Finite Group Databasebeta

← #29710877

Subgroups — #29710877

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#21448530C211yesmaximal, characteristic
2#21447382C211yesmaximal, characteristic, complemented
3#21447377C211yesmaximal, characteristic, complemented
4#21447228C211yesmaximal, characteristic, complemented
5#21447223C211yesmaximal, characteristic
6#21447060C211yesmaximal, characteristic
7#21447060C211yesmaximal, characteristic
8#21446967C211yesmaximal, characteristic, complemented
9#21446962C211yesmaximal, characteristic
10#21445652C211yesmaximal, characteristic, complemented
11#21445647C211yesmaximal, characteristic
12C4×#14519700C211yesmaximal, characteristic, complemented
13C4×#14519697C211yesmaximal, characteristic
14#21445571C211yesmaximal, characteristic, complemented
15C61×#2330C211yesmaximal, characteristic, complemented
16C2×#14519765C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519700C2211yescharacteristic
20C2×#14519700C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839668C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839654—21no—
32#16839654—21no—
33#16839636—21no—
34#16839636—21no—
35#16839632C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#125×C61C2211yescharacteristic
49C61×#389—21no—
50C61×#389—21no—
51C61×#389C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#387—21no—
54C61×#387—21no—
55C61×#384C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#376C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519765—22no—
67#14519765—22no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×D244—22no—
71C4×D244—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83#14519700—22no—
84#14519700—22no—
85#14519700—22no—
86#14519700—22no—
87#14519697—22no—
88#14519697—22no—
89#14519697—22no—
90#14519697—22no—
91#14519688—22no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—21no—
95#125×C61—22no—
96#125×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—21no—
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C23×D61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
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—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—81no—
159C4⋊C4×C61—81no—
160C4⋊C4×C61—81no—
161C4⋊C4×C61—81no—
162C4⋊C4×C61#4112yes—
163C4⋊C4×C61#4112yes—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—81no—
169C42×C61—81no—
170C42×C61#4112yes—
171C42×C61#4112yes—
172#11758198—24no—
173#11758198—24no—
174#11758198—42no—
175#11758198—81no—
176#11758198—81no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—81no—
182C4⋊Dic61—81no—
183#11758196—42no—
184C4×Dic61—42no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
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—22no—
214C2×C4×C61—22no—
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—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61#12412yes—
234C2×C4×C61#12412yes—
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—22no—
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—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—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—42no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61#52911yescharacteristic
289C4×C61#52911yescharacteristic
290C4×C61#52911yescharacteristic
291C4×C61#52911yescharacteristic
292Dic61—42no—
293Dic61—81no—
294Dic61—81no—
295#2330D6111yescharacteristic, complemented
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#1746—611no—
310C2×C61#239811yescharacteristic
311C2×C61#311911yescharacteristic
312C2×C61#312011yescharacteristic
313C2×C61#312411yescharacteristic
314C2×C61#312511yescharacteristic
315C2×C61#313611yescharacteristic
316C2×C61#313711yescharacteristic
317D61—82no—
318C2×#125C2×D6111yescharacteristic
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×#97—611no—
331C2×C4×C8—611no—
332#391—611no—
333#390—611no—
334#389—21no—
335#389—21no—
336#389C2×D6111yescharacteristic
337#389C2×D6111yescharacteristic
338#388—1221no—
339#388—1221no—
340#387—21no—
341#387—21no—
342#385—1221no—
343#385—1221no—
344#384—611no—
345#384C2×D6111yescharacteristic
346C4×C4⋊C4C2×D6111yescharacteristic
347C4×C4⋊C4C2×D6111yescharacteristic
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#336—611no—
358#336—611no—
359#336—611no—
360#336—611no—
361C61#2383211yescharacteristic, complemented, Sylow
362C22×Q8—611no—
363C22×D4—611no—
364C23×C4—1221no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369#125—22no—
370#125—22no—
371#124—1222no—
372#124—1222no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×D4—1222no—
376C4×D4—1222no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—21no—
398C2×C42—21no—
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—1222no—
412C4×C8—1222no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—1221no—
418#95—1221no—
419#95C22⋊D6111yescharacteristic
420#95C22⋊D6111yescharacteristic
421#95C22⋊D6111yescharacteristic
422#95C22⋊D6111yescharacteristic
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—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—4881no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C23×D6111yescharacteristic
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—2442no—
484C4⋊C4#1175822212yes—
485C4⋊C4#1175822212yes—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—4881no—
489C22⋊C4—4881no—
490C22⋊C4—2442no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—81no—
496C42—81no—
497C42—2442no—
498C42#1175822612yes—
499C42#1175822612yes—
500C23—1222no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23C22×C22⋊D6111yescharacteristic
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—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—1222no—
545C2×C4—4884no—
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#1451976612yes—
555C2×C4#1451976612yes—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4C2×#1175822211yescharacteristic
559C2×C4C2×#1175822211yescharacteristic
560C2×C4C2×#1175822611yescharacteristic
561C2×C4C2×#1175822611yescharacteristic
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22#1684026911yescharacteristic
574C22#1684027311yescharacteristic
575C22C2×#1451976611yescharacteristic
576C22#1684056511yescharacteristic
577C22#1684057011yescharacteristic
578C22#1684057311yescharacteristic
579C22#1684057711yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—4881no—
591C4—4881no—
592C4—2442no—
593C4#1684029911yescharacteristic
594C4#1684029911yescharacteristic
595C4#1684029911yescharacteristic
596C4#1684029911yescharacteristic
597C2—4882no—
598C2#2144910211yescharacteristic
599C2#2145601211yescharacteristic
600C2#2145602711yescharacteristic
601C2#2145608111yescharacteristic
602C2#2145609611yescharacteristic
603C2#2145617811yescharacteristic
604C2#2145618911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2330C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2330D61
hypocenterC61#23832
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #21447382C2semidirect
3. #21447377C2semidirect
4. #21447228C2semidirect
8. #21446967C2semidirect
10. #21445652C2semidirect
12. C4×#14519700C2semidirect
14. #21445571C2semidirect
15. C61×#2330C2semidirect
295. #2330D61semidirect
361. C61#23832semidirect

Maximal subgroups

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

subgroupindex
#214485302
#214473822
#214473772
#214472282
#214472232
#214470602
#214469672
#214469622
#214456522
#214456472
C4×#145197002
C4×#145196972
#214455712
C61×#23302
#2383261