Finite Group Databasebeta

← #29783144

Subgroups — #29783144

Order 15616.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454183C211yesmaximal, characteristic
2#21448290C211yesmaximal, characteristic, complemented
3#21447323C211yesmaximal, characteristic, complemented
4#21447215C211yesmaximal, characteristic, complemented
5#21447203C211yesmaximal, characteristic
6#21447135C211yesmaximal, characteristic
7#21446672C211yesmaximal, characteristic
8#21446549C211yesmaximal, characteristic, complemented
9#21446495C211yesmaximal, characteristic, complemented
10#21446442C211yesmaximal, characteristic
11#21446407C211yesmaximal, characteristic, complemented
12#21446171C211yesmaximal, characteristic
13#21446074C211yesmaximal, characteristic, complemented
14C4×#14519706C211yesmaximal, characteristic, complemented
15C61×#1934C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519770—21no—
18C2×#14519742C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839668C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839656—21no—
38#16839655—21no—
39#16839650—21no—
40#16839649—21no—
41#16839643—21no—
42C4⋊C4×Dic61—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44#16839607—21no—
45C42×Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#102×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×C4×C8×C61C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—22no—
62C2×#11758200—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519770—44no—
66#14519742—22no—
67#14519742—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519710—22no—
89#14519710—41no—
90#14519708—22no—
91#14519706—22no—
92#14519706—22no—
93#14519702—22no—
94#14519702—41no—
95#14519700—22no—
96#14519698—22no—
97C4×#92418—41no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C22×D4×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61C2311yescharacteristic
117#102×C61—22no—
118#102×C61—41no—
119C4×C8×C61—41no—
120#95×C61—21no—
121C24×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122#4112yes—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61#4112yes—
161C22⋊C4×C61—42no—
162C22⋊C4×C61—42no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168C22⋊Dic61—42no—
169#11758200—44no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61—42no—
178#11758196—24no—
179#11758196—42no—
180#11758196—42no—
181C4×Dic61—24no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—42no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418#4112yes—
196C2×#92418#4112yes—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61C22×D411yescharacteristic
205D4×C61—41no—
206D4×C61—41no—
207D4×C61—41no—
208D4×C61—41no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
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×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61#13712yes—
224C2×C4×C61C2×#4111yescharacteristic
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—21no—
235C2×Dic61—22no—
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—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61#13512yes—
253C2×Dic61#13512yes—
254Dic122—22no—
255Dic122—22no—
256Dic122—41no—
257Dic122—41no—
258C8×C61—41no—
259C8×C61—41no—
260#92418—22no—
261#92418—22no—
262#29435—611nomaximal, Sylow
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61#52511yescharacteristic
271C22×C61#53211yescharacteristic
272C22×C61#53711yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61C2×#13511yescharacteristic
275C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#55111yescharacteristic
283C4×C61#55211yescharacteristic
284C4×C61#55311yescharacteristic
285C4×C61#55611yescharacteristic
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—42no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294#2860—611no—
295#2295—611no—
296#2088—611no—
297#2068—611no—
298#2067—611no—
299#2055—611no—
300#1968—611no—
301#1942—611no—
302#1934D6111yescharacteristic
303#1933—611no—
304#1925—611no—
305#1920—611no—
306#1880—611no—
307#1863—611no—
308C4×C4⋊C8—611no—
309C2×C61—161no—
310C2×C61#260111yescharacteristic
311C2×C61#310611yescharacteristic
312C2×C61#310811yescharacteristic
313C2×C61#316811yescharacteristic
314C2×C61#318211yescharacteristic
315C2×C61#324711yescharacteristic
316C2×C61#324811yescharacteristic
317C22×SD16—611no—
318C2×#126—1221no—
319C2×C4×Q8—611no—
320C2×#107—611no—
321C2×#107—611no—
322C2×#106—611no—
323C2×C4⋊C8—611no—
324C2×C4⋊C8—611no—
325C2×#103—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#102—611no—
329C2×#102C2×D6111yescharacteristic
330C2×#102C2×D6111yescharacteristic
331C2×C4×C8—611no—
332C2×C4×C8C2×D6111yescharacteristic
333#391—611no—
334#391—611no—
335#390—611no—
336#390—611no—
337#390C2×D6111yescharacteristic
338#389—611no—
339#389—611no—
340#389—611no—
341#389—611no—
342#389C2×D6111yescharacteristic
343#388—1221no—
344#388—1221no—
345#388—1221no—
346#387—1221no—
347#387—1221no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—1221no—
350#376—1221no—
351C43—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340C2×D6111yescharacteristic
357#340C2×D6111yescharacteristic
358#336—611no—
359#336—611no—
360C61#2943511yescharacteristic, complemented, Sylow
361C22×Q8—611no—
362C22×D4C22×D6111yescharacteristic
363C2×SD16—1222no—
364C2×SD16—1222no—
365C2×SD16—2441no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8C22×D6111yescharacteristic
369C22×C8C22×D6111yescharacteristic
370#126—2444no—
371C4×Q8—1222no—
372C4×Q8—1222no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400#107—1222no—
401#107—2441no—
402#106—1222no—
403C4⋊C8—1222no—
404C4⋊C8—1222no—
405#103—1222no—
406#103—1222no—
407#103—2441no—
408#102—22no—
409#102—41no—
410#102—1222no—
411C4×C8—41no—
412C4×C8—2441no—
413#95—21no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423C24—21no—
424C2×Q8—612no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—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—1221no—
447C22×C4D4×D6111yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450SD16—2444no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—612no—
458C2×C8—612no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4#1175822212yes—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—1222no—
490C42—1222no—
491C42—1224no—
492C42—1221no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1222no—
497C42—1222no—
498C42—2442no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D4×D6111yescharacteristic
507Q8—1222no—
508Q8—1222no—
509Q8—2441no—
510Q8—2441no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
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—612no—
529C2×C4—612no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
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×C4—2441no—
552C2×C4#1451980112yes—
553C2×C4#1451980612yes—
554C2×C4#41×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#41×D6111yescharacteristic
557C2×C4C2×#1175822211yescharacteristic
558C2×C4#1451982211yescharacteristic
559C2×C4#1451982411yescharacteristic
560C8—41no—
561C8—41no—
562C8—1222no—
563C8—1222no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1684027111yescharacteristic
572C22#1684031111yescharacteristic
573C22#1684033911yescharacteristic
574C22#135×D6111yescharacteristic
575C22#135×D6111yescharacteristic
576C22C2×#1451980111yescharacteristic
577C22C2×#1451980611yescharacteristic
578C4—22no—
579C4—22no—
580C4—42no—
581C4—81no—
582C4—81no—
583C4—1222no—
584C4—1222no—
585C4—4881no—
586C4—4881no—
587C4—4881no—
588C4—2441no—
589C4—2441no—
590C4—2442no—
591C4#1684044011yescharacteristic
592C4#1684044811yescharacteristic
593C4#1684046211yescharacteristic
594C4#1684048011yescharacteristic
595C2—161no—
596C2#2145142311yescharacteristic
597C2#2145587511yescharacteristic
598C2#2145589711yescharacteristic
599C2#2145638811yescharacteristic
600C2#2145653611yescharacteristic
601C2#2145713811yescharacteristic
602C2#2145714911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1934C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1934D61
hypocenterC61#29435
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #21448290C2semidirect
3. #21447323C2semidirect
4. #21447215C2semidirect
8. #21446549C2semidirect
9. #21446495C2semidirect
11. #21446407C2semidirect
13. #21446074C2semidirect
14. C4×#14519706C2semidirect
360. C61#29435semidirect

Maximal subgroups

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

subgroupindex
#214541832
#214482902
#214473232
#214472152
#214472032
#214471352
#214466722
#214465492
#214464952
#214464422
#214464072
#214461712
#214460742
C4×#145197062
C61×#19342
#2943561