Finite Group Databasebeta

← #29681434

Subgroups — #29681434

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×#14519802C211yesmaximal, characteristic
2C2×#16840245C211yesmaximal, characteristic, complemented
3C22×#14519699C211yesmaximal, characteristic, complemented
4#21448399C211yesmaximal, characteristic, complemented
5#21446172C214yesmaximal, complemented
6#21446165C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445515C211yesmaximal, characteristic, complemented
9C2×C61×#416C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—24no—
12C2×#14519802—24no—
13C2×#14519802C418yescomplemented
14C22×#11758201C2212yes—
15C22×#11758199C2212yes—
16C22×#11758194C2211yescharacteristic
17#16840245—24no—
18#16840245—24no—
19C2×Q8×Dic61C2212yescomplemented
20C2×D4×Dic61—22no—
21C2×D4×Dic61C2212yescomplemented
22C2×#14519736—21no—
23C2×#14519736C2211yescharacteristic
24C22×C4⋊Dic61—21no—
25C22×C4⋊Dic61C2211yescharacteristic
26C2×#14519699—24no—
27C2×#14519699C2214yes—
28#16839664—21no—
29#16839664—21no—
30#16839663C2214yes—
31#16839617—22no—
32#16839524C2214yescomplemented
33#16839516C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C23×C8×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#416—24no—
38C2×#103×C61C2212yescomplemented
39C2×#102×C61C2212yescomplemented
40C24×D61—21no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244C2×C414yescomplemented
48C22×D244C2312yes—
49C23×Dic61—21no—
50C23×Dic61—22no—
51#14519802—232no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758201C2×C418yescomplemented
55C2×#11758199—24no—
56C2×#11758199—24no—
57C2×#11758199C2×C418yescomplemented
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194C2×C414yescomplemented
61C22×#92418C2312yes—
62Q8×Dic61—24no—
63Q8×Dic61—24no—
64D4×Dic61—216no—
65D4×Dic61—24no—
66D4×Dic61—24no—
67#14519736—28no—
68#14519736—22no—
69#14519736—22no—
70C2×C4⋊Dic61—28no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61D414yes—
77C2×C4⋊Dic61C2312yes—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61D412yes—
80C2×#11758198—24no—
81C2×#11758198—24no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2312yes—
84#14519699—28no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61C2×C414yes—
89C2×#41×C61D414yes—
90C2×#41×C61D414yes—
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61—21no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96C22×C8×C61—22no—
97C22×C8×C61—22no—
98C22×C8×C61—22no—
99C22×C8×C61C2312yes—
100#117×C61—22no—
101#117×C61—22no—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—21no—
105#103×C61—24no—
106#102×C61—24no—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109C23×D61—22no—
110C23×D61—24no—
111C23×D61—42no—
112C23×D61—42no—
113C23×D61—82no—
114C23×D61—82no—
115#41×C61—28no—
116#41×C61—28no—
117#41×C61—28no—
118#41×C61C22⋊C4116yescomplemented
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61C22×C412yes—
123C2×Q8×C61C2×D412yes—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C2×D4×C61C2×D412yes—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—22no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61—28no—
149C22×Dic61—24no—
150C22×Dic61—84no—
151C22×Dic61—82no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154C2×D244—28no—
155C2×D244—28no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—24no—
161C2×D244—22no—
162C2×D244—42no—
163C2×D244—44no—
164C2×D244—42no—
165C2×D244—44no—
166C2×D244C22×C414yes—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—44no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C22⋊C4×C61—42no—
174C42×C61—42no—
175C22⋊Dic61—28no—
176C22⋊Dic61—24no—
177#11758201—216no—
178#11758199—216no—
179#11758198—84no—
180#11758198—84no—
181C4⋊Dic61—28no—
182C4⋊Dic61—22no—
183C4⋊Dic61—24no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61C2×D414yes—
187C4×Dic61—28no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190#11758194—28no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418C22×C414yes—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C4×D412yes—
202C23×C61#12012yes—
203C23×C61#12112yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—24no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61—48no—
211C22×D61—88no—
212C22×D61—84no—
213C22×D61—84no—
214C22×D61—88no—
215Q8×C61—22no—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61C2×C22⋊C414yes—
219D4×C61—216no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—22no—
223D4×C61C2×C22⋊C414yes—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—216no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61C2×C22⋊C414yes—
232C2×C4×C61C4×D414yes—
233C2×C4×C61C4×D412yes—
234C2×C4×C61#12012yes—
235C2×C4×C61#12114yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#12314yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—44no—
250C2×Dic61—84no—
251C2×Dic61—88no—
252D244—24no—
253D244—24no—
254D244—42no—
255D244—42no—
256D244—48no—
257C8×C61—42no—
258#92418—24no—
259#20932—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61—44no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—44no—
269C22×C61#38614yes—
270C22×C61C2×C4×D411yescharacteristic
271C22×C61C2×#12011yescharacteristic
272C22×C61C2×#12111yescharacteristic
273C22×C61#57612yes—
274C22×C61#57712yes—
275C2×D61—48no—
276C2×D61—44no—
277C2×D61—816no—
278C2×D61—82no—
279C2×D61—82no—
280C2×D61—84no—
281C4×C61—24no—
282C4×C61—82no—
283C4×C61#38614yes—
284C4×C61C22×C22⋊C411yescharacteristic
285C4×C61C2×C4×D411yescharacteristic
286C4×C61C2×#12111yescharacteristic
287C4×C61C2×#12311yescharacteristic
288Dic61—42no—
289Dic61—44no—
290Dic61—42no—
291C22×AGL(1,ℤ/8)—611no—
292C2×#520—611no—
293C2×#416D6111yescharacteristic, complemented
294C22×#102—611no—
295#2309—611no—
296#1880—614no—
297#1879—614no—
298#1736—611no—
299#1729—611no—
300C2×C61—24no—
301C2×C61—44no—
302C2×C61C2×#38611yescharacteristic
303C2×C61#294612yes—
304C2×C61#300811yescharacteristic
305C2×C61#301011yescharacteristic
306C2×C61#304512yes—
307D61—84no—
308C22×#41C2×D6111yescharacteristic
309C23×D4—611no—
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×C8C2×D6111yescharacteristic
317#520—1224no—
318#520—1224no—
319C2×C4×Q8—612no—
320C2×C4×D4—612no—
321C2×C4×D4—1222no—
322C2×#117—611no—
323C2×#117—1221no—
324C2×#117C2×D6111yescharacteristic
325C22×C4⋊C4—611no—
326C22×C4⋊C4—1221no—
327#416—24no—
328C2×#103C2×D6112yescomplemented
329C2×#102—614no—
330C2×#102—1224no—
331C2×#102C2×D6112yescomplemented
332#390—614no—
333#390—1221no—
334#390—1221no—
335#379—1222no—
336#340—614no—
337#336—614no—
338C61#2093211yescharacteristic, complemented, Sylow
339C25—1221no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41C4×D6114yes—
344C2×#41D24414yes—
345C2×#41C22⋊D6114yes—
346C22×Q8C22×D6111yescharacteristic
347C22×D4—21no—
348C22×D4—614no—
349C22×D4—612no—
350C22×D4—1222no—
351C22×D4—1222no—
352C22×D4—1221no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4—1222no—
356C22×D4C22×D6111yescharacteristic
357C23×C4—21no—
358C23×C4—1222no—
359C23×C4—1221no—
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—22no—
372C22×C8—22no—
373C22×C8—22no—
374C22×C8—612no—
375C22×C8C22×D6112yes—
376C4×Q8—1224no—
377C4×Q8—1224no—
378C4×D4—1224no—
379C4×D4—12216no—
380C4×D4—1224no—
381#117—22no—
382#117—22no—
383#117—1228no—
384#117—1222no—
385#117—1222no—
386C2×C4⋊C4—614no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1228no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—612no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C42—21no—
401C2×C42—612no—
402C2×C42—1222no—
403#103—24no—
404#102—24no—
405#102—1228no—
406C24—21no—
407C24—1224no—
408C24—1222no—
409C24—4882no—
410C24—4882no—
411C24—2442no—
412C24—2442no—
413C24D4×D6111yescharacteristic
414#41—28no—
415#41—28no—
416#41—28no—
417#41#11758198116yescomplemented
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×C4×D6112yes—
422C2×Q8C2×D24412yes—
423C2×Q8C2×C22⋊D6112yes—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—614no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1228no—
437C2×D4—1228no—
438C2×D4—1224no—
439C2×D4—2442no—
440C2×D4—2444no—
441C2×D4—2442no—
442C2×D4—2444no—
443C2×D4C2×C4×D6112yes—
444C2×D4C2×D24412yes—
445C2×D4C2×C22⋊D6112yes—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—1228no—
454C22×C4—1224no—
455C22×C4—1222no—
456C22×C4—4884no—
457C22×C4—4882no—
458C22×C4C2×C4×D6112yes—
459C22×C4C2×D24412yes—
460C22×C4D4×D6112yes—
461C22×C4D4×D6111yescharacteristic
462C22×C4#1175822412yes—
463C22×C4#1175822412yes—
464C22×C4C2×C22⋊D6112yes—
465C22×C4C23×D6111yescharacteristic
466SD16—12216no—
467D8—12216no—
468M16—1228no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—44no—
473C2×C8—614no—
474C2×C8—1222no—
475C2×C8—1222no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—614no—
479C4⋊C4—1228no—
480C4⋊C4—1222no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C22⋊C4—42no—
485C22⋊C4—1228no—
486C22⋊C4—1224no—
487C22⋊C4—4884no—
488C22⋊C4—4884no—
489C42—42no—
490C42—1228no—
491C42—1222no—
492C42—1222no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—1224no—
501C23—4884no—
502C23—4884no—
503C23—4888no—
504C23—4888no—
505C23—2442no—
506C23—2442no—
507C23—2444no—
508C23—2444no—
509C23—2448no—
510C23#1451973712yes—
511C23#1451974712yes—
512C23#1451974912yes—
513C23C2×D4×D6111yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8C2×#1175819814yes—
518D4—216no—
519D4—22no—
520D4—22no—
521D4—22no—
522D4—1224no—
523D4—1224no—
524D4—2442no—
525D4—2442no—
526D4—2448no—
527D4C2×#1175819814yes—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—216no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—1222no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—4884no—
541C2×C4—4888no—
542C2×C4—2444no—
543C2×C4C2×#1175819814yes—
544C2×C4#1451973714yes—
545C2×C4#1451973712yes—
546C2×C4#1451974712yes—
547C2×C4#1451974914yes—
548C2×C4#1451974912yes—
549C2×C4#1451976114yes—
550C2×C4C22×C4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C8—42no—
557C8—1224no—
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—48816no—
568C22—4882no—
569C22—4882no—
570C22—4884no—
571C22—2444no—
572C22—2448no—
573C22#1683963914yes—
574C22C2×#1451973711yescharacteristic
575C22C2×#1451974711yescharacteristic
576C22C2×#1451974911yescharacteristic
577C22#1684056412yes—
578C22#1684056912yes—
579C4—24no—
580C4—82no—
581C4—2442no—
582C4—2442no—
583C4—2444no—
584C4#1683963914yes—
585C4C22×#1175819811yescharacteristic
586C4C2×#1451973711yescharacteristic
587C4C2×#1451974911yescharacteristic
588C4C2×#1451976111yescharacteristic
589C2—24no—
590C2—44no—
591C2—4884no—
592C2C2×#1683963911yescharacteristic
593C2#2145465412yes—
594C2#2145509011yescharacteristic
595C2#2145510811yescharacteristic
596C2#2145535612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC22×#14519802C2
cosocleC22×C4×C61C24
hypercenterC2×#416D61
hypocenterC61#20932
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#16840245C2semidirect
3. C22×#14519699C2semidirect
4. #21448399C2semidirect
5. #21446172C2semidirect
6. #21446165C2semidirect
7. #21445540C2semidirect
8. #21445515C2semidirect
9. C2×C61×#416C2semidirect
13. C2×#14519802C4semidirect
19. C2×Q8×Dic61C22semidirect
21. C2×D4×Dic61C22semidirect
32. #16839524C22semidirect
33. #16839516C22semidirect
38. C2×#103×C61C22semidirect
39. C2×#102×C61C22semidirect
47. C22×D244C2×C4semidirect
54. C2×#11758201C2×C4semidirect
57. C2×#11758199C2×C4semidirect
60. C2×#11758194C2×C4semidirect
118. #41×C61C22⋊C4semidirect
293. C2×#416D61semidirect
328. C2×#103C2×D61semidirect
331. C2×#102C2×D61semidirect
338. C61#20932semidirect
417. #41#11758198semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C2×#168402452
C22×#145196992
#214483992
#214461722
#214461652
#214455402
#214455152
C2×C61×#4162
#2093261