Finite Group Databasebeta

← #29714048

Subgroups — #29714048

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×#16840049C211yesmaximal, characteristic, complemented
2C2×#16839872C211yesmaximal, characteristic, complemented
3#21450720C211yesmaximal, characteristic, complemented
4#21447005C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446136C212yesmaximal, complemented
9#21446012C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#547C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16840049—24no—
16#16839872—24no—
17C2×#14519708C2211yescharacteristic
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839688—21no—
28#16839670—21no—
29#16839638—22no—
30#16839628C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C22×#41×C61—21no—
36C61×#547—24no—
37C61×#547—24no—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#123×C61—22no—
40C2×C22⋊Q8×C61—21no—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758199—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519751—24no—
55#14519751—42no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C22⋊Dic61—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63#14519708—24no—
64#14519708—22no—
65C4⋊#92418—22no—
66#14519698—22no—
67#14519698—42no—
68#14519697—24no—
69#14519697—22no—
70C22⋊#92418—24no—
71C22⋊#92418—42no—
72#14519688—22no—
73#14519684—22no—
74#14519684—21no—
75#14519684—21no—
76C2×#41×C61—28no—
77C2×#41×C61—48no—
78C22×Q8×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61D411yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—22no—
83C23×C4×C61—21no—
84C23×C4×C61C2312yes—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D414yescomplemented
87#123×C61—216no—
88C22⋊Q8×C61—28no—
89#121×C61—24no—
90#121×C61—24no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61D414yescomplemented
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61C2311yescharacteristic
106C24×C61—21no—
107C24×C61C2×D412yes—
108C23×D61—21no—
109#41×C61—432no—
110C2×Q8×C61—24no—
111C2×Q8×C61—44no—
112C2×D4×C61—24no—
113C2×D4×C61—28no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—44no—
121C2×D4×C61—44no—
122C2×D4×C61C2×D412yes—
123C22×C4×C61—28no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—44no—
134C22×C4×C61—44no—
135C22×C4×C61—44no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—44no—
141C2×C22⋊D61—44no—
142C2×C22⋊D61—82no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244#4112yescomplemented
151C2×C4×D61—42no—
152C2×C4×D61—82no—
153C4⋊C4×C61—28no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D412yes—
159C4⋊C4×C61C2×D414yes—
160C22⋊C4×C61—28no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165C22⋊Dic61—82no—
166#11758199—28no—
167#11758198—44no—
168#11758198—42no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—21no—
179C2×#92418—22no—
180C2×#92418—21no—
181C2×#92418#4112yescomplemented
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61AGL(1,ℤ/8)14yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200Q8×C61—48no—
201D4×C61—28no—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—416no—
205D4×C61—41no—
206D4×C61—41no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—48no—
215C2×C4×C61—416no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12012yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C22⋊D61—88no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237D244—22no—
238D244—22no—
239D244—41no—
240D244—41no—
241C4×D61—84no—
242#92418—22no—
243#92418—41no—
244#92418—42no—
245#24128—611nomaximal, Sylow
246C22×C61—24no—
247C22×C61—24no—
248C22×C61—24no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61C2×#12011yescharacteristic
259C22×C61#54611yescharacteristic
260C22×C61#54711yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)12yes—
262C22×C61#57712yes—
263C2×D61—42no—
264C2×D61—44no—
265C2×D61—82no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61—44no—
273C4×C61#52511yescharacteristic
274C4×C61#53711yescharacteristic
275C4×C61#54811yescharacteristic
276C4×C61#55011yescharacteristic
277Dic61—81no—
278Dic61—82no—
279C2×#547D6111yescharacteristic, complemented
280C2×#476—611no—
281C2×#442—611no—
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—44no—
291C2×C61—82no—
292C2×C61#240711yescharacteristic
293C2×C61#300311yescharacteristic
294C2×C61#301011yescharacteristic
295C2×C61#330612yes—
296C2×C61#334512yes—
297D61—82no—
298C22×#41—21no—
299C22×D8—611no—
300#547—24no—
301#547—24no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303C2×#123—22no—
304C2×C22⋊Q8—21no—
305C2×#121—612no—
306C2×#121C2×D6112yes—
307C2×C4×D4—611no—
308C2×C4×D4C2×D6111yescharacteristic
309C2×C4×D4C2×D6111yescharacteristic
310C22×C4⋊C4C2×D6112yes—
311#476—1224no—
312#442—1224no—
313C2×#106—611no—
314C2×#106—612no—
315C2×C4⋊C8—611no—
316C2×#102—611no—
317C2×#102—611no—
318C2×#102—612no—
319C2×#102—612no—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×#97—611no—
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—28no—
333C2×#41—48no—
334C22×Q8—21no—
335C22×D4—22no—
336C22×D4—611no—
337C22×D4—1222no—
338C22×D4C22⋊D6111yescharacteristic
339C22×D4C22×D6111yescharacteristic
340C23×C4—22no—
341C23×C4—21no—
342C23×C4—1221no—
343C23×C4C22×D6112yes—
344C2×D8—1222no—
345C2×D8—1222no—
346C2×D8—1222no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8—612no—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D6114yescomplemented
352#123—216no—
353C22⋊Q8—28no—
354#121—24no—
355#121—24no—
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—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4C22⋊D6111yescharacteristic
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4C22⋊D6112yes—
379C2×C42C22×D6111yescharacteristic
380#106—1222no—
381#106—1224no—
382C4⋊C8—1222no—
383#102—1222no—
384#102—1222no—
385#102—1224no—
386#102—2442no—
387C22⋊C8—1224no—
388C22⋊C8—2442no—
389#97—1222no—
390#95—1222no—
391#95—1221no—
392#95—1221no—
393C24—21no—
394C24—1221no—
395C24C2×C22⋊D6112yes—
396#41—432no—
397C2×Q8—24no—
398C2×Q8—44no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—612no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—4882no—
416C2×D4—2444no—
417C2×D4—2444no—
418C2×D4C2×C22⋊D6112yes—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—4882no—
435C22×C4—2442no—
436C22×C4C2×C22⋊D6112yes—
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440D8—1228no—
441C2×C8—612no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1221no—
448C2×C8—1222no—
449C2×C8—1221no—
450C4⋊C4—28no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4C2×C22⋊D6112yes—
460C4⋊C4C2×C22⋊D6114yes—
461C22⋊C4—28no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—4882no—
465C22⋊C4—2444no—
466C22⋊C4—2442no—
467C42C2×C22⋊D6112yes—
468C42C2×C22⋊D6112yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—1222no—
481C23—2441no—
482C23—2442no—
483C23—2442no—
484C23—2441no—
485C23#1451980314yes—
486C23C22×C22⋊D6111yescharacteristic
487Q8—48no—
488D4—28no—
489D4—22no—
490D4—22no—
491D4—416no—
492D4—41no—
493D4—41no—
494D4—1222no—
495D4—1222no—
496D4—4888no—
497D4—2441no—
498D4—2441no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—416no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—4884no—
516C2×C4—2441no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4#1451974612yes—
522C2×C4#1451974612yes—
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4#41×D6111yescharacteristic
526C2×C4#41×D6111yescharacteristic
527C2×C4#1451982311yescharacteristic
528C2×C4#1451982411yescharacteristic
529C8—1222no—
530C8—2441no—
531C8—2442no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—4881no—
545C22—4882no—
546C22—4881no—
547C22—2444no—
548C22—2442no—
549C22C2×#1451974611yescharacteristic
550C22#1684040011yescharacteristic
551C22#1684041111yescharacteristic
552C22C2×#1451980312yes—
553C22#1684057012yes—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—4881no—
560C4—4882no—
561C4#1684026911yescharacteristic
562C4#1684034011yescharacteristic
563C4#1684041911yescharacteristic
564C4#1684043111yescharacteristic
565C2—44no—
566C2—82no—
567C2—4882no—
568C2#2144915811yescharacteristic
569C2#2145504611yescharacteristic
570C2#2145510511yescharacteristic
571C2#2145777412yes—
572C2#2145828912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#547C2
socleC23×C61C22×D4
hypercenterC2×#547D61
hypocenterC61#24128
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
2. C2×#16839872C2semidirect
3. #21450720C2semidirect
4. #21447005C2semidirect
5. #21446815C2semidirect
6. #21446793C2semidirect
7. #21446618C2semidirect
8. #21446136C2semidirect
9. #21446012C2semidirect
10. #21445829C2semidirect
11. C2×C61×#547C2semidirect
86. C4⋊Q8×C61D4semidirect
94. C4×D4×C61D4semidirect
150. C2×D244#41semidirect
181. C2×#92418#41semidirect
279. C2×#547D61semidirect
331. C61#24128semidirect
351. C4⋊Q8C22⋊D61semidirect
363. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×#168398722
#214507202
#214470052
#214468152
#214467932
#214466182
#214461362
#214460122
#214458292
C2×C61×#5472
#2412861