C4⋊C8C22×D6113yes—
445C4⋊C8C22×D6113yes—
446C4⋊C8C22×D6116yes—
447C22⋊C8—611no—
448C22⋊C8—613no—
449C22⋊C8—611no—
450C22⋊C8—613no—
451C22⋊C8—616no—
452C22⋊C8—613no—
453C22⋊C8—613no—
454#97—613no—
455#97—613no—
456#97C22×D6113yes—
457#97C22×D6113yes—
458C4×C8—611no—
459C4×C8—613no—
460C4×C8—616no—
461C4×C8—613no—
462C4×C8—613no—
463C4×C8—616no—
464C4×C8C22×D6113yes—
465C4×C8C22×D6113yes—
466#41—1224no—
467#41—2446no—
468C2×Q8—24no—
469C2×Q8—613no—
470C2×Q8—613no—
471C2×Q8—1224no—
472C2×Q8C2×C4×D6113yes—
473C2×Q8C2×C4×D6111yescharacteristic
474C2×D4—611no—
475C2×D4—613no—
476C2×D4—613no—
477C2×D4—613no—
478C22×C4—613no—
479C22×C4—613no—
480C22×C4—613no—
481C22×C4—611no—
482C22×C4—613no—
483C22×C4—613no—
484C22×C4C2×C4×D6113yes—
485C22×C4C2×C4×D6113yes—
486C22×C4C23×D6111yescharacteristic
487M16—22no—
488M16—43no—
489M16—1222no—
490M16—2443no—
491M16—2446no—
492C2×C8—613no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—613no—
496C2×C8—1226no—
497C2×C8—2443no—
498C2×C8C23×D6113yes—
499C2×C8C23×D6113yes—
500C2×C8C23×D6111yescharacteristic
501C2×C8C23×D6111yescharacteristic
502C4⋊C4—616no—
503C4⋊C4—613no—
504C4⋊C4—611no—
505C4⋊C4—613no—
506C4⋊C4—613no—
507C4⋊C4—613no—
508C4⋊C4—616no—
509C4⋊C4—616no—
510C4⋊C4—613no—
511C4⋊C4—613no—
512C4⋊C4—613no—
513C4⋊C4C2×C4×D6113yes—
514C4⋊C4C2×C4×D6113yes—
515C4⋊C4C2×C4×D6116yes—
516C22⋊C4—611no—
517C22⋊C4—613no—
518C22⋊C4—611no—
519C22⋊C4—613no—
520C22⋊C4—616no—
521C22⋊C4—613no—
522C22⋊C4—613no—
523C22⋊C4—616no—
524C22⋊C4—616no—
525C42—613no—
526C42—613no—
527C42—613no—
528C42—613no—
529C42—611no—
530C42—613no—
531C42C2×C4×D6113yes—
532C42C2×C4×D6113yes—
533C42C23×D6113yes—
534C42C23×D6113yes—
535C23—613no—
536C23—611no—
537C23C22×C4×D6111yescharacteristic
538Q8—26no—
539Q8—1226no—
540Q8—1226no—
541Q8#1451978514yes—
542D4—1222no—
543D4—1226no—
544D4—2443no—
545D4—2443no—
546C2×C4—26no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—43no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—613no—
554C2×C4—613no—
555C2×C4—613no—
556C2×C4—1226no—
557C2×C4—1226no—
558C2×C4—1226no—
559C2×C4—1223no—
560C2×C4—1221no—
561C2×C4—1223no—
562C2×C4—1221no—
563C2×C4—2443no—
564C2×C4C22×C4×D6113yes—
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C22×C4×D6113yes—
567C2×C4C22×C4×D6113yes—
568C2×C4C22×C4×D6113yes—
569C2×C4C24×D6111yescharacteristic
570C8—21no—
571C8—23no—
572C8—23no—
573C8—1223no—
574C8—1221no—
575C8—1221no—
576C8—1223no—
577C8#1451982512yes—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—1221no—
582C22—1223no—
583C22—1223no—
584C22—1221no—
585C22—1223no—
586C22—1221no—
587C22C23×C4×D6111yescharacteristic
588C4—23no—
589C4—23no—
590C4—23no—
591C4—21no—
592C4—612no—
593C4—1221no—
594C4—1223no—
595C4—1223no—
596C4—1221no—
597C4—1223no—
598C4C2×#1451978516yes—
599C4C2×#1451982511yescharacteristic
600C4#1684059811yescharacteristic
601C2—41no—
602C2—2441no—
603C2—2443no—
604C2C4×#1451982511yescharacteristic
605C2C22×#1451978511yescharacteristic
606C2#568×D6111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D61
commutator, socleC22×C61C24×C4
FittingQ8×M16×C61C2
prime radical#21459436C2
cosocleC2×C4×C61C25
hypercenterQ8×M16D61
hypocenterC61#60345
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #21454971C2semidirect
3. #21454964C2semidirect
4. #21454950C2semidirect
5. #21454949C2semidirect
6. #21454917C2semidirect
7. #21454900C2semidirect
8. #21454898C2semidirect
9. #21454825C2semidirect
10. #21454823C2semidirect
11. #21454781C2semidirect
12. C8×#11758226C2semidirect
13. Q8×#11758189C2semidirect
14. #21454750C2semidirect
15. Q8×M16×C61C2semidirect
311. Q8×M16D61semidirect
377. C61#60345semidirect

Maximal subgroups

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

subgroupindex
#214594362
#214549712
#214549642
#214549502
#214549492
#214549172
#214549002
#214548982
#214548252
#214548232
#214547812
C8×#117582262
Q8×#117581892
#214547502
Q8×M16×C612
#6034561