Finite Group Databasebeta

← #29733351

Subgroups — #29733351

Order 15616.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840458C211yesmaximal, characteristic
2C2×#16840130C211yesmaximal, characteristic, complemented
3C2×#16839964C211yesmaximal, characteristic, complemented
4#21446820C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446132C212yesmaximal, complemented
9#21446127C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2542C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13#16840458—24no—
14#16840458—24no—
15C2×#14519772—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519751C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519736C2211yescharacteristic
24C22×C4⋊Dic61C2211yescharacteristic
25#16840130—24no—
26#16839964—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C22⋊#92418C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2212yes—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C61×#399—21no—
43C61×#399—21no—
44C61×#390—21no—
45C61×#384—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61—21no—
52C2×#11758201—24no—
53C2×#11758201—42no—
54C2×#11758201—42no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519772—28no—
58#14519769—22no—
59#14519769D414yes—
60#14519759—28no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751—22no—
65#14519749—28no—
66C4×D244—22no—
67C4×D244D414yes—
68#14519737—28no—
69#14519736—22no—
70#14519736—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84C4⋊#92418—22no—
85#14519701—24no—
86#14519701—24no—
87#14519701—24no—
88#14519697—24no—
89C22⋊#92418—42no—
90C22⋊#92418—42no—
91#14519688—22no—
92C22×Q8×C61C2312yes—
93C22×D4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—22no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—42no—
101C22⋊Q8×C61—42no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#95×C61—21no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—41no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—24no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—82no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—24no—
153C22×Dic61—82no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244C2×D412yes—
158C2×C4×D61—24no—
159C2×C4×D61—82no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—44no—
163C4⋊C4×C61—44no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61#4114yes—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—44no—
170C22⋊C4×C61—44no—
171C42×C61—42no—
172C42×C61C2×D412yes—
173C42×C61C2×D412yes—
174C22⋊Dic61—24no—
175#11758201—48no—
176#11758198—24no—
177#11758198—24no—
178#11758198—24no—
179C4⋊Dic61—24no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61C2×D412yes—
184C4⋊Dic61C2×D412yes—
185C4⋊Dic61C2×D412yes—
186#11758196—24no—
187#11758196—24no—
188C4×Dic61—24no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C23×C61—21no—
196C23×C61—22no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61#13712yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—42no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—42no—
211Q8×C61—42no—
212Q8×C61—42no—
213Q8×C61—42no—
214Q8×C61—42no—
215D4×C61—44no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—24no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61#12012yes—
236C2×C4×C61#12012yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4112yes—
240C2×C4×C61H(5,𝔽2)12yes—
241C22⋊D61—48no—
242C22⋊D61—84no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—84no—
251D244—22no—
252D244—22no—
253D244—42no—
254C4×D61—84no—
255#92418—42no—
256#92418—42no—
257#25582—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—24no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—42no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61C2×#12011yescharacteristic
269C22×C61#54612yes—
270C22×C61C2×#13711yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57712yes—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—82no—
277C2×D61—82no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—44no—
283C4×C61—82no—
284C4×C61#53211yescharacteristic
285C4×C61#53511yescharacteristic
286C4×C61#55012yes—
287Dic61—42no—
288Dic61—42no—
289Dic61—42no—
290C2×#553—611no—
291C2×#491—611no—
292C2×#461—611no—
293#2542D6111yescharacteristic, complemented
294#2000—612no—
295#2000—612no—
296#1989—611no—
297#1955—611no—
298#1873—612no—
299#1872—612no—
300#1762—611no—
301C2×C61—42no—
302C2×C61—81no—
303C2×C61#243111yescharacteristic
304C2×C61#300711yescharacteristic
305C2×C61#300911yescharacteristic
306C2×C61#330712yes—
307C2×C61#332012yes—
308D61—82no—
309C22×SD16—612no—
310#553—1224no—
311#553—1224no—
312C2×C4⋊Q8C2×D6111yescharacteristic
313C2×#126—1221no—
314C2×#125—611no—
315C2×#123—1221no—
316C2×C22⋊Q8C2×D6112yes—
317C2×C22⋊Q8C2×D6112yes—
318C2×#121—611no—
319C2×#121—611no—
320C2×#121—1221no—
321C2×C4×D4—611no—
322C2×C4×D4—1221no—
323C2×C4×D4C2×D6111yescharacteristic
324C2×#117—611no—
325C22×C4⋊C4—611no—
326#491—1224no—
327#461—1224no—
328C2×C4⋊C8—611no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#102—612no—
333C2×C22⋊C8—612no—
334C2×C22⋊C8—612no—
335C2×#97—611no—
336#399—21no—
337#399—21no—
338#390—21no—
339#384—21no—
340C4×C4⋊C4C2×D6111yescharacteristic
341#340—612no—
342#340—612no—
343#340—612no—
344C61#2558211yescharacteristic, complemented, Sylow
345C22×Q8C22×D6112yes—
346C22×D4—611no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4C22⋊D6111yescharacteristic
350C23×C4—1221no—
351C23×C4—1221no—
352C23×C4C22×D6111yescharacteristic
353C23×C4C22×D6111yescharacteristic
354C2×SD16—1224no—
355C2×SD16—2442no—
356C2×SD16—2442no—
357C22×C8—612no—
358C22×C8—612no—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361#126—1228no—
362#125—614no—
363#125—1222no—
364#123—1228no—
365C22⋊Q8—24no—
366C22⋊Q8—24no—
367C22⋊Q8—42no—
368C22⋊Q8—42no—
369#121—1222no—
370#121—1222no—
371#121—1222no—
372#121—1222no—
373#121—1228no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—614no—
377C4×D4—1228no—
378C4×D4—1222no—
379#117—1222no—
380#117—1222no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4C22⋊D6111yescharacteristic
402C2×C22⋊C4C22⋊D6111yescharacteristic
403C2×C42—21no—
404C2×C42—1221no—
405C2×C42C22×D6111yescharacteristic
406C4⋊C8—1222no—
407#103—1224no—
408#103—1224no—
409#103—1224no—
410#102—1224no—
411C22⋊C8—2442no—
412C22⋊C8—2442no—
413#97—1222no—
414#95—21no—
415C24—1221no—
416C24C2×C22⋊D6111yescharacteristic
417C24C2×C22⋊D6111yescharacteristic
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—41no—
427C2×D4—612no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1221no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—4882no—
434C2×D4—2442no—
435C2×D4—2442no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—1224no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1224no—
451C22×C4—4882no—
452C22×C4—4882no—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458SD16—2448no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—1224no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4#1175822414yes—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—1224no—
489C42—42no—
490C42—1224no—
491C42D4×D6112yes—
492C42D4×D6112yes—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—1222no—
503C23—2441no—
504C23—2441no—
505C23—2442no—
506C23—2442no—
507C23#1451980812yes—
508C23C22×C22⋊D6111yescharacteristic
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513D4—44no—
514D4—1222no—
515D4—1222no—
516D4—4884no—
517D4—2448no—
518D4—2442no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—4884no—
542C2×C4—4884no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974512yes—
548C2×C4#1451974512yes—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C2×#1175822412yes—
552C2×C4#1451982212yes—
553C8—2442no—
554C8—2442no—
555C22—24no—
556C22—22no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—4882no—
566C22—4882no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C2×#1451974511yescharacteristic
571C22#1684040612yes—
572C22C2×#1451980811yescharacteristic
573C22#1684056011yescharacteristic
574C22#1684057012yes—
575C4—22no—
576C4—22no—
577C4—44no—
578C4—41no—
579C4—41no—
580C4—82no—
581C4—2442no—
582C4—2442no—
583C4—2442no—
584C4#1684031011yescharacteristic
585C4#1684032511yescharacteristic
586C4#1684043212yes—
587C2—42no—
588C2—81no—
589C2—4882no—
590C2#2144943511yescharacteristic
591C2#2145507711yescharacteristic
592C2#2145509411yescharacteristic
593C2#2145779512yes—
594C2#2145799012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2542C2
socleC23×C61C22×D4
prime radicalC2×#16840458C2
hypercenter#2542D61
hypocenterC61#25582
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. C2×#16840130C2semidirect
3. C2×#16839964C2semidirect
4. #21446820C2semidirect
5. #21446820C2semidirect
6. #21446776C2semidirect
7. #21446618C2semidirect
8. #21446132C2semidirect
9. #21446127C2semidirect
10. #21445647C2semidirect
11. C61×#2542C2semidirect
293. #2542D61semidirect
344. C61#25582semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
C2×#168401302
C2×#168399642
#214468202
#214467762
#214466182
#214461322
#214461272
#214456472
C61×#25422
#2558261