Finite Group Databasebeta

← #30615723

Subgroups — #30615723

Order 15616.

2030 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
1C2×#16840829C211yesmaximal, characteristic, complemented
2C2×#16840741C212yesmaximal, complemented
3#21462292C211yesmaximal, characteristic, complemented
4#21461841C214yesmaximal, complemented
5#21461840C214yesmaximal, complemented
6#21461519C212yesmaximal, complemented
7C2×#16840328C211yesmaximal, characteristic
8#16840829—28no—
9C2×D4×#6923—24no—
10C2×#14519856C2212yes—
11C2×#14519856C2212yes—
12C22×C22⋊#6923C2214yes—
13#16840741—28no—
14C2×#14519836C2214yes—
15C2×#14519835C2214yes—
16#16840714—24no—
17#16840703C2214yes—
18#16840640C2218yes—
19C22×#11758222C2211yescharacteristic
20C22×D4×D61C2211yescharacteristic
21#16840328—24no—
22#16840328—24no—
23C2×#14519773C412yescomplemented
24C2×#14519763C412yescomplemented
25C2×#14519752C411yescharacteristic, complemented
26C2×#14519751C411yescharacteristic, complemented
27C2×#14519748C411yescharacteristic, complemented
28C2×#14519747C2212yes—
29C2×#14519745C2212yes—
30C2×#14519735C2211yescharacteristic
31C2×#121×C61C411yescharacteristic, complemented
32C23×#6923—24no—
33D4×#6923—232no—
34#14519856—24no—
35#14519856—24no—
36#14519856—24no—
37#14519856D418yescomplemented
38C2×C4⋊#6923—22no—
39C2×C4⋊#6923—22no—
40C2×C22⋊#6923—24no—
41C2×C22⋊#6923—216no—
42C2×C22⋊#6923—24no—
43C2×C22⋊#6923—24no—
44C2×C22⋊#6923C2314yes—
45C2×C22⋊#6923C2314yes—
46C2×C4×#6923—22no—
47C2×C4×#6923—22no—
48#14519836—28no—
49#14519835—28no—
50C24×D61C2312yes—
51C2×#11758222—24no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222—24no—
55C22×Dic122C2×C411yescharacteristic, complemented
56C22×D244C2×C411yescharacteristic
57C22×D244C2×C412yescomplemented
58C22×D244C2311yescharacteristic
59C22×C22⋊D61C2×C412yes—
60C22×C22⋊D61C2×C411yescharacteristic, complemented
61C22×C22⋊D61C2×C412yescomplemented
62C22×C22⋊D61C2×C412yescomplemented
63C22×C22⋊D61C2311yescharacteristic
64C2×D4×D61—24no—
65C2×D4×D61—24no—
66C2×D4×D61D414yes—
67C2×D4×D61D414yes—
68C22×C4×D61C2×C411yescharacteristic, complemented
69C22×C4×D61C2311yescharacteristic
70#14519773—28no—
71#14519763—28no—
72#14519752—24no—
73#14519751—24no—
74#14519748—24no—
75#14519747—24no—
76#14519747—24no—
77#14519745—24no—
78#14519745—24no—
79#14519735—22no—
80#14519735—22no—
81C2×C4⋊Dic61C2×C411yescharacteristic, complemented
82C2×C22⋊Dic61C2×C412yescomplemented
83C2×#11758198C2×C412yescomplemented
84C2×#11758198C2×C412yescomplemented
85C2×#11758198C2312yes—
86C2×C4×Dic61C2×C412yescomplemented
87C22×D4×C61C2×C411yescharacteristic
88C22×D4×C61C2×C412yescomplemented
89C23×C4×C61C2×C411yescharacteristic, complemented
90#121×C61—24no—
91C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
92C2×C22⋊C4×C61C2×C412yescomplemented
93C24×C61C22×C411yescharacteristic
94C24×C61C22×C412yes—
95C23×D61—24no—
96C23×D61—24no—
97C23×D61—82no—
98C23×D61—82no—
99C23×D61C22×C411yescharacteristic
100C23×D61C22×C412yes—
101C23×D61C2×D414yes—
102C23×D61C2×D414yes—
103C23×D61C2411yescharacteristic
104C22×#6923—24no—
105C22×#6923—216no—
106C22×#6923—24no—
107C22×#6923—84no—
108C22×#6923—84no—
109C2×D4×C61—24no—
110C2×D4×C61—24no—
111C2×D4×C61—82no—
112C2×D4×C61C22⋊C414yes—
113C22×C4×C61—22no—
114C22×C4×C61—24no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C412yes—
117C22×C4×C61C22×C411yescharacteristic
118C2×C22⋊D61—22no—
119C2×C22⋊D61—21no—
120C2×C22⋊D61—21no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—28no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—82no—
128C2×C22⋊D61—82no—
129C2×C22⋊D61C22⋊C418yes—
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C411yescharacteristic
133D4×D61—28no—
134D4×D61—28no—
135D4×D61—28no—
136D4×D61—28no—
137#11758222—44no—
138#11758222—44no—
139#11758222—44no—
140#11758222—44no—
141C2×D244—24no—
142C2×D244—22no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—22no—
146C2×D244—24no—
147C2×D244—82no—
148C2×D244C22⋊C414yes—
149C2×C4×D61—22no—
150C2×C4×D61—24no—
151C2×C4×D61C2×D414yes—
152C2×C4×D61C2×D412yes—
153C2×C4×D61C2×D412yes—
154C2×C4×D61#4114yes—
155C2×Dic122—22no—
156C2×Dic122—24no—
157C22⋊#6923—216no—
158C22⋊#6923—24no—
159C22⋊#6923—24no—
160C22⋊#6923—24no—
161C22⋊#6923—84no—
162C22⋊#6923—84no—
163C22⋊#6923C2×D418yes—
164C4⋊#6923—28no—
165C4⋊#6923—44no—
166C4×#6923—28no—
167C4×#6923—44no—
168C4⋊C4×C61—22no—
169C22⋊C4×C61—24no—
170C22⋊Dic61—24no—
171#11758198—24no—
172#11758198—24no—
173#11758198—22no—
174#11758198—22no—
175C4⋊Dic61—22no—
176C4×Dic61—24no—
177C23×C61—24no—
178C23×C61—21no—
179C23×C61—22no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61C2×C22⋊C414yes—
185C23×C61C23×C411yescharacteristic
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—28no—
190C22×D61—21no—
191C22×D61—22no—
192C22×D61—44no—
193C22×D61—44no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—44no—
197C22×D61—44no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—88no—
203C22×D61—88no—
204C22×D61C2×C22⋊C414yes—
205C22×D61C22×D411yescharacteristic
206C22×D61C22×D411yescharacteristic
207C22×D61C22×D412yes—
208C22×D61C2×#4112yes—
209C2×#6923—28no—
210C2×#6923—44no—
211C2×#6923—44no—
212C2×#6923—44no—
213C2×#6923—44no—
214C2×#6923—44no—
215C2×#6923—816no—
216D4×C61—28no—
217D4×C61—84no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C4×D414yes—
225C22⋊D61—216no—
226C22⋊D61—41no—
227C22⋊D61—42no—
228C22⋊D61—41no—
229C22⋊D61—44no—
230C22⋊D61—84no—
231C22⋊D61—84no—
232C2×Dic61—22no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—42no—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C4×D414yes—
238D244—28no—
239D244—41no—
240D244—42no—
241D244—41no—
242D244—84no—
243C4×D61—44no—
244C4×D61#12014yes—
245C4×D61#12114yes—
246C4×D61#12114yes—
247C4×D61#12314yes—
248Dic122—44no—
249#20985—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—84no—
259C22×C61—82no—
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61C2×C4×D412yes—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—24no—
267C2×D61—24no—
268C2×D61—24no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—48no—
272C2×D61—48no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—88no—
277C2×D61—84no—
278C2×D61—82no—
279C2×D61C2×#12011yescharacteristic
280C2×D61C2×#12112yes—
281C2×D61C2×#12311yescharacteristic
282#6923—48no—
283#6923—84no—
284C4×C61—42no—
285C4×C61—41no—
286C4×C61#38614yes—
287Dic61—42no—
288Dic61—41no—
289Dic61#38614yes—
290C2×#535—611no—
291C2×#518—611no—
292C2×#410—612no—
293#2309—611no—
294#1898—614no—
295#1898—614no—
296#1740—612no—
297C2×C61—22no—
298C2×C61—44no—
299C2×C61—81no—
300C2×C61C2×#38611yescharacteristic
301C2×C61#288311yescharacteristic
302C2×C61#288311yescharacteristic
303D61—21no—
304D61—22no—
305D61—21no—
306D61—44no—
307D61—81no—
308C22×#41—611no—
309C23×D4—611no—
310#535—1224no—
311#535—1224no—
312C2×#127—612no—
313C2×#124—612no—
314C2×#121—611no—
315C2×#121—611no—
316C2×#121—611no—
317C2×#121#692311yescharacteristic, complemented
318C2×#120—612no—
319C2×#120—612no—
320#518—1228no—
321C2×C4×D4—1224no—
322C2×#117—611no—
323C2×#117—612no—
324C2×#117—612no—
325C22×C22⋊C4—614no—
326#410—1228no—
327C2×#99—614no—
328C2×#99—614no—
329#390—1224no—
330#379—614no—
331#342—618no—
332C61#2098511yescharacteristic, complemented, Sylow
333C25—612no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41—1224no—
337C2×#41—1224no—
338C22×Q8—611no—
339C22×D4—612no—
340C22×D4—614no—
341C22×D4—614no—
342C22×D4—611no—
343C22×D4—611no—
344C22×D4—612no—
345C22×D4—612no—
346C22×D4—611no—
347C22×D4—611no—
348C22×D4—612no—
349C22×D4—1224no—
350C22×D4—1224no—
351C22×D4C2×#692311yescharacteristic
352C22×D4C2×#692312yescomplemented
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1224no—
356C23×C4C2×#692311yescharacteristic, complemented
357#127—1228no—
358#124—1228no—
359#121—24no—
360#121—1224no—
361#121—1224no—
362#121—1224no—
363#120—1224no—
364#120—1224no—
365#120—1224no—
366#120—1224no—
367C4×D4—12232no—
368#117—618no—
369#117—1222no—
370#117—1222no—
371#117—1224no—
372#117—1224no—
373#117—1224no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4C2×#692311yescharacteristic, complemented
378C2×C22⋊C4—614no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—614no—
384C2×C22⋊C4—12216no—
385C2×C22⋊C4—1224no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4C2×#692312yescomplemented
389C2×C42—612no—
390C2×C42—1222no—
391C2×C42—1222no—
392#99—1228no—
393#99—1228no—
394C24—611no—
395C24—614no—
396C24—614no—
397C24—612no—
398C24—611no—
399C24—1224no—
400C24—1224no—
401C24—4882no—
402C24—4882no—
403C24C22×#692311yescharacteristic
404C24C22×#692312yes—
405#41—2444no—
406#41—2444no—
407#41—2444no—
408#41—2444no—
409C2×Q8—1222no—
410C2×Q8—1224no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—82no—
414C2×D4—618no—
415C2×D4—614no—
416C2×D4—1224no—
417C2×D4—1222no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—1228no—
422C2×D4—1228no—
423C2×D4—1228no—
424C2×D4—1228no—
425C2×D4—1228no—
426C2×D4—1222no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1222no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1222no—
434C2×D4—1224no—
435C2×D4—4882no—
436C2×D4—4882no—
437C2×D4—4882no—
438C2×D4C22⋊#692314yes—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—611no—
447C22×C4—612no—
448C22×C4—12216no—
449C22×C4—1224no—
450C22×C4—1222no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—4884no—
454C22×C4—4884no—
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692312yes—
457C22×C4C22×#692311yescharacteristic
458C4⋊C4—22no—
459C4⋊C4—1228no—
460C4⋊C4—1222no—
461C4⋊C4—2444no—
462C22⋊C4—24no—
463C22⋊C4—618no—
464C22⋊C4—1222no—
465C22⋊C4—1222no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C22⋊C4—1224no—
469C22⋊C4—12216no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—4884no—
474C22⋊C4—4884no—
475C42—1224no—
476C42—1228no—
477C42—2444no—
478C23—24no—
479C23—21no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—611no—
486C23—611no—
487C23—612no—
488C23—612no—
489C23—614no—
490C23—1228no—
491C23—1224no—
492C23—1224no—
493C23—1224no—
494C23—1222no—
495C23—1221no—
496C23—4888no—
497C23—4888no—
498C23—2444no—
499C23—2444no—
500C23—2442no—
501C23—2442no—
502C23—2444no—
503C23—2444no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23C2×C22⋊#692314yes—
509C23C23×#692311yescharacteristic
510Q8—2444no—
511D4—28no—
512D4—84no—
513D4—12216no—
514D4—1228no—
515D4—4884no—
516D4—4884no—
517D4—4884no—
518D4—2441no—
519D4—2442no—
520D4—2441no—
521D4—2444no—
522D4—2441no—
523D4—2442no—
524D4—2441no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—612no—
531C2×C4—614no—
532C2×C4—614no—
533C2×C4—614no—
534C2×C4—614no—
535C2×C4—614no—
536C2×C4—1228no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1222no—
540C2×C4—48816no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2444no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2442no—
548C2×C4C2×C22⋊#692312yes—
549C2×C4D4×#692314yes—
550C22—21no—
551C22—21no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—84no—
559C22—82no—
560C22—611no—
561C22—612no—
562C22—611no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—1224no—
568C22—1224no—
569C22—1224no—
570C22—4882no—
571C22—4884no—
572C22—4888no—
573C22—2442no—
574C22—2442no—
575C22—2448no—
576C22—2448no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22C22×C22⋊#692311yescharacteristic
581C22C2×D4×#692312yes—
582C4—42no—
583C4—41no—
584C4—614no—
585C4—4884no—
586C4—2448no—
587C4—2442no—
588C4—2441no—
589C4#1684071014yes—
590C2—22no—
591C2—44no—
592C2—81no—
593C2—1222no—
594C2—1221no—
595C2—1221no—
596C2—4881no—
597C2—2444no—
598C2C2×#1684071011yescharacteristic
599C2#2146241811yescharacteristic
600C2#2146241911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#121×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840328C2
cosocleC23×D61C24
hypercenterC2×#121#6923
hypocenterC61#20985
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16840829C2semidirect
2. C2×#16840741C2semidirect
3. #21462292C2semidirect
4. #21461841C2semidirect
5. #21461840C2semidirect
6. #21461519C2semidirect
23. C2×#14519773C4semidirect
24. C2×#14519763C4semidirect
25. C2×#14519752C4semidirect
26. C2×#14519751C4semidirect
27. C2×#14519748C4semidirect
31. C2×#121×C61C4semidirect
37. #14519856D4semidirect
55. C22×Dic122C2×C4semidirect
57. C22×D244C2×C4semidirect
60. C22×C22⋊D61C2×C4semidirect
61. C22×C22⋊D61C2×C4semidirect
62. C22×C22⋊D61C2×C4semidirect
68. C22×C4×D61C2×C4semidirect
81. C2×C4⋊Dic61C2×C4semidirect
82. C2×C22⋊Dic61C2×C4semidirect
83. C2×#11758198C2×C4semidirect
84. C2×#11758198C2×C4semidirect
86. C2×C4×Dic61C2×C4semidirect
88. C22×D4×C61C2×C4semidirect
89. C23×C4×C61C2×C4semidirect
91. C2×C4⋊C4×C61C2×C4semidirect
92. C2×C22⋊C4×C61C2×C4semidirect
317. C2×#121#6923semidirect
332. C61#20985semidirect
352. C22×D4C2×#6923semidirect
356. C23×C4C2×#6923semidirect
377. C2×C4⋊C4C2×#6923semidirect
388. C2×C22⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408292
C2×#168407412
#214622922
#214618412
#214618402
#214615192
C2×#168403282
#2098561