Finite Group Databasebeta

← #29707105

Subgroups — #29707105

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
1C22×#14519807C211yesmaximal, characteristic, complemented
2C23×#11758190C211yesmaximal, characteristic, complemented
3C2×#16840326C211yesmaximal, characteristic
4C2×#16839766C211yesmaximal, characteristic, complemented
5#21448403C211yesmaximal, characteristic, complemented
6#21446914C214yesmaximal, complemented
7#21446908C214yesmaximal, complemented
8#21445515C211yesmaximal, characteristic, complemented
9C2×C61×#416C211yesmaximal, characteristic, complemented
10C23×Dic122C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519807—24no—
13C2×#14519807—24no—
14C2×#14519807—24no—
15C22×#11758202C2212yescomplemented
16C22×#11758190—24no—
17C22×#11758190—24no—
18C22×#11758190—24no—
19C22×#11758190C2214yes—
20C22×#11758200C2212yescomplemented
21C22×#11758194C2211yescharacteristic
22#16840326—24no—
23#16840326—24no—
24C2×#14519773C2212yes—
25C2×#14519766C2212yes—
26C2×#14519751—22no—
27C2×#14519745—22no—
28C2×#14519736C2211yescharacteristic
29#16839766—24no—
30C2×#14519697C2214yes—
31#16839668—22no—
32#16839667C2214yescomplemented
33#16839623—22no—
34#16839516C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C23×C8×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#416—24no—
39C2×#103×C61C2212yescomplemented
40C2×#102×C61C2212yescomplemented
41C24×D61—21no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—22no—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122—22no—
48C22×Dic122D414yescomplemented
49C22×Dic122C2312yes—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—22no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244—22no—
56C22×D244D414yes—
57C22×D244C2312yes—
58C22×C22⋊D61—24no—
59C23×Dic61—21no—
60#14519807—232no—
61C2×#11758202—24no—
62C2×#11758202—24no—
63C2×#11758202—24no—
64C2×#11758190—232no—
65C2×#11758190—28no—
66C2×#11758190—28no—
67C2×#11758190—28no—
68C2×#11758200—24no—
69C2×#11758200—24no—
70C2×#11758200—24no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194D414yescomplemented
74C22×#92418C2312yes—
75#14519773—24no—
76#14519773—24no—
77#14519766—24no—
78#14519766—24no—
79#14519751—216no—
80#14519745—216no—
81#14519736—22no—
82#14519736—22no—
83C2×C4⋊Dic61D412yes—
84C2×C22⋊Dic61D412yes—
85C2×#11758198—24no—
86C2×C4×Dic61C2312yes—
87#14519697—28no—
88#14519684—28no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61D414yes—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61D414yescomplemented
103C22×C8×C61C2312yes—
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109#103×C61—24no—
110#102×C61—24no—
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—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61C2×D412yes—
128C2×D4×C61—24no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—22no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—216no—
151C2×C22⋊D61—88no—
152C22×Dic61—24no—
153C22×Dic61—22no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C2×D414yes—
157C2×D244—28no—
158C2×D244—28no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—24no—
164C2×D244—22no—
165C2×D244—44no—
166C2×D244—44no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244C2×D414yes—
170C2×Dic122—28no—
171C2×Dic122—28no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122—24no—
177C2×Dic122—22no—
178C2×Dic122—42no—
179C2×Dic122—44no—
180C2×Dic122—44no—
181C2×Dic122—42no—
182C2×Dic122C2×D414yes—
183C2×C8×C61—28no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61C2×D414yes—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C22⋊C4×C61—42no—
190C42×C61—42no—
191C22⋊Dic61—24no—
192#11758202—216no—
193#11758200—216no—
194#11758198—216no—
195C4⋊Dic61—24no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198#11758194—28no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418C2×D414yes—
202#11758190—232no—
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—22no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—48no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—48no—
233C2×C4×C61—42no—
234C2×C4×C61—44no—
235C2×C4×C61#12014yes—
236C2×C4×C61#12014yes—
237C2×C4×C61#12012yes—
238C2×C4×C61#12114yes—
239C2×C4×C61#12114yes—
240C2×C4×C61#12112yes—
241C2×C4×C61#12712yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—816no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—28no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—48no—
255D244—24no—
256D244—24no—
257D244—42no—
258D244—42no—
259D244—48no—
260Dic122—24no—
261Dic122—24no—
262Dic122—42no—
263Dic122—42no—
264Dic122—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—82no—
287C4×C61#39214yes—
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12011yescharacteristic
290C4×C61C2×#12111yescharacteristic
291C4×C61C2×#12111yescharacteristic
292Dic61—44no—
293Dic61—44no—
294C22×#137—611no—
295C23×SD16—611no—
296C2×#535—611no—
297C2×#418—611no—
298C2×#416D6111yescharacteristic, complemented
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×Q8—611no—
312C23×D4—611no—
313C2×#137—1224no—
314C2×#137—1224no—
315C2×#137—1224no—
316C22×Q16—612no—
317C22×SD16—612no—
318C22×SD16—614no—
319C22×SD16—1224no—
320C22×SD16—1224no—
321C22×SD16—1224no—
322C22×M16—611no—
323C23×C8C2×D6111yescharacteristic
324#535—1224no—
325#535—1224no—
326C2×#127—612no—
327C2×#124—612no—
328C2×#121—1222no—
329C2×#120—1222no—
330C2×#117—611no—
331C2×#117C2×D6111yescharacteristic
332#418—1224no—
333#416—24no—
334C2×#103C2×D6112yescomplemented
335C2×#102—614no—
336C2×#102C2×D6112yescomplemented
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×#41D24414yes—
349C22×Q8—614no—
350C22×Q8—612no—
351C22×Q8—1222no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8—1222no—
357C22×Q8C22×D6111yescharacteristic
358C22×D4—21no—
359C22×D4—614no—
360C22×D4—612no—
361C22×D4—1224no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1221no—
367C22×D4—1222no—
368C22×D4C22×D6111yescharacteristic
369C23×C4—21no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372#137—12232no—
373C2×Q16—1224no—
374C2×Q16—1224no—
375C2×Q16—1224no—
376C2×SD16—1228no—
377C2×SD16—1228no—
378C2×SD16—12232no—
379C2×SD16—1224no—
380C2×SD16—1224no—
381C2×SD16—1224no—
382C2×SD16—1228no—
383C2×M16—614no—
384C2×M16—1222no—
385C2×M16—1222no—
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—612no—
389C22×C8C22⋊D6114yescomplemented
390C22×C8C22×D6112yes—
391#127—1224no—
392#127—1224no—
393#124—1224no—
394#124—1224no—
395#121—12216no—
396#120—12216no—
397#117—22no—
398#117—22no—
399#117—1222no—
400#117—1222no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—612no—
405C2×C22⋊C4—1224no—
406C2×C42—21no—
407C2×C42—612no—
408#103—24no—
409#102—24no—
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—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—614no—
429C2×Q8—1222no—
430C2×Q8—1224no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×Q8—1222no—
435C2×Q8—1228no—
436C2×Q8—1228no—
437C2×Q8—2442no—
438C2×Q8—2444no—
439C2×Q8—2444no—
440C2×Q8—2442no—
441C2×Q8C2×D24412yes—
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—2442no—
463C2×D4—2444no—
464C2×D4—2442no—
465C2×D4C2×D24412yes—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—614no—
472C22×C4—1222no—
473C22×C4—1224no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4C2×D24412yes—
477C22×C4#1175822212yes—
478C22×C4D4×D6112yes—
479C22×C4D4×D6112yes—
480C22×C4D4×D6112yes—
481C22×C4D4×D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483C22×C4C23×D6111yescharacteristic
484Q16—12216no—
485SD16—12232no—
486SD16—12216no—
487M16—1228no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—28no—
491C2×C8—614no—
492C2×C8—1222no—
493C2×C8—1222no—
494C2×C8C2×C22⋊D6114yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—1224no—
498C22⋊C4—42no—
499C22⋊C4—12216no—
500C22⋊C4—1224no—
501C42—42no—
502C42—1222no—
503C42—1222no—
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#1451974712yes—
518C23#1451975012yes—
519C23#1451977312yes—
520C23C2×D4×D6111yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—1224no—
526Q8—1224no—
527Q8—2442no—
528Q8—2442no—
529Q8—2448no—
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—41no—
543C2×C4—41no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1228no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2448no—
553C2×C4#1451974514yes—
554C2×C4#1451974714yes—
555C2×C4#1451974712yes—
556C2×C4#1451975014yes—
557C2×C4#1451975012yes—
558C2×C4#1451975114yes—
559C2×C4#1451977312yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C22×D24411yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
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#1683967114yes—
581C22C2×#1451974711yescharacteristic
582C22C2×#1451975011yescharacteristic
583C22C2×#1451977311yescharacteristic
584C22#1684056914yes—
585C4—42no—
586C4—82no—
587C4—2444no—
588C4—2444no—
589C4#1683967114yes—
590C4C2×#1451974511yescharacteristic
591C4C2×#1451974711yescharacteristic
592C4C2×#1451975011yescharacteristic
593C4C2×#1451975111yescharacteristic
594C2—24no—
595C2—82no—
596C2—4884no—
597C2C2×#1683967111yescharacteristic
598C2#2145509912yes—
599C2#2145534312yes—
600C2#2145629612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
hypercenterC2×#416D61
hypocenterC61#23423
LCSG ▷ C22×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
1. C22×#14519807C2semidirect
2. C23×#11758190C2semidirect
4. C2×#16839766C2semidirect
5. #21448403C2semidirect
6. #21446914C2semidirect
7. #21446908C2semidirect
8. #21445515C2semidirect
9. C2×C61×#416C2semidirect
15. C22×#11758202C22semidirect
20. C22×#11758200C22semidirect
32. #16839667C22semidirect
34. #16839516C22semidirect
39. C2×#103×C61C22semidirect
40. C2×#102×C61C22semidirect
48. C22×Dic122D4semidirect
73. C2×#11758194D4semidirect
102. C22×C8×C61D4semidirect
298. C2×#416D61semidirect
334. C2×#103C2×D61semidirect
336. C2×#102C2×D61semidirect
341. C61#23423semidirect
389. C22×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C23×#117581902
C2×#168403262
C2×#168397662
#214484032
#214469142
#214469082
#214455152
C2×C61×#4162
#2342361