Finite Group Databasebeta

← #29716144

Subgroups — #29716144

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519768C211yesmaximal, characteristic, complemented
2C2×#16840061C212yesmaximal, complemented
3#21448325C211yesmaximal, characteristic
4#21447159C212yesmaximal, complemented
5#21446964C212yesmaximal, complemented
6#21446775C212yesmaximal, complemented
7#21446136C211yesmaximal, characteristic
8#21446134C211yesmaximal, characteristic
9#21446006C211yesmaximal, characteristic, complemented
10#21445797C211yesmaximal, characteristic, complemented
11C61×#1853C211yesmaximal, characteristic, complemented
12C2×#14519768—28no—
13C2×#14519768—21no—
14C2×#14519768—21no—
15C2×#14519768C2212yes—
16C2×#14519768C2212yes—
17C2×#14519751C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519737—21no—
20C22×C4⋊Dic61—21no—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758196—21no—
23C22×C4×Dic61C2211yescharacteristic
24#16840061—28no—
25C2×#14519710C2212yes—
26C2×#14519706C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839651—21no—
33#16839639—21no—
34C4⋊C4×Dic61C2211yescharacteristic
35#16839609—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40C22×C4⋊C4×C61C2211yescharacteristic
41C2×#106×C61C2212yes—
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C61×#340C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×C4×D61—21no—
51C22×#92418C2312yes—
52#14519768—28no—
53#14519768—28no—
54#14519768—24no—
55#14519768—24no—
56#14519768—24no—
57#14519768D418yescomplemented
58#14519751—22no—
59#14519751Q814yes—
60#14519739—24no—
61#14519737—44no—
62C2×C4⋊Dic61—28no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—28no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—24no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83#14519710—24no—
84#14519706—24no—
85#14519699—24no—
86#14519697—24no—
87C22⋊#92418—22no—
88#14519684—21no—
89#14519684—21no—
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92C22×C8×C61C2312yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—24no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—21no—
99#106×C61—24no—
100C22⋊C8×C61—22no—
101#95×C61—21no—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—24no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×Q812yes—
114C22×C4×C61#4112yes—
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—24no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C22×Dic61—24no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—21no—
137C2×D244C2×Q812yes—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D414yes—
149C22⋊C4×C61—42no—
150#11758198—24no—
151#11758198—42no—
152#11758198—42no—
153#11758198—42no—
154C4⋊Dic61—28no—
155C4⋊Dic61—24no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—28no—
159C4⋊Dic61C2×D414yes—
160C4⋊Dic61C2×Q812yes—
161#11758196—28no—
162#11758196—28no—
163#11758196—42no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C4×Dic61—42no—
168C4×Dic61C2×D414yes—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61C22⋊Q812yes—
177C23×C61#13512yes—
178C23×C61AGL(1,ℤ/8)12yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—48no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61C22⋊Q812yes—
202C2×C4×C61C22×D412yes—
203C2×C4×C61C22×Q811yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C612−1+412yes—
206C22⋊D61—48no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—24no—
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—44no—
220C2×Dic61—44no—
221C2×Dic61—48no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61#12012yes—
227C2×Dic61#12012yes—
228D244—24no—
229D244—42no—
230C4×D61—84no—
231C8×C61—42no—
232#92418—42no—
233#24305—611nomaximal, Sylow
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61#47814yescomplemented
241C22×C61C2×#12011yescharacteristic
242C22×C61C2×C22⋊Q811yescharacteristic
243C22×C61#53611yescharacteristic
244C22×C61C2×#13511yescharacteristic
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61#57511yescharacteristic
247C22×C61#57511yescharacteristic
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C4×C61—22no—
256C4×C61—44no—
257C4×C61—81no—
258C4×C61—81no—
259C4×C61#54812yes—
260C4×C61D4×Q812yes—
261Dic61—22no—
262Dic61—22no—
263Dic61—42no—
264Dic61—41no—
265Dic61—41no—
266Dic61—44no—
267C22×#125—611no—
268C2×#478—612no—
269C4×#121—611no—
270#2059—612no—
271#2025—612no—
272#1989—612no—
273#1874—611no—
274#1874—611no—
275#1853D6111yescharacteristic, complemented
276#1852—611no—
277#1797—611no—
278C2×C61—24no—
279C2×C61#240911yescharacteristic
280C2×C61#300211yescharacteristic
281C2×C61#300611yescharacteristic
282C2×C61C2×#47811yescharacteristic
283C2×C61#308011yescharacteristic
284C2×C61#323211yescharacteristic
285C2×C61#323411yescharacteristic
286D61—82no—
287C2×#125—612no—
288C2×#125—612no—
289C2×#125—1221no—
290C2×#125—1221no—
291C2×#125—1228no—
292C2×#121—611no—
293C2×C4×D4—611no—
294C2×C4×D4—1221no—
295C22×C4⋊C4—611no—
296C22×C4⋊C4—1221no—
297C22×C4⋊C4—1221no—
298C22×C4⋊C4C2×D6111yescharacteristic
299C22×C42—611no—
300#478—1228no—
301C2×#107—612no—
302C2×#106C2×D6112yes—
303C2×C4⋊C8—612no—
304C2×#102—612no—
305C2×#102—612no—
306C2×C22⋊C8—611no—
307C2×C22⋊C8C2×D6111yescharacteristic
308#390—611no—
309#389—611no—
310#388—1221no—
311#386—1221no—
312#380C2×D6111yescharacteristic
313C4×C4⋊C4—611no—
314C4×C22⋊C4—1221no—
315#340—612no—
316#340—612no—
317#340—612no—
318#340C2×D6112yes—
319#336—612no—
320C61#2430511yescharacteristic, complemented, Sylow
321C22×D4—611no—
322C22×D4—1221no—
323C23×C4—21no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—1221no—
327C23×C4—1221no—
328C23×C4C22×D6111yescharacteristic
329C22×C8—612no—
330C22×C8C22×D6112yes—
331#125—618no—
332#125—1224no—
333#125—1224no—
334#125—1228no—
335#125—1228no—
336#125—1224no—
337#121—614no—
338#121—1222no—
339C4×D4—1224no—
340C4×D4—2444no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—1224no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1228no—
351C2×C4⋊C4—1228no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C4⋊C4C22×D6112yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C42—611no—
363C2×C42—611no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—1224no—
367C2×C42—1221no—
368#107—1224no—
369#106—24no—
370C4⋊C8—1224no—
371#102—1224no—
372#102—1224no—
373C22⋊C8—22no—
374C22⋊C8—1222no—
375#95—21no—
376#95—1221no—
377#95—1221no—
378C24—1221no—
379C24D4×D6111yescharacteristic
380C2×D4—612no—
381C2×D4—1224no—
382C2×D4—1222no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—2442no—
386C2×D4—2442no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1224no—
402C22×C4—1221no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4—2442no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4#1175822212yes—
414C22×C4D4×D6111yescharacteristic
415C22×C4Q8×D6112yes—
416C22×C4C23×D6111yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—614no—
428C4⋊C4—612no—
429C4⋊C4—1228no—
430C4⋊C4—1228no—
431C4⋊C4—1228no—
432C4⋊C4—1224no—
433C4⋊C4—1228no—
434C4⋊C4—1222no—
435C4⋊C4—1221no—
436C4⋊C4—2442no—
437C4⋊C4D4×D6114yes—
438C22⋊C4—42no—
439C22⋊C4—1224no—
440C22⋊C4—2442no—
441C22⋊C4—2442no—
442C22⋊C4—2442no—
443C42—614no—
444C42—1222no—
445C42—1222no—
446C42—1222no—
447C42—2442no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—1222no—
453C23—2441no—
454C23—2441no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23—2441no—
459C23#1451975512yes—
460C23#1451979812yes—
461C23#1451980412yes—
462C23C2×D4×D6111yescharacteristic
463D4—1224no—
464D4—2448no—
465D4—2442no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—48no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1224no—
489C2×C4—1222no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—4884no—
495C2×C4—2444no—
496C2×C4—2444no—
497C2×C4—2448no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451975512yes—
503C2×C4C2×D4×D6112yes—
504C2×C4C2×Q8×D6111yescharacteristic
505C2×C4C2×#1175822211yescharacteristic
506C2×C4#1451982412yes—
507C8—42no—
508C8—2442no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—4881no—
519C22—2442no—
520C22—2442no—
521C22—2442no—
522C22#1684006114yescomplemented
523C22#120×D6111yescharacteristic
524C22C2×#1451975511yescharacteristic
525C22#1684033411yescharacteristic
526C22C2×#1451979811yescharacteristic
527C22C2×#1451980411yescharacteristic
528C22#1684055811yescharacteristic
529C22#1684056111yescharacteristic
530C4—22no—
531C4—44no—
532C4—81no—
533C4—81no—
534C4—1222no—
535C4—1222no—
536C4—2441no—
537C4—2442no—
538C4—2441no—
539C4—2444no—
540C4#1684042112yes—
541C4#1684042512yes—
542C2—24no—
543C2—4882no—
544C2#2144918311yescharacteristic
545C2#2145503811yescharacteristic
546C2#2145507011yescharacteristic
547C2C2×#1684006111yescharacteristic
548C2#2145564011yescharacteristic
549C2#2145699011yescharacteristic
550C2#2145700811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1853C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1853D61
hypocenterC61#24305
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C22×#14519768C2semidirect
2. C2×#16840061C2semidirect
4. #21447159C2semidirect
5. #21446964C2semidirect
6. #21446775C2semidirect
9. #21446006C2semidirect
10. #21445797C2semidirect
11. C61×#1853C2semidirect
57. #14519768D4semidirect
240. C22×C61#478semidirect
275. #1853D61semidirect
320. C61#24305semidirect
522. C22#16840061semidirect

Maximal subgroups

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

subgroupindex
C22×#145197682
C2×#168400612
#214483252
#214471592
#214469642
#214467752
#214461362
#214461342
#214460062
#214457972
C61×#18532
#2430561