Finite Group Databasebeta

← #30615540

Subgroups — #30615540

Order 15616.

1458 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462799C211yesmaximal, characteristic
2#21462300C212yesmaximal, complemented
3#21462294C211yesmaximal, characteristic
4#21461790C218yesmaximal, complemented
5#21461517C212yesmaximal, complemented
6#410×D61C211yesmaximal, characteristic
7#16840831—22no—
8#16840831—22no—
9#16840830—21no—
10#16840829—21no—
11#16840829C2214yes—
12#41×#6923—24no—
13C2×#14519856—22no—
14C2×#14519856C2211yescharacteristic
15C2×#14519856C2212yes—
16C22×C4⋊#6923C2211yescharacteristic
17(C2×#6923)⋊Q8—21no—
18#16840714—21no—
19#16840713—24no—
20#16840710C2214yes—
21#16840709C2218yes—
22#16840708—22no—
23#16840706—22no—
24#16840704—22no—
25#16840639C2218yes—
26C2×#41×D61C2211yescharacteristic
27#117×D61C2212yes—
28#16839736C411yescharacteristic, complemented
29#16839735C412yescomplemented
30#16839734C414yescomplemented
31#99×D61C2214yes—
32C61×#410C411yescharacteristic, complemented
33C23×#6923C2312yes—
34Q8×#6923—24no—
35Q8×#6923—24no—
36D4×#6923—24no—
37D4×#6923—24no—
38D4×#6923—28no—
39D4×#6923—28no—
40#14519856—24no—
41#14519856—24no—
42#14519856—22no—
43#14519856—24no—
44#14519856—22no—
45#14519856—22no—
46#14519856—22no—
47#14519856—22no—
48#14519856—22no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923—22no—
55C2×C4⋊#6923—21no—
56C2×C4⋊#6923—28no—
57C2×C4⋊#6923D412yes—
58C2×C4⋊#6923D412yes—
59C2×C22⋊#6923—24no—
60C2×C22⋊#6923—24no—
61C2×C22⋊#6923C2312yes—
62C2×C22⋊#6923C2314yes—
63C2×C4×#6923—24no—
64C2×C4×#6923—21no—
65C2×C4×#6923D414yes—
66(C2×#6923)⋊C4—28no—
67(C2×#6923)⋊C4—24no—
68C2×#11758226Q811yescharacteristic
69C2×#11758222C2×C411yescharacteristic
70C2×#11758222Q812yes—
71C2×#11758224C2×C412yes—
72C2×#11758224Q811yescharacteristic
73#41×D61—22no—
74#41×D61—22no—
75#41×D61—22no—
76#41×D61—22no—
77C2×Q8×D61D411yescharacteristic
78C2×D4×D61D411yescharacteristic
79C2×D4×D61C2312yes—
80C22×C4×D61D412yes—
81C22×C4×D61C2311yescharacteristic
82#14519736C2×C412yescomplemented
83#14519735—22no—
84#14519734—22no—
85#14519733C2×C414yescomplemented
86C4⋊C4×D61—22no—
87C22⋊C4×D61C2314yes—
88C42×D61—22no—
89#14519692C2×C414yescomplemented
90#14519691C2×C418yescomplemented
91C2×#41×C61C2×C411yescharacteristic
92#117×C61C2×C412yescomplemented
93#99×C61C2×C414yescomplemented
94C23×D61C2×D412yes—
95C23×D61C2411yescharacteristic
96C22×#6923—24no—
97C22×#6923—24no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—24no—
101C22×#6923—82no—
102C22×#6923C2×D414yes—
103C22×#6923C2×D414yes—
104#41×C61—22no—
105#41×C61—22no—
106C2×Q8×C61C4⋊C411yescharacteristic, complemented
107C2×D4×C61C4⋊C411yescharacteristic, complemented
108C2×D4×C61C22×C412yes—
109C22×C4×C61C4⋊C412yescomplemented
110C22×C4×C61C22×C411yescharacteristic
111C2×C22⋊D61C4⋊C412yescomplemented
112C2×C22⋊D61C22×C412yes—
113C2×C22⋊D61C2×Q812yes—
114C22×Dic61C4⋊C412yescomplemented
115C22×Dic61C22×C411yescharacteristic
116#11758226—22no—
117#11758226—22no—
118Q8×D61—22no—
119Q8×D61—22no—
120#11758224—24no—
121#11758224—24no—
122#11758224—22no—
123#11758224—22no—
124D4×D61—24no—
125D4×D61—24no—
126D4×D61—22no—
127D4×D61—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—24no—
131#11758222—24no—
132C2×D244C4⋊C411yescharacteristic, complemented
133C2×D244C2×Q812yes—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—24no—
139C2×C4×D61C4⋊C412yescomplemented
140C2×C4×D61C4⋊C412yescomplemented
141C2×C4×D61C2×D412yes—
142C2×C4×D61C2×D412yes—
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D411yescharacteristic
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×Q811yescharacteristic
147C2×C4×D61C2×Q811yescharacteristic
148C2×C4×D61#4112yes—
149C2×C4×D61#4112yes—
150C2×Dic122C4⋊C412yescomplemented
151C2×Dic122C4⋊C411yescharacteristic, complemented
152C22⋊#6923—28no—
153C22⋊#6923—24no—
154C22⋊#6923—24no—
155C22⋊#6923—24no—
156C22⋊#6923—84no—
157C4⋊#6923—24no—
158C4⋊#6923—24no—
159C4⋊#6923—22no—
160C4⋊#6923—22no—
161C4⋊#6923—22no—
162C4⋊#6923—22no—
163C4⋊#6923—41no—
164C4⋊#6923—42no—
165C4⋊#6923—41no—
166C4⋊#6923—41no—
167C4⋊#6923—41no—
168C4⋊#6923—88no—
169C4×#6923—28no—
170C4×#6923—24no—
171C4×#6923—24no—
172C4×#6923—42no—
173C4⋊C4×C61—22no—
174C22⋊C4×C61C22×C414yes—
175C42×C61—22no—
176C22⋊Dic61C22×C414yes—
177#11758198—24no—
178C4⋊Dic61—22no—
179#11758196—24no—
180C4×Dic61—22no—
181C4×Dic61—24no—
182C23×C61C2×C4⋊C412yes—
183C23×C61C23×C411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61C2×C4⋊C412yes—
191C22×D61C22×D412yes—
192C22×D61C22×D411yescharacteristic
193C22×D61C22×Q811yescharacteristic
194C22×D61C2×#4112yes—
195C2×#6923—24no—
196C2×#6923—24no—
197C2×#6923—44no—
198C2×#6923—44no—
199C2×#6923—48no—
200C2×#6923—42no—
201C2×#6923—44no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—84no—
206C2×#6923—84no—
207Q8×C61—22no—
208D4×C61—24no—
209D4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61C2×C4⋊C412yes—
214C2×C4×C61C2×C4⋊C411yescharacteristic
215C2×C4×C61C2×C4⋊C411yescharacteristic
216C2×C4×C61C4×D412yes—
217C2×C4×C61C4×D412yes—
218C22⋊D61—24no—
219C22⋊D61—24no—
220C22⋊D61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61C2×C4⋊C412yes—
225C2×Dic61C2×C4⋊C411yescharacteristic
226C2×Dic61C2×C4⋊C411yescharacteristic
227C2×Dic61C4×D412yes—
228C2×Dic61C4×D412yes—
229D244—24no—
230D244—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—44no—
238C4×D61—44no—
239C4×D61#12011yescharacteristic
240C4×D61#12012yes—
241C4×D61#12011yescharacteristic
242C4×D61C22⋊Q812yes—
243C4×D61C22⋊Q812yes—
244C4×D61C22⋊Q812yes—
245C4×D61C22⋊Q812yes—
246C4×D61#12312yes—
247C4×D61#12311yescharacteristic
248C4×D61#12311yescharacteristic
249Dic122—22no—
250Dic122—24no—
251#20890—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61C22×C4⋊C411yescharacteristic
256C22×C61C2×C4×D412yes—
257C2×D61—22no—
258C2×D61—21no—
259C2×D61—22no—
260C2×D61—21no—
261C2×D61—22no—
262C2×D61—41no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—82no—
268C2×D61C2×#12011yescharacteristic
269C2×D61C2×C22⋊Q812yes—
270C2×D61C2×#12311yescharacteristic
271#6923—44no—
272#6923—44no—
273#6923—82no—
274#6923—82no—
275C4×C61—22no—
276C4×C61—44no—
277C4×C61#38512yes—
278C4×C61#38511yescharacteristic
279C4×C61#38511yescharacteristic
280Dic61—22no—
281Dic61—44no—
282Dic61#38512yes—
283Dic61#38511yescharacteristic
284Dic61#38511yescharacteristic
285#3428—611no—
286C2×#410—611no—
287#2317—612no—
288#2311—611no—
289#1867—618no—
290#1740—612no—
291C2×C61—22no—
292C2×C61—21no—
293C2×C61—42no—
294C2×C61C2×#38511yescharacteristic
295D61—21no—
296D61—22no—
297D61—42no—
298D61#302711yescharacteristic
299D61#302811yescharacteristic
300C22×#41—611no—
301#520—1222no—
302#520—1222no—
303#519—1221no—
304#518—614no—
305#518—1221no—
306C4×#41—1224no—
307C2×#117—612no—
308C2×#117—612no—
309C2×#117—611no—
310C2×#117—1222no—
311C22×C4⋊C4—611no—
312#410—611no—
313#410—612no—
314#410—614no—
315#410#692311yescharacteristic, complemented
316C2×#99—614no—
317#391—1221no—
318#390—1221no—
319#389—1224no—
320#386—614no—
321#385—618no—
322#384—1222no—
323#382—1222no—
324#380—1222no—
325#342—618no—
326C61#2089011yescharacteristic, complemented, Sylow
327C2×#41—611no—
328C2×#41—611no—
329C2×#41—612no—
330C2×#41—612no—
331C2×#41—611no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41C2×#692311yescharacteristic
337C22×Q8—611no—
338C22×D4—611no—
339C22×D4—612no—
340C23×C4—611no—
341C23×C4—612no—
342C23×C4—612no—
343C4×Q8—1224no—
344C4×Q8—1224no—
345C4×D4—1228no—
346C4×D4—1228no—
347C4×D4—1224no—
348C4×D4—1224no—
349#117—612no—
350#117—614no—
351#117—1222no—
352#117—1222no—
353#117—1222no—
354#117—1222no—
355#117—1222no—
356#117—1222no—
357#117—1224no—
358#117—1224no—
359#117—1224no—
360#117—1222no—
361#117—1222no—
362#117C2×#692312yescomplemented
363C2×C4⋊C4—612no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1228no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C22⋊C4—614no—
375C2×C22⋊C4—614no—
376C2×C22⋊C4—612no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1224no—
379C2×C42—614no—
380C2×C42—1222no—
381C2×C42—1224no—
382C2×C42—1221no—
383#99—614no—
384#99—618no—
385#99C2×#692314yescomplemented
386#95—1224no—
387#95—1228no—
388C24—611no—
389C24—612no—
390#41—22no—
391#41—22no—
392#41—1224no—
393#41—1224no—
394#41—1224no—
395#41—1224no—
396#41—1222no—
397#41—1222no—
398#41—1222no—
399#41—1222no—
400#41—1222no—
401#41—1222no—
402C2×Q8—612no—
403C2×Q8—611no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8C4⋊#692311yescharacteristic, complemented
407C2×D4—612no—
408C2×D4—612no—
409C2×D4—612no—
410C2×D4—611no—
411C2×D4—612no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1224no—
415C2×D4—1224no—
416C2×D4C4⋊#692311yescharacteristic, complemented
417C2×D4C22×#692312yes—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—614no—
432C22×C4—614no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1224no—
438C22×C4—1224no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—4882no—
444C22×C4C4⋊#692312yescomplemented
445C22×C4C22×#692311yescharacteristic
446C4⋊C4—22no—
447C4⋊C4—1224no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1222no—
455C4⋊C4—4888no—
456C4⋊C4—2441no—
457C4⋊C4—2442no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C22⋊C4—614no—
462C22⋊C4—1224no—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—1228no—
466C22⋊C4—1224no—
467C22⋊C4—4884no—
468C22⋊C4C22×#692314yes—
469C42—22no—
470C42—1224no—
471C42—1224no—
472C42—1224no—
473C42—1228no—
474C42—1222no—
475C42—2442no—
476C23—611no—
477C23—612no—
478C23—611no—
479C23—612no—
480C23—612no—
481C23—1222no—
482C23—1222no—
483C23—2441no—
484C23—2441no—
485C23—2442no—
486C23—2442no—
487C23C2×C4⋊#692312yes—
488C23C23×#692311yescharacteristic
489Q8—22no—
490Q8—1224no—
491Q8—1222no—
492D4—24no—
493D4—22no—
494D4—1224no—
495D4—1224no—
496D4—1224no—
497D4—1224no—
498D4—1222no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—611no—
507C2×C4—612no—
508C2×C4—611no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—612no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—4884no—
529C2×C4—4884no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4—2448no—
537C2×C4—2442no—
538C2×C4—2444no—
539C2×C4—2444no—
540C2×C4C2×C4⋊#692312yes—
541C2×C4C2×C4⋊#692311yescharacteristic
542C2×C4C2×C4⋊#692311yescharacteristic
543C2×C4D4×#692312yes—
544C2×C4D4×#692312yes—
545C22—22no—
546C22—41no—
547C22—42no—
548C22—611no—
549C22—611no—
550C22—612no—
551C22—1222no—
552C22—1221no—
553C22—1222no—
554C22—1221no—
555C22—1222no—
556C22—4882no—
557C22—2442no—
558C22—2441no—
559C22—2442no—
560C22—2442no—
561C22—2442no—
562C22C22×C4⋊#692311yescharacteristic
563C22C2×D4×#692312yes—
564C4—22no—
565C4—44no—
566C4—611no—
567C4—611no—
568C4—612no—
569C4—1222no—
570C4—4882no—
571C4—4882no—
572C4—2444no—
573C4—2444no—
574C4—2444no—
575C4#1684070911yescharacteristic
576C4#1684070911yescharacteristic
577C4#1684070912yes—
578C2—22no—
579C2—21no—
580C2—42no—
581C2—611no—
582C2—611no—
583C2—1221no—
584C2—1222no—
585C2—2442no—
586C2C2×#1684070911yescharacteristic
subgroupG/H
centerC2C2×#16840709
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC61×#410C4
socleC2×C61C2×#385
prime radicalC2×#41×D61C22
cosocleC23×D61C24
hypercenter#410#6923
hypocenterC61#20890
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #21462300C2semidirect
4. #21461790C2semidirect
5. #21461517C2semidirect
28. #16839736C4semidirect
29. #16839735C4semidirect
30. #16839734C4semidirect
32. C61×#410C4semidirect
82. #14519736C2×C4semidirect
85. #14519733C2×C4semidirect
89. #14519692C2×C4semidirect
90. #14519691C2×C4semidirect
92. #117×C61C2×C4semidirect
93. #99×C61C2×C4semidirect
106. C2×Q8×C61C4⋊C4semidirect
107. C2×D4×C61C4⋊C4semidirect
109. C22×C4×C61C4⋊C4semidirect
111. C2×C22⋊D61C4⋊C4semidirect
114. C22×Dic61C4⋊C4semidirect
132. C2×D244C4⋊C4semidirect
139. C2×C4×D61C4⋊C4semidirect
140. C2×C4×D61C4⋊C4semidirect
150. C2×Dic122C4⋊C4semidirect
151. C2×Dic122C4⋊C4semidirect
315. #410#6923semidirect
326. C61#20890semidirect
362. #117C2×#6923semidirect
385. #99C2×#6923semidirect
406. C2×Q8C4⋊#6923semidirect
416. C2×D4C4⋊#6923semidirect
444. C22×C4C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
#214627992
#214623002
#214622942
#214617902
#214615172
#410×D612
#2089061