Finite Group Databasebeta

← #29681439

Subgroups — #29681439

Order 15616.

1926 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519803C211yesmaximal, characteristic
2C2×#16840242C211yesmaximal, characteristic, complemented
3C2×#16839756C211yesmaximal, characteristic, complemented
4C22×#14519698C211yesmaximal, characteristic, complemented
5#21446177C214yesmaximal, complemented
6#21446170C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C61×#2309C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519803—24no—
12C2×#14519803—24no—
13C2×#14519803C418yescomplemented
14C22×#11758200C2212yes—
15C22×#11758199C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18#16840242—24no—
19#16840242—24no—
20C2×C4×Dic122C2212yescomplemented
21C2×C4×D244C2212yescomplemented
22C2×#14519737—22no—
23C2×#14519735—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C2×C4⋊C4×D61—21no—
26#16839756—24no—
27C2×#14519700C2212yescomplemented
28C2×#14519698—24no—
29C2×#14519698C2214yes—
30C2×#14519697C2212yescomplemented
31#16839524C2214yescomplemented
32#16839517C2214yescomplemented
33C23×D4×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#390—21no—
37C61×#390—21no—
38C61×#390C2214yes—
39C61×#379—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222C2×C414yes—
44C2×#11758222D414yes—
45C2×#11758222D414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C23×Dic61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—22no—
52#14519803—232no—
53C2×#11758200—24no—
54C2×#11758200—24no—
55C2×#11758200C2×C418yescomplemented
56C2×#11758199—24no—
57C2×#11758199—24no—
58C2×#11758199C2×C418yescomplemented
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194C2×C414yescomplemented
62C22×#92418—22no—
63C22×#92418—22no—
64C22×#92418—22no—
65C22×#92418C2312yes—
66C22×#92418C2312yes—
67C4×Dic122—24no—
68C4×Dic122—24no—
69C4×D244—24no—
70C4×D244—24no—
71#14519737—216no—
72#14519735—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61D414yes—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C4⋊C4×D61—28no—
79C2×#11758198—22no—
80C2×C4×Dic61—22no—
81#14519700—24no—
82#14519698—28no—
83#14519697—24no—
84C25×C61—21no—
85C22×D4×C61—22no—
86C22×D4×C61—22no—
87C22×D4×C61—22no—
88C22×D4×C61—21no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61C2×C414yescomplemented
92C22×D4×C61C2312yes—
93C23×C4×C61C2311yescharacteristic
94#117×C61—22no—
95#117×C61—22no—
96#117×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—24no—
103C2×C22⋊C4×C61—24no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61—21no—
106C2×C42×C61C2312yes—
107C24×C61—22no—
108C24×C61—24no—
109C24×C61—42no—
110C24×C61—42no—
111C24×C61—82no—
112C24×C61—82no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C2×D4×C61—28no—
116C2×D4×C61—28no—
117C2×D4×C61—24no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—42no—
124C2×D4×C61—44no—
125C2×D4×C61—42no—
126C2×D4×C61—44no—
127C2×D4×C61C22×C414yes—
128C22×C4×C61—22no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—24no—
141C2×C22⋊D61—24no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—82no—
146#11758222—28no—
147#11758222—28no—
148#11758222—28no—
149#11758222C22⋊C4116yescomplemented
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×D244C2×D412yes—
155C2×D244C2×D412yes—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—28no—
160C2×C4×D61—84no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C22×C412yes—
165C2×Dic122C2×D412yes—
166C2×Dic122C2×D412yes—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—42no—
173C22⋊C4×C61—84no—
174C22⋊C4×C61—84no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61—42no—
178#11758200—216no—
179#11758199—216no—
180#11758198—28no—
181C4⋊Dic61—24no—
182C4⋊Dic61—22no—
183C4⋊Dic61C2×D414yes—
184#11758196—28no—
185C4×Dic61—28no—
186#11758194—28no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—44no—
193C2×#92418C22×C414yes—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—24no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—44no—
200C23×C61—44no—
201C23×C61—48no—
202C23×C61—88no—
203C23×C61—84no—
204C23×C61—84no—
205C23×C61—88no—
206C23×C61C4×D412yes—
207C23×C61#12012yes—
208C23×C61#12112yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—42no—
214C22×D61—42no—
215D4×C61—24no—
216D4×C61—24no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—48no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C22⋊C414yes—
228C2×C4×C61C4×D414yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61#12012yes—
231C2×C4×C61#12114yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12314yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—216no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—24no—
245C2×Dic61—44no—
246C2×Dic61—84no—
247D244—22no—
248D244—22no—
249D244—22no—
250D244C2×C22⋊C414yes—
251C4×D61—216no—
252C4×D61—88no—
253Dic122—22no—
254Dic122—22no—
255Dic122—22no—
256Dic122C2×C22⋊C414yes—
257#92418—24no—
258#92418—42no—
259#20932—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—48no—
265C22×C61—44no—
266C22×C61—816no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—84no—
270C22×C61#38614yes—
271C22×C61C2×C4×D411yescharacteristic
272C22×C61C2×#12011yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61#57612yes—
275C22×C61#57712yes—
276C2×D61—24no—
277C2×D61—44no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—44no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—82no—
284C4×C61#38614yes—
285C4×C61C22×C22⋊C411yescharacteristic
286C4×C61C2×C4×D411yescharacteristic
287C4×C61C2×#12111yescharacteristic
288C4×C61C2×#12311yescharacteristic
289Dic61—24no—
290Dic61—44no—
291C22×AGL(1,ℤ/8)—611no—
292C2×#520—611no—
293C2×#416—611no—
294C22×#102—611no—
295#2309D6111yescharacteristic, complemented
296#1880—614no—
297#1879—614no—
298#1736—611no—
299#1729—611no—
300C2×C61—24no—
301C2×C61—84no—
302C2×C61C2×#38611yescharacteristic
303C2×C61#294612yes—
304C2×C61#300811yescharacteristic
305C2×C61#301011yescharacteristic
306C2×C61#304512yes—
307D61—44no—
308C22×#41—611no—
309C23×D4C2×D6111yescharacteristic
310C2×AGL(1,ℤ/8)—618no—
311C2×AGL(1,ℤ/8)—1224no—
312C2×AGL(1,ℤ/8)—1224no—
313C22×SD16—612no—
314C22×D8—612no—
315C22×M16—611no—
316C23×C8—611no—
317#520—1224no—
318#520—1224no—
319C2×C4×Q8—612no—
320C2×C4×D4—612no—
321C2×C4×D4—1222no—
322C2×#117—1221no—
323C2×#117C2×D6111yescharacteristic
324C2×#117C2×D6111yescharacteristic
325C22×C4⋊C4—611no—
326C22×C4⋊C4—1221no—
327#416—1224no—
328C2×#103—612no—
329C2×#102—614no—
330C2×#102—612no—
331C2×#102—1224no—
332#390—21no—
333#390—21no—
334#390C2×D6114yes—
335#379—22no—
336#340—614no—
337#336—614no—
338C61#2093211yescharacteristic, complemented, Sylow
339C25—21no—
340C2×#41—614no—
341C2×#41—614no—
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C22×Q8—611no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—611no—
354C22×D4—1221no—
355C22×D4C4×D6114yescomplemented
356C22×D4C22×D6112yes—
357C23×C4—1221no—
358C23×C4—1221no—
359C23×C4—1222no—
360C23×C4C22×D6111yescharacteristic
361AGL(1,ℤ/8)—12232no—
362C2×SD16—618no—
363C2×SD16—1224no—
364C2×SD16—1224no—
365C2×D8—618no—
366C2×D8—1224no—
367C2×D8—1224no—
368C2×M16—614no—
369C2×M16—1222no—
370C2×M16—1222no—
371C22×C8—612no—
372C22×C8—612no—
373C22×C8—1222no—
374C22×C8—1222no—
375C22×C8—1222no—
376C4×Q8—1224no—
377C4×Q8—1224no—
378C4×D4—1224no—
379C4×D4—1224no—
380C4×D4—12216no—
381#117—22no—
382#117—22no—
383#117—22no—
384#117—22no—
385#117—1228no—
386C2×C4⋊C4—614no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—1228no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4C22⋊D6111yescharacteristic
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4C22⋊D6112yes—
400C2×C42—21no—
401C2×C42—1222no—
402C2×C42C22×D6112yes—
403#103—1224no—
404#102—1224no—
405#102—1228no—
406C24—22no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24—82no—
411C24—82no—
412C24—1221no—
413C24C2×C22⋊D6111yescharacteristic
414#41—6116no—
415#41—1228no—
416#41—1228no—
417#41—1228no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—612no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—612no—
437C2×D4—612no—
438C2×D4—612no—
439C2×D4—1224no—
440C2×D4—1224no—
441C2×D4—1224no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4C2×C4×D6114yes—
446C22×C4—22no—
447C22×C4—1228no—
448C22×C4—1221no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—4884no—
455C22×C4—4882no—
456C22×C4C2×C4×D6112yes—
457C22×C4#1175822212yes—
458C22×C4D4×D6112yes—
459C22×C4D4×D6112yes—
460C22×C4D4×D6112yes—
461C22×C4#1175822412yes—
462C22×C4C2×C22⋊D6112yes—
463C22×C4C2×C22⋊D6112yes—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—12216no—
467D8—12216no—
468M16—1228no—
469C2×C8—614no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—2444no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—614no—
481C4⋊C4—1228no—
482C4⋊C4—1222no—
483C4⋊C4—1224no—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—1228no—
489C42—22no—
490C42—22no—
491C42—42no—
492C42—1228no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—48no—
501C23—88no—
502C23—84no—
503C23—88no—
504C23—84no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23—2442no—
509C23—2442no—
510C23C4×C22⋊D6112yes—
511C23#1451974512yes—
512C23#1451974812yes—
513C23C22×C22⋊D6111yescharacteristic
514Q8—614no—
515Q8—1222no—
516Q8—1222no—
517Q8—1222no—
518D4—24no—
519D4—24no—
520D4—42no—
521D4—42no—
522D4—48no—
523D4—614no—
524D4—12216no—
525D4—1222no—
526D4—1222no—
527D4—1222no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—12216no—
539C2×C4—1224no—
540C2×C4—4888no—
541C2×C4—4884no—
542C2×C4—2444no—
543C2×C4C22⋊C4×D6114yes—
544C2×C4#1451973714yes—
545C2×C4C4×C22⋊D6112yes—
546C2×C4#1451974512yes—
547C2×C4#1451974812yes—
548C2×C4#1451975014yes—
549C2×C4#1451975914yes—
550C2×C4C22×C4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4C2×#1175822211yescharacteristic
556C8—1224no—
557C8—2442no—
558C22—24no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—44no—
563C22—48no—
564C22—816no—
565C22—82no—
566C22—84no—
567C22—82no—
568C22—1224no—
569C22—2444no—
570C22—2442no—
571C22—2444no—
572C22—2442no—
573C22#1683963814yes—
574C22C2×C4×C22⋊D6111yescharacteristic
575C22C2×#1451974511yescharacteristic
576C22C2×#1451974811yescharacteristic
577C22#1684056512yes—
578C22#1684057312yes—
579C4—42no—
580C4—42no—
581C4—82no—
582C4—1224no—
583C4—2444no—
584C4#1683963814yes—
585C4C2×C22⋊C4×D6111yescharacteristic
586C4C2×#1451973711yescharacteristic
587C4C2×#1451975011yescharacteristic
588C4C2×#1451975911yescharacteristic
589C2—24no—
590C2—84no—
591C2—2444no—
592C2C2×#1683963811yescharacteristic
593C2#2145465112yes—
594C2#2145508511yescharacteristic
595C2#2145510311yescharacteristic
596C2#2145535412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2309C2
socleC23×C61C22×D4
prime radicalC22×#14519803C2
cosocleC22×C4×C61C24
hypercenter#2309D61
hypocenterC61#20932
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C2×#16840242C2semidirect
3. C2×#16839756C2semidirect
4. C22×#14519698C2semidirect
5. #21446177C2semidirect
6. #21446170C2semidirect
7. #21445540C2semidirect
8. #21445516C2semidirect
9. C61×#2309C2semidirect
13. C2×#14519803C4semidirect
20. C2×C4×Dic122C22semidirect
21. C2×C4×D244C22semidirect
27. C2×#14519700C22semidirect
30. C2×#14519697C22semidirect
31. #16839524C22semidirect
32. #16839517C22semidirect
55. C2×#11758200C2×C4semidirect
58. C2×#11758199C2×C4semidirect
61. C2×#11758194C2×C4semidirect
91. C22×D4×C61C2×C4semidirect
149. #11758222C22⋊C4semidirect
295. #2309D61semidirect
338. C61#20932semidirect
355. C22×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198032
C2×#168402422
C2×#168397562
C22×#145196982
#214461772
#214461702
#214455402
#214455162
C61×#23092
#2093261