Finite Group Databasebeta

← #30596950

Subgroups — #30596950

Order 15616.

1090 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C211yesmaximal, characteristic
2C2×C4×C22⋊#6923C211yesmaximal, characteristic
3#21460699C214yesmaximal, complemented
4C2×#16839725C211yesmaximal, characteristic
5C2×#14519856—21no—
6C22×C22⋊#6923—21no—
7C22×C4×#6923—21no—
8#16840705C412yescomplemented
9#16840704—21no—
10#16840704C412yescomplemented
11#16840703—21no—
12C4×C22⋊#6923—21no—
13C4×C22⋊#6923—21no—
14C4×C22⋊#6923—21no—
15C4×C22⋊#6923C412yescomplemented
16C4×C22⋊#6923C2212yes—
17C4×C22⋊#6923C2212yes—
18C2×(C2×#6923)⋊C4C412yescomplemented
19C23×C4×D61C2211yescharacteristic
20C22×#11758194C411yescharacteristic, complemented
21#16839725—21no—
22#16839725—21no—
23#16839725C412yescomplemented
24#16839725C2212yes—
25C2×#14519689—21no—
26C2×#14519689—21no—
27C22×M16×C61C411yescharacteristic, complemented
28C23×#6923—22no—
29#14519856—42no—
30#14519856—42no—
31C2×C4⋊#6923—22no—
32C2×C22⋊#6923—28no—
33C2×C22⋊#6923—22no—
34C2×C22⋊#6923—22no—
35C2×C22⋊#6923—22no—
36C2×C22⋊#6923—22no—
37C2×C22⋊#6923—22no—
38C2×C4×#6923—22no—
39C2×C4×#6923—22no—
40C2×C4×#6923—22no—
41C2×C4×#6923—42no—
42C2×C4×#6923—42no—
43(C2×#6923)⋊C4—22no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4C2×C414yes—
46(C2×#6923)⋊C4C2×C414yescomplemented
47C24×D61D411yescharacteristic
48C23×Dic61Q811yescharacteristic
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61Q811yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C2×#11758194—21no—
64C2×#11758194—42no—
65C2×#11758194C2×C412yescomplemented
66C22×#92418—21no—
67#14519689—22no—
68#14519689—22no—
69#14519689—22no—
70#14519689—22no—
71C23×C4×C61C2×C411yescharacteristic
72C2×M16×C61—21no—
73C2×M16×C61—42no—
74C2×M16×C61C2×C412yescomplemented
75C22×C8×C61—21no—
76C24×C61C4⋊C411yescharacteristic
77C23×D61—24no—
78C23×D61—42no—
79C23×D61—42no—
80C23×D61C22⋊C411yescharacteristic
81C23×D61C22⋊C411yescharacteristic
82C23×D61C22⋊C411yescharacteristic
83C23×D61C22⋊C411yescharacteristic
84C23×D61C4⋊C411yescharacteristic
85C23×D61C2×D411yescharacteristic
86C22×#6923—28no—
87C22×#6923—24no—
88C22×#6923—24no—
89C22×#6923—44no—
90C22×#6923—44no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61C4211yescharacteristic
94C22×C4×C61C4211yescharacteristic
95C22×C4×C61C22⋊C411yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C4⋊C411yescharacteristic
98C22×C4×C61C22×C411yescharacteristic
99C22×Dic61—42no—
100C22×Dic61C4⋊C411yescharacteristic
101C22×Dic61C4⋊C411yescharacteristic
102C22×Dic61C4⋊C411yescharacteristic
103C22×Dic61C4⋊C411yescharacteristic
104C22×Dic61C4⋊C411yescharacteristic
105C22×Dic61C2×Q811yescharacteristic
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—42no—
111C2×C4×D61—42no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—42no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—42no—
120C2×C4×D61—41no—
121C2×C4×D61—41no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61C4211yescharacteristic
125C2×C4×D61C4211yescharacteristic
126C2×C4×D61C22⋊C411yescharacteristic
127C2×C4×D61C22⋊C411yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C22⋊C411yescharacteristic
130C2×C4×D61C22⋊C411yescharacteristic
131C2×C4×D61C22⋊C411yescharacteristic
132C2×C4×D61C4⋊C411yescharacteristic
133C2×C4×D61C4⋊C411yescharacteristic
134C2×C4×D61C4⋊C411yescharacteristic
135C2×C4×D61C4⋊C411yescharacteristic
136C2×C4×D61C22×C411yescharacteristic
137C2×C4×D61C22×C411yescharacteristic
138C2×C4×D61C2×D411yescharacteristic
139C2×C4×D61C2×D411yescharacteristic
140C22⋊#6923—28no—
141C22⋊#6923—28no—
142C22⋊#6923—44no—
143C22⋊#6923—44no—
144C22⋊#6923—44no—
145C4⋊#6923—44no—
146C4×#6923—44no—
147C4×#6923—42no—
148C4×#6923—42no—
149C4×#6923—42no—
150C4×#6923—42no—
151M16×C61—42no—
152M16×C61—42no—
153C2×C8×C61—22no—
154C2×C8×C61—42no—
155#11758194—42no—
156#11758194—42no—
157C2×#92418—22no—
158C2×#92418—42no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—41no—
162C23×C61#9511yescharacteristic
163C23×C61#9511yescharacteristic
164C23×C61C2×C4⋊C411yescharacteristic
165C22×D61—22no—
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—416no—
183C22×D61#9511yescharacteristic
184C22×D61#9511yescharacteristic
185C22×D61#10414yes—
186C22×D61C2×C22⋊C411yescharacteristic
187C22×D61C2×C22⋊C411yescharacteristic
188C2×#6923—28no—
189C2×#6923—44no—
190C2×#6923—416no—
191C2×#6923—44no—
192C2×#6923—44no—
193C2×#6923—44no—
194C2×#6923—44no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61#9511yescharacteristic
203C2×C4×C61#9511yescharacteristic
204C2×C4×C61#9912yes—
205C2×C4×C61#9912yes—
206C2×C4×C61C2×C4211yescharacteristic
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61#9511yescharacteristic
215C2×Dic61#9511yescharacteristic
216C2×Dic61C2×C4⋊C411yescharacteristic
217C2×Dic61C2×C4⋊C411yescharacteristic
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—21no—
223C4×D61—21no—
224C4×D61—21no—
225C4×D61—21no—
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—41no—
237C4×D61—41no—
238C8×C61—42no—
239#92418—42no—
240#8051—611nomaximal, Sylow
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—44no—
249C22×C61#33912yes—
250C22×C61#34212yes—
251C22×C61C2×#9511yescharacteristic
252C2×D61—28no—
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—44no—
263C2×D61—44no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61#33912yes—
268C2×D61C2×#10412yes—
269C2×D61#42112yes—
270#6923—48no—
271#6923—84no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61#34212yes—
275C4×C61C2×#9911yescharacteristic
276C4×C61C2×#9911yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—21no—
281C2×#407—611no—
282#2280—611no—
283C2×C4×C22⋊C4—611no—
284#1297—614no—
285C2×C61—22no—
286C2×C61—22no—
287C2×C61—41no—
288C2×C61C2×#33911yescharacteristic
289C2×C61#173311yescharacteristic
290C2×C61C2×#34211yescharacteristic
291D61—22no—
292D61—22no—
293D61—22no—
294D61—41no—
295C24×C4—611no—
296C22×M16—611no—
297C22×M16#692311yescharacteristic, complemented
298C2×#117—1221no—
299C22×C22⋊C4—1221no—
300C22×C42—1221no—
301#407—612no—
302#407—612no—
303#407—1221no—
304#407—1221no—
305C2×C22⋊C8—1221no—
306C2×C22⋊C8—1221no—
307#381—612no—
308#380—612no—
309#380—1221no—
310#379—1221no—
311C4×C22⋊C4—612no—
312C4×C22⋊C4—612no—
313C4×C22⋊C4—612no—
314C4×C22⋊C4—1221no—
315C4×C22⋊C4—1221no—
316C4×C22⋊C4—1221no—
317C2×#95—612no—
318C61#805111yescharacteristic, complemented, Sylow
319C25—611no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—1222no—
334C23×C4—2442no—
335C23×C4—2442no—
336C23×C4C2×#692311yescharacteristic
337C2×M16—21no—
338C2×M16—42no—
339C2×M16—612no—
340C2×M16—1221no—
341C2×M16—2442no—
342C2×M16C2×#692312yescomplemented
343C22×C8—21no—
344C22×C8—1221no—
345#117—2442no—
346#117—2442no—
347C2×C4⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1228no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1222no—
354C2×C42—1222no—
355C2×C42—1222no—
356C2×C42—1222no—
357C2×C42—2442no—
358C2×C42—2442no—
359C22⋊C8—1222no—
360C22⋊C8—1222no—
361C22⋊C8—1222no—
362C22⋊C8—1222no—
363#95—614no—
364#95—614no—
365#95—1222no—
366#95—1222no—
367C24—611no—
368C24—611no—
369C24—611no—
370C24—611no—
371C24—611no—
372C24—611no—
373C24—1224no—
374C24—2442no—
375C24—2442no—
376C24C4⋊#692311yescharacteristic
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1224no—
406C22×C4—1228no—
407C22×C4—1224no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4—2441no—
413C22×C4—2441no—
414C22×C4—2441no—
415C22×C4—2441no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4—2444no—
424C22×C4—2444no—
425C22×C4C4×#692311yescharacteristic
426C22×C4C4×#692311yescharacteristic
427C22×C4C4⋊#692311yescharacteristic
428C22×C4C22⋊#692311yescharacteristic
429C22×C4C22⋊#692311yescharacteristic
430C22×C4C22×#692311yescharacteristic
431M16—42no—
432M16—42no—
433M16—2442no—
434M16—2442no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—1222no—
438C2×C8—2442no—
439C4⋊C4—2444no—
440C22⋊C4—1228no—
441C22⋊C4—1228no—
442C22⋊C4—2444no—
443C22⋊C4—2444no—
444C22⋊C4—2444no—
445C42—2442no—
446C42—2442no—
447C42—2444no—
448C42—2442no—
449C42—2442no—
450C23—22no—
451C23—22no—
452C23—41no—
453C23—611no—
454C23—611no—
455C23—611no—
456C23—614no—
457C23—611no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23—1222no—
465C23—1222no—
466C23—1222no—
467C23—1222no—
468C23—2441no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—24416no—
476C23(C2×#6923)⋊C411yescharacteristic
477C23(C2×#6923)⋊C411yescharacteristic
478C23C2×C4⋊#692311yescharacteristic
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1228no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2444no—
518C2×C4—2444no—
519C2×C4—2444no—
520C2×C4—2444no—
521C2×C4—2444no—
522C2×C4—24416no—
523C2×C4(C2×#6923)⋊C411yescharacteristic
524C2×C4(C2×#6923)⋊C411yescharacteristic
525C2×C4#1451983512yes—
526C2×C4#1451983612yes—
527C2×C4C2×C4×#692311yescharacteristic
528C2×C4C2×C22⋊#692311yescharacteristic
529C8—42no—
530C8—2442no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—44no—
539C22—612no—
540C22—612no—
541C22—612no—
542C22—1228no—
543C22—1221no—
544C22—1221no—
545C22—1221no—
546C22—1221no—
547C22—1221no—
548C22—1221no—
549C22—1221no—
550C22—1221no—
551C22—1221no—
552C22—2444no—
553C22—2444no—
554C22—2444no—
555C22—2444no—
556C22—2444no—
557C22#1684063412yes—
558C22#1684064012yes—
559C22C2×(C2×#6923)⋊C411yescharacteristic
560C4—21no—
561C4—21no—
562C4—1221no—
563C4—1221no—
564C4—1221no—
565C4—1221no—
566C4—4884no—
567C4—2448no—
568C4#1684064012yes—
569C4C2×#1451983511yescharacteristic
570C4C2×#1451983611yescharacteristic
571C2—22no—
572C2—22no—
573C2—41no—
574C2—1222no—
575C2—1222no—
576C2—1222no—
577C2—2441no—
578C2C2×#1684063411yescharacteristic
579C2#2146150111yescharacteristic
580C2C2×#1684064011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#8051
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. #21460699C2semidirect
8. #16840705C4semidirect
10. #16840704C4semidirect
15. C4×C22⋊#6923C4semidirect
18. C2×(C2×#6923)⋊C4C4semidirect
20. C22×#11758194C4semidirect
23. #16839725C4semidirect
27. C22×M16×C61C4semidirect
46. (C2×#6923)⋊C4C2×C4semidirect
65. C2×#11758194C2×C4semidirect
74. C2×M16×C61C2×C4semidirect
297. C22×M16#6923semidirect
318. C61#8051semidirect
342. C2×M16C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214622632
C2×C4×C22⋊#69232
#214606992
C2×#168397252
#805161