Finite Group Databasebeta

← #16023218

Subgroups — #16023218

Order 3328.

3118 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719533C211yesmaximal, characteristic
2C2×#11719532C211yesmaximal, characteristic, complemented
3C22×#88146C211yesmaximal, characteristic, complemented
4#13117879C214yesmaximal, complemented
5#13117868C214yesmaximal, complemented
6#13117780C214yesmaximal, complemented
7#13117767C214yesmaximal, complemented
8#13117499C214yesmaximal, complemented
9#13117460C211yesmaximal, characteristic, complemented
10#13117361C214yesmaximal, complemented
11#13117215C211yesmaximal, characteristic, complemented
12#13117192C211yesmaximal, characteristic, complemented
13C13×#2311C211yesmaximal, characteristic, complemented
14C22×C13×#41C2211yescharacteristic
15C23×C4×D13C2211yescharacteristic
16C2×C13×#117C2211yescharacteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18#11719533C4116yescomplemented
19#11719532—28no—
20C2×#88200C2212yes—
21C2×Q8×Dic13C2212yescomplemented
22C2×#88192C2212yes—
23C2×#88191C2212yes—
24C2×#88189C2212yes—
25C2×D4×Dic13C2214yes—
26C2×D4×Dic13C2212yescomplemented
27C2×C4×C22⋊D13C2214yes—
28C22×#6103C2212yes—
29C2×#88180C2211yescharacteristic
30C2×#88180C2211yescharacteristic
31C2×#88180C2212yescomplemented
32C22×C4⋊Dic13C2211yescharacteristic
33C22×C4⋊Dic13C2212yescomplemented
34C22×C4×Dic13C2212yes—
35C2×#88146—28no—
36C2×#88146C2214yes—
37C2×#88146C2214yes—
38C13×#391C2212yescomplemented
39C13×#390C2212yescomplemented
40C13×#386C2214yes—
41C13×#380C2214yescomplemented
42#11718871C2214yes—
43#11718648C2212yes—
44#11718648C2212yes—
45#11718638C2212yescomplemented
46#11718637C2212yescomplemented
47C4⋊C4×Dic13C2214yescomplemented
48#11718634C2214yescomplemented
49#11718619C2218yes—
50#11718618C2218yes—
51#11718616C2212yes—
52#11718606C2218yescomplemented
53#11718604C2212yes—
54C22⋊C4×Dic13C2214yes—
55C22⋊C4×Dic13C2218yescomplemented
56C4×#6103C2214yes—
57C4×C4⋊Dic13C2214yescomplemented
58#11718331C2218yes—
59#11718330C2218yes—
60#11718316C2218yescomplemented
61#11718310C2218yescomplemented
62C24×D13C2311yescharacteristic
63C2×C13×#41—24no—
64C2×C13×#41—24no—
65C2×C13×#41C2×C418yes—
66C22×Q8×C13C2311yescharacteristic
67C22×D4×C13C2312yes—
68C22×D4×C13C2311yescharacteristic
69C23×C4×C13C2312yes—
70C23×C4×C13C2312yes—
71C23×C4×C13C2311yescharacteristic
72C22×C22⋊D13C2314yes—
73C23×Dic13C2311yescharacteristic
74C23×Dic13C2314yes—
75C23×Dic13C2312yes—
76C22×C4×D13—24no—
77C22×C4×D13—24no—
78C22×C4×D13C2×C418yescomplemented
79C22×C4×D13C2312yes—
80C22×C4×D13C2312yes—
81C13×#117—24no—
82C2×C13×C4⋊C4—24no—
83C2×C13×C4⋊C4C2312yes—
84C2×C13×C4⋊C4C2312yes—
85C2×C13×C4⋊C4C2312yes—
86C2×C13×C22⋊C4C2312yes—
87C2×C13×C22⋊C4C2318yes—
88C2×C42×C13C2312yes—
89C2×C22⋊Dic13C2312yes—
90C2×C22⋊Dic13C2314yes—
91C2×C22⋊Dic13C2312yes—
92C2×C22⋊Dic13C2314yes—
93#88200C2×C4116yescomplemented
94Q8×Dic13—28no—
95#88192C2×C4116yescomplemented
96#88191C2×C4116yescomplemented
97#88189C2×C4116yescomplemented
98D4×Dic13—216no—
99D4×Dic13—28no—
100C4×C22⋊D13C2×C4132yescomplemented
101C2×#6103—28no—
102C2×#6103C2318yes—
103C2×#6103C2314yes—
104C2×#6103C2314yes—
105C2×#6103C2314yes—
106#88180—24no—
107#88180—28no—
108#88180C2×C418yescomplemented
109C2×C4⋊Dic13—28no—
110C2×C4⋊Dic13D418yescomplemented
111C2×C4⋊Dic13C2311yescharacteristic
112C2×C4⋊Dic13C2314yes—
113C2×C4⋊Dic13C2311yescharacteristic
114C2×C4⋊Dic13C2312yes—
115C2×C4⋊Dic13C2312yes—
116C2×C4⋊Dic13C2312yes—
117C2×#6101C2318yes—
118C2×#6101C2314yes—
119C2×C4×Dic13—28no—
120C2×C4×Dic13C2318yes—
121C2×C4×Dic13C2312yes—
122C2×C4×Dic13C2314yes—
123C2×C4×Dic13C2318yes—
124C2×C4×Dic13C2312yes—
125#88146—216no—
126#88146—216no—
127C13×#95C2318yes—
128#88071C2314yes—
129#88071C2314yes—
130#88071C2318yes—
131#88071C2318yes—
132#34201—131nomaximal, Sylow
133C24×C13C2412yes—
134C24×C13C2411yescharacteristic
135C23×D13—22no—
136C23×D13—22no—
137C23×D13—42no—
138C23×D13—42no—
139C23×D13C22×C414yes—
140C23×D13C2412yes—
141C13×#41—232no—
142C2×Q8×C13—22no—
143C2×Q8×C13—22no—
144C2×Q8×C13C22×C414yes—
145C2×D4×C13—24no—
146C2×D4×C13—24no—
147C2×D4×C13—22no—
148C2×D4×C13—22no—
149C2×D4×C13C22×C418yes—
150C2×D4×C13C22×C414yes—
151C22×C4×C13—24no—
152C22×C4×C13—24no—
153C22×C4×C13—24no—
154C22×C4×C13—44no—
155C22×C4×C13C22×C418yes—
156C22×C4×C13C22×C414yes—
157C22×C4×C13C2×D414yes—
158C22×C4×C13#4114yes—
159C22×C4×C13C2414yes—
160C22×C4×C13C2414yes—
161C22×C4×C13C2412yes—
162C22×C4×C13C2411yescharacteristic
163C22×C4×C13C2411yescharacteristic
164C2×C22⋊D13—44no—
165C2×C22⋊D13—44no—
166C2×C22⋊D13C22×C4116yes—
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—28no—
170C22×Dic13—44no—
171C22×Dic13—48no—
172C22×Dic13C22×C414yes—
173C22×Dic13C2×D418yes—
174C22×Dic13C2412yes—
175C22×Dic13C2412yes—
176C22×Dic13C2412yes—
177C22×Dic13C2414yes—
178C22×Dic13C2414yes—
179C2×C4×D13—232no—
180C2×C4×D13—24no—
181C2×C4×D13—24no—
182C2×C4×D13—24no—
183C2×C4×D13—24no—
184C2×C4×D13C22×C418yes—
185C2×C4×D13C22×C418yes—
186C13×C4⋊C4—24no—
187C13×C4⋊C4—24no—
188C13×C4⋊C4—24no—
189C13×C22⋊C4—24no—
190C13×C22⋊C4—48no—
191C42×C13—24no—
192C22⋊Dic13—24no—
193C22⋊Dic13—28no—
194C22⋊Dic13C22×C418yes—
195C22⋊Dic13C22×C4116yes—
196#6103—216no—
197#6103—44no—
198#6103—44no—
199#6103C22×C4116yes—
200C4⋊Dic13—22no—
201C4⋊Dic13—22no—
202C4⋊Dic13—28no—
203C4⋊Dic13C22×C418yes—
204C4⋊Dic13C2×D418yes—
205C4⋊Dic13C2×D418yes—
206#6101—48no—
207#6101C22×C4116yes—
208C4×Dic13—24no—
209C4×Dic13—216no—
210C4×Dic13—28no—
211C4×Dic13—48no—
212C4×Dic13C22×C418yes—
213C2×#525—131no—
214C2×#520—131no—
215C22×#117—131no—
216#2363—134no—
217#2362—134no—
218#2354—134no—
219#2352—134no—
220#2321—134no—
221#2317—131no—
222#2311D1311yescharacteristic, complemented
223C4×#123—134no—
224#2280—131no—
225#2274—131no—
226C23×C13—22no—
227C23×C13—22no—
228C23×C13—22no—
229C23×C13—22no—
230C23×C13—24no—
231C23×C13—24no—
232C23×C13C23×C414yes—
233C23×C13C23×C412yes—
234C23×C13C22×D412yes—
235C23×C13C2×#4112yes—
236C23×C13C2511yescharacteristic
237C22×D13—28no—
238C22×D13—22no—
239C22×D13—22no—
240C22×D13—22no—
241C22×D13—22no—
242C22×D13—22no—
243C22×D13—22no—
244C22×D13—48no—
245C22×D13—44no—
246C22×D13—44no—
247C22×D13—48no—
248C22×D13C23×C414yes—
249Q8×C13—28no—
250D4×C13—216no—
251D4×C13—28no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—216no—
255C2×C4×C13—24no—
256C2×C4×C13—24no—
257C2×C4×C13—24no—
258C2×C4×C13—28no—
259C2×C4×C13—24no—
260C2×C4×C13—48no—
261C2×C4×C13C4×D4116yes—
262C2×C4×C13C23×C414yes—
263C2×C4×C13C23×C412yes—
264C2×C4×C13C23×C412yes—
265C2×C4×C13C22×D412yes—
266C2×C4×C13C22×D412yes—
267C2×C4×C13C2×#4112yes—
268C2×C4×C13C2×#4112yes—
269C22⋊D13—416no—
270C2×Dic13—28no—
271C2×Dic13—22no—
272C2×Dic13—22no—
273C2×Dic13—24no—
274C2×Dic13—28no—
275C2×Dic13—24no—
276C2×Dic13—22no—
277C2×Dic13—22no—
278C2×Dic13—24no—
279C2×Dic13—24no—
280C2×Dic13—48no—
281C2×Dic13—88no—
282C2×Dic13C23×C418yes—
283C2×Dic13C23×C414yes—
284C2×Dic13C22×D418yes—
285C4×D13—216no—
286C4×D13—216no—
287C22×#41C2×D1311yescharacteristic
288C24×C4—131no—
289#525—1316no—
290C2×#123—132no—
291C2×C22⋊Q8—132no—
292C2×#121—132no—
293C2×#120—132no—
294#520—268no—
295C2×C4×Q8—132no—
296C2×C4×D4—134no—
297C2×C4×D4—134no—
298C2×C4×D4—132no—
299C2×#117—131no—
300C2×#117—131no—
301C2×#117—132no—
302C2×#117—134no—
303C2×#117—134no—
304C2×#117—268no—
305C2×#117C2×D1311yescharacteristic
306C22×C4⋊C4—131no—
307C22×C4⋊C4—132no—
308C22×C4⋊C4C2×D1311yescharacteristic
309C22×C22⋊C4—132no—
310C22×C42—132no—
311#391C2×D1312yescomplemented
312#390—138no—
313#390C2×D1312yescomplemented
314#389—134no—
315#388—138no—
316#388—138no—
317#388—138no—
318#387—138no—
319#386—138no—
320#386C2×D1314yes—
321#385—134no—
322#384—132no—
323#382—132no—
324#381—132no—
325#380—132no—
326#380—132no—
327#380C2×D1314yescomplemented
328#379—132no—
329C4×C4⋊C4—134no—
330C4×C4⋊C4—134no—
331C4×C4⋊C4—138no—
332C4×C22⋊C4—134no—
333C4×C22⋊C4—138no—
334C4×C22⋊C4—134no—
335C22×C13—24no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—24no—
339C22×C13—24no—
340C22×C13—28no—
341C22×C13—42no—
342C22×C13—42no—
343C22×C13—44no—
344C22×C13C2×C4×D414yes—
345C22×C13C24×C411yescharacteristic
346C22×C13C23×D411yescharacteristic
347C22×C13C22×#4111yescharacteristic
348C22×C13#58512yes—
349C22×C13#58512yes—
350C2×D13—24no—
351C2×D13—24no—
352C2×D13—24no—
353C2×D13—416no—
354C2×D13—42no—
355C2×D13—42no—
356C2×D13—42no—
357C2×D13—42no—
358C4×C13—24no—
359C4×C13—44no—
360C4×C13—44no—
361C4×C13C2×C4×D414yes—
362C4×C13C2×C4×D414yes—
363Dic13—24no—
364Dic13—28no—
365Dic13—28no—
366Dic13—42no—
367Dic13—42no—
368C25—131no—
369C2×#41—24no—
370C2×#41—24no—
371C2×#41C4×D1318yes—
372C22×Q8C22×D1311yescharacteristic
373C22×D4—134no—
374C22×D4C22×D1312yes—
375C22×D4C22×D1311yescharacteristic
376C23×C4—131no—
377C23×C4—132no—
378C23×C4—138no—
379C23×C4—132no—
380C23×C4—132no—
381C23×C4—134no—
382C23×C4—264no—
383C23×C4—264no—
384C23×C4C22×D1312yes—
385C23×C4C22×D1312yes—
386C23×C4C22×D1311yescharacteristic
387#123—1316no—
388C22⋊Q8—1316no—
389#121—1316no—
390#120—1316no—
391C4×Q8—268no—
392C4×D4—1332no—
393C4×D4—268no—
394C4×D4—2616no—
395#117—24no—
396#117—138no—
397#117—2616no—
398#117—2616no—
399#117—268no—
400#117—264no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—138no—
403C2×C4⋊C4—132no—
404C2×C4⋊C4—132no—
405C2×C4⋊C4—132no—
406C2×C4⋊C4—134no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—134no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—138no—
411C2×C4⋊C4—268no—
412C2×C4⋊C4C22×D1312yes—
413C2×C4⋊C4C22×D1312yes—
414C2×C4⋊C4C22×D1312yes—
415C2×C22⋊C4—132no—
416C2×C22⋊C4—134no—
417C2×C22⋊C4—134no—
418C2×C22⋊C4—132no—
419C2×C22⋊C4—134no—
420C2×C22⋊C4—138no—
421C2×C22⋊C4—134no—
422C2×C22⋊C4—134no—
423C2×C22⋊C4—268no—
424C2×C22⋊C4C22×D1312yes—
425C2×C22⋊C4C22×D1318yes—
426C2×C42—132no—
427C2×C42—138no—
428C2×C42—132no—
429C2×C42—134no—
430C2×C42—138no—
431C2×C42—268no—
432C2×C42C22×D1312yes—
433#95—134no—
434#95—134no—
435#95—138no—
436#95—138no—
437#95C22×D1318yes—
438C2×C13—24no—
439C2×C13—44no—
440C2×C13C22×C4×D411yescharacteristic
441C2×C13#342812yes—
442C2×C13#348511yescharacteristic
443C2×C13#348711yescharacteristic
444C2×C13#349312yes—
445D13—44no—
446C24—132no—
447C24—134no—
448C24—262no—
449C24—262no—
450C24—522no—
451C24—522no—
452C24C23×D1312yes—
453C24C23×D1311yescharacteristic
454#41—232no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8C2×C4×D1314yes—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—1316no—
463C2×D4—524no—
464C2×D4—524no—
465C2×D4C2×C4×D1318yes—
466C2×D4C2×C4×D1314yes—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—44no—
471C22×C4—134no—
472C22×C4—132no—
473C22×C4—134no—
474C22×C4—134no—
475C22×C4—138no—
476C22×C4—138no—
477C22×C4—138no—
478C22×C4—132no—
479C22×C4—132no—
480C22×C4—268no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—2632no—
484C22×C4—264no—
485C22×C4—264no—
486C22×C4—264no—
487C22×C4—264no—
488C22×C4—524no—
489C22×C4—528no—
490C22×C4C2×C4×D1318yes—
491C22×C4C2×C4×D1314yes—
492C22×C4D4×D1314yes—
493C22×C4#612914yes—
494C22×C4C23×D1314yes—
495C22×C4C23×D1314yes—
496C22×C4C23×D1312yes—
497C22×C4C23×D1311yescharacteristic
498C22×C4C23×D1311yescharacteristic
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—138no—
503C4⋊C4—1316no—
504C4⋊C4—138no—
505C4⋊C4—138no—
506C4⋊C4—268no—
507C4⋊C4—262no—
508C4⋊C4—262no—
509C4⋊C4—528no—
510C22⋊C4—24no—
511C22⋊C4—48no—
512C22⋊C4—138no—
513C22⋊C4—1316no—
514C22⋊C4—1316no—
515C22⋊C4—2616no—
516C22⋊C4—268no—
517C22⋊C4—264no—
518C22⋊C4—524no—
519C22⋊C4—524no—
520C42—24no—
521C42—138no—
522C42—2616no—
523C42—268no—
524C42—264no—
525C42—528no—
526C13#3420111yescharacteristic, complemented, Sylow
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—134no—
534C23—268no—
535C23—262no—
536C23—262no—
537C23—262no—
538C23—262no—
539C23—262no—
540C23—262no—
541C23—528no—
542C23—524no—
543C23—524no—
544C23—528no—
545C23C22×C4×D1314yes—
546C23C22×C4×D1312yes—
547C23C2×D4×D1312yes—
548C23C2×#612912yes—
549C23C24×D1311yescharacteristic
550Q8—28no—
551D4—216no—
552D4—28no—
553D4—5216no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—216no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—138no—
564C2×C4—134no—
565C2×C4—138no—
566C2×C4—268no—
567C2×C4—262no—
568C2×C4—262no—
569C2×C4—264no—
570C2×C4—264no—
571C2×C4—268no—
572C2×C4—2616no—
573C2×C4—2616no—
574C2×C4—262no—
575C2×C4—262no—
576C2×C4—264no—
577C2×C4—264no—
578C2×C4—528no—
579C2×C4—1048no—
580C2×C4#88135116yes—
581C2×C4C22×C4×D1314yes—
582C2×C4C22×C4×D1312yes—
583C2×C4C22×C4×D1312yes—
584C2×C4C2×D4×D1312yes—
585C2×C4C2×D4×D1312yes—
586C2×C4C2×#612912yes—
587C2×C4C2×#612912yes—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—28no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—264no—
598C22—264no—
599C22—264no—
600C22—5216no—
601C22—522no—
602C22—522no—
603C22—522no—
604C22—522no—
605C22C2×#8813514yes—
606C22C23×C4×D1311yescharacteristic
607C22C22×D4×D1311yescharacteristic
608C22C22×#612911yescharacteristic
609C22#1171965012yes—
610C22#1171965212yes—
611C4—24no—
612C4—44no—
613C4—44no—
614C4—268no—
615C4—264no—
616C4—268no—
617C4—522no—
618C4—522no—
619C4C2×#8813514yes—
620C4C2×#8813514yes—
621C2—24no—
622C2—44no—
623C2—524no—
624C2C22×#8813511yescharacteristic
625C2#1312803612yes—
626C2#1312847911yescharacteristic
627C2#1312849211yescharacteristic
628C2#1312855512yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC13×#2311C2
socle, cosocleC23×C13C25
prime radicalC2×#11719533C2
hypercenter#2311D13
hypocenterC13#34201
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#11719532C2semidirect
3. C22×#88146C2semidirect
4. #13117879C2semidirect
5. #13117868C2semidirect
6. #13117780C2semidirect
7. #13117767C2semidirect
8. #13117499C2semidirect
9. #13117460C2semidirect
10. #13117361C2semidirect
11. #13117215C2semidirect
12. #13117192C2semidirect
13. C13×#2311C2semidirect
18. #11719533C4semidirect
21. C2×Q8×Dic13C22semidirect
26. C2×D4×Dic13C22semidirect
31. C2×#88180C22semidirect
33. C22×C4⋊Dic13C22semidirect
38. C13×#391C22semidirect
39. C13×#390C22semidirect
41. C13×#380C22semidirect
45. #11718638C22semidirect
46. #11718637C22semidirect
47. C4⋊C4×Dic13C22semidirect
48. #11718634C22semidirect
52. #11718606C22semidirect
55. C22⋊C4×Dic13C22semidirect
57. C4×C4⋊Dic13C22semidirect
60. #11718316C22semidirect
61. #11718310C22semidirect
78. C22×C4×D13C2×C4semidirect
93. #88200C2×C4semidirect
95. #88192C2×C4semidirect
96. #88191C2×C4semidirect
97. #88189C2×C4semidirect
100. C4×C22⋊D13C2×C4semidirect
108. #88180C2×C4semidirect
110. C2×C4⋊Dic13D4semidirect
222. #2311D13semidirect
311. #391C2×D13semidirect
313. #390C2×D13semidirect
327. #380C2×D13semidirect
526. C13#34201semidirect

Maximal subgroups

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

subgroupindex
C2×#117195332
C2×#117195322
C22×#881462
#131178792
#131178682
#131177802
#131177672
#131174992
#131174602
#131173612
#131172152
#131171922
C13×#23112
#3420113