Finite Group Databasebeta

← #30571159

Subgroups — #30571159

Order 15616.

1698 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2C2×#16840491C211yesmaximal, characteristic, complemented
3#21458874C214yesmaximal, complemented
4#21458852C214yesmaximal, complemented
5#21458624C214yesmaximal, complemented
6#21458602C214yesmaximal, complemented
7#21456848C214yesmaximal, complemented
8#21456769C211yesmaximal, characteristic, complemented
9#21456746C211yesmaximal, characteristic
10#21455992C214yesmaximal, complemented
11#21455859C211yesmaximal, characteristic, complemented
12#21455851C211yesmaximal, characteristic
13C61×#2983C211yesmaximal, characteristic, complemented
14#16840497C2212yes—
15#16840491—22no—
16#16840491—22no—
17#16840491C2214yes—
18#16840491C2214yes—
19#16840490C2212yes—
20#16840485C2212yes—
21#16840484—22no—
22#16840472C2212yes—
23#16840453—22no—
24#16840359—22no—
25#16840336—21no—
26#16840319C2211yescharacteristic
27#16840312C2211yescharacteristic
28C2×#14519777C2211yescharacteristic
29C2×#14519777C2212yes—
30#16840296C2214yes—
31C2×#14519768—21no—
32C2×#14519768—22no—
33#16840243—21no—
34C2×Q8×Dic61—21no—
35C2×C4×Dic122C2212yes—
36C2×C4×Dic122C2211yescharacteristic
37#16840156C2218yes—
38#16840142C2212yes—
39#16840135C2212yes—
40#16840131C2218yes—
41#16840113C2212yes—
42#16840113C2212yes—
43#16840098C2212yes—
44#16840081C2212yes—
45#16840070C2218yes—
46#16840058C2212yes—
47#16840048C2212yes—
48#16839985C2218yes—
49#16839973C2214yes—
50#16839973C2212yes—
51#16839973C2218yes—
52#16839968C2214yes—
53#16839968C2218yes—
54#16839963C2212yes—
55#16839920C2214yes—
56#16839920C2212yes—
57#16839903C2214yes—
58#16839903C2212yes—
59#16839870C2218yes—
60#16839864C2218yes—
61C4×Dic244C2214yes—
62C4×#11758190C2214yes—
63C2×C4×Q8×C61C2211yescharacteristic
64C61×#446C2214yes—
65C61×#446C2214yes—
66C61×#432C2214yes—
67C61×#431C2211yescharacteristic
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C22×Dic122C2312yes—
71C22×Dic122C2311yescharacteristic
72C2×Dic244C2314yes—
73C2×Dic244C2318yes—
74C2×#11758190C2314yes—
75C2×#11758190C2318yes—
76#14519777—24no—
77#14519777—21no—
78#14519777—21no—
79#14519777C2312yes—
80#14519777C2318yes—
81#14519777C2311yescharacteristic
82#14519777C2312yes—
83#14519777C2311yescharacteristic
84#14519777C2311yescharacteristic
85#14519775—22no—
86#14519774C2311yescharacteristic
87#14519772—24no—
88#14519771—24no—
89#14519769—21no—
90#14519769C2314yes—
91#14519768—24no—
92#14519768—24no—
93#14519768—28no—
94#14519768—28no—
95#14519767—24no—
96#14519766—22no—
97#14519765—21no—
98#14519765C2314yes—
99#14519762—24no—
100#14519760—24no—
101#14519756—21no—
102#14519756C2312yes—
103#14519755—24no—
104#14519753—22no—
105#14519753—22no—
106#14519751—21no—
107#14519751C2312yes—
108Q8×Dic61—24no—
109Q8×Dic61—24no—
110C4×Dic122—24no—
111C4×Dic122—21no—
112C4×Dic122—21no—
113C4×Dic122—21no—
114C4×Dic122C2312yes—
115C4×Dic122C2312yes—
116C4×Dic122C2318yes—
117C4×Dic122C2314yes—
118C4×Dic122C2312yes—
119#14519742—22no—
120C4×D244—21no—
121C4×D244C2312yes—
122C4×D244C2314yes—
123#14519739—22no—
124C4×C22⋊D61—24no—
125#14519736—21no—
126#14519735—22no—
127#14519734—24no—
128C2×C4⋊Dic61—22no—
129C2×C4⋊Dic61—21no—
130C2×C4⋊Dic61—22no—
131C2×C4⋊Dic61—21no—
132C2×C4⋊Dic61C2311yescharacteristic
133C2×#11758196—22no—
134C2×#11758196—21no—
135C4⋊C4×D61—22no—
136C2×C4×Dic61—22no—
137C2×C4×Dic61—21no—
138#14519711C2314yes—
139#14519709C2314yes—
140#14519702C2314yes—
141#14519702C2318yes—
142#14519702C2318yes—
143#14519702C2314yes—
144#14519702C2314yes—
145#14519699C2314yes—
146C22×Q8×C61D411yescharacteristic
147C4×Q8×C61D412yes—
148C4×Q8×C61D412yes—
149C4×Q8×C61C2312yes—
150C4×Q8×C61C2312yes—
151C2×C4⋊C4×C61D411yescharacteristic
152C2×C4⋊C4×C61D412yes—
153C2×C42×C61C2312yes—
154C2×C42×C61C2311yescharacteristic
155C4⋊C8×C61C2312yes—
156C4⋊C8×C61C2312yes—
157C4⋊C8×C61C2318yes—
158C22⋊C8×C61C2312yes—
159C22⋊C8×C61C2312yes—
160#97×C61C2318yes—
161C4×C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C2×Q8×C61—24no—
164C2×Q8×C61C2×D412yes—
165C2×Q8×C61C2×D412yes—
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2411yescharacteristic
171C2×C22⋊D61—22no—
172C22×Dic61—22no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175#11758222—44no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244C2412yes—
179C2×C4×D61—22no—
180C2×C4×D61—24no—
181C2×Dic122—22no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—42no—
186C2×Dic122—44no—
187C2×Dic122C2412yes—
188C2×Dic122C2414yes—
189C2×Dic122C2412yes—
190C2×C8×C61C2414yes—
191C2×C8×C61C2414yes—
192C4⋊C4×C61C2×D412yes—
193C4⋊C4×C61C2×D412yes—
194C4⋊C4×C61C2×D414yes—
195C4⋊C4×C61C2×D414yes—
196C42×C61C2×D414yes—
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2411yescharacteristic
200C42×C61C2411yescharacteristic
201C42×C61C2414yes—
202C22⋊Dic61—22no—
203#11758198—22no—
204#11758198—24no—
205#11758198—24no—
206#11758198—22no—
207#11758198—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—28no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61—28no—
213C4⋊Dic61—22no—
214C4⋊Dic61—22no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61C2412yes—
220C4⋊Dic61C2412yes—
221C4⋊Dic61C2412yes—
222#11758196—24no—
223#11758196—28no—
224#11758196—22no—
225#11758196—22no—
226#11758196—22no—
227#11758196—24no—
228#11758196—24no—
229#11758196—22no—
230C4×Dic61—28no—
231C4×Dic61—22no—
232C4×Dic61—22no—
233C4×Dic61—24no—
234C4×Dic61—22no—
235Dic244—44no—
236Dic244—48no—
237#11758190—48no—
238#11758190—44no—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241Q8×C61—24no—
242Q8×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61#13714yes—
248C2×C4×C61C22×D414yes—
249C2×C4×C61C22×D414yes—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2511yescharacteristic
256C22⋊D61—44no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—42no—
264C2×Dic61—44no—
265C2×Dic61—42no—
266D244—24no—
267D244—42no—
268D244—81no—
269C4×D61—44no—
270C4×D61—84no—
271Dic122—22no—
272Dic122—28no—
273Dic122—24no—
274Dic122—22no—
275Dic122—42no—
276Dic122—44no—
277Dic122—42no—
278Dic122—81no—
279C8×C61—24no—
280C8×C61—24no—
281#61893—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61C2×#13712yes—
285C22×C61C23×D411yescharacteristic
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C4×C61—21no—
290C4×C61—24no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—24no—
294C4×C61C2×#13712yes—
295C4×C61C2×#13712yes—
296C4×C61#58512yes—
297Dic61—42no—
298Dic61—42no—
299Dic61—42no—
300Dic61—44no—
301Dic61—42no—
302Dic61—42no—
303#3507—611no—
304C2×#557—611no—
305#3400—614no—
306#3398—614no—
307#3374—614no—
308#3372—614no—
309#3218—614no—
310#3209—611no—
311#3206—611no—
312#3117—614no—
313#3104—611no—
314#3103—611no—
315#2983D6111yescharacteristic, complemented
316C2×C61—22no—
317C2×C61#348711yescharacteristic
318C2×C61C22×#13711yescharacteristic
319C2×C61#358611yescharacteristic
320D61—82no—
321#559—612no—
322#557—612no—
323#557—614no—
324#557—614no—
325#557—1222no—
326#557—1222no—
327#556—612no—
328#556—1222no—
329#554—612no—
330#552—1222no—
331#539—1222no—
332#536—1221no—
333#533—611no—
334#532—611no—
335C2×C4⋊Q8—611no—
336C2×C4⋊Q8—612no—
337#529—614no—
338C2×#125—1221no—
339C2×#125—1222no—
340#520—1221no—
341C2×C4×Q8—611no—
342C2×C4×Q8—612no—
343C2×C4×Q8—1221no—
344C2×C4×Q8C2×D6111yescharacteristic
345#497—618no—
346#494—612no—
347#492—612no—
348#491—618no—
349#487—612no—
350#487—612no—
351#484—612no—
352#481—612no—
353#479—618no—
354#477—612no—
355#475—612no—
356#464—618no—
357#462—612no—
358#462—618no—
359#462—614no—
360#461—614no—
361#461—618no—
362#460—612no—
363#451—612no—
364#451—614no—
365#448—612no—
366#448—614no—
367#446C2×D6114yes—
368#446C2×D6114yes—
369#441—618no—
370#440—618no—
371C4×Q16—614no—
372C4×SD16—614no—
373#432C2×D6114yes—
374#431C2×D6111yescharacteristic
375#431C2×D6111yescharacteristic
376C61#6189311yescharacteristic, complemented, Sylow
377C2×#41—1221no—
378C22×Q8—611no—
379C22×Q8—612no—
380C22×Q8D24411yescharacteristic
381C2×Q16—614no—
382C2×Q16—618no—
383C2×SD16—614no—
384C2×SD16—618no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387C4⋊Q8—612no—
388C4⋊Q8—618no—
389C4⋊Q8—612no—
390C4⋊Q8—611no—
391C4⋊Q8—1222no—
392C4⋊Q8—1224no—
393C4⋊Q8—1221no—
394C4⋊Q8—1221no—
395#127—611no—
396#126—1224no—
397#126—1224no—
398#125—614no—
399#125—1224no—
400#125—1224no—
401#125—1228no—
402#125—1228no—
403#125—1224no—
404#125—1221no—
405#124—614no—
406#124—1222no—
407#124—1221no—
408#123—1224no—
409#123—1224no—
410C22⋊Q8—612no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1221no—
414C22⋊Q8—1224no—
415#121—612no—
416#121—1221no—
417C4×Q8—612no—
418C4×Q8—612no—
419C4×Q8—618no—
420C4×Q8—614no—
421C4×Q8—612no—
422C4×Q8—1224no—
423C4×Q8—1224no—
424C4×Q8—1222no—
425C4×Q8—1224no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×Q8—1221no—
429C4×Q8D24412yes—
430C4×Q8D24412yes—
431C4×Q8C22×D6112yes—
432C4×Q8C22×D6112yes—
433C4×D4—614no—
434C4×D4—612no—
435C4×D4—1224no—
436C4×D4—1222no—
437C4×D4—1221no—
438#117—1221no—
439#117—1222no—
440#117—1224no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4D24411yescharacteristic
450C2×C4⋊C4D24412yes—
451C2×C42—1222no—
452C2×C42—1221no—
453C2×C42C22×D6112yes—
454C2×C42C22×D6111yescharacteristic
455#107—614no—
456#106—614no—
457C4⋊C8C22×D6112yes—
458C4⋊C8C22×D6112yes—
459C4⋊C8C22×D6118yes—
460#103—614no—
461#103—614no—
462#103—614no—
463#103—618no—
464#103—618no—
465#102—614no—
466C22⋊C8C22×D6112yes—
467C22⋊C8C22×D6112yes—
468#97C22×D6118yes—
469C4×C8C22×D6112yes—
470C4×C8C22×D6112yes—
471#41—2444no—
472C2×Q8—24no—
473C2×Q8—612no—
474C2×Q8—614no—
475C2×Q8—612no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1222no—
479C2×Q8—1221no—
480C2×Q8—2442no—
481C2×Q8—2444no—
482C2×Q8C2×D24412yes—
483C2×Q8C2×D24412yes—
484C2×D4—612no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4—1221no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1224no—
492C22×C4—1222no—
493C22×C4C2×D24412yes—
494C22×C4C2×D24412yes—
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C23×D6111yescharacteristic
498Q16—2444no—
499Q16—2448no—
500SD16—2448no—
501SD16—2444no—
502C2×C8C23×D6114yes—
503C2×C8C23×D6114yes—
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—612no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1224no—
510C4⋊C4—1228no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1228no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1228no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1224no—
521C4⋊C4—1222no—
522C4⋊C4—1224no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4C2×D24412yes—
527C4⋊C4C2×D24412yes—
528C4⋊C4C2×D24414yes—
529C4⋊C4C2×D24414yes—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1224no—
533C22⋊C4—1224no—
534C22⋊C4—1222no—
535C22⋊C4—1224no—
536C42—1228no—
537C42—1222no—
538C42—1222no—
539C42—1224no—
540C42—1222no—
541C42C2×D24414yes—
542C42C2×D24411yescharacteristic
543C42C2×D24411yescharacteristic
544C42C23×D6111yescharacteristic
545C42C23×D6111yescharacteristic
546C42C23×D6114yes—
547C23—1222no—
548C23C22×D24411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—1222no—
552Q8—1228no—
553Q8—1224no—
554Q8—1222no—
555Q8—4881no—
556Q8—2442no—
557Q8—2444no—
558Q8—2442no—
559D4—1224no—
560D4—4881no—
561D4—2444no—
562D4—2442no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1224no—
572C2×C4—1222no—
573C2×C4—4884no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4#1451981014yes—
579C2×C4C22×D24414yes—
580C2×C4C22×D24414yes—
581C2×C4C22×D24411yescharacteristic
582C2×C4C22×D24411yescharacteristic
583C2×C4C22×D24411yescharacteristic
584C2×C4C22×D24411yescharacteristic
585C2×C4C22×D24411yescharacteristic
586C2×C4C24×D6111yescharacteristic
587C8—24no—
588C8—24no—
589C22—21no—
590C22—21no—
591C22—2442no—
592C22—2442no—
593C22—2442no—
594C22C2×#1451981012yes—
595C22C23×D24411yescharacteristic
596C4—21no—
597C4—24no—
598C4—24no—
599C4—21no—
600C4—21no—
601C4—2442no—
602C4—2444no—
603C4—2442no—
604C4—2442no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451981012yes—
608C4C2×#1451981012yes—
609C4#1684059912yes—
610C2—22no—
611C2—4882no—
612C2#2145947811yescharacteristic
613C2C22×#1451981011yescharacteristic
614C2#2146060711yescharacteristic
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2983C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenter#2983D61
hypocenterC61#61893
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. C2×#16840491C2semidirect
3. #21458874C2semidirect
4. #21458852C2semidirect
5. #21458624C2semidirect
6. #21458602C2semidirect
7. #21456848C2semidirect
8. #21456769C2semidirect
10. #21455992C2semidirect
11. #21455859C2semidirect
13. C61×#2983C2semidirect
315. #2983D61semidirect
376. C61#61893semidirect

Maximal subgroups

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

subgroupindex
#214596842
C2×#168404912
#214588742
#214588522
#214586242
#214586022
#214568482
#214567692
#214567462
#214559922
#214558592
#214558512
C61×#29832
#6189361