Finite Group Databasebeta

← #29752732

Subgroups — #29752732

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450242C211yesmaximal, characteristic
2#21448820C211yesmaximal, characteristic
3#21447230C211yesmaximal, characteristic, complemented
4#21447196C211yesmaximal, characteristic, complemented
5#21447014C211yesmaximal, characteristic
6#21447013C211yesmaximal, characteristic
7#21446581C211yesmaximal, characteristic, complemented
8#21446544C211yesmaximal, characteristic, complemented
9#21446423C211yesmaximal, characteristic
10#21446405C211yesmaximal, characteristic, complemented
11#21446399C211yesmaximal, characteristic, complemented
12#21446010C211yesmaximal, characteristic
13#21445908C211yesmaximal, characteristic
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#1865C211yesmaximal, characteristic, complemented
16C2×#14519772—21no—
17C2×#14519753C2211yescharacteristic
18C2×C4⋊C4×D61C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28#16839704—21no—
29#16839691—21no—
30#16839669C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839657—21no—
34#16839646—21no—
35#16839643—21no—
36#16839633—21no—
37#16839631C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×#11758198—21no—
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519772—44no—
64#14519753—22no—
65#14519753—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C4⋊C4×D61—22no—
77C4⋊C4×D61—41no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198D411yescharacteristic
82C2×#11758198D411yescharacteristic
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86#14519709—22no—
87#14519708—22no—
88#14519708—41no—
89#14519706—22no—
90#14519701—22no—
91#14519701—22no—
92#14519689—22no—
93#14519689—41no—
94C4×#92418—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×Q8×C61—22no—
109C4×Q8×C61—41no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117#106×C61—22no—
118#103×C61—22no—
119#103×C61—41no—
120#95×C61—21no—
121C23×D61C2×D411yescharacteristic
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—81no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61#4112yes—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—42no—
168#11758198—24no—
169#11758198—41no—
170#11758198—42no—
171#11758198—42no—
172#11758198SD1614yes—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—42no—
181C4⋊Dic61#4112yes—
182#11758196—24no—
183#11758196—42no—
184#11758196—42no—
185#11758196—42no—
186#11758196—81no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—42no—
190C4×Dic61—81no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418#4112yes—
197C23×C61C22×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61C2×SD1612yes—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×C61—41no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C2×SD1612yes—
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C2×SD1612yes—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—84no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—22no—
262#92418—41no—
263#27034—611nomaximal, Sylow
264C22×C61#52511yescharacteristic
265C22×C61#54611yescharacteristic
266C22×C61#54711yescharacteristic
267C22×C61C22×SD1611yescharacteristic
268C22×C61#57511yescharacteristic
269C22×C61#57711yescharacteristic
270C22×C61#57811yescharacteristic
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61#52911yescharacteristic
285C4×C61#53911yescharacteristic
286C4×C61#55311yescharacteristic
287C4×C61#55611yescharacteristic
288Dic61—22no—
289Dic61—41no—
290Dic61—42no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295#2496—611no—
296#2360—611no—
297#2072—611no—
298#2066—611no—
299#2035—611no—
300#2035—611no—
301#1948—611no—
302#1941—611no—
303#1923—611no—
304#1920—611no—
305#1919—611no—
306#1865D6111yescharacteristic, complemented
307#1853—611no—
308#1824—611no—
309C4×#103—611no—
310C2×C61#246611yescharacteristic
311C2×C61#310311yescharacteristic
312C2×C61#312011yescharacteristic
313C2×C61#329811yescharacteristic
314C2×C61#331111yescharacteristic
315C2×C61#333511yescharacteristic
316C2×C61#335211yescharacteristic
317D61—42no—
318C2×#126—1221no—
319C2×C22⋊Q8—611no—
320C2×C4×Q8C2×D6111yescharacteristic
321C22×C4⋊C4—611no—
322C2×#106—611no—
323C2×#106—611no—
324C2×#106—611no—
325C2×#106C2×D6111yescharacteristic
326C2×C4⋊C8—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103C2×D6111yescharacteristic
330C2×#103C2×D6111yescharacteristic
331C2×C22⋊C8—611no—
332C2×C22⋊C8—611no—
333C2×C4×C8—611no—
334#400—1221no—
335#397—1221no—
336#391—611no—
337#389—611no—
338#389—611no—
339#389—1221no—
340#389C2×D6111yescharacteristic
341#387—1221no—
342#387—1221no—
343#385—1221no—
344#384—611no—
345#380—611no—
346#380—611no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349C4×C22⋊C4—611no—
350C4×C22⋊C4—1221no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340C2×D6111yescharacteristic
359#336—611no—
360#336C2×D6111yescharacteristic
361C61#2703411yescharacteristic, complemented, Sylow
362C22×Q8C22×D6111yescharacteristic
363C23×C4—611no—
364C23×C4—1221no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#126—2444no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C4×Q8—22no—
373C4×Q8—41no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—2441no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—611no—
392C2×C22⋊C4—611no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C42—21no—
397C2×C42—611no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#106—22no—
402#106—1222no—
403#106—1222no—
404#106—2441no—
405C4⋊C8—1222no—
406#103—22no—
407#103—41no—
408#103—1222no—
409#103—1222no—
410C22⋊C8—1222no—
411C22⋊C8—2441no—
412C4×C8—2441no—
413#95—21no—
414#95—611no—
415#95—611no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424C24—611no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#1175822212yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—4881no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—612no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—612no—
474C4⋊C4—1224no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—4881no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4#1175822412yes—
488C22⋊C4—614no—
489C22⋊C4—1224no—
490C22⋊C4—2441no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—1221no—
497C42—1221no—
498C42—4881no—
499C42—2442no—
500C23—612no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23C2×D4×D6111yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—612no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4884no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4SD16×D6112yes—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4C2×#1175822411yescharacteristic
559C2×C4C2×#1175822211yescharacteristic
560C2×C4#1451982211yescharacteristic
561C2×C4#1451982511yescharacteristic
562C8—41no—
563C8—41no—
564C8—1222no—
565C8—2441no—
566C22—1222no—
567C22—1222no—
568C22—1222no—
569C22—2441no—
570C22—2441no—
571C22—2441no—
572C22—2441no—
573C22#1684027111yescharacteristic
574C22#1684039911yescharacteristic
575C22#1684041411yescharacteristic
576C22C2×SD16×D6111yescharacteristic
577C22#1684055911yescharacteristic
578C22#1684057111yescharacteristic
579C22#1684057811yescharacteristic
580C4—22no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—1222no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C4—2441no—
591C4—2442no—
592C4—2441no—
593C4#1684029211yescharacteristic
594C4#1684035911yescharacteristic
595C4#1684045711yescharacteristic
596C4#1684048311yescharacteristic
597C2—2442no—
598C2#2144985711yescharacteristic
599C2#2145584811yescharacteristic
600C2#2145603111yescharacteristic
601C2#2145767711yescharacteristic
602C2#2145785911yescharacteristic
603C2#2145817911yescharacteristic
604C2#2145839111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1865D61
hypocenterC61#27034
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. #21447230C2semidirect
4. #21447196C2semidirect
7. #21446581C2semidirect
8. #21446544C2semidirect
10. #21446405C2semidirect
11. #21446399C2semidirect
14. C4×#14519701C2semidirect
15. C61×#1865C2semidirect
306. #1865D61semidirect
361. C61#27034semidirect

Maximal subgroups

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

subgroupindex
#214502422
#214488202
#214472302
#214471962
#214470142
#214470132
#214465812
#214465442
#214464232
#214464052
#214463992
#214460102
#214459082
C4×#145197012
C61×#18652
#2703461