Finite Group Databasebeta

← #29737055

Subgroups — #29737055

Order 15616.

1202 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450978C212yesmaximal, complemented
2#21450705C211yesmaximal, characteristic
3#21446985C212yesmaximal, complemented
4#21446982C212yesmaximal, complemented
5#21446805C212yesmaximal, complemented
6#21446232C212yesmaximal, complemented
7#21446227C212yesmaximal
8C4×#14519692C211yesmaximal, characteristic, complemented
9C61×#3501C211yesmaximal, characteristic, complemented
10C2×#14519758C2211yescharacteristic
11C2×#14519745C2211yescharacteristic
12C22×#11758198C2211yescharacteristic
13C2×#14519692C2212yes—
14#16839688—22no—
15#16839684—21no—
16#16839678—21no—
17#16839673—22no—
18#16839653—21no—
19#16839652—24no—
20#16839641—21no—
21#16839623C2211yescharacteristic
22#16839623C2214yes—
23#16839622C2211yescharacteristic
24#16839617C2211yescharacteristic
25#16839617C2214yes—
26C4×C22⋊Dic61C2212yes—
27#16839529C2212yes—
28#16839529C2214yes—
29#16839529C2214yes—
30C61×#554—22no—
31C61×#554—24no—
32D4×Q8×C61—22no—
33C61×#544—21no—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C24×D61C2311yescharacteristic
42C22×C22⋊D61—21no—
43C23×Dic61C2311yescharacteristic
44#14519758—24no—
45#14519745—24no—
46C2×C4⋊Dic61—21no—
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61C2314yes—
50C2×C22⋊Dic61C2311yescharacteristic
51C2×C22⋊Dic61C2311yescharacteristic
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—22no—
55C2×#11758198—24no—
56C2×#11758198—21no—
57C2×#11758198—22no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—24no—
61C2×C4×Dic61—22no—
62#14519692—24no—
63#14519684—21no—
64#14519684—22no—
65#14519684—24no—
66#14519684—21no—
67#14519684—24no—
68#14519684—21no—
69C22×Q8×C61—21no—
70C22×Q8×C61D412yescomplemented
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—24no—
77#125×C61—22no—
78#125×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—24no—
97#117×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D412yescomplemented
102C2×C4⋊C4×C61D414yescomplemented
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61C2314yes—
107C2×C42×C61D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C24×C61C2411yescharacteristic
110C23×D61—21no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—42no—
115C23×D61—81no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—82no—
121C2×Q8×C61—82no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61#4114yes—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—81no—
130C22×C4×C61—81no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D414yes—
135C22×C4×C61C2×D414yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—81no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—22no—
166C42×C61—24no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—24no—
170C22⋊Dic61—22no—
171C22⋊Dic61—22no—
172C22⋊Dic61—24no—
173C22⋊Dic61—42no—
174#11758198—48no—
175#11758198—42no—
176#11758198—41no—
177#11758198—41no—
178#11758198—42no—
179#11758198—42no—
180#11758198—84no—
181#11758198—81no—
182C4⋊Dic61—81no—
183#11758196—42no—
184C4×Dic61—82no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61#12412yes—
192C23×C61C22×D411yescharacteristic
193C23×C61C22×D412yes—
194C23×C61C2×#4112yes—
195C23×C612−1+412yes—
196C22×D61—24no—
197C22×D61—22no—
198C22×D61—44no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—44no—
207C22×D61—84no—
208C22×D61—84no—
209Q8×C61—44no—
210Q8×C61—82no—
211Q8×C61—82no—
212Q8×C61—82no—
213D4×C61—28no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—21no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—44no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—44no—
229C2×C4×C61—81no—
230C2×C4×C61—82no—
231C2×C4×C61—82no—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12012yes—
235C2×C4×C61#12412yes—
236C22⋊D61—84no—
237C2×Dic61—44no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—44no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—44no—
249C2×Dic61—44no—
250#25949—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—21no—
253C22×C61—22no—
254C22×C61—21no—
255C22×C61—22no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—81no—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×#12411yescharacteristic
264C22×C61#53611yescharacteristic
265C22×C61#54814yes—
266C2×D61—42no—
267C2×D61—48no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—81no—
271C2×D61—84no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—22no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—22no—
279C4×C61—44no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—44no—
283Dic61—81no—
284Dic61—84no—
285Dic61—81no—
286Dic61—81no—
287#3501D6111yescharacteristic, complemented
288#2561—612no—
289#2534—611no—
290#2030—612no—
291#2029—612no—
292#1996—612no—
293#1890—612no—
294#1889—612no—
295C4×#99—611no—
296C2×C61—21no—
297C2×C61—24no—
298C2×C61—21no—
299C2×C61—42no—
300C2×C61#243611yescharacteristic
301C2×C61#302511yescharacteristic
302C2×C61#302611yescharacteristic
303D61—82no—
304#554—24no—
305#554—22no—
306D4×Q8—22no—
307#544—21no—
308#543—21no—
309#543C2×D6112yes—
310#533—21no—
311C2×#123—611no—
312C2×C22⋊Q8—21no—
313C2×C22⋊Q8C2×D6112yes—
314C2×#120—611no—
315#518C2×D6112yes—
316C2×C4×D4C2×D6111yescharacteristic
317C22×C22⋊C4—611no—
318C2×#99—612no—
319#397—1222no—
320#396—1221no—
321#394—1221no—
322#393—1222no—
323#388—1224no—
324#388—1221no—
325#386—1221no—
326#381—614no—
327#381—611no—
328#380—611no—
329#379—614no—
330#379—611no—
331C4×C22⋊C4—612no—
332#342—614no—
333#342—614no—
334#342—612no—
335C61#2594911yescharacteristic, complemented, Sylow
336C25—611no—
337C22×Q8—21no—
338C22×Q8C22⋊D6112yescomplemented
339C22×D4—1221no—
340C22×D4C22×D6111yescharacteristic
341C23×C4—611no—
342C23×C4C22⋊D6111yescharacteristic
343C23×C4C22×D6111yescharacteristic
344C4⋊Q8—21no—
345C4⋊Q8—21no—
346C4⋊Q8—24no—
347#125—22no—
348#125—24no—
349#123—1224no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362#120—1224no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—24no—
368C4×D4—24no—
369#117—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4C22⋊D6111yescharacteristic
375C2×C4⋊C4C22⋊D6114yescomplemented
376C2×C4⋊C4C22⋊D6112yescomplemented
377C2×C22⋊C4—22no—
378C2×C22⋊C4—611no—
379C2×C22⋊C4—611no—
380C2×C22⋊C4—614no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1224no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4C22⋊D6111yescharacteristic
392C2×C22⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4C22×D6114yes—
394C2×C42—1222no—
395C2×C42C22⋊D6111yescharacteristic
396#99—1224no—
397#95—1222no—
398#95—1221no—
399#95—1221no—
400#95—1224no—
401#95—1221no—
402#95—1224no—
403C24—1222no—
404C24—1221no—
405C24—1222no—
406C24—1222no—
407C24—4881no—
408C24—2442no—
409C24C2×C22⋊D6111yescharacteristic
410C24C23×D6111yescharacteristic
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—4881no—
420C2×D4—2442no—
421C2×D4—2442no—
422C2×D4#1175822614yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6114yes—
440C22×C4C2×C22⋊D6114yes—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6112yes—
444C22×C4C2×C22⋊D6111yescharacteristic
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—4881no—
457C4⋊C4—2442no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—1224no—
464C22⋊C4—1222no—
465C22⋊C4—1222no—
466C22⋊C4—1224no—
467C22⋊C4—4881no—
468C22⋊C4—4884no—
469C22⋊C4—2448no—
470C22⋊C4—2442no—
471C22⋊C4—2442no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C42—24no—
477C42—21no—
478C42—21no—
479C42—4882no—
480C23—21no—
481C23—21no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—1224no—
487C23—1222no—
488C23—4884no—
489C23—4884no—
490C23—2444no—
491C23—2441no—
492C23—2441no—
493C23—2442no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23—2444no—
499C23#1451976612yes—
500C23C22×C22⋊D6111yescharacteristic
501C23C22×C22⋊D6112yes—
502C23C2×#1175822612yes—
503C23#1451982512yes—
504Q8—44no—
505Q8—82no—
506Q8—82no—
507Q8—82no—
508D4—28no—
509D4—4884no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—2444no—
529C2×C4—2441no—
530C2×C4—2444no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2444no—
540C2×C4—2444no—
541C2×C4#1451974611yescharacteristic
542C2×C4#1451974611yescharacteristic
543C2×C4#1451974612yes—
544C2×C4#1451976612yes—
545C22—22no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—4881no—
557C22—4884no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2442no—
562C22—2448no—
563C22—2442no—
564C22—2442no—
565C22C2×#1451974611yescharacteristic
566C22C2×#1451976611yescharacteristic
567C22#1684033611yescharacteristic
568C22#1684042314yes—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—22no—
573C4—44no—
574C4—41no—
575C4—41no—
576C4—44no—
577C4—4881no—
578C4—4884no—
579C4—4881no—
580C4—4881no—
581C2—21no—
582C2—24no—
583C2—21no—
584C2—42no—
585C2—4882no—
586C2#2144947311yescharacteristic
587C2#2145521111yescharacteristic
588C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC2×#14519745C22
hypercenter#3501D61
hypocenterC61#25949
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21450978C2semidirect
3. #21446985C2semidirect
4. #21446982C2semidirect
5. #21446805C2semidirect
6. #21446232C2semidirect
8. C4×#14519692C2semidirect
9. C61×#3501C2semidirect
70. C22×Q8×C61D4semidirect
101. C2×C4⋊C4×C61D4semidirect
102. C2×C4⋊C4×C61D4semidirect
287. #3501D61semidirect
335. C61#25949semidirect
338. C22×Q8C22⋊D61semidirect
375. C2×C4⋊C4C22⋊D61semidirect
376. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214509782
#214507052
#214469852
#214469822
#214468052
#214462322
#214462272
C4×#145196922
C61×#35012
#2594961