Finite Group Databasebeta

← #30616257

Subgroups — #30616257

Order 15616.

1858 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519836C212yesmaximal, complemented
2C2×D61⋊#99C212yesmaximal, complemented
3#21462298C211yesmaximal, characteristic, complemented
4#21462255C211yesmaximal, characteristic, complemented
5#21461960C214yesmaximal, complemented
6#21461959C214yesmaximal, complemented
7C2×#16840323C211yesmaximal, characteristic
8C22×C22⋊#6923C2212yes—
9C22×C22⋊#6923C2212yes—
10C2×#14519836—28no—
11C2×#14519836C2214yescomplemented
12C2×#14519836C2218yescomplemented
13D61⋊#99—28no—
14C2×#14519835C2214yescomplemented
15#16840709—24no—
16#16840704C2214yescomplemented
17C22⋊C4×#6923—24no—
18C4×C22⋊#6923C2214yescomplemented
19C23×D244C2211yescharacteristic
20C22×D4×D61C2211yescharacteristic
21#16840323—24no—
22#16840323—24no—
23C2×#14519762C412yescomplemented
24C2×#14519760C411yescharacteristic, complemented
25C2×#14519752C2212yes—
26C2×#14519750C2212yes—
27C2×#14519747C411yescharacteristic, complemented
28C2×#14519747C412yescomplemented
29C2×#14519745C411yescharacteristic, complemented
30C2×C22⋊C4×D61C2211yescharacteristic
31C2×#123×C61C411yescharacteristic, complemented
32C23×#6923—22no—
33C23×#6923—22no—
34C2×C4⋊#6923—24no—
35C2×C22⋊#6923—22no—
36C2×C22⋊#6923—24no—
37C2×C22⋊#6923—24no—
38C2×C22⋊#6923—24no—
39C2×C22⋊#6923—22no—
40C2×C22⋊#6923D418yescomplemented
41C2×C22⋊#6923C2314yes—
42C2×C22⋊#6923C2314yes—
43C2×C4×#6923—24no—
44#14519836—28no—
45#14519836—216no—
46#14519835—28no—
47(C2×#6923)⋊C4—24no—
48(C2×#6923)⋊C4—24no—
49C24×D61C2×C411yescharacteristic
50C24×D61C2×C411yescharacteristic
51C24×D61C2311yescharacteristic
52C22×D244—24no—
53C22×D244—22no—
54C22×D244—22no—
55C22×D244—24no—
56C22×D244C2×C412yes—
57C22×D244C2×C412yescomplemented
58C22×D244C2312yes—
59C22×D244C2311yescharacteristic
60C22×C22⋊D61C2×C411yescharacteristic
61C22×C22⋊D61C2×C412yescomplemented
62C22×C22⋊D61C2311yescharacteristic
63C2×D4×D61—24no—
64C2×D4×D61—22no—
65C2×D4×D61—22no—
66C2×D4×D61—24no—
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61C2312yes—
69#14519762—28no—
70#14519760—24no—
71#14519752—24no—
72#14519752—24no—
73#14519750—24no—
74#14519750—24no—
75#14519747—24no—
76#14519747—28no—
77#14519745—24no—
78C2×#11758196C2×C412yescomplemented
79C2×C22⋊Dic61C2×C411yescharacteristic
80C2×#11758198C2×C412yes—
81C2×#11758198C2×C412yescomplemented
82C2×#11758198C2×C412yescomplemented
83C2×#11758198C2×C412yescomplemented
84C22⋊C4×D61D414yes—
85C22⋊C4×D61D414yes—
86C22×D4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88#123×C61—24no—
89C2×C4⋊C4×C61C2×C412yescomplemented
90C2×C22⋊C4×C61C2×C411yescharacteristic
91C2×C22⋊C4×C61C2×C412yescomplemented
92C24×C61C22×C411yescharacteristic
93C24×C61C22×C411yescharacteristic
94C23×D61—22no—
95C23×D61—24no—
96C23×D61—22no—
97C23×D61—24no—
98C23×D61—82no—
99C23×D61—82no—
100C23×D61—82no—
101C23×D61C22×C411yescharacteristic
102C23×D61C22×C412yes—
103C23×D61C22×C411yescharacteristic
104C23×D61C2×D412yes—
105C23×D61C2×D412yes—
106C23×D61C2×D414yes—
107C23×D61#4114yes—
108C23×D61C2411yescharacteristic
109C22×#6923—24no—
110C22×#6923—28no—
111C22×#6923—24no—
112C22×#6923—44no—
113C22×#6923—84no—
114C22×#6923—84no—
115C2×D4×C61—22no—
116C2×D4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—24no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—21no—
126C2×C22⋊D61—21no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—82no—
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C412yes—
132D4×D61—44no—
133D4×D61—44no—
134D4×D61—44no—
135D4×D61—44no—
136C2×D244—24no—
137C2×D244—28no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—24no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—24no—
146C2×D244—24no—
147C2×D244—44no—
148C2×D244—44no—
149C2×D244—44no—
150C2×D244—44no—
151C2×D244—82no—
152C2×C4×D61—22no—
153C2×C4×D61—24no—
154C2×C4×D61—82no—
155C2×C4×D61C2×D414yes—
156C2×C4×D61C2×D414yes—
157C22⋊#6923—28no—
158C22⋊#6923—24no—
159C22⋊#6923—24no—
160C22⋊#6923—24no—
161C22⋊#6923—44no—
162C22⋊#6923C2×D418yes—
163C4⋊#6923—84no—
164C4×#6923—84no—
165C4⋊C4×C61—24no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61C22⋊C414yes—
168C22⋊Dic61C22⋊C414yes—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172#11758198C22⋊C418yes—
173#11758196—24no—
174C23×C61—21no—
175C23×C61—22no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61C2×C22⋊C412yes—
179C23×C61C4×D414yes—
180C23×C61C23×C411yescharacteristic
181C22×D61—21no—
182C22×D61—22no—
183C22×D61—24no—
184C22×D61—22no—
185C22×D61—42no—
186C22×D61—44no—
187C22×D61—44no—
188C22×D61—42no—
189C22×D61—44no—
190C22×D61—44no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—84no—
195C22×D61—84no—
196C22×D61—84no—
197C22×D61—84no—
198C22×D61—88no—
199C22×D61—88no—
200C22×D61C2×C22⋊C412yes—
201C22×D61C4×D414yes—
202C22×D61#12114yes—
203C22×D61#12114yes—
204C22×D61#12414yes—
205C22×D61#12714yes—
206C22×D61C22×D411yescharacteristic
207C22×D61C22×D411yescharacteristic
208C22×D61C22×D412yes—
209C22×D61C2×#4112yes—
210C2×#6923—28no—
211C2×#6923—48no—
212C2×#6923—48no—
213C2×#6923—44no—
214C2×#6923—88no—
215C2×#6923—88no—
216D4×C61—44no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61C2×C22⋊C414yes—
224C22⋊D61—44no—
225C22⋊D61—42no—
226C22⋊D61—41no—
227C22⋊D61—41no—
228C22⋊D61—84no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—42no—
232C2×Dic61C2×C22⋊C414yes—
233D244—44no—
234D244—42no—
235D244—42no—
236D244—41no—
237D244—42no—
238D244—41no—
239D244—48no—
240D244—84no—
241C4×D61—44no—
242C4×D61—44no—
243C4×D61—44no—
244#21213—611nomaximal, Sylow
245C22×C61—22no—
246C22×C61—24no—
247C22×C61—44no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—82no—
251C22×C61#39014yes—
252C22×C61C22×C22⋊C411yescharacteristic
253C22×C61C2×C4×D412yes—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—24no—
257C2×D61—44no—
258C2×D61—44no—
259C2×D61—44no—
260C2×D61—44no—
261C2×D61—44no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—41no—
269C2×D61—88no—
270C2×D61—88no—
271C2×D61—84no—
272C2×D61—82no—
273C2×D61#39014yes—
274C2×D61C2×#12112yes—
275C2×D61C2×#12411yescharacteristic
276C2×D61C2×#12711yescharacteristic
277#6923—48no—
278#6923—84no—
279C4×C61—24no—
280C4×C61—41no—
281C4×C61—42no—
282Dic61—24no—
283Dic61—41no—
284C2×#534—611no—
285C22×#99—612no—
286C22×#99—612no—
287#2315—611no—
288#2273—611no—
289#1964—614no—
290#1964—614no—
291C2×C61—22no—
292C2×C61—24no—
293C2×C61—81no—
294C2×C61C2×#39011yescharacteristic
295C2×C61#288312yes—
296D61—22no—
297D61—21no—
298D61—24no—
299D61—21no—
300D61—81no—
301D61—82no—
302C23×D4—611no—
303C23×D4—611no—
304#534—1224no—
305#534—1224no—
306C2×#123—611no—
307C2×#123—612no—
308C2×#123#692311yescharacteristic, complemented
309C2×#121—612no—
310C2×#121—612no—
311C2×#120—611no—
312C2×#120—611no—
313C2×#120—612no—
314C22×C22⋊C4—611no—
315C22×C22⋊C4—612no—
316C22×C22⋊C4—612no—
317C2×#99—614no—
318C2×#99—614no—
319C2×#99—618no—
320C2×#99—1228no—
321C2×#99—1228no—
322#385—1224no—
323#380—614no—
324C4×C22⋊C4—614no—
325C4×C22⋊C4—1224no—
326C61#2121311yescharacteristic, complemented, Sylow
327C25—611no—
328C25—611no—
329C25—611no—
330C22×D4—611no—
331C22×D4—611no—
332C22×D4—612no—
333C22×D4—612no—
334C22×D4—611no—
335C22×D4—612no—
336C22×D4—612no—
337C22×D4—1222no—
338C22×D4—1222no—
339C22×D4—1224no—
340C22×D4—1224no—
341C22×D4—1222no—
342C22×D4—1222no—
343C22×D4—1224no—
344C22×D4—1224no—
345C22×D4C2×#692311yescharacteristic
346C23×C4—612no—
347C23×C4—611no—
348C23×C4—1222no—
349C23×C4—1222no—
350C23×C4C2×#692311yescharacteristic
351#123—24no—
352#123—1228no—
353#123—1224no—
354#121—1224no—
355#121—1224no—
356#121—1224no—
357#121—1224no—
358#120—1228no—
359#120—1224no—
360#120—1224no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—1224no—
363C2×C4⋊C4C2×#692312yescomplemented
364C2×C22⋊C4—611no—
365C2×C22⋊C4—612no—
366C2×C22⋊C4—614no—
367C2×C22⋊C4—614no—
368C2×C22⋊C4—618no—
369C2×C22⋊C4—614no—
370C2×C22⋊C4—612no—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—612no—
373C2×C22⋊C4—614no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4C2×#692311yescharacteristic
380C2×C22⋊C4C2×#692312yescomplemented
381C2×C42—1224no—
382#99—1228no—
383#99—1228no—
384#99—12216no—
385#95—1224no—
386#95—1224no—
387C24—614no—
388C24—614no—
389C24—612no—
390C24—612no—
391C24—611no—
392C24—611no—
393C24—611no—
394C24—612no—
395C24—1222no—
396C24—1224no—
397C24—1222no—
398C24—1224no—
399C24—4882no—
400C24—4882no—
401C24—4882no—
402C24C22×#692311yescharacteristic
403C24C22×#692311yescharacteristic
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—1224no—
407C2×D4—1222no—
408C2×D4—1224no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1228no—
414C2×D4—1224no—
415C2×D4—1222no—
416C2×D4—1224no—
417C2×D4—1222no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—4882no—
425C2×D4—4882no—
426C2×D4—2444no—
427C2×D4—2444no—
428C2×D4—2444no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4—2444no—
433C2×D4—2444no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—612no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—611no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—1228no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—4882no—
446C22×C4—4884no—
447C22×C4—4884no—
448C22×C4—2444no—
449C22×C4C22×#692311yescharacteristic
450C22×C4C22×#692312yes—
451C22×C4C22×#692312yes—
452C4⋊C4—24no—
453C4⋊C4—1224no—
454C4⋊C4—4884no—
455C22⋊C4—24no—
456C22⋊C4—614no—
457C22⋊C4—618no—
458C22⋊C4—618no—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1228no—
462C22⋊C4—1224no—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—2444no—
467C22⋊C4C22⋊#692314yes—
468C42—4884no—
469C23—21no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—612no—
474C23—611no—
475C23—611no—
476C23—614no—
477C23—614no—
478C23—614no—
479C23—614no—
480C23—614no—
481C23—612no—
482C23—612no—
483C23—1224no—
484C23—1222no—
485C23—1221no—
486C23—1222no—
487C23—4884no—
488C23—4884no—
489C23—4888no—
490C23—4884no—
491C23—4884no—
492C23—4888no—
493C23—2442no—
494C23—2444no—
495C23—2444no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23—2444no—
501C23—2444no—
502C23C2×C22⋊#692312yes—
503C23D4×#692314yes—
504C23C23×#692311yescharacteristic
505D4—44no—
506D4—4884no—
507D4—4884no—
508D4—2444no—
509D4—2442no—
510D4—2448no—
511D4—2442no—
512D4—2442no—
513D4—2441no—
514D4—2441no—
515D4—2442no—
516D4—2441no—
517D4—2441no—
518D4—2444no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—614no—
526C2×C4—1228no—
527C2×C4—1221no—
528C2×C4—1222no—
529C2×C4—4888no—
530C2×C4—4888no—
531C2×C4—2444no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4—2448no—
537C2×C4—2448no—
538C2×C4C2×C22⋊#692314yes—
539C22—22no—
540C22—24no—
541C22—44no—
542C22—41no—
543C22—42no—
544C22—82no—
545C22—612no—
546C22—611no—
547C22—614no—
548C22—611no—
549C22—1222no—
550C22—1222no—
551C22—1224no—
552C22—4884no—
553C22—4882no—
554C22—4888no—
555C22—4888no—
556C22—2444no—
557C22—2444no—
558C22—2444no—
559C22—2442no—
560C22—2442no—
561C22—2444no—
562C22—2444no—
563C22—2442no—
564C22—2441no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22#1684071414yes—
569C22C22×C22⋊#692311yescharacteristic
570C22C2×D4×#692312yes—
571C4—24no—
572C4—42no—
573C4—41no—
574C4—1224no—
575C4—4884no—
576C4—2448no—
577C4—2441no—
578C2—22no—
579C2—24no—
580C2—81no—
581C2—1221no—
582C2—1222no—
583C2—1224no—
584C2—1221no—
585C2—4882no—
586C2—4881no—
587C2C2×#1684071411yescharacteristic
588C2#2146241912yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840323C2
cosocleC23×D61C24
hypercenterC2×#123#6923
hypocenterC61#21213
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#14519836C2semidirect
2. C2×D61⋊#99C2semidirect
3. #21462298C2semidirect
4. #21462255C2semidirect
5. #21461960C2semidirect
6. #21461959C2semidirect
11. C2×#14519836C22semidirect
12. C2×#14519836C22semidirect
14. C2×#14519835C22semidirect
16. #16840704C22semidirect
18. C4×C22⋊#6923C22semidirect
23. C2×#14519762C4semidirect
24. C2×#14519760C4semidirect
27. C2×#14519747C4semidirect
28. C2×#14519747C4semidirect
29. C2×#14519745C4semidirect
31. C2×#123×C61C4semidirect
40. C2×C22⋊#6923D4semidirect
57. C22×D244C2×C4semidirect
61. C22×C22⋊D61C2×C4semidirect
78. C2×#11758196C2×C4semidirect
81. C2×#11758198C2×C4semidirect
82. C2×#11758198C2×C4semidirect
83. C2×#11758198C2×C4semidirect
89. C2×C4⋊C4×C61C2×C4semidirect
91. C2×C22⋊C4×C61C2×C4semidirect
308. C2×#123#6923semidirect
326. C61#21213semidirect
363. C2×C4⋊C4C2×#6923semidirect
380. C2×C22⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C22×#145198362
C2×D61⋊#992
#214622982
#214622552
#214619602
#214619592
C2×#168403232
#2121361