Finite Group Databasebeta

← #29713292

Subgroups — #29713292

Order 15616.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840083C212yesmaximal, complemented
2#21448302C211yesmaximal, characteristic, complemented
3#21447149C211yesmaximal, characteristic, complemented
4#21447144C211yesmaximal, characteristic, complemented
5#21447050C211yesmaximal, characteristic, complemented
6#21447050C211yesmaximal, characteristic, complemented
7#21446142C212yesmaximal, complemented
8#21446080C212yesmaximal, complemented
9#21446066C212yesmaximal, complemented
10#21445887C211yesmaximal, characteristic, complemented
11C2×C61×#532C211yesmaximal, characteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16840083—28no—
15C2×#14519710C2212yes—
16C2×#14519708C2212yes—
17C2×#14519698C2212yes—
18C2×#14519698C2212yes—
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×C22⋊#92418C2212yes—
22#16839660C2211yescharacteristic
23#16839660C2211yescharacteristic
24#16839660C2211yescharacteristic
25#16839660C2211yescharacteristic
26#16839635—21no—
27#16839635—21no—
28C4×#11758196—21no—
29C4×#11758196—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2211yescharacteristic
33#16839517C2211yescharacteristic
34#16839517C2212yes—
35C22×#41×C61—21no—
36C61×#532—28no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#127×C61C2211yescharacteristic
39C2×#124×C61—21no—
40C2×#121×C61—21no—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2212yes—
44C22×C42×C61C2211yescharacteristic
45C22×#92418C2312yes—
46C22×#92418C2312yes—
47#14519736—22no—
48#14519736—22no—
49#14519736—24no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2311yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61—21no—
62#14519710—24no—
63#14519708—24no—
64#14519698—24no—
65#14519698—24no—
66#14519698—24no—
67#14519698—24no—
68C22⋊#92418—24no—
69#14519684—21no—
70#14519684—21no—
71#14519684—21no—
72#14519684—21no—
73C2×#41×C61—28no—
74C2×#41×C61—44no—
75C2×#41×C61—44no—
76C22×Q8×C61—21no—
77C22×D4×C61—22no—
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61C2312yes—
81C23×C4×C61—22no—
82C23×C4×C61D411yescharacteristic
83C23×C4×C61D411yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C4⋊Q8×C61—24no—
86#127×C61—24no—
87#124×C61—28no—
88#121×C61—28no—
89#121×C61—24no—
90#121×C61D418yes—
91C4×D4×C61—28no—
92C4×D4×C61—24no—
93C4×D4×C61—24no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C42×C61—24no—
99C2×C42×C61D411yescharacteristic
100C2×C42×C61D411yescharacteristic
101C2×C42×C61C2311yescharacteristic
102C2×C42×C61C2311yescharacteristic
103C24×C61—22no—
104C24×C61C2×D411yescharacteristic
105#41×C61—432no—
106C2×Q8×C61—24no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C2×D4×C61—24no—
110C2×D4×C61—24no—
111C2×D4×C61—28no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—44no—
120C2×D4×C61—44no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61C2×D414yes—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—28no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—41no—
135C22×C4×C61—44no—
136C22×C4×C61—44no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D414yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D414yes—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—28no—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61—22no—
156C42×C61—22no—
157C22⋊Dic61—24no—
158C22⋊Dic61—42no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61#4112yescomplemented
164C4⋊Dic61#4112yescomplemented
165#11758196—42no—
166#11758196—42no—
167#11758196—42no—
168#11758196—42no—
169C4×Dic61—24no—
170C4×Dic61—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—24no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61#12012yes—
191C23×C61C22×D411yescharacteristic
192Q8×C61—48no—
193D4×C61—24no—
194D4×C61—24no—
195D4×C61—48no—
196D4×C61—48no—
197D4×C61—416no—
198D4×C61—42no—
199D4×C61—42no—
200C2×C4×C61—28no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—21no—
212C2×C4×C61—24no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—44no—
216C2×C4×C61—416no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61#12012yes—
220C2×C4×C61#13512yes—
221C2×C4×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61C22×D412yes—
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C612−1+412yes—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239#92418—42no—
240#92418—42no—
241#24062—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—44no—
250C22×C61—44no—
251C22×C61—44no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61C2×#12011yescharacteristic
257C22×C61#52511yescharacteristic
258C22×C61#53611yescharacteristic
259C22×C61C2×#13511yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61#57511yescharacteristic
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—41no—
268C4×C61—44no—
269C4×C61#54712yes—
270C4×C61#54712yes—
271Dic61—42no—
272Dic61—42no—
273Dic61—81no—
274Dic61—81no—
275C2×#532D6111yescharacteristic
276C2×#482—612no—
277#2296—611no—
278#2058—611no—
279#2057—611no—
280#2041—611no—
281#2041—611no—
282#1875—612no—
283#1864—612no—
284#1862—612no—
285#1817—611no—
286C2×C61—42no—
287C2×C61—84no—
288C2×C61#240611yescharacteristic
289C2×C61#300211yescharacteristic
290C2×C61#300311yescharacteristic
291C2×C61#310611yescharacteristic
292C2×C61#311511yescharacteristic
293C2×C61#322811yescharacteristic
294C2×C61#323511yescharacteristic
295C22×#41—21no—
296#532—28no—
297C2×C4⋊Q8C2×D6111yescharacteristic
298C2×#127C2×D6111yescharacteristic
299C2×#124—21no—
300C2×#121—21no—
301C2×#121C2×D6112yes—
302C2×C4×D4—21no—
303C2×C4×D4C2×D6112yes—
304C2×#117—611no—
305C2×#117—611no—
306C22×C42C2×D6111yescharacteristic
307#482—1228no—
308C2×#107—612no—
309C2×#106—612no—
310C2×#102—612no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×C22⋊C8—612no—
315#389—611no—
316#389—611no—
317#389—611no—
318#389—611no—
319#385—1221no—
320#385—1221no—
321C4×C4⋊C4—1221no—
322C4×C4⋊C4—1221no—
323#340—612no—
324#340—612no—
325#336—611no—
326#336—611no—
327#336—612no—
328C61#2406211yescharacteristic, complemented, Sylow
329C2×#41—28no—
330C2×#41—44no—
331C2×#41—44no—
332C22×Q8—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C22×D6112yes—
337C23×C4—22no—
338C23×C4C22⋊D6111yescharacteristic
339C23×C4C22⋊D6111yescharacteristic
340C23×C4C22×D6111yescharacteristic
341C22×C8—612no—
342C22×C8—612no—
343C4⋊Q8—24no—
344#127—24no—
345#124—28no—
346#121—28no—
347#121—24no—
348#121C22⋊D6118yes—
349C4×D4—28no—
350C4×D4—24no—
351C4×D4—24no—
352#117—1222no—
353#117—1222no—
354#117—1224no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C42—24no—
370C2×C42—1221no—
371C2×C42—1221no—
372C2×C42C22⋊D6111yescharacteristic
373C2×C42C22⋊D6111yescharacteristic
374C2×C42C22×D6111yescharacteristic
375C2×C42C22×D6111yescharacteristic
376#107—1224no—
377#106—1224no—
378#102—1224no—
379#102—1224no—
380#102—1224no—
381#102—1224no—
382C22⋊C8—1224no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386#95—1221no—
387C24—22no—
388C24C2×C22⋊D6111yescharacteristic
389#41—432no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C2×C22⋊D6114yes—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6114yes—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—612no—
441C4⋊C4—612no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—2442no—
447C4⋊C4—2442no—
448C4⋊C4—2442no—
449C4⋊C4—2442no—
450C4⋊C4C2×C22⋊D6114yes—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—1224no—
454C22⋊C4—2442no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—1224no—
460C42—2442no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23#1451974612yes—
475C23C22×C22⋊D6111yescharacteristic
476Q8—48no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—48no—
481D4—416no—
482D4—42no—
483D4—42no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—44no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4#1451974612yes—
516C2×C4#1451979712yes—
517C2×C4#1451980312yes—
518C2×C4#1451980312yes—
519C2×C4C22×C22⋊D6112yes—
520C2×C4#41×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C2×C4#1451982412yes—
523C8—2442no—
524C8—2442no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22C2×#1451974611yescharacteristic
540C22#1684026911yescharacteristic
541C22#1684033311yescharacteristic
542C22C2×#1451979711yescharacteristic
543C22C2×#1451980311yescharacteristic
544C22C2×#1451980311yescharacteristic
545C22#1684056011yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—44no—
552C4—4881no—
553C4—4881no—
554C4—2442no—
555C4—2442no—
556C4#1684041112yes—
557C4#1684041112yes—
558C2—42no—
559C2—84no—
560C2#2144915011yescharacteristic
561C2#2145503711yescharacteristic
562C2#2145504611yescharacteristic
563C2#2145587311yescharacteristic
564C2#2145596411yescharacteristic
565C2#2145695111yescharacteristic
566C2#2145701311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#532C2
socleC23×C61C22×D4
hypercenterC2×#532D61
hypocenterC61#24062
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16840083C2semidirect
2. #21448302C2semidirect
3. #21447149C2semidirect
4. #21447144C2semidirect
5. #21447050C2semidirect
6. #21447050C2semidirect
7. #21446142C2semidirect
8. #21446080C2semidirect
9. #21446066C2semidirect
10. #21445887C2semidirect
163. C4⋊Dic61#41semidirect
164. C4⋊Dic61#41semidirect
328. C61#24062semidirect

Maximal subgroups

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

subgroupindex
C2×#168400832
#214483022
#214471492
#214471442
#214470502
#214461422
#214460802
#214460662
#214458872
C2×C61×#5322
#2406261