Finite Group Databasebeta

← #29712986

Subgroups — #29712986

Order 15616.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840418C211yesmaximal, characteristic
2C22×#14519700C211yesmaximal, characteristic, complemented
3#21450121C211yesmaximal, characteristic, complemented
4#21447058C212yesmaximal, complemented
5#21447024C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446109C211yesmaximal, characteristic, complemented
8#21446070C212yesmaximal, complemented
9#21445756C212yesmaximal, complemented
10#21445755C211yesmaximal, characteristic, complemented
11C2×C61×#480C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840418—24no—
16#16840418—24no—
17C2×#14519764C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519760—21no—
20C2×#14519757C2211yescharacteristic
21C2×#14519754—21no—
22C2×#14519749—21no—
23C2×#14519742C2211yescharacteristic
24C2×D4×Dic61—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700—24no—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519698C2212yes—
32C2×#14519689C2212yes—
33#16839693—21no—
34#16839688—21no—
35#16839658C2212yes—
36#16839625—21no—
37#16839622C2211yescharacteristic
38#16839619C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×#121×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#480—24no—
44C2×#107×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—24no—
50C2×#11758224—42no—
51C22×Dic122—24no—
52C22×Dic122—21no—
53C22×Dic122—22no—
54C22×Dic122—42no—
55C22×Dic122D414yes—
56C22×Dic122C2311yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C22⋊D61D411yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×C4×D61C2311yescharacteristic
63C22×#92418—22no—
64C22×#92418—42no—
65C22×#92418C2312yes—
66#14519764—22no—
67#14519764—22no—
68#14519763—28no—
69#14519760—44no—
70#14519757—22no—
71#14519757—22no—
72#14519754—28no—
73#14519749—28no—
74#14519742—22no—
75#14519742—22no—
76D4×Dic61—28no—
77C4×C22⋊D61—22no—
78C4×C22⋊D61—22no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2312yes—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61D411yescharacteristic
88C2×#11758198—21no—
89C2×#11758198—22no—
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519702—24no—
94#14519702—42no—
95#14519700—24no—
96#14519700—24no—
97#14519698—42no—
98#14519689—42no—
99#14519684—21no—
100#14519684—22no—
101#14519684—22no—
102C22×D4×C61—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61C2312yes—
107#121×C61—22no—
108#121×C61D414yes—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2312yes—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115#107×C61—24no—
116#102×C61—24no—
117C22⋊C8×C61—22no—
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120C23×D61C2×D411yescharacteristic
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—22no—
141C2×C22⋊D61—22no—
142C2×C22⋊D61—22no—
143C2×C22⋊D61—44no—
144C22×Dic61—21no—
145C22×Dic61—24no—
146C22×Dic61—24no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—44no—
150C22×Dic61—44no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154#11758224—48no—
155#11758224—44no—
156#11758224—44no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—28no—
166C2×Dic122—24no—
167C2×Dic122—21no—
168C2×Dic122—44no—
169C2×Dic122—48no—
170C2×Dic122—44no—
171C2×Dic122C2×D412yes—
172C2×Dic122C2×D412yes—
173C2×Dic122#4114yes—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—24no—
183C22⋊Dic61—24no—
184C22⋊Dic61—24no—
185C22⋊Dic61—22no—
186C22⋊Dic61—42no—
187#11758198—24no—
188#11758198—22no—
189#11758198—44no—
190C4⋊Dic61—24no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193#11758196—24no—
194#11758196—22no—
195#11758196—44no—
196C4×Dic61—24no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—44no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—42no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—42no—
210C23×C61#12012yes—
211C23×C61#13512yes—
212C23×C61AGL(1,ℤ/8)12yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—22no—
219D4×C61—24no—
220D4×C61—48no—
221D4×C61—41no—
222D4×C61—41no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—84no—
234C2×C4×C61#12012yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4112yes—
238C2×C4×C61H(5,𝔽2)12yes—
239C22⋊D61—48no—
240C22⋊D61—44no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—48no—
252C2×Dic61—48no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—42no—
260Dic122—22no—
261Dic122—24no—
262Dic122—22no—
263Dic122—22no—
264Dic122—48no—
265Dic122—42no—
266Dic122#12314yescomplemented
267C8×C61—42no—
268#92418—42no—
269#24035—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61#44914yes—
280C22×C61C2×#12011yescharacteristic
281C22×C61#54612yes—
282C22×C61C2×#13511yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61#57812yes—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—42no—
289C2×D61—42no—
290C4×C61—22no—
291C4×C61—42no—
292C4×C61—82no—
293C4×C61C2×#12311yescharacteristic
294C4×C61#53411yescharacteristic
295C4×C61#54612yes—
296Dic61—24no—
297Dic61—42no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61—82no—
302C2×#548—611no—
303C2×#480D6111yescharacteristic, complemented
304C22×#103—611no—
305#2486—611no—
306#2042—612no—
307#2036—612no—
308#1893—611no—
309#1869—611no—
310#1863—612no—
311#1788—612no—
312#1787—611no—
313C2×C61—24no—
314C2×C61—82no—
315C2×C61#240511yescharacteristic
316C2×C61C2×#44911yescharacteristic
317C2×C61#301111yescharacteristic
318C2×C61#330012yes—
319C2×C61#330812yes—
320D61—81no—
321C22×#41—611no—
322C23×Q8—611no—
323C23×C8—611no—
324#548—1224no—
325#548—1224no—
326C2×#124—611no—
327C2×#124—1221no—
328C2×#123—1221no—
329C2×C22⋊Q8—611no—
330C2×C22⋊Q8—1221no—
331C2×#121—1221no—
332C2×#121C2×D6111yescharacteristic
333C2×C4×Q8—611no—
334C2×C4×D4—611no—
335C2×C4×D4—1221no—
336C22×C4⋊C4C2×D6111yescharacteristic
337#480—24no—
338C2×#107C2×D6112yes—
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103—612no—
343C2×#103—1224no—
344C2×#102—612no—
345C2×#102C2×D6112yes—
346C2×C22⋊C8—612no—
347C2×C22⋊C8C2×D6111yescharacteristic
348#398—1221no—
349#397—1221no—
350#389—612no—
351#381—1221no—
352#380—611no—
353#380—612no—
354#340—612no—
355#336—612no—
356C61#2403511yescharacteristic, complemented, Sylow
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—1224no—
360C2×#41—2442no—
361C22×Q8—614no—
362C22×Q8—611no—
363C22×Q8—611no—
364C22×Q8—1224no—
365C22×Q8—1221no—
366C22×Q8—1222no—
367C22×Q8—2442no—
368C22×D4—21no—
369C22×D4—611no—
370C22×D4—1221no—
371C22×D4C22×D6111yescharacteristic
372C23×C4—21no—
373C23×C4—611no—
374C23×C4—1221no—
375C23×C4—1221no—
376C23×C4C22×D6111yescharacteristic
377C22×C8—612no—
378C22×C8—1222no—
379C22×C8—2442no—
380C22×C8C22×D6112yes—
381#124—1228no—
382#124—1222no—
383#124—1222no—
384#123—2444no—
385C22⋊Q8—1228no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388#121—22no—
389#121—1228no—
390#121C22⋊D6114yes—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×D4—1228no—
394C4×D4—1222no—
395C4×D4—1222no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4C22×D6112yes—
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4—611no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—1221no—
416#107—24no—
417#103—1224no—
418#103—1224no—
419#103—1224no—
420#103—2442no—
421#102—24no—
422#102—2442no—
423C22⋊C8—22no—
424C22⋊C8—2442no—
425#95—1222no—
426#95—1222no—
427#95—1221no—
428C24—21no—
429C24—611no—
430C24D4×D6111yescharacteristic
431#41—2448no—
432#41—2444no—
433#41—2444no—
434C2×Q8—612no—
435C2×Q8—612no—
436C2×Q8—614no—
437C2×Q8—1228no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1222no—
441C2×Q8—1224no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8—2444no—
445C2×Q8—2444no—
446C2×Q8—2448no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—1224no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—2444no—
458C2×D4C2×C22⋊D6112yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1224no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—2444no—
479C22×C4—2444no—
480C22×C4D4×D6112yes—
481C22×C4D4×D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483C22×C4C23×D6111yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—2444no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—1224no—
496C4⋊C4—1224no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—2444no—
501C4⋊C4C2×C22⋊D6112yes—
502C22⋊C4—24no—
503C22⋊C4—1224no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—2444no—
509C22⋊C4—2442no—
510C42—1224no—
511C42—1221no—
512C42—1221no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—1221no—
521C23—1221no—
522C23—1221no—
523C23—1222no—
524C23—1222no—
525C23#1451974512yes—
526C23#1451979612yes—
527C23#1451980112yes—
528C23C2×D4×D6111yescharacteristic
529Q8—614no—
530Q8—1222no—
531Q8—1224no—
532Q8—1222no—
533Q8—1222no—
534Q8—2448no—
535Q8—2442no—
536D4—24no—
537D4—48no—
538D4—41no—
539D4—41no—
540D4—2448no—
541D4—2444no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—1222no—
554C2×C4—1224no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2448no—
564C2×C4—2448no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2442no—
572C2×C4#1451974512yes—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#41×D6112yes—
576C2×C4#1451982312yes—
577C8—42no—
578C8—2442no—
579C22—22no—
580C22—24no—
581C22—24no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—2441no—
589C22—2442no—
590C22—2442no—
591C22—2441no—
592C22—2441no—
593C22#1683990414yes—
594C22C2×#1451974511yescharacteristic
595C22#1684040412yes—
596C22C2×#1451979611yescharacteristic
597C22C2×#1451980111yescharacteristic
598C22#1684057812yes—
599C4—22no—
600C4—42no—
601C4—82no—
602C4—1224no—
603C4—4882no—
604C4—2441no—
605C4—2441no—
606C4—2441no—
607C4—2442no—
608C4#123×D6111yescharacteristic
609C4#1684032311yescharacteristic
610C4#1684040012yes—
611C2—24no—
612C2—82no—
613C2—4881no—
614C2#2144914311yescharacteristic
615C2C2×#1683990411yescharacteristic
616C2#2145511311yescharacteristic
617C2#2145770412yes—
618C2#2145780812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#480C2
socleC23×C61C22×D4
prime radicalC2×#16840418C2
hypercenterC2×#480D61
hypocenterC61#24035
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C22×#14519700C2semidirect
3. #21450121C2semidirect
4. #21447058C2semidirect
5. #21447024C2semidirect
6. #21446248C2semidirect
7. #21446109C2semidirect
8. #21446070C2semidirect
9. #21445756C2semidirect
10. #21445755C2semidirect
11. C2×C61×#480C2semidirect
266. Dic122#123semidirect
303. C2×#480D61semidirect
356. C61#24035semidirect

Maximal subgroups

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

subgroupindex
C2×#168404182
C22×#145197002
#214501212
#214470582
#214470242
#214462482
#214461092
#214460702
#214457562
#214457552
C2×C61×#4802
#2403561