Finite Group Databasebeta

← #29710879

Subgroups — #29710879

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#21447381C211yesmaximal, characteristic, complemented
3#21447376C211yesmaximal, characteristic, complemented
4#21447227C211yesmaximal, characteristic, complemented
5#21447222C211yesmaximal, characteristic
6#21447055C211yesmaximal, characteristic
7#21447055C211yesmaximal, characteristic
8#21446970C211yesmaximal, characteristic, complemented
9#21446965C211yesmaximal, characteristic
10#21445649C211yesmaximal, characteristic, complemented
11#21445644C211yesmaximal, characteristic
12C4×#14519701C211yesmaximal, characteristic, complemented
13C4×#14519698C211yesmaximal, characteristic
14#21445571C211yesmaximal, characteristic, complemented
15C61×#2326C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519701C2211yescharacteristic
18C2×#14519701C2211yescharacteristic
19C2×#14519701C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839660C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839659—21no—
30#16839659—21no—
31#16839649—21no—
32#16839649—21no—
33#16839632C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C4⋊Dic61C2211yescharacteristic
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×#124×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#390C2211yescharacteristic
53C61×#388—21no—
54C61×#388—21no—
55C61×#385—21no—
56C61×#385—21no—
57C61×#384C2211yescharacteristic
58C61×#376C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519769—22no—
64#14519769—22no—
65C2×C4⋊Dic61—21no—
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×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77#14519701—22no—
78#14519701—22no—
79#14519701—22no—
80#14519701—22no—
81#14519698—22no—
82#14519698—22no—
83#14519698—22no—
84#14519698—22no—
85#14519688—22no—
86C4×#92418—22no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94#124×C61—22no—
95#124×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—21no—
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C24×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—81no—
138C22×C4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—81no—
156C4⋊C4×C61—81no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—81no—
161C22⋊C4×C61—81no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—42no—
167C42×C61#4112yes—
168C42×C61#4112yes—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—81no—
176C4⋊Dic61—81no—
177C4⋊Dic61#4112yes—
178C4⋊Dic61#4112yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—81no—
182#11758196—81no—
183C4×Dic61—81no—
184C4×Dic61—81no—
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×C61—22no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61C22×D411yescharacteristic
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×C61—41no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
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—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—84no—
239C2×C4×C61#12412yes—
240C2×C4×C61#12412yes—
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#23832—611nomaximal, Sylow
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61#52511yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61C2×#12411yescharacteristic
274C22×C61#57611yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57811yescharacteristic
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61#52911yescharacteristic
288C4×C61#52911yescharacteristic
289C4×C61#52911yescharacteristic
290C4×C61#52911yescharacteristic
291Dic61—42no—
292Dic61—42no—
293Dic61—81no—
294Dic61—81no—
295#2330—611no—
296#2326D6111yescharacteristic
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—82no—
311C2×C61#239811yescharacteristic
312C2×C61#311911yescharacteristic
313C2×C61#312011yescharacteristic
314C2×C61#312411yescharacteristic
315C2×C61#312511yescharacteristic
316C2×C61#313611yescharacteristic
317C2×C61#313711yescharacteristic
318C2×#125—611no—
319C2×#124C2×D6111yescharacteristic
320C2×C4×Q8C2×D6111yescharacteristic
321C2×C4×D4C2×D6111yescharacteristic
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#391C2×D6111yescharacteristic
333#390C2×D6111yescharacteristic
334#389—611no—
335#389—611no—
336#389—1221no—
337#389—1221no—
338#388—21no—
339#388—21no—
340#387—1221no—
341#387—1221no—
342#385—21no—
343#385—21no—
344#384—611no—
345#384C2×D6111yescharacteristic
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#336—611no—
358#336—611no—
359#336—611no—
360#336—611no—
361C61#2383211yescharacteristic, complemented, Sylow
362C22×Q8C22×D6111yescharacteristic
363C22×D4C22×D6111yescharacteristic
364C23×C4—21no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369#125—1222no—
370#125—1222no—
371#124—22no—
372#124—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C42—21no—
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—1222no—
412C4×C8—1222no—
413#95—21no—
414#95—21no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95C22⋊D6111yescharacteristic
420#95C22⋊D6111yescharacteristic
421#95C22⋊D6111yescharacteristic
422#95C22⋊D6111yescharacteristic
423C24—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1175822212yes—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4#1175822212yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
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—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—4881no—
483C4⋊C4—4881no—
484C4⋊C4—4881no—
485C4⋊C4—4881no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—4881no—
497C42—4881no—
498C42#1175822412yes—
499C42#1175822412yes—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×C22⋊D6111yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
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—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—1222no—
543C2×C4—1222no—
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#1451976412yes—
555C2×C4#1451976412yes—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4C2×#1175822411yescharacteristic
559C2×C4C2×#1175822411yescharacteristic
560C2×C4C2×#1175822211yescharacteristic
561C2×C4C2×#1175822211yescharacteristic
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#1684026911yescharacteristic
574C22#1684027311yescharacteristic
575C22C2×#1451976411yescharacteristic
576C22#1684056511yescharacteristic
577C22#1684057011yescharacteristic
578C22#1684057311yescharacteristic
579C22#1684057711yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—4881no—
590C4—4881no—
591C4—2442no—
592C4—2442no—
593C4#1684029211yescharacteristic
594C4#1684029211yescharacteristic
595C4#1684029211yescharacteristic
596C4#1684029211yescharacteristic
597C2—82no—
598C2#2144910311yescharacteristic
599C2#2145601811yescharacteristic
600C2#2145603311yescharacteristic
601C2#2145607511yescharacteristic
602C2#2145609011yescharacteristic
603C2#2145617311yescharacteristic
604C2#2145618411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2326C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2326D61
hypocenterC61#23832
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #21448560C2semidirect
2. #21447381C2semidirect
3. #21447376C2semidirect
4. #21447227C2semidirect
8. #21446970C2semidirect
10. #21445649C2semidirect
12. C4×#14519701C2semidirect
14. #21445571C2semidirect
361. C61#23832semidirect

Maximal subgroups

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

subgroupindex
#214485602
#214473812
#214473762
#214472272
#214472222
#214470552
#214469702
#214469652
#214456492
#214456442
C4×#145197012
C4×#145196982
#214455712
C61×#23262
#2383261