Finite Group Databasebeta

← #29713229

Subgroups — #29713229

Order 15616.

1358 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×#16840417C211yesmaximal, characteristic
2C2×#16840097C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4#21447065C212yesmaximal, complemented
5#21447022C212yesmaximal, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446117C211yesmaximal, characteristic, complemented
8#21446065C212yesmaximal, complemented
9#21445757C211yesmaximal, characteristic, complemented
10#21445757C212yesmaximal, complemented
11C61×#2486C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758222C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15#16840417—24no—
16#16840417—24no—
17C2×#14519777C2211yescharacteristic
18C2×#14519759—21no—
19C2×#14519759—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519756C2211yescharacteristic
22C2×#14519748—21no—
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840097—24no—
26C2×#14519710C2212yes—
27#16839761—24no—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2211yescharacteristic
35C2×C22⋊#92418C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C2×#123×C61—21no—
39C2×C4×D4×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#398—21no—
42C61×#397—21no—
43C61×#389C2212yes—
44C61×#381—21no—
45C61×#380C2211yescharacteristic
46C61×#380C2212yes—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—24no—
50C2×#11758222—24no—
51C2×#11758222—22no—
52C2×#11758222—42no—
53C2×#11758222—42no—
54C2×#11758222D414yes—
55C22×Dic122C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C2×#11758194—22no—
64C2×#11758194—42no—
65C22×#92418C2312yes—
66C22×#92418C2312yes—
67#14519777—22no—
68#14519777—22no—
69#14519759—28no—
70#14519759—28no—
71#14519756—22no—
72#14519756—22no—
73#14519756—22no—
74#14519756D414yes—
75#14519748—28no—
76C4×D244—22no—
77C4×D244—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C4⋊C4×D61—28no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87#14519710—24no—
88#14519700—24no—
89#14519700—24no—
90#14519700—42no—
91#14519700—42no—
92#14519697—24no—
93#14519697—24no—
94C22⋊#92418—22no—
95C22⋊#92418—42no—
96C22×D4×C61D411yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100#123×C61—44no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#95×C61—22no—
116#95×C61—21no—
117#95×C61—22no—
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—41no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—44no—
145C2×C22⋊D61—44no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149#11758222—28no—
150#11758222—44no—
151#11758222—48no—
152#11758222—48no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—21no—
156C2×D244—22no—
157C2×D244C2×D412yes—
158C2×D244#4114yes—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C2×D412yes—
173C2×Dic122C2×D412yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61—44no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—44no—
185C42×C61—21no—
186C42×C61—21no—
187C22⋊Dic61—24no—
188C22⋊Dic61—24no—
189#11758198—24no—
190C4⋊Dic61—24no—
191C4⋊Dic61—24no—
192C4⋊Dic61—21no—
193C4⋊Dic61C2×D412yes—
194#11758196—24no—
195#11758196—24no—
196#11758194—44no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61#12012yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—24no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—42no—
219D4×C61—44no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—84no—
240C2×C4×C61#12012yes—
241C2×C4×C61AGL(1,ℤ/8)12yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4112yes—
245C2×C4×C61H(5,𝔽2)12yes—
246C22⋊D61—48no—
247C22⋊D61—48no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255D244—22no—
256D244—24no—
257D244#12314yescomplemented
258C4×D61—48no—
259C4×D61—48no—
260C4×D61—48no—
261Dic122—22no—
262Dic122—22no—
263Dic122—24no—
264Dic122—41no—
265Dic122—41no—
266Dic122—42no—
267#92418—42no—
268#92418—42no—
269#24054—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—42no—
280C22×C61C2×#12011yescharacteristic
281C22×C61#54612yes—
282C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57812yes—
285C2×D61—24no—
286C2×D61—44no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—44no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—82no—
295C4×C61—82no—
296C4×C61C2×#12311yescharacteristic
297C4×C61#53411yescharacteristic
298C4×C61#54612yes—
299Dic61—42no—
300Dic61—42no—
301Dic61—42no—
302C2×#547—611no—
303C2×#484—611no—
304C2×#417—611no—
305#2486D6111yescharacteristic, complemented
306#2043—612no—
307#2036—612no—
308#1894—611no—
309#1870—611no—
310#1862—612no—
311#1788—611no—
312#1788—612no—
313C2×C61—42no—
314C2×C61—81no—
315C2×C61#240511yescharacteristic
316C2×C61#300611yescharacteristic
317C2×C61#301111yescharacteristic
318C2×C61#331512yes—
319C2×C61#333012yes—
320D61—44no—
321C22×#41—611no—
322C22×#41—611no—
323C22×M16—611no—
324#547—1224no—
325#547—1224no—
326C2×C4⋊Q8—611no—
327C2×#123—21no—
328C2×#123—1221no—
329C2×#123—1221no—
330C2×C22⋊Q8—611no—
331C2×C22⋊Q8—611no—
332C2×#121—1221no—
333C2×C4×D4—611no—
334C2×C4×D4C2×D6111yescharacteristic
335C22×C4⋊C4—1221no—
336C22×C4⋊C4C2×D6111yescharacteristic
337#484—1224no—
338C2×#107—612no—
339#417—1224no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103—612no—
343C2×#103—612no—
344C2×#102—612no—
345C2×#102—612no—
346C2×C22⋊C8—611no—
347C2×C22⋊C8—612no—
348#398—21no—
349#397—21no—
350#389C2×D6112yes—
351#381—21no—
352#380C2×D6111yescharacteristic
353#380C2×D6112yes—
354#340—612no—
355#336—612no—
356C61#2405411yescharacteristic, complemented, Sylow
357C2×#41—614no—
358C2×#41—1222no—
359C2×#41—1222no—
360C2×#41—1224no—
361C2×#41—1224no—
362C2×#41—1222no—
363C2×#41—2442no—
364C2×#41—2442no—
365C22×Q8—611no—
366C22×Q8—611no—
367C22×D4—611no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4C22⋊D6111yescharacteristic
371C23×C4—21no—
372C23×C4—1221no—
373C23×C4—1221no—
374C23×C4—1221no—
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C2×M16—1222no—
378C2×M16—2442no—
379C22×C8—612no—
380C22×C8—612no—
381C4⋊Q8—1222no—
382C4⋊Q8—1222no—
383#123—44no—
384#123—1228no—
385#123—1228no—
386C22⋊Q8—614no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390#121—1228no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—1222no—
394C4×D4—1222no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1228no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22⋊D6111yescharacteristic
406C2×C4⋊C4C22×D6112yes—
407C2×C4⋊C4C22×D6112yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C22⋊C4C22⋊D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416#107—1224no—
417#103—1224no—
418#103—1224no—
419#103—2442no—
420#103—2442no—
421#102—1224no—
422#102—1224no—
423C22⋊C8—1222no—
424C22⋊C8—2442no—
425#95—22no—
426#95—21no—
427#95—22no—
428C24—1221no—
429C24C2×C22⋊D6111yescharacteristic
430C24C2×C22⋊D6111yescharacteristic
431#41—1228no—
432#41—2444no—
433#41—2448no—
434#41—2444no—
435#41—2448no—
436#41—2444no—
437C2×Q8—612no—
438C2×Q8—612no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—41no—
448C2×D4—612no—
449C2×D4—614no—
450C2×D4—1224no—
451C2×D4—1224no—
452C2×D4—1221no—
453C2×D4—1222no—
454C2×D4—2444no—
455C2×D4—2444no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1224no—
468C22×C4—1224no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—2444no—
474C22×C4—2444no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4D4×D6112yes—
478C22×C4D4×D6112yes—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484M16—2444no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—612no—
499C4⋊C4—1224no—
500C4⋊C4—1224no—
501C4⋊C4—1224no—
502C4⋊C4—1221no—
503C4⋊C4—1224no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C42—21no—
512C42—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—1222no—
523C23—1224no—
524C23—2441no—
525C23—2441no—
526C23—2442no—
527C23#1451974512yes—
528C23C22×C22⋊D6111yescharacteristic
529Q8—1222no—
530Q8—1222no—
531Q8—1224no—
532Q8—2441no—
533Q8—2441no—
534Q8—2442no—
535D4—44no—
536D4—614no—
537D4—1222no—
538D4—1224no—
539D4—2448no—
540D4—2448no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2448no—
565C2×C4—2448no—
566C2×C4—2448no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4#1451974512yes—
572C2×C4#1451980312yes—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4#41×D6112yes—
576C2×C4#1451982312yes—
577C8—2442no—
578C8—2442no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—1224no—
590C22—2442no—
591C22—2444no—
592C22—2442no—
593C22—2444no—
594C22C2×#1451974511yescharacteristic
595C22#1684040012yes—
596C22C2×#1451980311yescharacteristic
597C22#1684056011yescharacteristic
598C22#1684057712yes—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4—82no—
605C4—2442no—
606C4—2442no—
607C4—2442no—
608C4#123×D6111yescharacteristic
609C4#1684032311yescharacteristic
610C4#1684040412yes—
611C2—42no—
612C2—81no—
613C2—2444no—
614C2#2144914311yescharacteristic
615C2#2145506911yescharacteristic
616C2#2145511211yescharacteristic
617C2#2145790912yes—
618C2#2145811812yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2486C2
socleC23×C61C22×D4
prime radicalC2×#16840417C2
hypercenter#2486D61
hypocenterC61#24054
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
3. C2×#16839761C2semidirect
4. #21447065C2semidirect
5. #21447022C2semidirect
6. #21446250C2semidirect
7. #21446117C2semidirect
8. #21446065C2semidirect
9. #21445757C2semidirect
10. #21445757C2semidirect
11. C61×#2486C2semidirect
257. D244#123semidirect
305. #2486D61semidirect
356. C61#24054semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168400972
C2×#168397612
#214470652
#214470222
#214462502
#214461172
#214460652
#214457572
C61×#24862
#2405461