Finite Group Databasebeta

← #30615962

Subgroups — #30615962

Order 15616.

1338 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×#16840754C211yesmaximal, characteristic
2C2×M16×#6923C211yesmaximal, characteristic, complemented
3#21462279C211yesmaximal, characteristic, complemented
4C2×#16840714C211yesmaximal, characteristic
5#21461911C214yesmaximal, complemented
6#21461909C214yesmaximal, complemented
7#21461508C211yesmaximal, characteristic, complemented
8#21461507C211yesmaximal, characteristic, complemented
9#418×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13C22×C22⋊#6923—22no—
14C22×C4×#6923C2211yescharacteristic
15#16840754—24no—
16C2×#14519839C2214yes—
17M16×#6923—24no—
18C2×#14519831C2212yescomplemented
19C2×C8×#6923C2212yescomplemented
20#16840714—22no—
21#16840714C2212yes—
22#16840714C2212yes—
23#16840713C2212yescomplemented
24#16840713C2212yescomplemented
25#16840709—22no—
26C4×C4⋊#6923—22no—
27#16840636C2214yescomplemented
28#16840635C2214yescomplemented
29C22×D4×D61C2211yescharacteristic
30C2×M16×D61C2211yescharacteristic
31#117×D61C2211yescharacteristic
32#16839768C411yescharacteristic, complemented
33#16839767C411yescharacteristic, complemented
34#16839766C411yescharacteristic, complemented
35#16839765C414yescomplemented
36#102×D61C2214yes—
37C61×#418C411yescharacteristic, complemented
38C23×#6923—21no—
39C23×#6923D412yes—
40C2×#11758213—21no—
41D61⋊M16—22no—
42D61⋊M16D414yes—
43C2×#11758217—21no—
44C2×D61⋊C8C2312yes—
45#14519856—22no—
46#14519856—22no—
47C2×C4⋊#6923—22no—
48C2×C4⋊#6923—22no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923C2312yes—
53C2×C4⋊#6923C2312yes—
54C2×C22⋊#6923—21no—
55C2×C22⋊#6923—24no—
56C2×C22⋊#6923—24no—
57C2×C22⋊#6923—48no—
58C2×C4×#6923—22no—
59C2×C4×#6923—22no—
60C2×C4×#6923—21no—
61C2×C4×#6923D412yes—
62C2×C4×#6923C2311yescharacteristic
63C2×C4×#6923C2311yescharacteristic
64#14519839—28no—
65#14519831—24no—
66C8×#6923—24no—
67(C2×#6923)⋊C4—22no—
68(C2×#6923)⋊C4—22no—
69C24×D61—21no—
70C22×D244C2×C411yescharacteristic
71C22×C22⋊D61—21no—
72C2×D4×D61—22no—
73C2×D4×D61—21no—
74C2×D4×D61—22no—
75C2×D4×D61—22no—
76C2×D4×D61—22no—
77C2×D4×D61—21no—
78C2×D4×D61C2×C412yes—
79C2×D4×D61C2312yes—
80C22×C4×D61C2311yescharacteristic
81C2×#11758194C2×C411yescharacteristic
82C2×#11758189C2×C412yes—
83M16×D61—22no—
84M16×D61—22no—
85C2×C8×D61C2312yes—
86#14519736C2×C411yescharacteristic
87#14519735C2×C412yes—
88#14519734—21no—
89#14519733—21no—
90C4⋊C4×D61C2312yes—
91C22⋊C4×D61—21no—
92C42×D61—21no—
93#14519699C2×C414yescomplemented
94#14519698C2×C414yescomplemented
95#14519697C2×C414yescomplemented
96C22×D4×C61C2×C411yescharacteristic
97C2×M16×C61C2×C411yescharacteristic
98#117×C61C2×C411yescharacteristic
99#102×C61C2×C414yescomplemented
100C24×C61—21no—
101C23×D61—21no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—44no—
105C23×D61—44no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61C2×D411yescharacteristic
109C22×#6923—24no—
110C22×#6923—24no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—42no—
114C22×#6923—44no—
115C22×#6923—42no—
116C22×#6923C2×D412yes—
117C22×#6923C2×D412yes—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61#4111yescharacteristic
128D4×D61—24no—
129D4×D61—24no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—22no—
133D4×D61—22no—
134D4×D61—42no—
135D4×D61—42no—
136D4×D61—42no—
137D4×D61—42no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—21no—
141C2×D244C22×C412yes—
142C2×C4×D61—22no—
143C2×C4×D61C22×C411yescharacteristic
144C2×C4×D61C2×D412yes—
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61#4112yes—
147C2×C4×D61#4112yes—
148C2×C4×D61#4112yes—
149C2×C4×D61#4111yescharacteristic
150C2×C4×D61C2411yescharacteristic
151C22⋊#6923—48no—
152C22⋊#6923—42no—
153C22⋊#6923—48no—
154#11758217—24no—
155C2×#92420—22no—
156C4⋊#6923—22no—
157C4⋊#6923—22no—
158C4⋊#6923—22no—
159C4⋊#6923—22no—
160C4⋊#6923—44no—
161C4⋊#6923—44no—
162C4⋊#6923—42no—
163C4×#6923—24no—
164C4×#6923—21no—
165C4×#6923—21no—
166C4×#6923—21no—
167C4×#6923—21no—
168C4×#6923—42no—
169#11758213—24no—
170D61⋊C8—22no—
171D61⋊C8C2×D414yes—
172M16×C61—22no—
173C2×C8×C61C22×C412yes—
174C4⋊C4×C61C22×C412yes—
175C22⋊C4×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—21no—
178#11758198—22no—
179C4⋊Dic61C22×C412yes—
180#11758196—22no—
181C4×Dic61—21no—
182C4×Dic61—22no—
183#11758194—22no—
184C2×#92418C22×C412yes—
185#11758189—24no—
186C8×D61—22no—
187C8×D61—22no—
188C23×C61—22no—
189C23×C61—44no—
190C23×C61—42no—
191C23×C61C4×D411yescharacteristic
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—48no—
196C22×D61—44no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—44no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61—84no—
204C22×D61—84no—
205C22×D61—88no—
206C22×D61C4×D411yescharacteristic
207C22×D61#12112yes—
208C22×D61#12412yes—
209C22×D61C22×D411yescharacteristic
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—22no—
213C2×#6923—22no—
214C2×#6923—22no—
215C2×#6923—48no—
216C2×#6923—42no—
217C2×#6923—42no—
218C2×#6923—42no—
219C2×#6923—44no—
220C2×#6923—42no—
221C2×#6923—84no—
222C2×#6923AGL(1,ℤ/8)14yes—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—42no—
226C2×C4×C61—22no—
227C2×C4×C61#11712yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D411yescharacteristic
230C2×C4×C61C23×C411yescharacteristic
231C22⋊D61—42no—
232C22⋊D61—42no—
233C22⋊D61—44no—
234C2×Dic61—22no—
235C2×Dic61C4×D411yescharacteristic
236C2×Dic61#12312yes—
237C2×Dic61#12612yes—
238C2×Dic61C2×#4111yescharacteristic
239D244—22no—
240D244—22no—
241D244—42no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61#11712yes—
245C4×D61C4×D412yes—
246C4×D61#12112yes—
247C4×D61#12312yes—
248C4×D61#12412yes—
249C4×D61#12612yes—
250C4×D61C22×D411yescharacteristic
251C4×D61C2×#4111yescharacteristic
252C4×D61C2×#4111yescharacteristic
253C4×D61C2×#4111yescharacteristic
254#92420—24no—
255C8×C61—22no—
256#92418—22no—
257#21095—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—44no—
260C22×C61—42no—
261C22×C61—84no—
262C22×C61#38812yes—
263C22×C61C2×C4×D411yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—24no—
268C2×D61—44no—
269C2×D61—42no—
270C2×D61—88no—
271C2×D61—84no—
272C2×D61—82no—
273C2×D61—82no—
274C2×D61—84no—
275C2×D61#38812yes—
276C2×D61C2×#12111yescharacteristic
277C2×D61C2×#12411yescharacteristic
278C2×D61C2×AGL(1,ℤ/8)12yes—
279C2×D61#57512yes—
280#6923—24no—
281#6923—42no—
282#6923—82no—
283#6923—82no—
284C4×C61—42no—
285C4×C61#38812yes—
286C4×C61C2×#11711yescharacteristic
287C4×C61C2×C4×D411yescharacteristic
288Dic61—42no—
289Dic61#38812yes—
290Dic61C2×#12311yescharacteristic
291Dic61C2×#12611yescharacteristic
292C2×#418—611no—
293C2×#418—611no—
294C2×C4×M16—611no—
295#2296—611no—
296C2×#390—611no—
297#1936—614no—
298#1934—614no—
299#1736—611no—
300#1736—611no—
301C2×C61—22no—
302C2×C61—82no—
303C2×C61C2×#38811yescharacteristic
304C2×C61C4×AGL(1,ℤ/8)12yes—
305D61—22no—
306D61—82no—
307D61#305512yes—
308D61#313811yescharacteristic
309D61#314011yescharacteristic
310C23×D4—611no—
311C22×M16—611no—
312C22×M16—611no—
313C2×#117—611no—
314C2×#117—611no—
315C22×C4⋊C4—611no—
316C22×C22⋊C4—1222no—
317C22×C42—611no—
318#418—611no—
319#418—611no—
320#418—611no—
321#418—614no—
322#418—1224no—
323#418#692311yescharacteristic, complemented
324C2×#102—614no—
325C2×#102—614no—
326C4×M16—1224no—
327C2×#97—612no—
328C2×C4×C8—612no—
329#390—612no—
330#390—612no—
331#390—1222no—
332#389—612no—
333#389—612no—
334#385—1222no—
335C4×C4⋊C4—1222no—
336#340—614no—
337#340—614no—
338C61#2109511yescharacteristic, complemented, Sylow
339C25—1221no—
340C22×D4—612no—
341C22×D4—612no—
342C22×D4—611no—
343C22×D4—1222no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4—1221no—
350C22×D4C2×#692311yescharacteristic
351C23×C4—611no—
352C23×C4—612no—
353C23×C4—1221no—
354C2×M16—614no—
355C2×M16—611no—
356C2×M16—612no—
357C2×M16—1222no—
358C2×M16—1222no—
359C2×M16—1221no—
360C2×M16—1222no—
361C2×M16—1221no—
362C2×M16C2×#692311yescharacteristic
363C22×C8—612no—
364C22×C8—612no—
365#117—611no—
366#117—612no—
367#117—1221no—
368#117—1221no—
369#117—1222no—
370#117—1222no—
371#117C2×#692311yescharacteristic
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—1224no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—2448no—
385C2×C42—612no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—1221no—
389C2×C42—1222no—
390C2×C42—1222no—
391C2×C42—1221no—
392#102—614no—
393#102—614no—
394#102—614no—
395#102—1228no—
396#102C2×#692314yescomplemented
397#97—1224no—
398C4×C8—1224no—
399#95—1222no—
400#95—1222no—
401C24—21no—
402C24—611no—
403C24—1222no—
404C24—1222no—
405C24—1221no—
406C24—2442no—
407C24—2442no—
408C24—2444no—
409C24—2444no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—612no—
414C2×D4—1222no—
415C2×D4—1224no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1221no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4—2442no—
430C2×D4—2442no—
431C2×D4C22×#692312yes—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—1222no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4—2444no—
451C22×C4C22×#692311yescharacteristic
452M16—22no—
453M16—1224no—
454M16—1224no—
455M16—1224no—
456M16—1222no—
457C2×C8—614no—
458C2×C8—612no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8C22×#692312yes—
464C4⋊C4—612no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—2442no—
471C4⋊C4—2444no—
472C4⋊C4—2444no—
473C4⋊C4C22×#692312yes—
474C22⋊C4—21no—
475C22⋊C4—1222no—
476C22⋊C4—1221no—
477C22⋊C4—2448no—
478C22⋊C4—2448no—
479C22⋊C4—2442no—
480C42—21no—
481C42—1222no—
482C42—1221no—
483C42—1221no—
484C42—1224no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C42—2442no—
489C23—22no—
490C23—44no—
491C23—42no—
492C23—611no—
493C23—612no—
494C23—612no—
495C23—611no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—4884no—
500C23—4888no—
501C23—4884no—
502C23—2444no—
503C23—2444no—
504C23—2442no—
505C23—2442no—
506C23—2444no—
507C23—2444no—
508C23—2448no—
509C23—2442no—
510C23D4×#692311yescharacteristic
511D4—22no—
512D4—22no—
513D4—42no—
514D4—1222no—
515D4—1222no—
516D4—2442no—
517D4—2442no—
518D4—2444no—
519D4—2442no—
520C2×C4—22no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—614no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—4884no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2448no—
546C2×C4—2442no—
547C2×C4—2444no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4#1451985612yes—
552C2×C4D4×#692311yescharacteristic
553C2×C4D4×#692312yes—
554C2×C4C23×#692311yescharacteristic
555C8—22no—
556C8—1224no—
557C8—1222no—
558C22—22no—
559C22—42no—
560C22—44no—
561C22—84no—
562C22—611no—
563C22—611no—
564C22—612no—
565C22—612no—
566C22—612no—
567C22—1222no—
568C22—1224no—
569C22—1222no—
570C22—1222no—
571C22—4884no—
572C22—4884no—
573C22—4882no—
574C22—4882no—
575C22—4888no—
576C22—2442no—
577C22—2444no—
578C22#1684071212yes—
579C22C2×D4×#692311yescharacteristic
580C4—42no—
581C4—611no—
582C4—611no—
583C4—612no—
584C4—1224no—
585C4—4882no—
586C4—4882no—
587C4—2442no—
588C4—2442no—
589C4#1684071212yes—
590C4C2×#1451985611yescharacteristic
591C4C2×D4×#692311yescharacteristic
592C2—22no—
593C2—82no—
594C2—612no—
595C2—611no—
596C2—611no—
597C2—1222no—
598C2—4882no—
599C2C2×#1684071211yescharacteristic
600C2AGL(1,ℤ/8)×#692312yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#418C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenter#418#6923
hypocenterC61#21095
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×M16×#6923C2semidirect
3. #21462279C2semidirect
5. #21461911C2semidirect
6. #21461909C2semidirect
7. #21461508C2semidirect
8. #21461507C2semidirect
18. C2×#14519831C22semidirect
19. C2×C8×#6923C22semidirect
23. #16840713C22semidirect
24. #16840713C22semidirect
27. #16840636C22semidirect
28. #16840635C22semidirect
32. #16839768C4semidirect
33. #16839767C4semidirect
34. #16839766C4semidirect
35. #16839765C4semidirect
37. C61×#418C4semidirect
93. #14519699C2×C4semidirect
94. #14519698C2×C4semidirect
95. #14519697C2×C4semidirect
99. #102×C61C2×C4semidirect
323. #418#6923semidirect
338. C61#21095semidirect
396. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407542
C2×M16×#69232
#214622792
C2×#168407142
#214619112
#214619092
#214615082
#214615072
#418×D612
#2109561