Finite Group Databasebeta

← #29736560

Subgroups — #29736560

Order 15616.

1302 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840421C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840044C211yesmaximal, characteristic, complemented
4#21447015C212yesmaximal, complemented
5#21447008C212yesmaximal, complemented
6#21446751C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21445757C212yesmaximal, complemented
9#21445753C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2563C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14#16840421—24no—
15#16840421—24no—
16C2×#14519765C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519751C2211yescharacteristic
21C2×#14519749—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519737—21no—
24#16840124—24no—
25#16840044—24no—
26C2×#14519708C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2212yes—
35C2×C22⋊#92418C2212yes—
36C2×#14519688C2211yescharacteristic
37#16839524C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#393—21no—
41C61×#393—21no—
42C61×#390—21no—
43C61×#382—21no—
44C61×#380C2212yes—
45C61×#380C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—24no—
50C2×#11758222—42no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122—22no—
54C22×Dic122—24no—
55C22×Dic122—42no—
56C22×Dic122C2311yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C22⋊D61—21no—
61C23×Dic61—21no—
62C22×C4×D61—21no—
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65#14519765—22no—
66#14519765D414yes—
67#14519763—28no—
68#14519756—22no—
69#14519756—22no—
70#14519755—28no—
71#14519751—22no—
72#14519751—22no—
73#14519749—28no—
74C4×Dic122—22no—
75C4×Dic122D414yes—
76#14519737—28no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85#14519708—24no—
86C4⋊#92418—22no—
87#14519700—24no—
88#14519700—24no—
89#14519700—24no—
90#14519700—42no—
91#14519697—24no—
92#14519697—42no—
93C22⋊#92418—42no—
94C22⋊#92418—42no—
95#14519688—22no—
96C22×Q8×C61—21no—
97C22×D4×C61D411yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C42×C61—21no—
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—42no—
121C2×Q8×C61—82no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—41no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61—44no—
143C2×C22⋊D61—42no—
144C2×C22⋊D61—42no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—44no—
150#11758222—48no—
151#11758222—44no—
152#11758222—44no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—21no—
156C2×D244C2×D412yes—
157C2×C4×D61—24no—
158C2×C4×D61—44no—
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—24no—
164C2×Dic122—21no—
165C2×Dic122—48no—
166C2×Dic122—44no—
167C2×Dic122—44no—
168C2×Dic122—44no—
169C2×Dic122C2×D412yes—
170C2×Dic122C2×D412yes—
171C2×Dic122#4114yes—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—42no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—82no—
181C22⋊C4×C61—82no—
182C42×C61—42no—
183C42×C61C2×D412yes—
184C42×C61C2×D412yes—
185C22⋊Dic61—24no—
186#11758198—24no—
187#11758198—24no—
188#11758198—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61C2×D412yes—
191#11758196—24no—
192#11758196—24no—
193C4×Dic61—24no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61#13712yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—42no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—42no—
216Q8×C61—84no—
217D4×C61—44no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61#12012yes—
238C2×C4×C61#12012yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4112yes—
242C2×C4×C61H(5,𝔽2)12yes—
243C22⋊D61—48no—
244C22⋊D61—48no—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—48no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253D244—24no—
254D244—41no—
255D244—41no—
256C4×D61—48no—
257Dic122—22no—
258Dic122—24no—
259Dic122—24no—
260Dic122—48no—
261Dic122—41no—
262Dic122—41no—
263Dic122#12414yes—
264#92418—42no—
265#92418—42no—
266#25895—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—24no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61C2×#12011yescharacteristic
278C22×C61#54612yes—
279C22×C61C2×#13711yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57712yes—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—82no—
292C4×C61—82no—
293C4×C61C2×#12411yescharacteristic
294C4×C61#53511yescharacteristic
295C4×C61#54612yes—
296Dic61—24no—
297Dic61—42no—
298Dic61—42no—
299C2×#548—611no—
300C2×#489—611no—
301C2×#475—611no—
302#2563D6111yescharacteristic, complemented
303#2035—612no—
304#2034—612no—
305#1983—611no—
306#1955—611no—
307#1788—612no—
308#1787—612no—
309#1763—611no—
310C2×C61—42no—
311C2×C61—81no—
312C2×C61#243511yescharacteristic
313C2×C61#300711yescharacteristic
314C2×C61#301011yescharacteristic
315C2×C61#331612yes—
316C2×C61#332112yes—
317D61—82no—
318C22×#41—611no—
319C23×Q8—611no—
320#548—1224no—
321#548—1224no—
322C2×C4⋊Q8C2×D6111yescharacteristic
323C2×#124—611no—
324C2×#124—1221no—
325C2×C22⋊Q8—611no—
326C2×C22⋊Q8—1221no—
327C2×#121—611no—
328C2×#121—1221no—
329C2×C4×Q8—611no—
330C2×C4×D4—1221no—
331C2×C4×D4C2×D6111yescharacteristic
332#489—1224no—
333#475—1224no—
334C2×#106—612no—
335C2×C4⋊C8—611no—
336C2×#103—612no—
337C2×#103—612no—
338C2×#103—612no—
339C2×#103—612no—
340C2×#102—612no—
341C2×#102—612no—
342C2×C22⋊C8—612no—
343C2×C22⋊C8—612no—
344C2×#97—611no—
345#393—21no—
346#393—21no—
347#390—21no—
348#382—21no—
349#380C2×D6112yes—
350#380C2×D6112yes—
351C4×C4⋊C4C2×D6111yescharacteristic
352#340—612no—
353C61#2589511yescharacteristic, complemented, Sylow
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41—1224no—
357C2×#41—2442no—
358C22×Q8—21no—
359C22×Q8—611no—
360C22×Q8—611no—
361C22×Q8—1222no—
362C22×Q8—1222no—
363C22×Q8—1224no—
364C22×Q8—1221no—
365C22×Q8—2442no—
366C22×D4—611no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4C22⋊D6111yescharacteristic
370C23×C4—1221no—
371C23×C4—1221no—
372C23×C4C22×D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C22×C8—612no—
375C22×C8—612no—
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378#124—614no—
379#124—1228no—
380#124—1222no—
381C22⋊Q8—1228no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384#121—1222no—
385#121—1222no—
386#121—1228no—
387C4×Q8—614no—
388C4×Q8—1222no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—1228no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22⋊D6111yescharacteristic
397C2×C4⋊C4C22×D6112yes—
398C2×C4⋊C4C22×D6112yes—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4C22⋊D6111yescharacteristic
406C2×C22⋊C4C22⋊D6111yescharacteristic
407C2×C42—21no—
408C2×C42—1221no—
409C2×C42C22×D6111yescharacteristic
410#106—1224no—
411C4⋊C8—1222no—
412#103—1224no—
413#103—1224no—
414#103—1224no—
415#103—2442no—
416#102—1224no—
417#102—2442no—
418C22⋊C8—2442no—
419C22⋊C8—2442no—
420#97—1222no—
421#95—21no—
422#95—22no—
423#95—22no—
424C24—1221no—
425C24C2×C22⋊D6111yescharacteristic
426C24C2×C22⋊D6111yescharacteristic
427#41—2448no—
428#41—2444no—
429#41—2444no—
430C2×Q8—42no—
431C2×Q8—82no—
432C2×Q8—612no—
433C2×Q8—612no—
434C2×Q8—614no—
435C2×Q8—1224no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8—1222no—
441C2×Q8—2444no—
442C2×Q8—2444no—
443C2×Q8—2448no—
444C2×Q8—2444no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—41no—
448C2×D4—612no—
449C2×D4—1224no—
450C2×D4—1224no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—2444no—
455C2×D4—2442no—
456C2×D4—2442no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—1224no—
468C22×C4—1221no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—2444no—
473C22×C4—2444no—
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C23×D6111yescharacteristic
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1222no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—612no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1224no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—1224no—
500C22⋊C4—1224no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—42no—
504C42—1224no—
505C42D4×D6112yes—
506C42D4×D6112yes—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—1222no—
517C23—2441no—
518C23—2441no—
519C23—2442no—
520C23—2442no—
521C23#1451980812yes—
522C23C22×C22⋊D6111yescharacteristic
523Q8—84no—
524Q8—614no—
525Q8—1222no—
526Q8—1224no—
527Q8—1224no—
528Q8—2448no—
529Q8—2441no—
530Q8—2441no—
531D4—44no—
532D4—1224no—
533D4—2448no—
534D4—2448no—
535D4—2441no—
536D4—2441no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—2448no—
561C2×C4—2448no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2442no—
565C2×C4#1451974512yes—
566C2×C4#1451974512yes—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4#41×D6112yes—
570C2×C4#1451982312yes—
571C8—2442no—
572C8—2442no—
573C22—24no—
574C22—22no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—4882no—
584C22—4882no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22C2×#1451974511yescharacteristic
589C22#1684040012yes—
590C22C2×#1451980811yescharacteristic
591C22#1684056011yescharacteristic
592C22#1684057312yes—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—82no—
598C4—82no—
599C4—1224no—
600C4—2442no—
601C4—2442no—
602C4#124×D6111yescharacteristic
603C4#1684033011yescharacteristic
604C4#1684040512yes—
605C2—42no—
606C2—81no—
607C2—4882no—
608C2#2144946711yescharacteristic
609C2#2145507711yescharacteristic
610C2#2145510311yescharacteristic
611C2#2145792412yes—
612C2#2145799912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2563C2
socleC23×C61C22×D4
prime radicalC2×#16840421C2
hypercenter#2563D61
hypocenterC61#25895
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. C2×#16840124C2semidirect
3. C2×#16840044C2semidirect
4. #21447015C2semidirect
5. #21447008C2semidirect
6. #21446751C2semidirect
7. #21446618C2semidirect
8. #21445757C2semidirect
9. #21445753C2semidirect
10. #21445652C2semidirect
11. C61×#2563C2semidirect
302. #2563D61semidirect
353. C61#25895semidirect

Maximal subgroups

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

subgroupindex
C2×#168404212
C2×#168401242
C2×#168400442
#214470152
#214470082
#214467512
#214466182
#214457572
#214457532
#214456522
C61×#25632
#2589561