Finite Group Databasebeta

← #29682028

Subgroups — #29682028

Order 15616.

1748 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2#21459022C211yesmaximal, characteristic, complemented
3C2×#16839735C212yesmaximal, complemented
4#21446359C218yesmaximal, complemented
5#21445552C212yesmaximal, complemented
6C2×C61×#410C211yesmaximal, characteristic, complemented
7C22×#11758222C2211yescharacteristic
8#16840340—22no—
9#16840336—21no—
10#16840333C412yescomplemented
11#16840329—21no—
12#16840326C412yescomplemented
13#16840321C2214yes—
14#16840308—22no—
15#16840273—22no—
16#16840269C414yescomplemented
17#16840245—24no—
18#16840240—21no—
19#16840237—21no—
20#16840237C2214yes—
21#41×Dic61—24no—
22C2×#14519736—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2212yes—
25#16839735—24no—
26#16839735—24no—
27C2×#14519691C2218yes—
28#16839529C2218yes—
29C22×#41×C61C2211yescharacteristic
30C2×#117×C61C2212yes—
31C61×#410—22no—
32C61×#410—22no—
33C2×#99×C61C2214yes—
34C2×#11758222—22no—
35C2×#11758222—22no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C22×Dic122C2×C411yescharacteristic, complemented
43C22×D244C2×C411yescharacteristic, complemented
44C22×C22⋊D61C2312yes—
45C22×C4×D61C2×C412yescomplemented
46#14519775—22no—
47#14519773—22no—
48#14519766—22no—
49#14519764—22no—
50#14519760—24no—
51#14519758C2×C418yescomplemented
52#14519756—24no—
53#14519753—24no—
54#14519751—24no—
55#14519749—28no—
56#14519748—24no—
57#14519746—24no—
58#14519745C2×C418yescomplemented
59Q8×Dic61—28no—
60D4×Dic61—28no—
61D4×Dic61—216no—
62C4×C22⋊D61—28no—
63#14519736—24no—
64#14519736—24no—
65#14519736—22no—
66#14519736—24no—
67#14519736—22no—
68#14519736—21no—
69#14519736—21no—
70#14519736C2×C412yescomplemented
71#14519736C2×C412yescomplemented
72#14519736D414yescomplemented
73#14519736D414yescomplemented
74C2×C4⋊Dic61—24no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C22⋊Dic61—24no—
78C2×C22⋊Dic61C2312yes—
79C2×C22⋊Dic61C2314yes—
80C2×C4×Dic61—24no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83#14519691—216no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61C2×C412yes—
89C2×#41×C61C2×C412yes—
90C2×#41×C61D412yes—
91C2×#41×C61D412yes—
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61D411yescharacteristic
94C22×D4×C61C2312yes—
95C23×C4×C61D412yes—
96C23×C4×C61C2311yescharacteristic
97#117×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61—22no—
102C2×C22⋊C4×C61C2314yes—
103C2×C42×C61—22no—
104#99×C61—28no—
105C24×C61C2×D412yes—
106C24×C61C2411yescharacteristic
107C23×D61C22×C412yes—
108#41×C61—24no—
109#41×C61—24no—
110#41×C61—24no—
111#41×C61—24no—
112#41×C61—24no—
113#41×C61—24no—
114#41×C61—24no—
115#41×C61—24no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61C22⋊C412yes—
119C2×Q8×C61C22⋊C412yes—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61C22⋊C412yes—
125C2×D4×C61C22⋊C412yes—
126C2×D4×C61C22×C414yes—
127C2×D4×C61C2×D414yes—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61C22⋊C414yes—
134C22×C4×C61C22⋊C414yes—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C2×C22⋊D61—24no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—28no—
151C22×Dic61—28no—
152C22×Dic61—24no—
153C22×Dic61—22no—
154C22×Dic61C22×C412yes—
155#11758222—41no—
156#11758222—42no—
157#11758222—41no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161#11758222—42no—
162#11758222—41no—
163#11758222—41no—
164#11758222—42no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—22no—
168C2×D244—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—24no—
172C2×C4×D61—24no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—22no—
176C2×Dic122—22no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—24no—
181C42×C61—42no—
182C42×C61—42no—
183C22⋊Dic61—28no—
184C22⋊Dic61—28no—
185C22⋊Dic61—24no—
186C22⋊Dic61—22no—
187C22⋊Dic61C22×C414yes—
188C22⋊Dic61C2×D418yes—
189#11758198—28no—
190C4⋊Dic61—28no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—24no—
194C4⋊Dic61—24no—
195#11758196—28no—
196C4×Dic61—28no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—24no—
200C4×Dic61—24no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—81no—
204C23×C61—82no—
205C23×C61C2×C22⋊C414yes—
206C23×C61C23×C411yescharacteristic
207C23×C61C22×D411yescharacteristic
208C23×C61C22×D411yescharacteristic
209C23×C61C22×D412yes—
210C23×C61C2×#4112yes—
211C22×D61—22no—
212C22×D61—24no—
213C22×D61—82no—
214Q8×C61—24no—
215Q8×C61—24no—
216D4×C61—28no—
217D4×C61—28no—
218D4×C61—24no—
219D4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—44no—
227C2×C4×C61—84no—
228C2×C4×C61C2×C22⋊C414yes—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C4×D412yes—
234C2×C4×C61C4×D412yes—
235C2×C4×C61C4×D412yes—
236C2×C4×C61C4×D412yes—
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12012yes—
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12112yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12311yescharacteristic
245C2×C4×C61#12311yescharacteristic
246C2×C4×C61#12312yes—
247C22⋊D61—42no—
248C22⋊D61—44no—
249C22⋊D61—42no—
250C22⋊D61—88no—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—28no—
255C2×Dic61—84no—
256C2×Dic61—88no—
257C2×Dic61—82no—
258D244—41no—
259D244—41no—
260D244—42no—
261C4×D61—42no—
262C4×D61—44no—
263C4×D61—42no—
264Dic122—41no—
265Dic122—41no—
266Dic122—42no—
267#21014—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—42no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—82no—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61C2×C4×D412yes—
279C22×C61C2×#12011yescharacteristic
280C22×C61C2×#12112yes—
281C22×C61C2×#12311yescharacteristic
282C2×D61—44no—
283C2×D61—42no—
284C2×D61—84no—
285C4×C61—24no—
286C4×C61—84no—
287C4×C61#38611yescharacteristic
288C4×C61#38611yescharacteristic
289C4×C61#38612yes—
290C4×C61#38611yescharacteristic
291C4×C61#38612yes—
292C4×C61#38611yescharacteristic
293Dic61—48no—
294Dic61—44no—
295Dic61—42no—
296#3449—611no—
297#3428—611no—
298C2×#410—612no—
299C2×#410D6111yescharacteristic, complemented
300#1913—618no—
301#1740—612no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—22no—
305C2×C61—22no—
306C2×C61—21no—
307C2×C61—44no—
308C2×C61C2×#38611yescharacteristic
309D61—82no—
310C22×#41—611no—
311C22×#41C2×D6111yescharacteristic
312#537—1222no—
313#536—612no—
314#536—1221no—
315#535—612no—
316#535—1221no—
317#534—614no—
318#532—1222no—
319#525—614no—
320#525—1222no—
321#520—1224no—
322#519—1221no—
323#518—614no—
324#518—1221no—
325C4×#41—1224no—
326C2×#117—612no—
327C2×#117—611no—
328C2×#117—1222no—
329C2×#117C2×D6112yes—
330#410—22no—
331#410—22no—
332#410—1224no—
333#410—1224no—
334C2×#99—618no—
335C2×#99C2×D6114yes—
336#342—618no—
337C61#2101411yescharacteristic, complemented, Sylow
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C2×#41—1221no—
349C2×#41—1221no—
350C2×#41C4×D6112yes—
351C2×#41C4×D6112yes—
352C2×#41C22⋊D6112yes—
353C2×#41C22⋊D6112yes—
354C22×Q8—611no—
355C22×Q8D24411yescharacteristic
356C22×D4—612no—
357C22×D4—611no—
358C22×D4D24411yescharacteristic
359C22×D4C22×D6112yes—
360C23×C4—612no—
361C23×C4D24412yes—
362C23×C4C22×D6111yescharacteristic
363C4⋊Q8—1222no—
364#127—1222no—
365#124—1222no—
366#124—1222no—
367#123—618no—
368#123—1224no—
369C22⋊Q8—1224no—
370C22⋊Q8—1224no—
371#121—1224no—
372#121—1228no—
373#121—1224no—
374#120—618no—
375#120—1224no—
376C4×Q8—1228no—
377C4×D4—12216no—
378C4×D4—1228no—
379C4×D4—1228no—
380#117—22no—
381#117—22no—
382#117—22no—
383#117—22no—
384#117—612no—
385#117—612no—
386#117—614no—
387#117—614no—
388#117—1224no—
389#117—1224no—
390#117—1222no—
391#117—1222no—
392#117—1221no—
393#117—1221no—
394#117—1224no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C22⋊C4—614no—
400C2×C22⋊C4—612no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4C22×D6114yes—
403C2×C42—22no—
404C2×C42—1224no—
405C2×C42—1222no—
406C2×C42—1221no—
407#99—28no—
408#99—12216no—
409C24—612no—
410C24C2×D24412yes—
411C24C23×D6111yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41—24no—
420#41—2442no—
421#41—2441no—
422#41—2441no—
423#41—2442no—
424#41—2442no—
425#41—2442no—
426#41—2442no—
427#41—2441no—
428#41—2442no—
429#41—2441no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8#1175819812yes—
437C2×Q8#1175819812yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—1228no—
443C2×D4—1224no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1224no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4#1175819812yes—
452C2×D4#1175819812yes—
453C2×D4C2×C4×D6114yes—
454C2×D4C2×C22⋊D6114yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—612no—
461C22×C4—1228no—
462C22×C4—1224no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1224no—
467C22×C4—1224no—
468C22×C4#1175819814yes—
469C22×C4#1175819814yes—
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4C2×C4×D6111yescharacteristic
472C22×C4C2×D24412yes—
473C22×C4C2×D24411yescharacteristic
474C22×C4C2×D24411yescharacteristic
475C22×C4D4×D6112yes—
476C22×C4D4×D6112yes—
477C22×C4#1175822412yes—
478C22×C4#1175822412yes—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—1228no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1228no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—614no—
492C22⋊C4—618no—
493C22⋊C4—1228no—
494C22⋊C4—1224no—
495C22⋊C4—1222no—
496C22⋊C4—1228no—
497C22⋊C4—1228no—
498C42—42no—
499C42—42no—
500C42—1228no—
501C42—1222no—
502C42—1222no—
503C42—1224no—
504C42—1224no—
505C23—22no—
506C23—22no—
507C23—81no—
508C23—82no—
509C23—1224no—
510C23—1222no—
511C23—4882no—
512C23C2×#1175819814yes—
513C23C22×C4×D6111yescharacteristic
514C23C2×D4×D6112yes—
515C23C22×C22⋊D6111yescharacteristic
516C23C22×D24411yescharacteristic
517C23C2×#1175822412yes—
518Q8—24no—
519Q8—24no—
520Q8—2441no—
521Q8—2441no—
522Q8—2442no—
523D4—28no—
524D4—28no—
525D4—24no—
526D4—24no—
527D4—4888no—
528D4—2442no—
529D4—2444no—
530D4—2442no—
531D4—2441no—
532D4—2441no—
533D4—2442no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—44no—
541C2×C4—84no—
542C2×C4—1224no—
543C2×C4—1222no—
544C2×C4—1224no—
545C2×C4—1228no—
546C2×C4—4884no—
547C2×C4—4888no—
548C2×C4—4882no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4—2442no—
552C2×C4C2×#1175819814yes—
553C2×C4C2×#1175819811yescharacteristic
554C2×C4C2×#1175819811yescharacteristic
555C2×C4C2×#1175819811yescharacteristic
556C2×C4C2×#1175819811yescharacteristic
557C2×C4#1451973712yes—
558C2×C4#1451973712yes—
559C2×C4#1451973712yes—
560C2×C4#1451973712yes—
561C2×C4#1451974712yes—
562C2×C4#1451974711yescharacteristic
563C2×C4#1451974711yescharacteristic
564C2×C4#1451974912yes—
565C2×C4#1451974912yes—
566C2×C4#1451974912yes—
567C2×C4#1451974912yes—
568C2×C4#1451976111yescharacteristic
569C2×C4#1451976112yes—
570C2×C4#1451976111yescharacteristic
571C22—24no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—82no—
580C22—4884no—
581C22—2444no—
582C22—2442no—
583C22C22×#1175819811yescharacteristic
584C22C2×#1451973712yes—
585C22C2×#1451974711yescharacteristic
586C22C2×#1451974912yes—
587C22C2×#1451976111yescharacteristic
588C4—24no—
589C4—84no—
590C4—2448no—
591C4—2444no—
592C4—2442no—
593C4#1683963911yescharacteristic
594C4#1683963911yescharacteristic
595C4#1683963911yescharacteristic
596C4#1683963912yes—
597C4#1683963912yes—
598C4#1683963911yescharacteristic
599C2—22no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—22no—
604C2—44no—
605C2—4882no—
606C2C2×#1683963911yescharacteristic
subgroupG/H
centerC4#16839639
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC2×C61×#410C2
socleC2×C61C2×#386
prime radical#21459148C2
cosocleC24×C61C24
hypercenterC2×#410D61
hypocenterC61#21014
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. #21459022C2semidirect
3. C2×#16839735C2semidirect
4. #21446359C2semidirect
5. #21445552C2semidirect
6. C2×C61×#410C2semidirect
10. #16840333C4semidirect
12. #16840326C4semidirect
16. #16840269C4semidirect
42. C22×Dic122C2×C4semidirect
43. C22×D244C2×C4semidirect
45. C22×C4×D61C2×C4semidirect
51. #14519758C2×C4semidirect
58. #14519745C2×C4semidirect
70. #14519736C2×C4semidirect
71. #14519736C2×C4semidirect
72. #14519736D4semidirect
73. #14519736D4semidirect
299. C2×#410D61semidirect
337. C61#21014semidirect

Maximal subgroups

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

subgroupindex
#214591482
#214590222
C2×#168397352
#214463592
#214455522
C2×C61×#4102
#2101461