Finite Group Databasebeta

← #29737669

Subgroups — #29737669

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449517C211yesmaximal, characteristic
2#21448695C211yesmaximal, characteristic
3#21447380C211yesmaximal, characteristic, complemented
4#21447321C211yesmaximal, characteristic, complemented
5#21447319C211yesmaximal, characteristic, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446579C211yesmaximal, characteristic, complemented
8#21446399C211yesmaximal, characteristic, complemented
9#21446398C211yesmaximal, characteristic, complemented
10#21446010C211yesmaximal, characteristic
11#21445998C211yesmaximal, characteristic
12#21445908C211yesmaximal, characteristic
13#21445841C211yesmaximal, characteristic
14#340×D61C211yesmaximal, characteristic
15C61×#2058C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519753—21no—
18C2×C4⋊C4×D61C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25#16839695—21no—
26#16839681—21no—
27#16839661C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839649—21no—
31#16839643—21no—
32#16839633—21no—
33#16839631C2211yescharacteristic
34#16839619—21no—
35#16839619C2211yescharacteristic
36#16839619C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C4×#11758198C2211yescharacteristic
40#16839609—21no—
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C2×C8×D61—22no—
65C2×C8×D61—22no—
66C2×C8×D61—41no—
67#14519753—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C4⋊C4×D61—22no—
80C4⋊C4×D61—22no—
81C4⋊C4×D61—41no—
82C4⋊C4×D61Q814yescomplemented
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198D411yescharacteristic
86C2×#11758198D411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519709—22no—
91#14519708—41no—
92C4⋊#92418—41no—
93#14519706—22no—
94#14519689—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C4⋊Q8×C61—22no—
108C4⋊Q8×C61—41no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116#106×C61—22no—
117#106×C61—41no—
118#95×C61—21no—
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—42no—
121C2×Q8×C61—42no—
122C2×Q8×C61—81no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—81no—
145C2×C4×D61—81no—
146C2×C4×D61—81no—
147C2×C4×D61C2×Q812yes—
148C2×C4×D61#4112yes—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61#4112yes—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—81no—
163C4⋊C4×C61C2×Q812yes—
164C4⋊C4×C61#4112yes—
165C42×C61—21no—
166C42×C61—21no—
167#11758198—22no—
168#11758198—42no—
169#11758198—42no—
170#11758198SD1614yes—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—42no—
179C4⋊Dic61—81no—
180C4⋊Dic61C2×Q812yes—
181#11758196—24no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185#11758196—81no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—81no—
189C4×Dic61—81no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C8×D61—44no—
197C23×C61C22×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61C22⋊Q812yes—
203C22×D61C2×SD1612yes—
204C22×D61#13712yes—
205Q8×C61—84no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
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×C61C2×SD1612yes—
224C2×C4×C61C22×Q811yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
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×Dic61C22⋊Q812yes—
250C2×Dic61C2×SD1612yes—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—84no—
256C4×D61—84no—
257C8×C61—22no—
258C8×C61—41no—
259#92418—41no—
260#92418—41no—
261#25992—611nomaximal, Sylow
262C22×C61C2×C22⋊Q811yescharacteristic
263C22×C61#54611yescharacteristic
264C22×C61#54711yescharacteristic
265C22×C61C22×SD1611yescharacteristic
266C22×C61C2×#13711yescharacteristic
267C22×C61#57611yescharacteristic
268C22×C61#57711yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61#47514yes—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61—41no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#53311yescharacteristic
283C4×C61#54411yescharacteristic
284C4×C61#55111yescharacteristic
285C4×C61#55211yescharacteristic
286Dic61—22no—
287Dic61—22no—
288Dic61—42no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293#2440—611no—
294#2345—611no—
295#2099—611no—
296#2088—611no—
297#2088—611no—
298#2058D6111yescharacteristic, complemented
299#1988—611no—
300#1948—611no—
301#1919—611no—
302#1919—611no—
303#1853—611no—
304#1850—611no—
305#1824—611no—
306#1807—611no—
307C2×#340—611no—
308C2×C61#243811yescharacteristic
309C2×C61C2×#47511yescharacteristic
310C2×C61#308111yescharacteristic
311C2×C61#329511yescharacteristic
312C2×C61#332011yescharacteristic
313C2×C61#333511yescharacteristic
314C2×C61#334611yescharacteristic
315D61—24no—
316C23×C8—611no—
317C2×C4⋊Q8C2×D6111yescharacteristic
318C2×C22⋊Q8—1221no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321C2×#106—611no—
322C2×#106—611no—
323C2×#106C2×D6111yescharacteristic
324C2×#106C2×D6111yescharacteristic
325C2×C4⋊C8—611no—
326C2×C4⋊C8—611no—
327C2×C22⋊C8—611no—
328#398—1221no—
329#395—1221no—
330#389—611no—
331#389—611no—
332#389—611no—
333#389C2×D6111yescharacteristic
334#387—1221no—
335#387—1221no—
336#385—1221no—
337#384—611no—
338#380—611no—
339#380—611no—
340#380—1221no—
341C4×C4⋊C4—611no—
342C4×C4⋊C4—611no—
343C4×C22⋊C4—611no—
344C4×C22⋊C4—1221no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340C2×D6111yescharacteristic
356#340C2×D6111yescharacteristic
357#336—611no—
358#336C2×D6111yescharacteristic
359C61#2599211yescharacteristic, complemented, Sylow
360C22×Q8—21no—
361C23×C4—611no—
362C23×C4—1221no—
363C23×C4—1221no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1222no—
367C22×C8—1222no—
368C22×C8—2441no—
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—41no—
373C22⋊Q8—2444no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—614no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1222no—
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—2441no—
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—611no—
396C2×C22⋊C4—611no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—611no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42C22×D6111yescharacteristic
403#106—22no—
404#106—41no—
405#106—1222no—
406#106—2441no—
407C4⋊C8—1222no—
408C4⋊C8—2441no—
409C22⋊C8—1222no—
410#95—21no—
411#95—611no—
412#95—611no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—611no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—81no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—1224no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—4881no—
443C22×C4—4881no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4D4×D6111yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—2444no—
462C2×C8#1175822612yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—612no—
472C4⋊C4—1224no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—4881no—
481C4⋊C4—4881no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4#1175822412yes—
487C4⋊C4Q8×D6112yes—
488C22⋊C4—614no—
489C22⋊C4—1222no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C42—21no—
493C42—21no—
494C42—1221no—
495C42—1221no—
496C42—4881no—
497C42—4881no—
498C23—612no—
499C23—612no—
500C23—612no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23C2×D4×D6111yescharacteristic
506Q8—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—4884no—
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×C4SD16×D6112yes—
552C2×C4C2×Q8×D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4C2×#1175822611yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—22no—
559C8—41no—
560C8—2441no—
561C8—2441no—
562C22—614no—
563C22—1222no—
564C22—1222no—
565C22—1222no—
566C22—1222no—
567C22—1222no—
568C22—1222no—
569C22C22⋊Q8×D6111yescharacteristic
570C22#1684039911yescharacteristic
571C22#1684041511yescharacteristic
572C22C2×SD16×D6111yescharacteristic
573C22#137×D6111yescharacteristic
574C22#1684056611yescharacteristic
575C22#1684056811yescharacteristic
576C4—22no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—1222no—
583C4—1222no—
584C4—4881no—
585C4—4881no—
586C4—4881no—
587C4—2442no—
588C4—2441no—
589C4#1684031711yescharacteristic
590C4#1684038911yescharacteristic
591C4#1684043911yescharacteristic
592C4#1684044611yescharacteristic
593C2—1224no—
594C2#2144948811yescharacteristic
595C2#475×D6111yescharacteristic
596C2#2145564511yescharacteristic
597C2#2145763611yescharacteristic
598C2#2145798311yescharacteristic
599C2#2145817511yescharacteristic
600C2#2145830311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2058C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2058D61
hypocenterC61#25992
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
3. #21447380C2semidirect
4. #21447321C2semidirect
5. #21447319C2semidirect
6. #21446772C2semidirect
7. #21446579C2semidirect
8. #21446399C2semidirect
9. #21446398C2semidirect
15. C61×#2058C2semidirect
82. C4⋊C4×D61Q8semidirect
298. #2058D61semidirect
359. C61#25992semidirect

Maximal subgroups

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

subgroupindex
#214495172
#214486952
#214473802
#214473212
#214473192
#214467722
#214465792
#214463992
#214463982
#214460102
#214459982
#214459082
#214458412
#340×D612
C61×#20582
#2599261