Finite Group Databasebeta

← #28319161

Subgroups — #28319161

Order 13568.

1602 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291187C212yesmaximal
2#20290613C212yesmaximal, complemented
3#20290612C212yesmaximal, complemented
4#20290611C212yesmaximal, complemented
5#20290610C212yesmaximal, complemented
6#20290412C212yesmaximal, complemented
7C2×#16811501C212yesmaximal, complemented
8C2×#16811095C211yesmaximal, characteristic
9C22×C22⋊#6157C2214yes—
10#16811571—28no—
11#16811566C2214yes—
12#16811564C2214yes—
13#16811501C2214yes—
14#16811501C2214yes—
15#16811501C2214yes—
16#16811501C2214yes—
17C22×D4×D53C2211yescharacteristic
18#16811095—28no—
19C2×C4×D212C411yescharacteristic, complemented
20C2×D4×Dic53C411yescharacteristic, complemented
21C2×#13133179C411yescharacteristic, complemented
22C2×C4×C22⋊D53C411yescharacteristic, complemented
23C2×C4×C22⋊D53C411yescharacteristic, complemented
24C2×#13133177C411yescharacteristic, complemented
25C2×#13133177C411yescharacteristic, complemented
26C2×#13133175C2211yescharacteristic
27C2×#13133174C2211yescharacteristic
28C22×#11719921C2211yescharacteristic
29C22×#11719921C2211yescharacteristic
30C2×C22⋊C4×D53C2211yescharacteristic
31C2×C22⋊C4×D53C2211yescharacteristic
32C2×C4×D4×C53C411yescharacteristic, complemented
33C23×#6157—24no—
34C2×C22⋊#6157—216no—
35C2×C22⋊#6157—24no—
36C2×C22⋊#6157—24no—
37C2×C22⋊#6157—24no—
38C2×C22⋊#6157C2318yes—
39(C2×#6157)⋊C4—28no—
40C24×D53C2311yescharacteristic
41C24×D53C2311yescharacteristic
42C22×D212C2×C411yescharacteristic
43C22×C22⋊D53C2×C411yescharacteristic
44C22×C22⋊D53C2×C411yescharacteristic
45C2×D4×D53—28no—
46C2×D4×D53D418yes—
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic, complemented
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic, complemented
52C22×C4×D53C2×C411yescharacteristic, complemented
53C22×C4×D53C2311yescharacteristic
54C4×D212—24no—
55D4×Dic53—24no—
56#13133179—24no—
57C4×C22⋊D53—24no—
58C4×C22⋊D53—24no—
59#13133177—24no—
60#13133177—24no—
61#13133175—24no—
62#13133174—24no—
63C2×C4⋊Dic53C2×C411yescharacteristic, complemented
64C2×#11719919C2×C411yescharacteristic, complemented
65C2×#11719919C2×C411yescharacteristic, complemented
66C2×C22⋊Dic53C2×C411yescharacteristic
67C2×C22⋊Dic53C2×C411yescharacteristic, complemented
68C2×#11719921—24no—
69C2×#11719921—24no—
70C2×#11719921C2×C411yescharacteristic
71C2×#11719921C2×C411yescharacteristic
72C2×#11719921C2×C411yescharacteristic, complemented
73C2×#11719921C2×C411yescharacteristic, complemented
74C2×#11719921C2311yescharacteristic
75C2×#11719921C2311yescharacteristic
76C2×#11719921C2311yescharacteristic
77C2×#11719921C2311yescharacteristic
78C22⋊C4×D53—24no—
79C22⋊C4×D53—24no—
80C2×C4×Dic53C2×C411yescharacteristic, complemented
81C2×C4×Dic53C2×C411yescharacteristic, complemented
82C2×C4×Dic53C2×C411yescharacteristic, complemented
83C22×D4×C53C2×C411yescharacteristic
84C23×C4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic, complemented
86C4×D4×C53—24no—
87C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
88C2×C22⋊C4×C53C2×C411yescharacteristic
89C2×C22⋊C4×C53C2×C411yescharacteristic, complemented
90C2×C42×C53C2×C411yescharacteristic, complemented
91C24×C53C22×C411yescharacteristic
92C24×C53C22×C411yescharacteristic
93C23×D53—24no—
94C23×D53—81no—
95C23×D53—81no—
96C23×D53—81no—
97C23×D53—81no—
98C23×D53C22×C411yescharacteristic
99C23×D53C22×C411yescharacteristic
100C23×D53C2×D414yes—
101C23×D53C2×D414yes—
102C23×D53#4118yes—
103C23×D53C2411yescharacteristic
104C22×#6157—28no—
105C22×#6157—48no—
106C22×#6157—84no—
107C22×#6157—84no—
108C2×D4×C53—24no—
109C2×D4×C53C22⋊C414yes—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—81no—
113C22×C4×C53—81no—
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C2×C22⋊D53—24no—
120C2×C22⋊D53—24no—
121C2×C22⋊D53C22⋊C414yes—
122C2×C22⋊D53C22⋊C414yes—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—81no—
126C22×Dic53—81no—
127C22×Dic53C22×C411yescharacteristic
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C22×C411yescharacteristic
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132D4×D53—216no—
133D4×D53—216no—
134C2×D212—24no—
135C2×D212C22⋊C414yes—
136C2×C4×D53—24no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—81no—
142C2×C4×D53—81no—
143C2×C4×D53—81no—
144C2×C4×D53—81no—
145C2×C4×D53C2×D414yes—
146C22⋊#6157—216no—
147C22⋊#6157—48no—
148C22⋊#6157—84no—
149C22⋊#6157—84no—
150C4⋊C4×C53—22no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C42×C53—22no—
154C22⋊Dic53—22no—
155C22⋊Dic53—22no—
156#11719921—22no—
157#11719921—22no—
158#11719921—22no—
159#11719921—22no—
160#11719921—22no—
161#11719921—22no—
162#11719921—22no—
163#11719921—22no—
164C4⋊Dic53—22no—
165#11719919—22no—
166#11719919—22no—
167C4×Dic53—22no—
168C4×Dic53—22no—
169C4×Dic53—22no—
170C23×C53—22no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53C2×C22⋊C412yes—
176C23×C53C2×C22⋊C412yes—
177C23×C53#11714yes—
178C23×C53C23×C411yescharacteristic
179C22×D53—22no—
180C22×D53—28no—
181C22×D53—41no—
182C22×D53—41no—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—44no—
186C22×D53—44no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—44no—
190C22×D53—44no—
191C22×D53—88no—
192C22×D53—88no—
193C22×D53C2×C22⋊C412yes—
194C22×D53C2×C22⋊C412yes—
195C22×D53#11714yes—
196C22×D53#12318yes—
197C22×D53#12318yes—
198C22×D53C22×D412yes—
199C22×D53C2×#4114yes—
200C2×#6157—48no—
201C2×#6157—48no—
202C2×#6157—48no—
203C2×#6157—816no—
204D4×C53—28no—
205C2×C4×C53—22no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—82no—
215C2×C4×C53—82no—
216C2×C4×C53C2×C22⋊C412yes—
217C22⋊D53—28no—
218C22⋊D53—28no—
219C2×Dic53—22no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—82no—
229C2×Dic53—82no—
230C2×Dic53C2×C22⋊C412yes—
231D212—28no—
232C4×D53—28no—
233C4×D53—82no—
234C4×D53—82no—
235C4×D53—82no—
236C4×D53—82no—
237#20689—531nomaximal, Sylow
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—44no—
246C22×C53—44no—
247C22×C53#38114yes—
248C22×C53C22×C22⋊C411yescharacteristic
249C22×C53C2×#11712yes—
250C2×D53—22no—
251C2×D53—22no—
252C2×D53—22no—
253C2×D53—22no—
254C2×D53—22no—
255C2×D53—22no—
256C2×D53—22no—
257C2×D53—48no—
258C2×D53—44no—
259C2×D53—44no—
260C2×D53—48no—
261C2×D53—44no—
262C2×D53—88no—
263C2×D53#38114yes—
264C2×D53C2×#12314yes—
265#6157—88no—
266C4×C53—22no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—41no—
271Dic53—22no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276C2×#518—531no—
277#2308—532no—
278#1820—532no—
279#1820—532no—
280#1819—532no—
281#1819—532no—
282C2×#342—532no—
283C2×#342—532no—
284C2×C53—22no—
285C2×C53—24no—
286C2×C53—42no—
287C2×C53C2×#38111yescharacteristic
288C2×C53#288311yescharacteristic
289C2×C53#288311yescharacteristic
290D53—24no—
291D53—24no—
292D53—42no—
293C23×D4—531no—
294#518—1068no—
295C2×C4×D4—531no—
296C2×C4×D4—531no—
297C2×C4×D4—531no—
298C2×C4×D4—531no—
299C2×C4×D4—531no—
300C2×C4×D4—531no—
301C2×C4×D4—531no—
302C2×C4×D4#615711yescharacteristic, complemented
303C2×#117—531no—
304C2×#117—531no—
305C22×C22⋊C4—531no—
306C22×C22⋊C4—531no—
307C22×C22⋊C4—531no—
308C22×C22⋊C4—531no—
309C22×C22⋊C4—534no—
310#386—1068no—
311#381—534no—
312#379—534no—
313#342—534no—
314#342—534no—
315#342—534no—
316#342—534no—
317C53#2068911yescharacteristic, complemented, Sylow
318C25—531no—
319C25—531no—
320C22×D4—531no—
321C22×D4—531no—
322C22×D4—538no—
323C22×D4—531no—
324C22×D4—1068no—
325C22×D4C2×#615711yescharacteristic
326C23×C4—531no—
327C23×C4—531no—
328C23×C4—531no—
329C23×C4—531no—
330C23×C4—531no—
331C23×C4—531no—
332C23×C4—531no—
333C23×C4—1064no—
334C23×C4C2×#615711yescharacteristic
335C23×C4C2×#615711yescharacteristic, complemented
336C4×D4—24no—
337C4×D4—1064no—
338C4×D4—1064no—
339C4×D4—1064no—
340C4×D4—1064no—
341C4×D4—1064no—
342C4×D4—1064no—
343C4×D4—1064no—
344#117—1064no—
345#117—1064no—
346C2×C4⋊C4—531no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4C2×#615711yescharacteristic, complemented
350C2×C22⋊C4—531no—
351C2×C22⋊C4—531no—
352C2×C22⋊C4—531no—
353C2×C22⋊C4—531no—
354C2×C22⋊C4—531no—
355C2×C22⋊C4—531no—
356C2×C22⋊C4—531no—
357C2×C22⋊C4—531no—
358C2×C22⋊C4—531no—
359C2×C22⋊C4—531no—
360C2×C22⋊C4—538no—
361C2×C22⋊C4—1064no—
362C2×C22⋊C4—1064no—
363C2×C22⋊C4—1064no—
364C2×C22⋊C4—1064no—
365C2×C22⋊C4—10616no—
366C2×C22⋊C4—1064no—
367C2×C22⋊C4—1064no—
368C2×C22⋊C4—1064no—
369C2×C22⋊C4C2×#615711yescharacteristic
370C2×C22⋊C4C2×#615711yescharacteristic, complemented
371C2×C42—531no—
372C2×C42—531no—
373C2×C42—531no—
374C2×C42C2×#615711yescharacteristic, complemented
375#95—1068no—
376C24—538no—
377C24—534no—
378C24—531no—
379C24—531no—
380C24—534no—
381C24—531no—
382C24—1064no—
383C24—4241no—
384C24—4241no—
385C24—4241no—
386C24—4241no—
387C24C22×#615711yescharacteristic
388C24C22×#615711yescharacteristic
389C2×D4—24no—
390C2×D4—534no—
391C2×D4—534no—
392C2×D4—534no—
393C2×D4—1064no—
394C2×D4—10616no—
395C2×D4—10616no—
396C2×D4—1064no—
397C2×D4—1064no—
398C2×D4C22⋊#615714yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—531no—
404C22×C4—534no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—1064no—
410C22×C4—1062no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4—1062no—
416C22×C4—1068no—
417C22×C4—4241no—
418C22×C4—4241no—
419C22×C4—4241no—
420C22×C4—4241no—
421C22×C4—4244no—
422C22×C4—4244no—
423C22×C4—4241no—
424C22×C4—4241no—
425C22×C4—2128no—
426C22×C4C22×#615711yescharacteristic
427C22×C4C22×#615711yescharacteristic
428C22×C4C22×#615711yescharacteristic
429C22×C4C22×#615711yescharacteristic
430C22×C4C22×#615711yescharacteristic
431C4⋊C4—22no—
432C4⋊C4—1062no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—1062no—
438C22⋊C4—1062no—
439C22⋊C4—1062no—
440C22⋊C4—1062no—
441C22⋊C4—1062no—
442C22⋊C4—1062no—
443C22⋊C4—1062no—
444C22⋊C4—1062no—
445C22⋊C4—1062no—
446C22⋊C4—1062no—
447C22⋊C4—10616no—
448C22⋊C4—4244no—
449C22⋊C4—4244no—
450C22⋊C4—2128no—
451C42—22no—
452C42—1062no—
453C42—1062no—
454C42—1062no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—532no—
461C23—532no—
462C23—538no—
463C23—538no—
464C23—534no—
465C23—534no—
466C23—532no—
467C23—1068no—
468C23—1062no—
469C23—4248no—
470C23—4248no—
471C23—2124no—
472C23—2124no—
473C23—2121no—
474C23—2121no—
475C23—2124no—
476C23—2121no—
477C23—2121no—
478C23—2124no—
479C23—2121no—
480C23—2121no—
481C23C2×C22⋊#615712yes—
482C23C2×C22⋊#615712yes—
483C23#1313329614yes—
484C23C23×#615711yescharacteristic
485D4—28no—
486D4—1068no—
487D4—1068no—
488D4—1068no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—532no—
501C2×C4—1068no—
502C2×C4—1062no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—4242no—
508C2×C4—4242no—
509C2×C4—4242no—
510C2×C4—4242no—
511C2×C4—42416no—
512C2×C4—4242no—
513C2×C4—4242no—
514C2×C4—2128no—
515C2×C4—2128no—
516C2×C4—2128no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4C2×C22⋊#615712yes—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—44no—
530C22—44no—
531C22—534no—
532C22—534no—
533C22—1062no—
534C22—1062no—
535C22—1062no—
536C22—1062no—
537C22—1062no—
538C22—1062no—
539C22—1062no—
540C22—4248no—
541C22—2128no—
542C22—2124no—
543C22—2124no—
544C22—2128no—
545C22—2124no—
546C22#1681156614yes—
547C22C22×C22⋊#615711yescharacteristic
548C22C2×#1313329612yes—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—1062no—
555C4—4248no—
556C4—2121no—
557C4—2121no—
558C4—2121no—
559C4—2121no—
560C2—22no—
561C2—24no—
562C2—42no—
563C2—1064no—
564C2—1064no—
565C2—2122no—
566C2C2×#1681156611yescharacteristic
567C2#2029131411yescharacteristic
568C2#2029131511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C22
cosocleC23×D53C24
hypercenterC2×C4×D4#6157
hypocenterC53#20689
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #20290613C2semidirect
3. #20290612C2semidirect
4. #20290611C2semidirect
5. #20290610C2semidirect
6. #20290412C2semidirect
7. C2×#16811501C2semidirect
19. C2×C4×D212C4semidirect
20. C2×D4×Dic53C4semidirect
21. C2×#13133179C4semidirect
22. C2×C4×C22⋊D53C4semidirect
23. C2×C4×C22⋊D53C4semidirect
24. C2×#13133177C4semidirect
25. C2×#13133177C4semidirect
32. C2×C4×D4×C53C4semidirect
48. C23×Dic53C2×C4semidirect
51. C22×C4×D53C2×C4semidirect
52. C22×C4×D53C2×C4semidirect
63. C2×C4⋊Dic53C2×C4semidirect
64. C2×#11719919C2×C4semidirect
65. C2×#11719919C2×C4semidirect
67. C2×C22⋊Dic53C2×C4semidirect
72. C2×#11719921C2×C4semidirect
73. C2×#11719921C2×C4semidirect
80. C2×C4×Dic53C2×C4semidirect
81. C2×C4×Dic53C2×C4semidirect
82. C2×C4×Dic53C2×C4semidirect
85. C23×C4×C53C2×C4semidirect
87. C2×C4⋊C4×C53C2×C4semidirect
89. C2×C22⋊C4×C53C2×C4semidirect
90. C2×C42×C53C2×C4semidirect
302. C2×C4×D4#6157semidirect
317. C53#20689semidirect
335. C23×C4C2×#6157semidirect
349. C2×C4⋊C4C2×#6157semidirect
370. C2×C22⋊C4C2×#6157semidirect
374. C2×C42C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202911872
#202906132
#202906122
#202906112
#202906102
#202904122
C2×#168115012
C2×#168110952
#2068953