Finite Group Databasebeta

← #29731597

Subgroups — #29731597

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840166C211yesmaximal, characteristic, complemented
2C2×#16840050C211yesmaximal, characteristic
3#21450990C211yesmaximal, characteristic
4#21448906C211yesmaximal, characteristic, complemented
5#21447303C211yesmaximal, characteristic, complemented
6#21447302C211yesmaximal, characteristic, complemented
7#21447207C211yesmaximal, characteristic
8#21447203C211yesmaximal, characteristic
9#21446765C211yesmaximal, characteristic, complemented
10#21446587C211yesmaximal, characteristic, complemented
11#21446586C211yesmaximal, characteristic, complemented
12#21446536C211yesmaximal, characteristic, complemented
13#21446441C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#1925C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16840166—24no—
20#16840050—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839674—21no—
33#16839674—21no—
34#16839668—21no—
35#16839666—21no—
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839659—21no—
40#16839628C2211yescharacteristic
41#16839627C2211yescharacteristic
42#16839620—21no—
43#16839620—21no—
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776—22no—
68#14519776—22no—
69#14519776D414yes—
70D4×Dic61—22no—
71D4×Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519710—22no—
91#14519710—22no—
92#14519709—22no—
93#14519708—22no—
94C4⋊#92418—22no—
95#14519706—22no—
96#14519706—22no—
97#14519698—22no—
98#14519698—22no—
99C8×Dic61—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#107×C61—22no—
121#102×C61—22no—
122#102×C61Q814yescomplemented
123#95×C61—21no—
124C24×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C2×Q812yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×Dic122—24no—
149C2×Dic122—42no—
150C2×Dic122—81no—
151C2×Dic122—81no—
152C2×Dic122—81no—
153C2×Dic122—81no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C2×Q812yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×Q812yes—
166C22⋊C4×C61—81no—
167C22⋊C4×C61—81no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—81no—
171C22⋊Dic61—42no—
172C22⋊Dic61—81no—
173C22⋊Dic61—81no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—81no—
182C4⋊Dic61C2×D412yes—
183#11758196—24no—
184#11758196—22no—
185#11758196—22no—
186#11758196—42no—
187#11758196—81no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—42no—
191C4×Dic61—81no—
192C4×Dic61C2×D412yes—
193C4×Dic61#4112yes—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418#4112yes—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61C22×D411yescharacteristic
208D4×C61—22no—
209D4×C61—22no—
210D4×C61—41no—
211D4×C61—41no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
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×C61—41no—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×Q811yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
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×Dic61—84no—
254C2×Dic61#12112yes—
255C2×Dic61#12112yes—
256C2×Dic61#13512yes—
257C2×Dic61#13712yes—
258Dic122—84no—
259Dic122—84no—
260C8×C61—22no—
261C8×C61—41no—
262#92418—22no—
263#92418—41no—
264#25450—611nomaximal, Sylow
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61C2×#12111yescharacteristic
273C22×C61#54711yescharacteristic
274C22×C61D4×Q811yescharacteristic
275C22×C61C2×#13511yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C22×C61#57611yescharacteristic
278C22×C61#57611yescharacteristic
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61#53311yescharacteristic
285C4×C61#54111yescharacteristic
286C4×C61#54811yescharacteristic
287C4×C61D4×Q811yescharacteristic
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61—42no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295Dic61#46414yes—
296C2×#500—611no—
297C2×#476—611no—
298#2563—611no—
299#2369—611no—
300#2085—611no—
301#2085—611no—
302#2067—611no—
303#2067—611no—
304#1987—611no—
305#1949—611no—
306#1949—611no—
307#1939—611no—
308#1925—611no—
309#1925D6111yescharacteristic
310C4×#102—611no—
311C2×C61—82no—
312C2×C61#242911yescharacteristic
313C2×C61C2×#46411yescharacteristic
314C2×C61#304111yescharacteristic
315C2×C61#334111yescharacteristic
316C2×C61#335711yescharacteristic
317C2×C61#338011yescharacteristic
318C2×C61#338511yescharacteristic
319C2×C4⋊Q8—611no—
320C2×C4⋊Q8—611no—
321C2×C4×D4—611no—
322#500—1224no—
323#476—1224no—
324C2×#107—611no—
325C2×#107—611no—
326C2×#107—611no—
327C2×#107C2×D6111yescharacteristic
328C2×#106—611no—
329C2×#106—611no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102C2×D6111yescharacteristic
336C2×#102C2×D6111yescharacteristic
337C2×C4×C8—611no—
338#393—1221no—
339#393—1221no—
340#391—1221no—
341#391—1221no—
342#390—611no—
343#390C2×D6111yescharacteristic
344#389—611no—
345#389—611no—
346#389—1221no—
347#382—611no—
348#382—611no—
349#380—1221no—
350#380—1221no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—1221no—
355C4×C4⋊C4C2×D6111yescharacteristic
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340C2×D6111yescharacteristic
362#336—611no—
363#336C2×D6111yescharacteristic
364C61#2545011yescharacteristic, complemented, Sylow
365C22×Q8—1221no—
366C22×Q8—1221no—
367C22×D4C22×D6111yescharacteristic
368C23×C4—1221no—
369C22×C8—611no—
370C22×C8—611no—
371C22×C8C22×D6111yescharacteristic
372C22×C8C22×D6111yescharacteristic
373C4⋊Q8—614no—
374C4⋊Q8—1222no—
375C4⋊Q8—1222no—
376C4⋊Q8—1222no—
377C4×D4—1222no—
378C4×D4—1222no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—21no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42C22×D6111yescharacteristic
403#107—22no—
404#107—1222no—
405#107—1222no—
406#107—1222no—
407#106—1222no—
408#106—1222no—
409C4⋊C8—1222no—
410C4⋊C8—1222no—
411C4⋊C8—1222no—
412#102—22no—
413#102—1222no—
414#102—1222no—
415#102Dic12214yescomplemented
416C4×C8—1222no—
417#95—21no—
418#95—611no—
419#95—611no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428C24—21no—
429C2×Q8—1224no—
430C2×Q8—4881no—
431C2×Q8—4881no—
432C2×Q8—4881no—
433C2×Q8—4881no—
434C2×Q8—2442no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×Dic12212yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—612no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8C2×Dic12212yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—612no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1224no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—4881no—
486C4⋊C4—4881no—
487C4⋊C4—2442no—
488C4⋊C4C2×Dic12212yes—
489C4⋊C4D4×D6112yes—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—4881no—
493C22⋊C4—4881no—
494C22⋊C4—2442no—
495C42—21no—
496C42—21no—
497C42—81no—
498C42—612no—
499C42—612no—
500C42—1221no—
501C42—1221no—
502C42—4881no—
503C42—2442no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D6111yescharacteristic
512Q8—4884no—
513Q8—4884no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1222no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—4884no—
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×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C22×Dic12211yescharacteristic
560C2×C4#41×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#1451982411yescharacteristic
563C2×C4#1451982411yescharacteristic
564C8—22no—
565C8—41no—
566C8—1222no—
567C8—2441no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#121×D6111yescharacteristic
576C22#1684041011yescharacteristic
577C22D4×Dic12211yescharacteristic
578C22#135×D6111yescharacteristic
579C22#137×D6111yescharacteristic
580C22#1684056311yescharacteristic
581C22#1684056611yescharacteristic
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—614no—
588C4—1222no—
589C4—4881no—
590C4—4881no—
591C4—4881no—
592C4—2441no—
593C4—2441no—
594C4—2442no—
595C4#1684031511yescharacteristic
596C4#1684036911yescharacteristic
597C4#1684042011yescharacteristic
598C4D4×Dic12211yescharacteristic
599C2—82no—
600C2#2144941511yescharacteristic
601C2#464×D6111yescharacteristic
602C2#2145531711yescharacteristic
603C2#2145823611yescharacteristic
604C2#2145843811yescharacteristic
605C2#2145868611yescharacteristic
606C2#2145873011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1925D61
hypocenterC61#25450
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
1. C2×#16840166C2semidirect
4. #21448906C2semidirect
5. #21447303C2semidirect
6. #21447302C2semidirect
9. #21446765C2semidirect
10. #21446587C2semidirect
11. #21446586C2semidirect
12. #21446536C2semidirect
122. #102×C61Q8semidirect
364. C61#25450semidirect
415. #102Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168401662
C2×#168400502
#214509902
#214489062
#214473032
#214473022
#214472072
#214472032
#214467652
#214465872
#214465862
#214465362
#214464412
#102×Dic612
C61×#19252
#2545061