Finite Group Databasebeta

← #29710881

Subgroups — #29710881

Order 15616.

730 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#21448560C211yesmaximal, characteristic, complemented
2#21448530C211yesmaximal, characteristic
3#21447383C211yesmaximal, characteristic, complemented
4#21447378C211yesmaximal, characteristic, complemented
5#21447229C211yesmaximal, characteristic, complemented
6#21447224C211yesmaximal, characteristic
7#21447061C211yesmaximal, characteristic
8#21447061C211yesmaximal, characteristic
9#21446971C211yesmaximal, characteristic, complemented
10#21446966C211yesmaximal, characteristic
11#21445650C211yesmaximal, characteristic, complemented
12#21445645C211yesmaximal, characteristic
13C4×#14519702C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1746C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28#16839668C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839659—21no—
33#16839659—21no—
34#16839654—21no—
35#16839654—21no—
36#16839649—21no—
37#16839649—21no—
38#16839636—21no—
39#16839636—21no—
40#16839632C2211yescharacteristic
41#16839632C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×#97×C61C2211yescharacteristic
53C2×C4×C8×C61C2211yescharacteristic
54C61×#376C2211yescharacteristic
55C61×#336C2211yescharacteristic
56C61×#336C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62#14519769—22no—
63#14519769—22no—
64#14519765—22no—
65#14519765—22no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68C4×D244—22no—
69C4×D244—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93#14519702—22no—
94#14519702—22no—
95#14519702—22no—
96#14519702—22no—
97#14519699—22no—
98#14519699—22no—
99#14519699—22no—
100#14519699—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#97×C61—22no—
115C4×C8×C61—22no—
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C23×D61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244#4112yes—
141C2×C4×D61—42no—
142C2×C4×D61—81no—
143C2×C4×D61—81no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122#4112yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61#4112yes—
167C42×C61#4112yes—
168#11758198—24no—
169#11758198—24no—
170#11758198—42no—
171#11758198—81no—
172#11758198—81no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—81no—
184C4⋊Dic61—81no—
185C4⋊Dic61—81no—
186C4⋊Dic61—81no—
187C4⋊Dic61#4112yes—
188C4⋊Dic61#4112yes—
189#11758196—24no—
190#11758196—24no—
191#11758196—42no—
192#11758196—81no—
193#11758196—81no—
194C4×Dic61—42no—
195C4×Dic61—81no—
196C4×Dic61—81no—
197C23×C61C22×D411yescharacteristic
198C22×D61—22no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61#12412yes—
222C2×C4×C61#12412yes—
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250D244—22no—
251D244—22no—
252D244—41no—
253D244—41no—
254C4×D61—84no—
255Dic122—22no—
256Dic122—22no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—41no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61—41no—
263#23832—611nomaximal, Sylow
264C22×C61#52511yescharacteristic
265C22×C61#52511yescharacteristic
266C22×C61C2×#12411yescharacteristic
267C22×C61#57611yescharacteristic
268C22×C61#57711yescharacteristic
269C22×C61#57711yescharacteristic
270C22×C61#57811yescharacteristic
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61#52911yescharacteristic
285C4×C61#52911yescharacteristic
286C4×C61#52911yescharacteristic
287C4×C61#52911yescharacteristic
288Dic61—42no—
289Dic61—42no—
290Dic61—42no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295#2330—611no—
296#2326—611no—
297#2100—611no—
298#2099—611no—
299#2072—611no—
300#2071—611no—
301#2042—611no—
302#2042—611no—
303#2026—611no—
304#2025—611no—
305#1763—611no—
306#1762—611no—
307C4×#103—611no—
308C4×#102—611no—
309#1746D6111yescharacteristic, complemented
310C2×C61#239811yescharacteristic
311C2×C61#311911yescharacteristic
312C2×C61#312011yescharacteristic
313C2×C61#312411yescharacteristic
314C2×C61#312511yescharacteristic
315C2×C61#313611yescharacteristic
316C2×C61#313711yescharacteristic
317D61—82no—
318C2×#125—611no—
319C2×#124—611no—
320C2×C4×Q8—611no—
321C2×C4×D4—611no—
322C2×#103—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102—611no—
330C2×#97C2×D6111yescharacteristic
331C2×C4×C8C2×D6111yescharacteristic
332#391—611no—
333#390—611no—
334#389—611no—
335#389—611no—
336#389—1221no—
337#389—1221no—
338#388—1221no—
339#388—1221no—
340#387—1221no—
341#387—1221no—
342#385—1221no—
343#385—1221no—
344#384—611no—
345#384—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348#376C2×D6111yescharacteristic
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#336C2×D6111yescharacteristic
358#336C2×D6111yescharacteristic
359#336C2×D6111yescharacteristic
360#336C2×D6111yescharacteristic
361C61#2383211yescharacteristic, complemented, Sylow
362C22×Q8—611no—
363C22×D4—611no—
364C23×C4—1221no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#125—1222no—
370#125—1222no—
371#124—1222no—
372#124—1222no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×D4—1222no—
376C4×D4—1222no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401C2×C42C22×D6111yescharacteristic
402C2×C42C22×D6111yescharacteristic
403#103—1222no—
404#103—1222no—
405#103—1222no—
406#103—1222no—
407#102—1222no—
408#102—1222no—
409#102—1222no—
410#102—1222no—
411#97—22no—
412C4×C8—22no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95D24411yescharacteristic
420#95D24411yescharacteristic
421#95D24411yescharacteristic
422#95D24411yescharacteristic
423C24—1221no—
424C2×Q8—612no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×D4—612no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—4881no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4C2×D24411yescharacteristic
447C22×C4C2×D24411yescharacteristic
448C22×C4C2×D24411yescharacteristic
449C22×C4C2×D24411yescharacteristic
450C22×C4C2×D24411yescharacteristic
451C22×C4C2×D24411yescharacteristic
452C22×C4C23×D6111yescharacteristic
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—4881no—
480C4⋊C4—4881no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—4881no—
484C4⋊C4—4881no—
485C4⋊C4—2442no—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—4881no—
489C22⋊C4—4881no—
490C22⋊C4—2442no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—4881no—
496C42—4881no—
497C42—2442no—
498C42#1175822212yes—
499C42#1175822212yes—
500C23—1222no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23C22×D24411yescharacteristic
508Q8—1222no—
509Q8—1222no—
510Q8—2441no—
511Q8—2441no—
512D4—1222no—
513D4—1222no—
514D4—2441no—
515D4—2441no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—4884no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
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#1451976512yes—
555C2×C4#1451976512yes—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4#41×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4C2×#1175822211yescharacteristic
561C2×C4C2×#1175822211yescharacteristic
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22#1684027411yescharacteristic
574C22#1684027411yescharacteristic
575C22C2×#1451976511yescharacteristic
576C22#1684056711yescharacteristic
577C22#1684057411yescharacteristic
578C22#1684057411yescharacteristic
579C22#1684057911yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—4881no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C4—2442no—
591C4—2442no—
592C4—2442no—
593C4#1684029011yescharacteristic
594C4#1684029011yescharacteristic
595C4#1684029011yescharacteristic
596C4#1684029011yescharacteristic
597C2—4882no—
598C2#2144910511yescharacteristic
599C2#2145602011yescharacteristic
600C2#2145603511yescharacteristic
601C2#2145608311yescharacteristic
602C2#2145609811yescharacteristic
603C2#2145617511yescharacteristic
604C2#2145618611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1746C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1746D61
hypocenterC61#23832
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
1. #21448560C2semidirect
3. #21447383C2semidirect
4. #21447378C2semidirect
5. #21447229C2semidirect
9. #21446971C2semidirect
11. #21445650C2semidirect
13. C4×#14519702C2semidirect
15. C61×#1746C2semidirect
309. #1746D61semidirect
361. C61#23832semidirect

Maximal subgroups

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

subgroupindex
#214485602
#214485302
#214473832
#214473782
#214472292
#214472242
#214470612
#214469712
#214469662
#214456502
#214456452
C4×#145197022
C4×#145196992
C61×#17462
#2383261