Finite Group Databasebeta

← #29714054

Subgroups — #29714054

Order 15616.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3#21450716C211yesmaximal, characteristic, complemented
4#21447006C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446792C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446010C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#442C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840417—24no—
14#16840417—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519759—22no—
17C2×#14519756—21no—
18C2×#14519748C2212yes—
19C2×#14519748C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61C2212yes—
22#16840052—24no—
23C2×#14519708C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29#16839687—21no—
30#16839671—21no—
31#16839641—22no—
32#16839628C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×D8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#442—24no—
41C2×#106×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×#97×C61C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—48no—
47C22×Dic122—21no—
48C22×D244D411yescharacteristic
49C22×C22⋊D61—22no—
50C22×C22⋊D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61C2312yes—
53C22×#92418C2312yes—
54#14519777—22no—
55#14519777D414yescomplemented
56#14519759—216no—
57#14519756—28no—
58#14519748—24no—
59#14519748—24no—
60#14519748—24no—
61#14519748—42no—
62C4×D244—22no—
63C4×D244—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C4⋊C4×D61—24no—
70C4⋊C4×D61—24no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—22no—
74C2×#11758198D412yes—
75#14519708—24no—
76C4⋊#92418—22no—
77#14519698—24no—
78#14519698—42no—
79#14519689—24no—
80#14519689—42no—
81#14519684—21no—
82#14519684—21no—
83#14519684—22no—
84C22×D4×C61C2311yescharacteristic
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C2×D8×C61—22no—
89C2×D8×C61—22no—
90C2×D8×C61—22no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61D414yescomplemented
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#106×C61—22no—
103#102×C61—22no—
104#102×C61—22no—
105#97×C61—22no—
106C24×C61—21no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61C2×D412yes—
120C2×D4×C61#4112yescomplemented
121C22×C4×C61—21no—
122C22×C4×C61—24no—
123C22×C4×C61—21no—
124C22×C4×C61—44no—
125C22×C4×C61—42no—
126C22×C4×C61—82no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—28no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61C2×D412yes—
139#11758222—432no—
140C2×D244—24no—
141C2×D244—22no—
142C2×C4×D61—28no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—44no—
150C2×C4×D61—44no—
151C2×Dic122—24no—
152C2×Dic122—44no—
153D8×C61—28no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61#4112yescomplemented
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—42no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—28no—
169C22⋊Dic61—44no—
170#11758198—24no—
171#11758198—82no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D414yes—
178#11758196—28no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61AGL(1,ℤ/8)14yes—
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—41no—
205D4×C61—41no—
206D4×C61—41no—
207D4×C61—41no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—48no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—84no—
216C2×C4×C61#12012yes—
217C2×C4×C61#12012yes—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C22⋊D61—416no—
225C22⋊D61—88no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—28no—
236C4×D61—416no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241Dic122—48no—
242C8×C61—22no—
243C8×C61—41no—
244#92418—42no—
245#24128—611nomaximal, Sylow
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—44no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—82no—
256C22×C61C2×#12011yescharacteristic
257C22×C61#54611yescharacteristic
258C22×C61#54711yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)12yes—
260C22×C61#57712yes—
261C2×D61—24no—
262C2×D61—24no—
263C2×D61—24no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—42no—
271C4×C61—81no—
272C4×C61#52511yescharacteristic
273C4×C61#53711yescharacteristic
274C4×C61#54811yescharacteristic
275C4×C61#55011yescharacteristic
276Dic61—42no—
277Dic61—44no—
278Dic61—82no—
279C2×#547—611no—
280C2×#476—611no—
281C2×#442D6111yescharacteristic, complemented
282#2535—611no—
283#2034—612no—
284#1999—612no—
285#1993—611no—
286#1955—611no—
287#1874—612no—
288#1853—612no—
289#1804—611no—
290C2×C61—82no—
291C2×C61—82no—
292C2×C61#240711yescharacteristic
293C2×C61#300311yescharacteristic
294C2×C61#301011yescharacteristic
295C2×C61#330612yes—
296C2×C61#334512yes—
297D61—44no—
298C22×#41—1221no—
299C22×D8C2×D6111yescharacteristic
300#547—1224no—
301#547—1224no—
302C2×C4⋊Q8—611no—
303C2×#123—1222no—
304C2×C22⋊Q8—1221no—
305C2×#121—612no—
306C2×#121—612no—
307C2×C4×D4—611no—
308C2×C4×D4C2×D6111yescharacteristic
309C2×C4×D4C2×D6111yescharacteristic
310C22×C4⋊C4—612no—
311#476—1224no—
312#442—24no—
313C2×#106—612no—
314C2×#106C2×D6111yescharacteristic
315C2×C4⋊C8—611no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6111yescharacteristic
319C2×#102C2×D6111yescharacteristic
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×#97C2×D6111yescharacteristic
323#397—1221no—
324#392—1221no—
325#386—1222no—
326#382—611no—
327#380—612no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2412811yescharacteristic, complemented, Sylow
332C2×#41—1228no—
333C2×#41—2448no—
334C22×Q8—1221no—
335C22×D4—611no—
336C22×D4—1222no—
337C22×D4—1222no—
338C22×D4C22×D6111yescharacteristic
339C22×D4C22×D6111yescharacteristic
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—612no—
343C23×C4—1222no—
344C2×D8—22no—
345C2×D8—22no—
346C2×D8—22no—
347C22×C8—612no—
348C22×C8C22×D6111yescharacteristic
349C22×C8C22×D6111yescharacteristic
350C4⋊Q8—614no—
351C4⋊Q8—1222no—
352#123—12216no—
353C22⋊Q8—1228no—
354#121—1224no—
355#121—1224no—
356#121—1224no—
357#121—2442no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—1222no—
362C4×D4—1222no—
363C4×D4C22⋊D6114yescomplemented
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1224no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—612no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C42C22×D6111yescharacteristic
380#106—22no—
381#106—1224no—
382C4⋊C8—1222no—
383#102—22no—
384#102—22no—
385#102—1224no—
386#102—2442no—
387C22⋊C8—1224no—
388C22⋊C8—2442no—
389#97—22no—
390#95—1222no—
391#95—1221no—
392#95—1221no—
393C24—21no—
394C24—21no—
395C24—612no—
396#41—24432no—
397C2×Q8—1224no—
398C2×Q8—2444no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—1224no—
410C2×D4—1228no—
411C2×D4—1222no—
412C2×D4—4882no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4—2444no—
416C2×D4—2444no—
417C2×D4#1175822212yescomplemented
418C2×D4C2×C22⋊D6112yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—82no—
425C22×C4—612no—
426C22×C4—1228no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—2444no—
436C22×C4—2444no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440D8—28no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8#1175822212yescomplemented
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—614no—
454C4⋊C4—1228no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4C2×C22⋊D6112yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—1228no—
464C22⋊C4—1224no—
465C22⋊C4—4882no—
466C22⋊C4—2444no—
467C42D4×D6112yes—
468C42C2×C22⋊D6112yes—
469C23—22no—
470C23—22no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—614no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23C2×D4×D6111yescharacteristic
487Q8—2448no—
488D4—22no—
489D4—22no—
490D4—22no—
491D4—22no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—1228no—
497D4—4888no—
498D4—24416no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—1222no—
508C2×C4—1224no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—24416no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4#1451974512yes—
522C2×C4#1451974512yes—
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4C2×#1175822211yescharacteristic
527C2×C4#1451982311yescharacteristic
528C2×C4#1451982411yescharacteristic
529C8—22no—
530C8—41no—
531C8—2442no—
532C22—42no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—81no—
542C22—1224no—
543C22—1224no—
544C22—1224no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22C2×#1451974511yescharacteristic
550C22#1684040811yescharacteristic
551C22#1684041711yescharacteristic
552C22AGL(1,ℤ/8)×D6112yes—
553C22#1684057212yes—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—81no—
558C4—4882no—
559C4—2444no—
560C4—2442no—
561C4#1684027311yescharacteristic
562C4#1684034011yescharacteristic
563C4#1684042111yescharacteristic
564C4#1684043311yescharacteristic
565C2—82no—
566C2—82no—
567C2—2444no—
568C2#2144916411yescharacteristic
569C2#2145504411yescharacteristic
570C2#2145510411yescharacteristic
571C2#2145778512yes—
572C2#2145829412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#442C2
socleC23×C61C22×D4
hypercenterC2×#442D61
hypocenterC61#24128
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#16840417C2semidirect
2. C2×#16840052C2semidirect
3. #21450716C2semidirect
4. #21447006C2semidirect
5. #21446814C2semidirect
6. #21446792C2semidirect
7. #21446617C2semidirect
8. #21446133C2semidirect
9. #21446010C2semidirect
10. #21445829C2semidirect
11. C2×C61×#442C2semidirect
55. #14519777D4semidirect
96. C4×D4×C61D4semidirect
120. C2×D4×C61#41semidirect
159. C2×C8×C61#41semidirect
281. C2×#442D61semidirect
331. C61#24128semidirect
363. C4×D4C22⋊D61semidirect
417. C2×D4#11758222semidirect
449. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168400522
#214507162
#214470062
#214468142
#214467922
#214466172
#214461332
#214460102
#214458292
C2×C61×#4422
#2412861