Finite Group Databasebeta

← #29707106

Subgroups — #29707106

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×#14519810C211yesmaximal, characteristic, complemented
2C23×#11758200C211yesmaximal, characteristic, complemented
3C2×#16840329C211yesmaximal, characteristic
4C2×#16839756C211yesmaximal, characteristic, complemented
5#21448403C211yesmaximal, characteristic, complemented
6#21446915C214yesmaximal, complemented
7#21446908C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C2×C61×#418C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12C2×#14519810—24no—
13C2×#14519810—24no—
14C2×#14519810—24no—
15C22×Dic244C2212yescomplemented
16C22×#11758190C2212yescomplemented
17C22×#11758200—24no—
18C22×#11758200—24no—
19C22×#11758200—24no—
20C22×#11758200C2214yes—
21C23×#92418C2211yescharacteristic
22#16840329—24no—
23#16840329—24no—
24C2×#14519773C2212yes—
25C2×#14519764C2212yes—
26C2×#14519748—22no—
27C2×#14519746—22no—
28C2×#14519736C2211yescharacteristic
29#16839756—24no—
30C2×#14519700C2212yescomplemented
31C2×#14519697C2212yescomplemented
32#16839668—22no—
33#16839667C2214yescomplemented
34#16839623—22no—
35#16839516C2214yescomplemented
36C23×D4×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#418—24no—
40C2×#102×C61C2214yes—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222D414yes—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—22no—
50C22×Dic122—22no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122D414yescomplemented
54C22×Dic122C2311yescharacteristic
55C22×Dic122C2312yes—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C22×C22⋊D61—24no—
59C23×Dic61—21no—
60C22×C4×D61—21no—
61#14519810—232no—
62C2×Dic244—24no—
63C2×Dic244—24no—
64C2×Dic244—24no—
65C2×#11758190—24no—
66C2×#11758190—24no—
67C2×#11758190—24no—
68C2×#11758200—232no—
69C2×#11758200—28no—
70C2×#11758200—28no—
71C2×#11758200—28no—
72C22×#92418—22no—
73C22×#92418—22no—
74C22×#92418D414yescomplemented
75C22×#92418C2312yes—
76#14519773—24no—
77#14519773—24no—
78#14519764—24no—
79#14519764—24no—
80#14519748—216no—
81#14519746—216no—
82#14519736—22no—
83#14519736—22no—
84C2×C4⋊Dic61D412yes—
85C2×C22⋊Dic61—24no—
86C2×C22⋊Dic61D412yes—
87C2×C4×Dic61C2312yes—
88#14519700—24no—
89#14519697—24no—
90#14519684—28no—
91C25×C61—21no—
92C22×D4×C61—22no—
93C22×D4×C61—22no—
94C22×D4×C61—22no—
95C22×D4×C61—21no—
96C22×D4×C61—22no—
97C22×D4×C61—22no—
98C22×D4×C61D414yes—
99C22×D4×C61C2312yes—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—22no—
102C2×M16×C61—22no—
103C2×M16×C61D414yescomplemented
104C22×C8×C61C2312yes—
105#117×C61—22no—
106#117×C61—22no—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110#102×C61—28no—
111C24×C61—22no—
112C24×C61—24no—
113C24×C61—42no—
114C24×C61—42no—
115C24×C61—84no—
116C24×C61C2×D411yescharacteristic
117C23×D61—21no—
118C2×D4×C61—28no—
119C2×D4×C61—24no—
120C2×D4×C61—28no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—44no—
127C2×D4×C61—44no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61C2×D414yes—
131C22×C4×C61—22no—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—216no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C2×C22⋊D61—88no—
145C22×Dic61—24no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61C2×D414yes—
151#11758222—28no—
152#11758222—28no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C2×D412yes—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×Dic122—28no—
165C2×Dic122—28no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—24no—
175C2×Dic122—21no—
176C2×Dic122—22no—
177C2×Dic122—42no—
178C2×Dic122—44no—
179C2×Dic122—44no—
180C2×Dic122—42no—
181C2×Dic122C2×D414yes—
182C2×Dic122C2×D412yes—
183M16×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—216no—
192C22⋊Dic61—24no—
193#11758200—232no—
194C4⋊Dic61—24no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C2×#92418—28no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418C2×D414yes—
201Dic244—216no—
202#11758190—216no—
203C23×C61—24no—
204C23×C61—24no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—48no—
208C23×C61—48no—
209C23×C61—88no—
210C23×C61—88no—
211C23×C61—88no—
212C23×C61#12012yes—
213C23×C61#12112yes—
214C23×C61#12712yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—44no—
220D4×C61—24no—
221D4×C61—24no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—48no—
225C2×C4×C61—42no—
226C2×C4×C61—44no—
227C2×C4×C61#12014yes—
228C2×C4×C61#12014yes—
229C2×C4×C61#12012yes—
230C2×C4×C61#12114yes—
231C2×C4×C61#12114yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12712yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—48no—
241C22⋊D61—816no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—24no—
245C2×Dic61—28no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248C2×Dic61—42no—
249C2×Dic61—41no—
250C2×Dic61—48no—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—48no—
256Dic122—24no—
257Dic122—22no—
258Dic122—24no—
259Dic122—22no—
260Dic122—41no—
261Dic122—42no—
262Dic122—42no—
263Dic122—41no—
264Dic122—48no—
265C8×C61—24no—
266#92418—24no—
267#23423—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—24no—
271C22×C61—44no—
272C22×C61—48no—
273C22×C61—816no—
274C22×C61—84no—
275C22×C61—84no—
276C22×C61#39214yes—
277C22×C61C2×#12011yescharacteristic
278C22×C61C2×#12111yescharacteristic
279C22×C61C2×#12711yescharacteristic
280C22×C61#57714yes—
281C2×D61—42no—
282C2×D61—44no—
283C2×D61—82no—
284C2×D61—82no—
285C4×C61—82no—
286C4×C61#39214yes—
287C4×C61C2×#12011yescharacteristic
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12111yescharacteristic
290C4×C61C2×#12111yescharacteristic
291Dic61—44no—
292Dic61—42no—
293Dic61—44no—
294C22×#137—611no—
295C23×SD16—611no—
296C2×#535—611no—
297C2×#418D6111yescharacteristic, complemented
298C2×#416—611no—
299#2310—611no—
300#2015—614no—
301#2014—614no—
302#1729—611no—
303C2×C61—24no—
304C2×C61—84no—
305C2×C61C2×#39211yescharacteristic
306C2×C61#300912yes—
307C2×C61#304312yes—
308C2×C61#315412yes—
309D61—82no—
310C22×#41—611no—
311C23×Q8—611no—
312C23×D4C2×D6111yescharacteristic
313C2×#137—1224no—
314C2×#137—1224no—
315C2×#137—1224no—
316C22×Q16—612no—
317C22×SD16—614no—
318C22×SD16—612no—
319C22×SD16—1224no—
320C22×SD16—1224no—
321C22×SD16—1224no—
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×#117C2×D6111yescharacteristic
332#418—24no—
333#416—1224no—
334C2×#103—612no—
335C2×#102—612no—
336C2×#102C2×D6114yes—
337#391—614no—
338#391—1222no—
339#381—1222no—
340#336—614no—
341C61#2342311yescharacteristic, complemented, Sylow
342C25—21no—
343C2×#41—614no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C2×#41—1222no—
349C22×Q8—614no—
350C22×Q8—611no—
351C22×Q8—612no—
352C22×Q8—1222no—
353C22×Q8—1221no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8—1222no—
357C22×Q8—1222no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—21no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—611no—
365C22×D4—1224no—
366C22×D4—1221no—
367C22×D4D24414yes—
368C22×D4C22×D6112yes—
369C23×C4—1221no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372#137—12232no—
373C2×Q16—1224no—
374C2×Q16—1224no—
375C2×Q16—1224no—
376C2×SD16—1224no—
377C2×SD16—1224no—
378C2×SD16—1228no—
379C2×SD16—1228no—
380C2×SD16—12232no—
381C2×SD16—1228no—
382C2×SD16—1224no—
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—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—1224no—
409#102—28no—
410#102—1224no—
411#95—1228no—
412C24—22no—
413C24—24no—
414C24—42no—
415C24—42no—
416C24—84no—
417C24—1221no—
418C24D4×D6111yescharacteristic
419#41—1228no—
420#41—1228no—
421#41—2444no—
422#41—2444no—
423C2×Q8—612no—
424C2×Q8—614no—
425C2×Q8—1221no—
426C2×Q8—1222no—
427C2×Q8—1224no—
428C2×Q8—1221no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1221no—
432C2×Q8—1222no—
433C2×Q8—1221no—
434C2×Q8—1222no—
435C2×Q8—1221no—
436C2×Q8—1228no—
437C2×Q8—1228no—
438C2×Q8—2442no—
439C2×Q8—2444no—
440C2×Q8—2444no—
441C2×Q8—2442no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—28no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—612no—
455C2×D4—12216no—
456C2×D4—1224no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—4888no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4C2×D24414yes—
466C22×C4—22no—
467C22×C4—614no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—2442no—
473C22×C4—2442no—
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—12216no—
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—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—24no—
506C23—42no—
507C23—42no—
508C23—48no—
509C23—48no—
510C23—88no—
511C23—88no—
512C23—88no—
513C23—1222no—
514C23—2441no—
515C23—2441no—
516C23—2444no—
517C23#1451974712yes—
518C23#1451975012yes—
519C23#1451977312yes—
520C23C2×D4×D6111yescharacteristic
521Q8—1224no—
522Q8—1222no—
523Q8—1224no—
524Q8—1222no—
525Q8—2441no—
526Q8—2442no—
527Q8—2442no—
528Q8—2441no—
529Q8—2448no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—48no—
535D4—1222no—
536D4—1222no—
537D4—48816no—
538D4—2448no—
539D4—2441no—
540D4—2441no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1228no—
547C2×C4—2442no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2442no—
551C2×C4—2448no—
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—44no—
572C22—48no—
573C22—816no—
574C22—84no—
575C22—84no—
576C22—4882no—
577C22—4882no—
578C22—2442no—
579C22—2444no—
580C22#1683967114yes—
581C22C2×#1451974711yescharacteristic
582C22C2×#1451975011yescharacteristic
583C22C2×#1451977311yescharacteristic
584C22#1684057214yes—
585C4—82no—
586C4—2442no—
587C4—2444no—
588C4—2444no—
589C4#1683967114yes—
590C4C2×#1451974511yescharacteristic
591C4C2×#1451974711yescharacteristic
592C4C2×#1451975011yescharacteristic
593C4C2×#1451975111yescharacteristic
594C2—24no—
595C2—84no—
596C2—4882no—
597C2C2×#1683967111yescharacteristic
598C2#2145509712yes—
599C2#2145533912yes—
600C2#2145629312yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC23×C61C22×D4
prime radicalC2×#16840329C2
hypercenterC2×#418D61
hypocenterC61#23423
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
1. C22×#14519810C2semidirect
2. C23×#11758200C2semidirect
4. C2×#16839756C2semidirect
5. #21448403C2semidirect
6. #21446915C2semidirect
7. #21446908C2semidirect
8. #21445517C2semidirect
9. C2×C61×#418C2semidirect
15. C22×Dic244C22semidirect
16. C22×#11758190C22semidirect
30. C2×#14519700C22semidirect
31. C2×#14519697C22semidirect
33. #16839667C22semidirect
35. #16839516C22semidirect
53. C22×Dic122D4semidirect
74. C22×#92418D4semidirect
103. C2×M16×C61D4semidirect
297. C2×#418D61semidirect
341. C61#23423semidirect
385. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198102
C23×#117582002
C2×#168403292
C2×#168397562
#214484032
#214469152
#214469082
#214455172
C2×C61×#4182
#2342361