Finite Group Databasebeta

← #29707100

Subgroups — #29707100

Order 15616.

2130 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758201C211yesmaximal, characteristic, complemented
2C2×#16840326C211yesmaximal, characteristic
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839758C211yesmaximal, characteristic, complemented
5#21448405C211yesmaximal, characteristic, complemented
6#21446910C214yesmaximal, complemented
7#21446905C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C22×#137×C61C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C22×#11758201—24no—
12C22×#11758201—24no—
13C22×#11758201—24no—
14C22×#11758201C2214yes—
15C23×#92418C2211yescharacteristic
16#16840326—24no—
17#16840326—24no—
18C2×#14519773C2212yes—
19C2×#14519766C2212yes—
20C2×#14519751—22no—
21C2×#14519745—22no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24#16839768—24no—
25#16839758—24no—
26C2×#14519701C2212yescomplemented
27C2×#14519699C2214yes—
28C2×#14519698C2212yescomplemented
29#16839669—22no—
30#16839669C2214yescomplemented
31#16839625—22no—
32#16839516C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C23×Q8×C61C2211yescharacteristic
35C2×#137×C61—24no—
36C2×#137×C61—24no—
37C2×#137×C61—24no—
38C22×Q16×C61C2212yescomplemented
39C22×SD16×C61C2212yescomplemented
40C22×M16×C61C2211yescharacteristic
41C24×D61—21no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244D414yes—
49C22×D244C2312yes—
50C22×C22⋊D61—24no—
51C2×#11758201—232no—
52C2×#11758201—28no—
53C2×#11758201—28no—
54C2×#11758201—28no—
55C22×#92418—22no—
56C22×#92418—22no—
57C22×#92418D414yescomplemented
58C22×#92418C2312yes—
59#14519773—24no—
60#14519773—24no—
61#14519766—24no—
62#14519766—24no—
63#14519751—216no—
64#14519745—216no—
65#14519736—22no—
66#14519736—22no—
67#14519736—22no—
68#14519736—22no—
69C2×C4⋊Dic61D412yes—
70C2×C4⋊Dic61C2312yes—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61D412yes—
73C2×#11758198—24no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61C2312yes—
76#14519701—24no—
77#14519699—28no—
78#14519698—24no—
79#14519684—28no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61D414yes—
86C22×Q8×C61—21no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61D414yescomplemented
93C22×Q8×C61C2312yes—
94C22×Q8×C61C2311yescharacteristic
95C22×D4×C61—21no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100#137×C61—232no—
101C2×Q16×C61—24no—
102C2×Q16×C61—24no—
103C2×Q16×C61—24no—
104C2×SD16×C61—24no—
105C2×SD16×C61—24no—
106C2×SD16×C61—24no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61D414yescomplemented
110C22×C8×C61C2312yes—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—22no—
114C23×D61—24no—
115C23×D61—42no—
116C23×D61—42no—
117C23×D61—84no—
118#41×C61—28no—
119#41×C61—28no—
120#41×C61—44no—
121#41×C61—44no—
122C2×Q8×C61—24no—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—28no—
126C2×Q8×C61—28no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61—44no—
137C2×Q8×C61—44no—
138C2×Q8×C61—42no—
139C2×Q8×C61C2×D414yes—
140C2×Q8×C61C2×D412yes—
141C2×D4×C61—24no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61C2×D412yes—
150C22×C4×C61—24no—
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—24no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—42no—
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D412yes—
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C2×C22⋊D61—216no—
167C2×C22⋊D61—88no—
168C22×Dic61—22no—
169C22×Dic61C2×D414yes—
170C2×D244—28no—
171C2×D244—28no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—24no—
177C2×D244—22no—
178C2×D244—44no—
179C2×D244—44no—
180C2×D244—42no—
181C2×D244—42no—
182C2×D244C2×D414yes—
183Q16×C61—216no—
184SD16×C61—216no—
185M16×C61—28no—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61C2×D414yes—
189C22⋊Dic61—24no—
190C22⋊Dic61—42no—
191#11758201—232no—
192#11758198—216no—
193C4⋊Dic61—24no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—42no—
199C2×#92418—28no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418C2×D414yes—
203C23×C61—24no—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—44no—
208C23×C61#12012yes—
209C23×C61#12112yes—
210C23×C61#12712yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—24no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—48no—
216C22×D61—48no—
217C22×D61—88no—
218C22×D61—88no—
219C22×D61—88no—
220Q8×C61—24no—
221Q8×C61—24no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—42no—
226Q8×C61—42no—
227Q8×C61—41no—
228Q8×C61—48no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—48no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—48no—
241C2×C4×C61—48no—
242C2×C4×C61#12014yes—
243C2×C4×C61#12014yes—
244C2×C4×C61#12012yes—
245C2×C4×C61#12114yes—
246C2×C4×C61#12114yes—
247C2×C4×C61#12112yes—
248C2×C4×C61#12712yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C22⋊D61—816no—
256C2×Dic61—24no—
257C2×Dic61—28no—
258C2×Dic61—42no—
259C2×Dic61—44no—
260D244—24no—
261D244—24no—
262D244—42no—
263D244—42no—
264D244—48no—
265C8×C61—24no—
266#92418—24no—
267#23423—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—24no—
271C22×C61—42no—
272C22×C61—44no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61#39214yes—
276C22×C61C2×#12011yescharacteristic
277C22×C61C2×#12111yescharacteristic
278C22×C61C2×#12711yescharacteristic
279C22×C61#57714yes—
280C2×D61—44no—
281C2×D61—48no—
282C2×D61—816no—
283C2×D61—84no—
284C2×D61—84no—
285C4×C61—42no—
286C4×C61—44no—
287C4×C61#39214yes—
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12011yescharacteristic
290C4×C61C2×#12111yescharacteristic
291C4×C61C2×#12111yescharacteristic
292Dic61—44no—
293Dic61—82no—
294C22×#137D6111yescharacteristic, complemented
295C23×SD16—611no—
296C2×#535—611no—
297C2×#418—611no—
298C2×#416—611no—
299#2310—611no—
300#2015—614no—
301#2014—614no—
302#1729—611no—
303C2×C61—24no—
304C2×C61—82no—
305C2×C61C2×#39211yescharacteristic
306C2×C61#300912yes—
307C2×C61#304312yes—
308C2×C61#315412yes—
309D61—84no—
310C22×#41C2×D6111yescharacteristic
311C23×Q8C2×D6111yescharacteristic
312C23×D4—611no—
313C2×#137—24no—
314C2×#137—24no—
315C2×#137—24no—
316C22×Q16C2×D6112yescomplemented
317C22×SD16—614no—
318C22×SD16—1224no—
319C22×SD16—1224no—
320C22×SD16—1224no—
321C22×SD16C2×D6112yescomplemented
322C22×M16C2×D6111yescharacteristic
323C23×C8—611no—
324#535—1224no—
325#535—1224no—
326C2×#127—612no—
327C2×#124—612no—
328C2×#121—1222no—
329C2×#120—1222no—
330C2×#117—611no—
331C2×#117—611no—
332#418—1224no—
333#416—1224no—
334C2×#103—612no—
335C2×#102—612no—
336C2×#102—614no—
337#391—614no—
338#391—1222no—
339#381—1222no—
340#336—614no—
341C61#2342311yescharacteristic, complemented, Sylow
342C25—1221no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41C22⋊D6114yes—
349C22×Q8—21no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8C22⋊D6114yescomplemented
356C22×Q8C22×D6111yescharacteristic
357C22×Q8C22×D6112yes—
358C22×D4—21no—
359C22×D4—614no—
360C22×D4—612no—
361C22×D4—1224no—
362C22×D4—1222no—
363C22×D4—1221no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4C22×D6111yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4C22×D6111yescharacteristic
372#137—232no—
373C2×Q16—24no—
374C2×Q16—24no—
375C2×Q16—24no—
376C2×SD16—24no—
377C2×SD16—24no—
378C2×SD16—24no—
379C2×SD16—1228no—
380C2×SD16—1228no—
381C2×SD16—12232no—
382C2×SD16—1228no—
383C2×M16—22no—
384C2×M16—22no—
385C2×M16C22⋊D6114yescomplemented
386C22×C8—614no—
387C22×C8—612no—
388C22×C8—1222no—
389C22×C8—1222no—
390C22×C8C22×D6112yes—
391#127—1224no—
392#127—1224no—
393#124—1224no—
394#124—1224no—
395#121—12216no—
396#120—12216no—
397#117—1222no—
398#117—1222no—
399#117—1222no—
400#117—1222no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—612no—
403C2×C22⋊C4—612no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1221no—
406C2×C42—612no—
407C2×C42—1221no—
408#103—1224no—
409#102—1224no—
410#102—1228no—
411#95—1228no—
412C24—21no—
413C24—1224no—
414C24—1222no—
415C24—4884no—
416C24—2442no—
417C24—2442no—
418C24D4×D6111yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8C2×C22⋊D6112yes—
441C2×Q8C2×C22⋊D6114yes—
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—614no—
451C2×D4—12216no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1228no—
458C2×D4—1224no—
459C2×D4—1228no—
460C2×D4—4888no—
461C2×D4—2444no—
462C2×D4—2444no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4C2×C22⋊D6112yes—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—614no—
475C22×C4—1222no—
476C22×C4D4×D6112yes—
477C22×C4D4×D6112yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4#1175822412yes—
480C22×C4C2×C22⋊D6112yes—
481C22×C4C2×C22⋊D6112yes—
482C22×C4C2×C22⋊D6112yes—
483C22×C4C23×D6111yescharacteristic
484Q16—216no—
485SD16—216no—
486SD16—12232no—
487M16—28no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—614no—
491C2×C8—1222no—
492C2×C8—1222no—
493C2×C8—1228no—
494C2×C8C2×C22⋊D6114yes—
495C4⋊C4—1224no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C22⋊C4—12216no—
499C22⋊C4—1224no—
500C22⋊C4—2442no—
501C42—1222no—
502C42—1222no—
503C42—2442no—
504C23—24no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—44no—
509C23—1224no—
510C23—4888no—
511C23—4888no—
512C23—4888no—
513C23—2442no—
514C23—2442no—
515C23—2448no—
516C23—2448no—
517C23#1451974512yes—
518C23#1451974912yes—
519C23#1451977312yes—
520C23C2×D4×D6111yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—42no—
527Q8—41no—
528Q8—42no—
529Q8—48no—
530D4—22no—
531D4—22no—
532D4—48no—
533D4—41no—
534D4—41no—
535D4—1224no—
536D4—1224no—
537D4—48816no—
538D4—2442no—
539D4—2442no—
540D4—2448no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—1224no—
550C2×C4—1228no—
551C2×C4—2442no—
552C2×C4—2444no—
553C2×C4#1451974514yes—
554C2×C4#1451974512yes—
555C2×C4#1451974614yes—
556C2×C4#1451974814yes—
557C2×C4#1451974914yes—
558C2×C4#1451974912yes—
559C2×C4#1451977312yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C8—24no—
567C8—1224no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—42no—
572C22—44no—
573C22—82no—
574C22—82no—
575C22—48816no—
576C22—4884no—
577C22—4884no—
578C22—2444no—
579C22—2448no—
580C22#1683967014yes—
581C22C2×#1451974511yescharacteristic
582C22C2×#1451974911yescharacteristic
583C22C2×#1451977311yescharacteristic
584C22#1684057114yes—
585C4—42no—
586C4—44no—
587C4—4882no—
588C4—2444no—
589C4#1683967014yes—
590C4C2×#1451974511yescharacteristic
591C4C2×#1451974611yescharacteristic
592C4C2×#1451974811yescharacteristic
593C4C2×#1451974911yescharacteristic
594C2—24no—
595C2—82no—
596C2—4884no—
597C2C2×#1683967011yescharacteristic
598C2#2145509512yes—
599C2#2145534212yes—
600C2#2145629812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
hypercenterC22×#137D61
hypocenterC61#23423
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C23×#11758201C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839758C2semidirect
5. #21448405C2semidirect
6. #21446910C2semidirect
7. #21446905C2semidirect
8. #21445517C2semidirect
9. C22×#137×C61C2semidirect
26. C2×#14519701C22semidirect
28. C2×#14519698C22semidirect
30. #16839669C22semidirect
32. #16839516C22semidirect
38. C22×Q16×C61C22semidirect
39. C22×SD16×C61C22semidirect
57. C22×#92418D4semidirect
92. C22×Q8×C61D4semidirect
109. C2×M16×C61D4semidirect
294. C22×#137D61semidirect
316. C22×Q16C2×D61semidirect
321. C22×SD16C2×D61semidirect
341. C61#23423semidirect
355. C22×Q8C22⋊D61semidirect
385. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C23×#117582012
C2×#168403262
C2×#168397682
C2×#168397582
#214484052
#214469102
#214469052
#214455172
C22×#137×C612
#2342361