Finite Group Databasebeta

← #29731819

Subgroups — #29731819

Order 15616.

756 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
1C2×#16840170C211yesmaximal, characteristic, complemented
2C2×#16840061C211yesmaximal, characteristic
3#21450991C211yesmaximal, characteristic
4#21448911C211yesmaximal, characteristic, complemented
5#21447304C211yesmaximal, characteristic, complemented
6#21447301C211yesmaximal, characteristic, complemented
7#21447202C211yesmaximal, characteristic
8#21447192C211yesmaximal, characteristic
9#21446769C211yesmaximal, characteristic, complemented
10#21446586C211yesmaximal, characteristic, complemented
11#21446539C211yesmaximal, characteristic, complemented
12#21446444C211yesmaximal, characteristic
13#21446439C211yesmaximal, characteristic
14#21445643C211yesmaximal, characteristic
15C61×#1949C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19#16840170—24no—
20#16840061—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32#16839675—21no—
33#16839675—21no—
34#16839664C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839660—21no—
38#16839660C2211yescharacteristic
39#16839649—21no—
40#16839645—21no—
41#16839627C2211yescharacteristic
42#16839621—21no—
43#16839621—21no—
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776—22no—
68#14519768—22no—
69#14519768D414yes—
70#14519739—22no—
71#14519739—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519710—22no—
92#14519710—22no—
93#14519709—22no—
94#14519708—22no—
95#14519706—22no—
96#14519706—22no—
97#14519699—22no—
98#14519699—22no—
99#14519697—22no—
100#14519697Q814yescomplemented
101#14519687—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#106×C61—22no—
120C4⋊C8×C61—22no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124C23×D61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C2×Q812yes—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×Dic122—81no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C2×D412yes—
164C4⋊C4×C61C2×Q812yes—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—81no—
168#11758198—42no—
169#11758198—81no—
170#11758198—81no—
171#11758198—81no—
172#11758198—81no—
173C4⋊Dic61—21no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—81no—
180C4⋊Dic61C2×D412yes—
181#11758196—24no—
182#11758196—41no—
183#11758196—41no—
184#11758196—42no—
185#11758196—42no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—42no—
189C4×Dic61—81no—
190C4×Dic61C2×D412yes—
191C4×Dic61#4112yes—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418C2×Q812yes—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
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×C61—41no—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×Q811yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C612−1+411yescharacteristic
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—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
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#12112yes—
250C2×Dic61#12112yes—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—84no—
256Dic122—84no—
257C8×C61—22no—
258C8×C61—41no—
259#92418—22no—
260#92418—41no—
261#25466—611nomaximal, Sylow
262C22×C61C2×#12111yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61D4×Q811yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57611yescharacteristic
268C22×C61#57711yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—42no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#53311yescharacteristic
283C4×C61#54111yescharacteristic
284C4×C61#54811yescharacteristic
285C4×C61D4×Q811yescharacteristic
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290Dic61—42no—
291Dic61—81no—
292Dic61—81no—
293C2×#501—611no—
294C2×#478—611no—
295#2563—611no—
296#2370—611no—
297#2085—611no—
298#2085—611no—
299#2067—611no—
300#2065—611no—
301#1988—611no—
302#1949—611no—
303#1949D6111yescharacteristic, complemented
304#1940—611no—
305#1926—611no—
306#1925—611no—
307#1762—611no—
308C2×C61#242911yescharacteristic
309C2×C61#304011yescharacteristic
310C2×C61#304511yescharacteristic
311C2×C61#334711yescharacteristic
312C2×C61#334811yescharacteristic
313C2×C61#338811yescharacteristic
314C2×C61#338911yescharacteristic
315D61—82no—
316C2×C4⋊Q8—611no—
317C2×#125—611no—
318C2×C4×D4—611no—
319#501—1224no—
320#478—1224no—
321C2×#107—611no—
322C2×#107—611no—
323C2×#107C2×D6111yescharacteristic
324C2×#106—611no—
325C2×#106—611no—
326C2×#106C2×D6111yescharacteristic
327C2×C4⋊C8—611no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8C2×D6111yescharacteristic
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#102—611no—
334C2×#97—611no—
335#393—1221no—
336#393—1221no—
337#390—611no—
338#390—611no—
339#389—611no—
340#389—611no—
341#389—1221no—
342#387—1221no—
343#387—1221no—
344#382—611no—
345#382C2×D6111yescharacteristic
346#380—1221no—
347#380—1221no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—1221no—
352C4×C4⋊C4C2×D6111yescharacteristic
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340C2×D6111yescharacteristic
358#340C2×D6111yescharacteristic
359#336—611no—
360#336—611no—
361C61#2546611yescharacteristic, complemented, Sylow
362C22×Q8—1221no—
363C22×D4—611no—
364C23×C4—1221no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369C4⋊Q8—1222no—
370C4⋊Q8—1222no—
371#125—614no—
372#125—1222no—
373C4×D4—1222no—
374C4×D4—1222no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—21no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400#107—22no—
401#107—1222no—
402#107—1222no—
403#106—22no—
404#106—1222no—
405#106—1222no—
406C4⋊C8—22no—
407C4⋊C8—1222no—
408C4⋊C8—1222no—
409#102—614no—
410#102—1222no—
411#102—1222no—
412#102—1222no—
413#97—1222no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—611no—
418#95—611no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425C24—1221no—
426C2×Q8—4881no—
427C2×Q8—4881no—
428C2×Q8—2442no—
429C2×D4—612no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—612no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8#1175822212yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—612no—
471C4⋊C4—1224no—
472C4⋊C4—1224no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—4881no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2442no—
482C4⋊C4—2442no—
483C4⋊C4D4×D6112yes—
484C4⋊C4Q8×D6112yes—
485C22⋊C4—4881no—
486C22⋊C4—4881no—
487C22⋊C4—4881no—
488C22⋊C4—4881no—
489C22⋊C4—2442no—
490C42—21no—
491C42—21no—
492C42—81no—
493C42—612no—
494C42—612no—
495C42—1221no—
496C42—1221no—
497C42—4881no—
498C42—2442no—
499C23—1222no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23C2×D4×D6111yescharacteristic
507Q8—4884no—
508D4—1222no—
509D4—1222no—
510D4—2441no—
511D4—2441no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—4884no—
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×C4C2×D4×D6111yescharacteristic
553C2×C4C2×Q8×D6111yescharacteristic
554C2×C4#41×D6111yescharacteristic
555C2×C4C2×#1175822211yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—22no—
559C8—41no—
560C8—1222no—
561C8—2441no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22#121×D6111yescharacteristic
570C22#1684041311yescharacteristic
571C22#1684042511yescharacteristic
572C22#1684055911yescharacteristic
573C22#1684056111yescharacteristic
574C22#1684056611yescharacteristic
575C22#1684056911yescharacteristic
576C4—42no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—1222no—
583C4—1222no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4—2441no—
588C4—2442no—
589C4#1684031611yescharacteristic
590C4#1684037111yescharacteristic
591C4#1684042111yescharacteristic
592C4#1684042711yescharacteristic
593C2—4882no—
594C2#2144941711yescharacteristic
595C2#2145531011yescharacteristic
596C2#2145535911yescharacteristic
597C2#2145832011yescharacteristic
598C2#2145833011yescharacteristic
599C2#2145876711yescharacteristic
600C2#2145877711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1949C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1949D61
hypocenterC61#25466
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
1. C2×#16840170C2semidirect
4. #21448911C2semidirect
5. #21447304C2semidirect
6. #21447301C2semidirect
9. #21446769C2semidirect
10. #21446586C2semidirect
11. #21446539C2semidirect
15. C61×#1949C2semidirect
100. #14519697Q8semidirect
303. #1949D61semidirect
361. C61#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168401702
C2×#168400612
#214509912
#214489112
#214473042
#214473012
#214472022
#214471922
#214467692
#214465862
#214465392
#214464442
#214464392
#214456432
C61×#19492
#2546661