Finite Group Databasebeta

← #29685951

Subgroups — #29685951

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519825C211yesmaximal, characteristic, complemented
2C2×#16839934C211yesmaximal, characteristic
3C4×#14519803C212yesmaximal, complemented
4#21446223C211yesmaximal, characteristic, complemented
5#21446198C212yesmaximal, complemented
6#21446044C212yesmaximal, complemented
7#21446033C211yesmaximal, characteristic, complemented
8#21445777C211yesmaximal, characteristic, complemented
9#21445630C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C61×#2326C211yesmaximal, characteristic, complemented
12C2×#14519825C2211yescharacteristic
13C2×#14519803C2212yes—
14#16840244—22no—
15#16840243—22no—
16#16840243C2212yes—
17#16840240C2211yescharacteristic
18#16840238—22no—
19C4×#11758226—22no—
20C4×#11758222—22no—
21C4×#11758222C2212yes—
22C4×Q8×D61—22no—
23#16839934C418yescomplemented
24#16839872—22no—
25#16839862—22no—
26C4×#11758200—22no—
27C4×#11758199—22no—
28#16839802C2212yes—
29C2×#14519705C2212yes—
30C2×#14519705C2212yes—
31#16839767C2212yes—
32#16839761C2212yes—
33C2×#14519696C2211yescharacteristic
34C2×#14519696C2211yescharacteristic
35C4×#11758194C2212yes—
36#16839531C2212yes—
37#16839522C2212yes—
38#16839522C2212yes—
39C2×#124×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#391C2211yescharacteristic
43C61×#390C2211yescharacteristic
44C61×#388—22no—
45C61×#385—22no—
46C61×#384C2211yescharacteristic
47C61×#376C2211yescharacteristic
48#14519825—22no—
49#14519825—24no—
50#14519825C2×C414yescomplemented
51C2×#11758226—22no—
52C2×#11758222—22no—
53C2×#11758222C2312yes—
54C2×Q8×D61—22no—
55#14519803—24no—
56#14519803C2×C418yescomplemented
57C2×#11758200—22no—
58C2×#11758199—22no—
59C2×#11758194C2312yes—
60C2×#11758194C2312yes—
61Q8×Dic61—22no—
62Q8×Dic61—22no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×Dic122D412yes—
66C4×Dic122D412yes—
67#14519742—24no—
68#14519742—22no—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244D412yes—
72C4×D244D412yes—
73#14519739—24no—
74#14519739—22no—
75C4×C22⋊D61—22no—
76C4×C22⋊D61—24no—
77C4×C22⋊D61—22no—
78#14519736—22no—
79#14519736C2312yes—
80#14519735—24no—
81#14519735—22no—
82#14519734—24no—
83#14519734—22no—
84C4⋊C4×D61—24no—
85C4⋊C4×D61—22no—
86C42×D61—24no—
87C42×D61—22no—
88#14519710—22no—
89#14519708—22no—
90#14519705—24no—
91#14519705C2×C418yescomplemented
92#14519700—22no—
93#14519698—22no—
94#14519698—22no—
95#14519697—22no—
96#14519696—22no—
97#14519696C2×C414yescomplemented
98#14519688—22no—
99C4×#92418—22no—
100C22×Q8×C61C2311yescharacteristic
101C22×D4×C61C2311yescharacteristic
102C23×C4×C61—21no—
103#124×C61—22no—
104#124×C61C2×C414yescomplemented
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61D412yes—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—22no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61D412yes—
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—22no—
123C24×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61C22×C412yes—
127C2×Q8×C61#4112yes—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—42no—
136C22×C4×C61—82no—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—22no—
143#11758226—28no—
144#11758226—28no—
145Q8×D61—28no—
146Q8×D61—28no—
147#11758222—28no—
148#11758222—28no—
149#11758222—22no—
150#11758222—22no—
151#11758222C22×C414yes—
152#11758222#4114yes—
153C2×D244—22no—
154C2×D244C2×D412yes—
155C2×C4×D61—24no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—24no—
159C2×C4×D61—22no—
160C2×Dic122—22no—
161C2×Dic122C2×D412yes—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—82no—
166C22⋊C4×C61—28no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—82no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—21no—
173C42×C61—22no—
174C42×C61C22×C412yes—
175C42×C61C2×D412yes—
176C42×C61C2×D412yes—
177C22⋊Dic61—22no—
178#11758200—28no—
179#11758199—28no—
180#11758198—24no—
181#11758198—22no—
182#11758198—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
185C4⋊Dic61C2×D412yes—
186C4⋊Dic61C2×D412yes—
187#11758196—24no—
188#11758196—22no—
189#11758196—22no—
190C4×Dic61—24no—
191C4×Dic61—24no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195#11758194—22no—
196#11758194—22no—
197#11758194—22no—
198#11758194C22×C414yes—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—22no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208Q8×C61—24no—
209Q8×C61—24no—
210D4×C61—24no—
211D4×C61—42no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—84no—
227C2×C4×C61C4×D414yes—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12012yes—
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D412yes—
233C2×C4×C61C2×#4112yes—
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—24no—
236C22⋊D61—24no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240D244—28no—
241D244—22no—
242D244C4×D414yes—
243C4×D61—216no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—44no—
247C4×D61—82no—
248C4×D61—84no—
249Dic122—28no—
250Dic122—22no—
251Dic122C4×D414yes—
252#92418—24no—
253#92418—42no—
254#21475—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61—82no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C2×C4×D411yescharacteristic
263C22×C61C2×#12011yescharacteristic
264C22×C61#52511yescharacteristic
265C2×D61—24no—
266C2×D61—42no—
267C2×D61—42no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—24no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×C4×D412yes—
277C4×C61#52512yes—
278Dic61—24no—
279Dic61—42no—
280Dic61—42no—
281C4×2−1+4—611no—
282C2×#454—611no—
283C4×AGL(1,ℤ/8)—612no—
284#2326D6111yescharacteristic, complemented
285#1888—611no—
286#1883—612no—
287#1859—612no—
288#1857—611no—
289#1793—611no—
290#1759—612no—
291C4×#101—611no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—82no—
295C2×C61C4×#12011yescharacteristic
296C2×C61#302011yescharacteristic
297C2×C61#302111yescharacteristic
298D61—44no—
299C2×2−1+4—611no—
300C2×AGL(1,ℤ/8)—612no—
301C2×#124C2×D6111yescharacteristic
302#520—612no—
303#520—1222no—
304#520—1222no—
305#519—611no—
306#519—1222no—
307C4×#41—612no—
308C4×#41—1222no—
309C4×#41—1222no—
310C2×C4×Q8—1222no—
311C2×C4×Q8C2×D6111yescharacteristic
312C2×C4×D4C2×D6111yescharacteristic
313#454—618no—
314#442—1222no—
315#440—1222no—
316C4×SD16—1222no—
317C4×D8—1222no—
318#428—612no—
319C2×#104—612no—
320C2×#104—612no—
321#418—612no—
322#417—612no—
323C2×#101—611no—
324C2×#101—611no—
325C4×M16—612no—
326#391C2×D6111yescharacteristic
327#390C2×D6111yescharacteristic
328#388—22no—
329#385—22no—
330#384C2×D6111yescharacteristic
331#376C2×D6111yescharacteristic
332#343—612no—
333#339—612no—
334#339—612no—
335C61#2147511yescharacteristic, complemented, Sylow
3362−1+4—614no—
3372−1+4—1224no—
3382−1+4—1222no—
339C2×#41—612no—
340C2×#41—1222no—
341C2×#41—1222no—
342C22×Q8—1222no—
343C22×Q8C22×D6111yescharacteristic
344C22×D4C22×D6111yescharacteristic
345C23×C4—21no—
346AGL(1,ℤ/8)—618no—
347AGL(1,ℤ/8)—1224no—
348C2×SD16—1222no—
349C2×D8—1222no—
350C2×M16—612no—
351C2×M16—612no—
352#124—22no—
353#124C4×D6114yescomplemented
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—612no—
357C4×Q8—612no—
358C4×Q8—1222no—
359C4×Q8—1222no—
360C4×Q8—1222no—
361C4×Q8—1224no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—612no—
367C4×D4—612no—
368C4×D4—1222no—
369C4×D4—1222no—
370C4×D4—1222no—
371C4×D4—1224no—
372C4×D4—1222no—
373C4×D4—1222no—
374C4×D4—1224no—
375#117—612no—
376#117—1224no—
377#117—1222no—
378#117—1224no—
379#117—1222no—
380#117—1222no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1224no—
384C2×C4⋊C4C22⋊D6112yes—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C42—1224no—
391C2×C42—1222no—
392C2×C42C22⋊D6112yes—
393C2×C42C22×D6111yescharacteristic
394C2×C42C22×D6111yescharacteristic
395C2×C42C22×D6111yescharacteristic
396#107—1222no—
397#106—1222no—
398#104—618no—
399#104—1224no—
400#103—1222no—
401#102—1222no—
402#102—1222no—
403#102—1222no—
404#101—614no—
405#101—1222no—
406#97—1222no—
407C4×C8—1222no—
408#95—21no—
409#95—21no—
410#95—22no—
411C24—21no—
412#41—614no—
413#41—614no—
414#41—1228no—
415#41—1228no—
416#41—1228no—
417#41—1222no—
418#41—1222no—
419#41—1228no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—612no—
423C2×Q8—1222no—
424C2×Q8—1228no—
425C2×Q8—1228no—
426C2×Q8C2×C4×D6112yes—
427C2×Q8#1175822212yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—612no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4C2×C4×D6112yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6112yes—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450SD16—1228no—
451D8—1228no—
452M16—614no—
453M16—1222no—
454M16—1222no—
455M16—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—82no—
462C4⋊C4—612no—
463C4⋊C4—612no—
464C4⋊C4—1224no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C22⋊C4—28no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C22⋊C4—1224no—
474C22⋊C4—1222no—
475C22⋊C4—1222no—
476C22⋊C4—1222no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—1224no—
482C42—1224no—
483C42—1222no—
484C42—1222no—
485C42—1222no—
486C42C2×C4×D6112yes—
487C42D4×D6112yes—
488C42D4×D6112yes—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—1222no—
495C23C22×C22⋊D6111yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—614no—
499Q8—1228no—
500Q8—1222no—
501D4—24no—
502D4—42no—
503D4—614no—
504D4—1228no—
505D4—1224no—
506D4—1224no—
507D4—1222no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4—12216no—
524C2×C4—1222no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—4882no—
530C2×C4—4884no—
531C2×C4—2444no—
532C2×C4C4×C22⋊D6114yes—
533C2×C4#1451974511yescharacteristic
534C2×C4#1451974511yescharacteristic
535C2×C4#1451974512yes—
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C2×D4×D6112yes—
538C2×C4#41×D6112yes—
539C2×C4C2×#1175822211yescharacteristic
540C8—1224no—
541C8—2442no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—42no—
546C22—82no—
547C22—81no—
548C22—81no—
549C22—1224no—
550C22—2442no—
551C22—2442no—
552C22C2×C4×C22⋊D6111yescharacteristic
553C22C2×#1451974511yescharacteristic
554C22#1684026911yescharacteristic
555C4—21no—
556C4—21no—
557C4—22no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—1224no—
564C4—2442no—
565C4—2442no—
566C4C4×D4×D6112yes—
567C4#1684027112yes—
568C2—21no—
569C2—21no—
570C2—82no—
571C2—2444no—
572C2C4×#1451974511yescharacteristic
573C2#2145518211yescharacteristic
574C2#2145519011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2326C2
socleC22×C61C2×#120
prime radicalC2×#16839934C2
cosocleC22×C4×C61C24
hypercenter#2326D61
hypocenterC61#21475
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. C4×#14519825C2semidirect
3. C4×#14519803C2semidirect
4. #21446223C2semidirect
5. #21446198C2semidirect
6. #21446044C2semidirect
7. #21446033C2semidirect
8. #21445777C2semidirect
9. #21445630C2semidirect
10. C4×#14519696C2semidirect
11. C61×#2326C2semidirect
23. #16839934C4semidirect
50. #14519825C2×C4semidirect
56. #14519803C2×C4semidirect
91. #14519705C2×C4semidirect
97. #14519696C2×C4semidirect
104. #124×C61C2×C4semidirect
284. #2326D61semidirect
335. C61#21475semidirect
353. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C4×#145198252
C2×#168399342
C4×#145198032
#214462232
#214461982
#214460442
#214460332
#214457772
#214456302
C4×#145196962
C61×#23262
#2147561