Finite Group Databasebeta

← #29783142

Subgroups — #29783142

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#21454187C211yesmaximal, characteristic
2#21448294C211yesmaximal, characteristic, complemented
3#21447322C211yesmaximal, characteristic, complemented
4#21447210C211yesmaximal, characteristic, complemented
5#21447208C211yesmaximal, characteristic
6#21447133C211yesmaximal, characteristic
7#21446676C211yesmaximal, characteristic
8#21446549C211yesmaximal, characteristic, complemented
9#21446502C211yesmaximal, characteristic
10#21446437C211yesmaximal, characteristic
11#21446402C211yesmaximal, characteristic, complemented
12#21446172C211yesmaximal, characteristic
13#21446071C211yesmaximal, characteristic, complemented
14C4×#14519706C211yesmaximal, characteristic, complemented
15C61×#1933C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519772—21no—
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839669C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839663C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839654—21no—
38#16839652—21no—
39#16839651—21no—
40#16839647—21no—
41#16839645—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43#16839613—21no—
44#16839607—21no—
45C42×Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#103×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×C4×C8×C61C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C2×#11758201—22no—
61C2×#11758201—22no—
62C2×#11758201—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519772—44no—
66Q8×Dic61—22no—
67Q8×Dic61—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×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—22no—
91#14519710—41no—
92#14519708—22no—
93#14519706—22no—
94#14519706—22no—
95#14519701—22no—
96#14519699—22no—
97#14519699—41no—
98#14519697—22no—
99C4×#92418—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C22×Q8×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×C42×C61C2311yescharacteristic
115#103×C61—22no—
116#103×C61—41no—
117C4×C8×C61—41no—
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121C23×D61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
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—21no—
160C4⋊C4×C61—42no—
161C42×C61—21no—
162C42×C61—21no—
163#11758201—44no—
164#11758198—42no—
165#11758198—42no—
166#11758198—42no—
167#11758198—42no—
168#11758198—42no—
169#11758198—42no—
170C4⋊Dic61—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61#4112yes—
178#11758196—42no—
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×C61C22×D411yescharacteristic
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×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×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61AGL(1,ℤ/8)12yes—
226C2×C4×C61#13712yes—
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
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—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—41no—
258C8×C61—41no—
259C8×C61—41no—
260#92418—22no—
261#92418—22no—
262#29435—611nomaximal, Sylow
263C22×C61#52511yescharacteristic
264C22×C61#53211yescharacteristic
265C22×C61#53711yescharacteristic
266C22×C61C2×#13511yescharacteristic
267C22×C61C2×#13511yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C61C2×#13711yescharacteristic
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61#55111yescharacteristic
284C4×C61#55211yescharacteristic
285C4×C61#55311yescharacteristic
286C4×C61#55611yescharacteristic
287Dic61—22no—
288Dic61—22no—
289Dic61—41no—
290Dic61—42no—
291Dic61—41no—
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#1934—611no—
303#1933D6111yescharacteristic, complemented
304#1925—611no—
305#1920—611no—
306#1880—611no—
307#1863—611no—
308C4×C4⋊C8—611no—
309C2×C61#260111yescharacteristic
310C2×C61#310611yescharacteristic
311C2×C61#310811yescharacteristic
312C2×C61#316811yescharacteristic
313C2×C61#318211yescharacteristic
314C2×C61#324711yescharacteristic
315C2×C61#324811yescharacteristic
316D61—161no—
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×#103C2×D6111yescharacteristic
327C2×#103C2×D6111yescharacteristic
328C2×#102—611no—
329C2×#102—611no—
330C2×#102—611no—
331C2×C4×C8—611no—
332C2×C4×C8C2×D6111yescharacteristic
333#391—611no—
334#391C2×D6111yescharacteristic
335#390—611no—
336#390—611no—
337#390—611no—
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×Q8C22×D6111yescharacteristic
362C22×D4—611no—
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—1221no—
386C2×C22⋊C4—1221no—
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—22no—
406#103—41no—
407#103—1222no—
408#102—1222no—
409#102—1222no—
410#102—2441no—
411C4×C8—41no—
412C4×C8—2441no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423C24—1221no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1175822212yes—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
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—21no—
469C4⋊C4—42no—
470C4⋊C4—612no—
471C4⋊C4—1224no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—1222no—
490C42—1222no—
491C42—1224no—
492C42—1222no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—1222no—
498C42—2442no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23C2×D4×D6111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—2441no—
512D4—2441no—
513D4—2441no—
514D4—2441no—
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—41no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
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#1451980412yes—
553C2×C4#1451980912yes—
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—4881no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22—4881no—
571C22#1684027111yescharacteristic
572C22#1684031111yescharacteristic
573C22#1684033911yescharacteristic
574C22#135×D6111yescharacteristic
575C22#135×D6111yescharacteristic
576C22C2×#1451980411yescharacteristic
577C22C2×#1451980911yescharacteristic
578C4—22no—
579C4—22no—
580C4—42no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—1222no—
585C4—1222no—
586C4—4881no—
587C4—4881no—
588C4—2441no—
589C4—2442no—
590C4—2441no—
591C4#1684044111yescharacteristic
592C4#1684044511yescharacteristic
593C4#1684045711yescharacteristic
594C4#1684047911yescharacteristic
595C2—9761no—
596C2#2145142111yescharacteristic
597C2#2145587711yescharacteristic
598C2#2145590311yescharacteristic
599C2#2145638811yescharacteristic
600C2#2145654011yescharacteristic
601C2#2145714011yescharacteristic
602C2#2145715111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1933C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1933D61
hypocenterC61#29435
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. #21448294C2semidirect
3. #21447322C2semidirect
4. #21447210C2semidirect
8. #21446549C2semidirect
11. #21446402C2semidirect
13. #21446071C2semidirect
14. C4×#14519706C2semidirect
15. C61×#1933C2semidirect
303. #1933D61semidirect
360. C61#29435semidirect

Maximal subgroups

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

subgroupindex
#214541872
#214482942
#214473222
#214472102
#214472082
#214471332
#214466762
#214465492
#214465022
#214464372
#214464022
#214461722
#214460712
C4×#145197062
C61×#19332
#2943561