Finite Group Databasebeta

← #489×#6923 #30617408

Subgroups — #489×#6923 #30617408

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊Q8×#6923C211yesmaximal, characteristic, complemented
2#124×#6923C211yesmaximal, characteristic
3#21462003C211yesmaximal, characteristic, complemented
4#21461949C211yesmaximal, characteristic
5#21461911C212yesmaximal
6#21461910C212yesmaximal, complemented
7#21461742C211yesmaximal, characteristic, complemented
8#103×#6923C212yesmaximal, complemented
9#102×#6923C212yesmaximal
10#97×#6923C211yesmaximal, characteristic, complemented
11#489×D61C211yesmaximal, characteristic
12C2×Q8×#6923—21no—
13C2×Q8×#6923C2211yescharacteristic
14C2×D4×#6923C2211yescharacteristic
15C2×#14519840C2212yes—
16C2×#14519839C2212yes—
17#16840721C2211yescharacteristic
18C2×#14519831C2212yes—
19C2×C8×#6923C2212yes—
20(C2×#6923)⋊Q8—21no—
21(C2×#6923)⋊Q8C2211yescharacteristic
22#16840714C2211yescharacteristic
23#16840713—21no—
24#16840713C2212yes—
25#16840712—21no—
26#16840712—21no—
27#16840708C2211yescharacteristic
28#16840706C2211yescharacteristic
29C4⋊C4×#6923—21no—
30C4⋊C4×#6923C2212yes—
31C22⋊C4×#6923—21no—
32C22⋊C4×#6923—21no—
33C42×#6923C2211yescharacteristic
34#16840636C2212yes—
35#16840636C2212yes—
36#16840635C2212yes—
37#16840635C2212yes—
38C4⋊Q8×D61C2211yescharacteristic
39#124×D61C2211yescharacteristic
40#16840124C411yescharacteristic, complemented
41#16840123C412yescomplemented
42#16840122C411yescharacteristic, complemented
43#16840121C411yescharacteristic, complemented
44#16840120C412yescomplemented
45#103×D61C2212yes—
46#102×D61C2212yes—
47#97×D61C2211yescharacteristic
48C61×#489C411yescharacteristic, complemented
49C23×#6923—21no—
50C2×D61⋊C8C2312yes—
51Q8×#6923—24no—
52Q8×#6923—44no—
53D4×#6923—24no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923—22no—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923C2311yescharacteristic
59C2×C4⋊#6923C2311yescharacteristic
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—21no—
62C2×C22⋊#6923—21no—
63C2×C22⋊#6923—21no—
64C2×C4×#6923—22no—
65C2×C4×#6923—21no—
66C2×C4×#6923D411yescharacteristic
67C2×C4×#6923D411yescharacteristic
68C2×C4×#6923D411yescharacteristic
69C2×C4×#6923D411yescharacteristic
70C2×C4×#6923C2311yescharacteristic
71C2×C4×#6923C2311yescharacteristic
72#14519840—24no—
73#14519839—24no—
74#14519832—21no—
75#14519832—21no—
76#14519831—24no—
77C8×#6923—24no—
78(C2×#6923)⋊C4—21no—
79(C2×#6923)⋊C4—21no—
80(C2×#6923)⋊C4—22no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—21no—
83(C2×#6923)⋊C4—21no—
84C2×Q8×D61—21no—
85C2×Q8×D61C2311yescharacteristic
86C2×D4×D61C2311yescharacteristic
87C2×C8×D61C2312yes—
88#14519777C2×C411yescharacteristic, complemented
89#14519776—21no—
90#14519776C2×C412yescomplemented
91#14519775—21no—
92#14519766C2×C411yescharacteristic
93#14519765C2×C411yescharacteristic
94#14519764C2×C411yescharacteristic
95#14519763—21no—
96#14519763—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61C2312yes—
99C22⋊C4×D61—21no—
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519702C2×C412yescomplemented
103#14519701C2×C412yescomplemented
104#14519700C2×C412yescomplemented
105#14519699C2×C412yes—
106#14519698C2×C412yes—
107#14519697C2×C412yes—
108#14519688C2×C411yescharacteristic, complemented
109#14519687C2×C412yescomplemented
110C4⋊Q8×C61C2×C411yescharacteristic, complemented
111#124×C61C2×C411yescharacteristic
112#103×C61C2×C412yescomplemented
113#102×C61C2×C412yes—
114#97×C61C2×C411yescharacteristic, complemented
115C23×D61—21no—
116C22×#6923—21no—
117C22×#6923—21no—
118C22×#6923—22no—
119C22×#6923—42no—
120C22×#6923—42no—
121C22×#6923—42no—
122C22×#6923C2×D411yescharacteristic
123C22×#6923C2×D411yescharacteristic
124C22×#6923C2×D411yescharacteristic
125C22×#6923C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61C22×C411yescharacteristic
128C2×D4×C61C22×C411yescharacteristic
129C2×C22⋊D61—21no—
130Q8×D61—22no—
131Q8×D61—22no—
132Q8×D61—42no—
133Q8×D61—42no—
134D4×D61—22no—
135D4×D61—22no—
136C2×D244C22×C411yescharacteristic
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2411yescharacteristic
142C2×Dic122—21no—
143C2×Dic122—22no—
144C2×Dic122—21no—
145C2×Dic122C22×C411yescharacteristic
146C22⋊#6923—42no—
147C22⋊#6923—42no—
148C22⋊#6923—42no—
149C2×#92420—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—22no—
153C4⋊#6923—22no—
154C4⋊#6923—42no—
155C4⋊#6923—42no—
156C4⋊#6923—44no—
157C4⋊#6923—42no—
158C4×#6923—22no—
159C4×#6923—22no—
160C4×#6923—22no—
161C4×#6923—41no—
162C4×#6923—41no—
163C4×#6923—42no—
164C4×#6923—44no—
165C4×#6923#4114yes—
166D61⋊C8—22no—
167D61⋊C8#4114yes—
168C2×C8×C61C22×C412yes—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C42×C61C22×C411yescharacteristic
174C22⋊Dic61—21no—
175C22⋊Dic61—21no—
176#11758198—21no—
177#11758198—21no—
178#11758198—21no—
179#11758198—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22×C412yes—
182#11758196—22no—
183#11758196—22no—
184C4×Dic61C22×C411yescharacteristic
185C4×Dic61#4111yescharacteristic
186C4×Dic61#4111yescharacteristic
187C2×#92418C22×C412yes—
188C8×D61—22no—
189C8×D61—22no—
190C23×C61—21no—
191C22×D61—21no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61C22×D411yescharacteristic
199C2×#6923—21no—
200C2×#6923—21no—
201C2×#6923—21no—
202C2×#6923—21no—
203C2×#6923—21no—
204C2×#6923—21no—
205C2×#6923—21no—
206C2×#6923—21no—
207C2×#6923—41no—
208C2×#6923—41no—
209C2×#6923—42no—
210C2×#6923—42no—
211C2×#6923—41no—
212C2×#6923—41no—
213C2×#6923—42no—
214C2×#6923—41no—
215C2×#6923—41no—
216C2×#6923—84no—
217C2×#6923#12412yes—
218C2×#6923#12412yes—
219C2×#6923AGL(1,ℤ/8)12yes—
220C2×#6923#13712yes—
221Q8×C61—22no—
222Q8×C61—42no—
223D4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C22⋊D61—42no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61C4×D411yescharacteristic
234C2×Dic61C2×#4111yescharacteristic
235D244—22no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61C2×#4112yes—
245C4×D61C2×#4112yes—
246Dic122—22no—
247Dic122—44no—
248Dic122—42no—
249Dic122—42no—
250#92420—24no—
251C8×C61—22no—
252#92418—22no—
253C4×#489—611nomaximal, Sylow
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61C2×C4×D411yescharacteristic
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61C2×#12411yescharacteristic
266C2×D61#53211yescharacteristic
267C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
268C2×D61C2×#13711yescharacteristic
269C2×D61#57511yescharacteristic
270C2×D61#57511yescharacteristic
271#6923—22no—
272#6923—41no—
273#6923—41no—
274#6923—81no—
275#6923—81no—
276#6923—82no—
277#6923#48914yesdirect product factor
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61C4×#4112yes—
283Dic61—21no—
284Dic61—21no—
285Dic61—42no—
286Dic61—41no—
287Dic61#52912yes—
288C2×#489—611no—
289C4×C4⋊Q8—611no—
290C4×#124—611no—
291#1993—611no—
292#1959—611no—
293#1936—612no—
294#1935—612no—
295#1836—611no—
296C4×#103—612no—
297C4×#102—612no—
298C4×#97—611no—
299C2×C61—81no—
300C2×C61C4×#12411yescharacteristic
301C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C61C4×#13711yescharacteristic
303D61—81no—
304D61C2×#48911yescharacteristic
305D61#313511yescharacteristic
306D61#317111yescharacteristic
307D61#317211yescharacteristic
308C2×C4⋊Q8—611no—
309C2×#124—611no—
310C2×C4×Q8—611no—
311C2×C4×Q8—1221no—
312C2×C4×D4—611no—
313#489—611no—
314#489—611no—
315#489—612no—
316#489—612no—
317#489—611no—
318#489#692311yescharacteristic, direct product factor
319C2×#103—612no—
320C2×#103—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#97—611no—
324C2×#97—612no—
325C2×#97—611no—
326C2×C4×C8—612no—
327#391—611no—
328#391—1221no—
329#390—611no—
330#389—612no—
331#389—1221no—
332#388—1221no—
333#388—1221no—
334#384—611no—
335#382—611no—
336C4×C4⋊C4—612no—
337C4×C4⋊C4—1221no—
338C4×C22⋊C4—1221no—
339C4×C22⋊C4—1221no—
340C43—611no—
341#340—612no—
342#340—612no—
343#340—612no—
344#340—612no—
345C61C4×#48911yescharacteristic, complemented, Sylow
346C22×Q8—611no—
347C22×Q8—1221no—
348C22×D4—611no—
349C23×C4—1221no—
350C22×C8—612no—
351C22×C8—612no—
352C4⋊Q8—612no—
353C4⋊Q8—611no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8C2×#692311yescharacteristic, complemented
357#124—611no—
358#124—611no—
359#124—611no—
360#124—1221no—
361#124—1221no—
362#124C2×#692311yescharacteristic
363C4×Q8—1224no—
364C4×Q8—2444no—
365C4×D4—1224no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1221no—
388C2×C42—1222no—
389#103—612no—
390#103—612no—
391#103—612no—
392#103—1224no—
393#103C2×#692312yescomplemented
394#102—612no—
395#102—612no—
396#102—612no—
397#102—1224no—
398#102C2×#692312yes—
399#97—612no—
400#97—611no—
401#97—1221no—
402#97—1221no—
403#97—1224no—
404#97C2×#692311yescharacteristic, complemented
405C4×C8—1224no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1222no—
411#95—1222no—
412C24—1221no—
413C2×Q8—21no—
414C2×Q8—611no—
415C2×Q8—1221no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8—1221no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×Q8C22×#692311yescharacteristic
423C2×D4—611no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1221no—
427C2×D4C22×#692311yescharacteristic
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C2×C8—614no—
444C2×C8—612no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8C22×#692312yes—
450C4⋊C4—21no—
451C4⋊C4—612no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1221no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C4⋊C4—2444no—
462C4⋊C4—2442no—
463C4⋊C4C22×#692312yes—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—1221no—
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C22⋊C4—1221no—
470C22⋊C4—1221no—
471C22⋊C4—1221no—
472C22⋊C4—2442no—
473C22⋊C4—2442no—
474C22⋊C4—2442no—
475C42—614no—
476C42—611no—
477C42—611no—
478C42—611no—
479C42—1222no—
480C42—1222no—
481C42—1222no—
482C42—2442no—
483C42—2444no—
484C42—2441no—
485C42—2441no—
486C42C22×#692311yescharacteristic
487C23—21no—
488C23—611no—
489C23—1221no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496Q8—22no—
497Q8—42no—
498Q8—1222no—
499Q8—2442no—
500Q8—2442no—
501Q8—2444no—
502D4—22no—
503D4—1222no—
504D4—2442no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—611no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1222no—
530C2×C4—4884no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4D4×#692311yescharacteristic
545C2×C4D4×#692311yescharacteristic
546C2×C4C23×#692311yescharacteristic
547C8—22no—
548C8—1224no—
549C8—1222no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—611no—
554C22—611no—
555C22—611no—
556C22—611no—
557C22—611no—
558C22—611no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—4881no—
563C22—2441no—
564C22—2441no—
565C22—2441no—
566C22C2×D4×#692311yescharacteristic
567C4—21no—
568C4—21no—
569C4—41no—
570C4—42no—
571C4—614no—
572C4—612no—
573C4—1222no—
574C4—1221no—
575C4—1221no—
576C4—4881no—
577C4—4881no—
578C4—4882no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C4—2442no—
583C4#41×#692312yes—
584C2—81no—
585C2—611no—
586C2—611no—
587C2—611no—
588C2—611no—
589C2—4881no—
590C2#124×#692311yescharacteristic
591C2AGL(1,ℤ/8)×#692311yescharacteristic
592C2#137×#692311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#489C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#489#6923
hypocenterC61C4×#489
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×#6923C2semidirect
3. #21462003C2semidirect
6. #21461910C2semidirect
7. #21461742C2semidirect
8. #103×#6923C2semidirect
10. #97×#6923C2semidirect
40. #16840124C4semidirect
41. #16840123C4semidirect
42. #16840122C4semidirect
43. #16840121C4semidirect
44. #16840120C4semidirect
48. C61×#489C4semidirect
88. #14519777C2×C4semidirect
90. #14519776C2×C4semidirect
102. #14519702C2×C4semidirect
103. #14519701C2×C4semidirect
104. #14519700C2×C4semidirect
108. #14519688C2×C4semidirect
109. #14519687C2×C4semidirect
110. C4⋊Q8×C61C2×C4semidirect
112. #103×C61C2×C4semidirect
114. #97×C61C2×C4semidirect
277. #6923#489direct (N × H)
318. #489#6923direct (N × H)
345. C61C4×#489semidirect
356. C4⋊Q8C2×#6923semidirect
393. #103C2×#6923semidirect
404. #97C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×#69232
#124×#69232
#214620032
#214619492
#214619112
#214619102
#214617422
#103×#69232
#102×#69232
#97×#69232
#489×D612
C4×#48961