Finite Group Databasebeta

← #29712215

Subgroups — #29712215

Order 15616.

1304 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×#16840458C211yesmaximal, characteristic
2C2×#16839901C211yesmaximal, characteristic, complemented
3C22×#14519701C211yesmaximal, characteristic, complemented
4#21446970C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446245C211yesmaximal, characteristic, complemented
7#21446156C212yesmaximal, complemented
8#21446132C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445907C211yesmaximal, characteristic, complemented
11C61×#2409C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C23×#92418C2211yescharacteristic
14#16840458—24no—
15#16840458—24no—
16C2×#14519772—21no—
17C2×#14519769C2211yescharacteristic
18C2×#14519759—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×#14519749—21no—
22C2×C4×D244C2211yescharacteristic
23C2×#14519737—21no—
24C2×#14519736C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26#16839901—24no—
27C2×#14519701—24no—
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C23×Q8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#395—21no—
42C61×#393—21no—
43C61×#391C2212yes—
44C61×#386—21no—
45C61×#385—21no—
46C61×#380C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61—21no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758201—42no—
55C22×#92418—22no—
56C22×#92418—42no—
57C22×#92418C2312yes—
58C22×#92418C2312yes—
59#14519772—28no—
60#14519769—22no—
61#14519769—22no—
62#14519759—28no—
63#14519751—22no—
64#14519751—22no—
65#14519751—22no—
66#14519751D414yes—
67#14519749—28no—
68C4×D244—22no—
69C4×D244—22no—
70#14519737—28no—
71#14519736—22no—
72#14519736—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D414yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86#14519701—24no—
87#14519701—24no—
88#14519701—24no—
89#14519697—42no—
90C22⋊#92418—22no—
91C22⋊#92418—42no—
92C22×Q8×C61—21no—
93C22×Q8×C61—24no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61—42no—
97C22×Q8×C61C2312yes—
98C22×D4×C61D411yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—42no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—22no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—28no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61—48no—
128C2×Q8×C61—44no—
129C2×Q8×C61—42no—
130C2×Q8×C61#4114yes—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—44no—
143C22×C4×C61—42no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—24no—
152C2×C22⋊D61—24no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—82no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—24no—
160C22×Dic61—82no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244C2×D412yes—
165C2×C4×D61—24no—
166C2×C4×D61—82no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—44no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—44no—
174C22⋊C4×C61—42no—
175C42×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—24no—
178#11758201—48no—
179#11758198—24no—
180#11758198—24no—
181#11758198—24no—
182C4⋊Dic61—24no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C2×D412yes—
187C4⋊Dic61C2×D412yes—
188C4⋊Dic61C2×D412yes—
189#11758196—24no—
190#11758196—24no—
191C4×Dic61—24no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—44no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61#12012yes—
207C23×C61#13512yes—
208C23×C61#13712yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—42no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—42no—
215Q8×C61—24no—
216Q8×C61—24no—
217Q8×C61—48no—
218Q8×C61—42no—
219Q8×C61—42no—
220D4×C61—44no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61#12012yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4112yes—
246C2×C4×C61H(5,𝔽2)12yes—
247C22⋊D61—48no—
248C22⋊D61—84no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—84no—
257D244—24no—
258D244—41no—
259D244—41no—
260C4×D61—84no—
261#92418—42no—
262#92418—42no—
263#23964—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61#45214yes—
274C22×C61C2×#12011yescharacteristic
275C22×C61#54612yes—
276C22×C61C2×#13511yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57712yes—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—82no—
283C2×D61—82no—
284C4×C61—22no—
285C4×C61—41no—
286C4×C61—44no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—82no—
290C4×C61#52511yescharacteristic
291C4×C61#53511yescharacteristic
292C4×C61#54612yes—
293Dic61—42no—
294Dic61—42no—
295Dic61—42no—
296C2×#553—611no—
297C2×#448—611no—
298C22×#103—611no—
299#2409D6111yescharacteristic, complemented
300#2026—612no—
301#2000—612no—
302#1893—611no—
303#1877—612no—
304#1873—612no—
305#1869—611no—
306#1824—611no—
307C2×C61—24no—
308C2×C61—81no—
309C2×C61#240411yescharacteristic
310C2×C61C2×#45211yescharacteristic
311C2×C61#300911yescharacteristic
312C2×C61#330112yes—
313C2×C61#332012yes—
314D61—82no—
315C23×Q8C2×D6111yescharacteristic
316C22×SD16—612no—
317C23×C8—611no—
318#553—1224no—
319#553—1224no—
320C2×#126—1221no—
321C2×#125—611no—
322C2×#123—1221no—
323C2×C22⋊Q8C2×D6112yes—
324C2×#121—611no—
325C2×#121—611no—
326C2×#121—1221no—
327C2×C4×D4—611no—
328C2×C4×D4—1221no—
329C2×C4×D4C2×D6111yescharacteristic
330C2×#117—611no—
331C22×C4⋊C4—611no—
332#448—1224no—
333C2×#103—612no—
334C2×#103—612no—
335C2×#103—612no—
336C2×#103—1224no—
337C2×#102—612no—
338C2×C22⋊C8—611no—
339C2×C22⋊C8—612no—
340#395—21no—
341#393—21no—
342#391C2×D6112yes—
343#386—21no—
344#385—21no—
345#380C2×D6111yescharacteristic
346#340—612no—
347#340—612no—
348#340—612no—
349#336—612no—
350C61#2396411yescharacteristic, complemented, Sylow
351C22×Q8—21no—
352C22×Q8—24no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—42no—
356C22×Q8C22×D6112yes—
357C22×D4—611no—
358C22×D4—1221no—
359C22×D4—1221no—
360C22×D4C22⋊D6111yescharacteristic
361C23×C4—21no—
362C23×C4—1221no—
363C23×C4—1221no—
364C23×C4C22×D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C2×SD16—1224no—
367C2×SD16—1224no—
368C2×SD16—2442no—
369C22×C8—612no—
370C22×C8—612no—
371C22×C8—1222no—
372C22×C8—2442no—
373#126—1228no—
374#125—1222no—
375#125—1222no—
376#123—1228no—
377C22⋊Q8—24no—
378C22⋊Q8—42no—
379#121—614no—
380#121—1222no—
381#121—1222no—
382#121—1228no—
383#121—1222no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1228no—
389#117—1222no—
390#117—1222no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—614no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22⋊D6111yescharacteristic
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22⋊D6111yescharacteristic
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412#103—1224no—
413#103—1224no—
414#103—1224no—
415#102—2442no—
416C22⋊C8—1222no—
417C22⋊C8—2442no—
418#95—21no—
419#95—22no—
420#95—22no—
421C24—1221no—
422C24C2×C22⋊D6111yescharacteristic
423C24C2×C22⋊D6111yescharacteristic
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8#1175822414yes—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—612no—
439C2×D4—1224no—
440C2×D4—1224no—
441C2×D4—1221no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—4882no—
445C2×D4—2442no—
446C2×D4—2442no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—4882no—
463C22×C4—4882no—
464C22×C4D4×D6112yes—
465C22×C4D4×D6112yes—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471SD16—2448no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—2444no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—612no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1224no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C42—21no—
501C42—21no—
502C42—1224no—
503C23—22no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2442no—
514C23—2442no—
515C23#1451974512yes—
516C23#1451979712yes—
517C23#1451980812yes—
518C23C22×C22⋊D6111yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524D4—44no—
525D4—1224no—
526D4—4884no—
527D4—2448no—
528D4—2441no—
529D4—2441no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—4884no—
555C2×C4—4884no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4#1451974512yes—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×#1175822412yes—
564C2×C4#1451982212yes—
565C8—2442no—
566C8—2442no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—4882no—
577C22—4882no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22#1683992214yes—
582C22C2×#1451974511yescharacteristic
583C22#1684040612yes—
584C22C2×#1451979711yescharacteristic
585C22C2×#1451980811yescharacteristic
586C22#1684057012yes—
587C4—22no—
588C4—41no—
589C4—44no—
590C4—41no—
591C4—41no—
592C4—82no—
593C4—2442no—
594C4—2442no—
595C4—2442no—
596C4#1684026811yescharacteristic
597C4#1684032711yescharacteristic
598C4#1684039912yes—
599C2—24no—
600C2—81no—
601C2—4882no—
602C2#2144913511yescharacteristic
603C2C2×#1683992211yescharacteristic
604C2#2145509411yescharacteristic
605C2#2145772112yes—
606C2#2145799012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2409C2
socleC23×C61C22×D4
prime radicalC2×#16840458C2
hypercenter#2409D61
hypocenterC61#23964
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. C2×#16839901C2semidirect
3. C22×#14519701C2semidirect
4. #21446970C2semidirect
5. #21446820C2semidirect
6. #21446245C2semidirect
7. #21446156C2semidirect
8. #21446132C2semidirect
9. #21446112C2semidirect
10. #21445907C2semidirect
11. C61×#2409C2semidirect
299. #2409D61semidirect
350. C61#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
C2×#168399012
C22×#145197012
#214469702
#214468202
#214462452
#214461562
#214461322
#214461122
#214459072
C61×#24092
#2396461