Finite Group Databasebeta

← #16797665

Subgroups — #16797665

Order 3328.

990 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719076C211yesmaximal, characteristic
2C2×#11719075C211yesmaximal, characteristic
3#13130635C211yesmaximal, characteristic, complemented
4#13130628C211yesmaximal, characteristic, complemented
5#13130624C211yesmaximal, characteristic, complemented
6#13130577C211yesmaximal, characteristic, complemented
7#13130577C211yesmaximal, characteristic, complemented
8#13130577C211yesmaximal, characteristic, complemented
9#13130570C211yesmaximal, characteristic, complemented
10#13130565C211yesmaximal, characteristic, complemented
11C4×#88111C211yesmaximal, characteristic, complemented
12#13130555C211yesmaximal, characteristic, complemented
13#13130503C211yesmaximal, characteristic, complemented
14#13130500C211yesmaximal, characteristic, complemented
15#13117274C211yesmaximal, characteristic
16C22×#6118C2211yescharacteristic
17C22×#6118C2211yescharacteristic
18C22×#6118C2211yescharacteristic
19C22×D13⋊C8C2211yescharacteristic
20C2×#88147C2211yescharacteristic
21C2×#88146C2211yescharacteristic
22C2×C42×D13C2211yescharacteristic
23C2×#88112C2211yescharacteristic
24C2×#88111C2211yescharacteristic
25C2×#88111C2211yescharacteristic
26C2×#88111C2211yescharacteristic
27C2×#88111C2211yescharacteristic
28C2×#88110C2211yescharacteristic
29C2×#88110C2211yescharacteristic
30C2×#88110C2211yescharacteristic
31C2×C4⋊#1057C2211yescharacteristic
32C2×C4×#1057C2211yescharacteristic
33C2×C4×#1057C2211yescharacteristic
34C2×C4×#1057C2211yescharacteristic
35C2×C4×#1057C2211yescharacteristic
36C2×C4×#1057C2211yescharacteristic
37#11719076C418yescomplemented
38#11719075C418yescomplemented
39C13×#387C411yescharacteristic
40#11718650C411yescharacteristic
41#11718642C411yescharacteristic
42#11718601C411yescharacteristic
43#11718330C411yescharacteristic
44#11718329C411yescharacteristic
45#11718327C2211yescharacteristic
46#11718327C2211yescharacteristic
47#11718326C411yescharacteristic
48#11718326C411yescharacteristic
49#11718310C2211yescharacteristic
50#11718309C2211yescharacteristic
51#11718290C2211yescharacteristic
52#11718290C2211yescharacteristic
53#11718290C2211yescharacteristic
54#11718290C2211yescharacteristic
55#11718290C2211yescharacteristic
56#11718290C2211yescharacteristic
57#11718290C2211yescharacteristic
58#11718290C2211yescharacteristic
59#11718290C2211yescharacteristic
60#11718290C2211yescharacteristic
61C22×D52C2×C411yescharacteristic
62C22×C4×D13C2×C411yescharacteristic
63C22×C4×D13C2×C411yescharacteristic
64C22×C4×D13C2311yescharacteristic
65C22×C4×D13C2311yescharacteristic
66C22×#1057C2311yescharacteristic
67C22×#1057C2311yescharacteristic
68C22×#1057C2311yescharacteristic
69C22×#1057C2311yescharacteristic
70C22×#1057C2311yescharacteristic
71C22×#1057C2311yescharacteristic
72C22×#1057C2311yescharacteristic
73C22×#1057C2311yescharacteristic
74C2×#6118—41no—
75C2×#6118—41no—
76C2×#6118—41no—
77C2×#6118—41no—
78C2×#6118—41no—
79C2×#6118—41no—
80C2×#6118C2×C414yes—
81C2×#6118C2×C414yes—
82C2×#6118C2×C414yescomplemented
83C2×D13⋊C8—41no—
84C2×D13⋊C8—41no—
85C2×D13⋊C8C2×C414yescomplemented
86C2×C13×C4⋊C4C2×C411yescharacteristic
87C2×C13×C4⋊C4C2×C411yescharacteristic
88C2×C42×C13C2×C411yescharacteristic
89C2×#6103C2×C411yescharacteristic
90C2×#6103C2×C411yescharacteristic
91C2×#6103C2×C411yescharacteristic
92C2×#6103C2×C411yescharacteristic
93C2×#6103C2×C411yescharacteristic
94C2×#6103C2×C411yescharacteristic
95C2×#6103C2×C411yescharacteristic
96C2×#6103C2×C411yescharacteristic
97C2×C4⋊Dic13C2×C411yescharacteristic
98C2×#6101C2×C411yescharacteristic
99C2×#6101C2×C411yescharacteristic
100C2×C4×Dic13C2×C411yescharacteristic
101C2×C4×Dic13C2311yescharacteristic
102C2×C4×Dic13C2311yescharacteristic
103C2×C4×Dic13C2311yescharacteristic
104C2×C4×Dic13C2311yescharacteristic
105C2×C4×Dic13C2311yescharacteristic
106#88147C2×C414yes—
107#88147C2×C414yes—
108#88146—41no—
109#88146—41no—
110C42×D13—41no—
111C42×D13—41no—
112#88112C2×C414yescomplemented
113#88111C2×C414yes—
114#88111C2×C414yes—
115#88111C2×C414yescomplemented
116#88111C2×C414yescomplemented
117#88110—41no—
118#88110—41no—
119#88110C2×C414yescomplemented
120C4⋊#1057C2×C414yescomplemented
121C4×#1057—41no—
122C4×#1057—41no—
123C4×#1057—41no—
124C4×#1057—41no—
125C4×#1057C2×C414yescomplemented
126C13×#95C2×C411yescharacteristic
127C13×#95C2×C411yescharacteristic
128C13×#95C2×C411yescharacteristic
129C13×#95C2×C411yescharacteristic
130#88071C2×C411yescharacteristic
131#88071C2×C411yescharacteristic
132#88071C2×C411yescharacteristic
133#88071C2×C411yescharacteristic
134#88071C2×C411yescharacteristic
135#88071C2×C411yescharacteristic
136#20561—131nomaximal, Sylow
137C23×D13C22×C411yescharacteristic
138C23×D13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×C4×C13C22×C411yescharacteristic
142C22×C4×C13C22×C411yescharacteristic
143C22×C4×C13C22×C411yescharacteristic
144C22×C4×C13C22×C411yescharacteristic
145C22×C4×C13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C2411yescharacteristic
152C2×D52—41no—
153C2×D52—41no—
154C2×D52C4214yes—
155C2×C4×D13—21no—
156C2×C4×D13—21no—
157C2×C4×D13—21no—
158C2×C4×D13—21no—
159C2×C4×D13—21no—
160C2×C4×D13—21no—
161C2×C4×D13—21no—
162C2×C4×D13—21no—
163C2×C4×D13—22no—
164C2×C4×D13—22no—
165C2×C4×D13—41no—
166C2×C4×D13—41no—
167C2×C4×D13—41no—
168C2×C4×D13—41no—
169C2×C4×D13C22×C412yes—
170C2×C4×D13C22×C412yes—
171C2×C4×D13C22×C412yes—
172C2×C4×D13C22×C412yes—
173C2×#1057—21no—
174C2×#1057—21no—
175C2×#1057—21no—
176C2×#1057—21no—
177C2×#1057—21no—
178C2×#1057—21no—
179C2×#1057—21no—
180C2×#1057—21no—
181C2×#1057—21no—
182C2×#1057—21no—
183C2×#1057—21no—
184C2×#1057—21no—
185C2×#1057—21no—
186C2×#1057—21no—
187C2×#1057—21no—
188C2×#1057—21no—
189C2×#1057C22×C412yes—
190C2×#1057C22×C412yes—
191C2×#1057C22×C412yes—
192C2×#1057C22×C412yes—
193C2×#1057C22×C412yes—
194C2×#1057C22×C412yes—
195C2×#1057C22×C412yes—
196C2×#1057C22×C412yes—
197#6118—44no—
198#6118—44no—
199#6118—44no—
200D13⋊C8—44no—
201C13×C4⋊C4—41no—
202C13×C4⋊C4C4214yes—
203C42×C13—41no—
204#6103—22no—
205#6103—22no—
206#6103—41no—
207#6103—41no—
208#6103—41no—
209#6103—41no—
210#6103C4214yes—
211#6103C4214yes—
212C4⋊Dic13—41no—
213#6101—22no—
214#6101—22no—
215C4×Dic13—21no—
216C4×Dic13—21no—
217C4×Dic13—21no—
218C4×Dic13—21no—
219C4×Dic13—21no—
220C4×Dic13—21no—
221C4×Dic13—21no—
222C4×Dic13—21no—
223C4×Dic13—41no—
224C4×Dic13C22×C412yes—
225C4×Dic13C22×C412yes—
226C2×#436—131no—
227C2×#435—131no—
228#2290—131no—
229#1771—131no—
230C8×C4⋊C4—131no—
231#1768—131no—
232#1753—131no—
233#1753—131no—
234#1753—131no—
235C8×C22⋊C4—131no—
236#1750—131no—
237C4×C22⋊C8—131no—
238#1746—131no—
239#1729—131no—
240#1728—131no—
241C23×C13C23×C411yescharacteristic
242C22×D13—21no—
243C22×D13—21no—
244C22×D13—21no—
245C22×D13—21no—
246C22×D13—21no—
247C22×D13—21no—
248C22×D13—21no—
249C22×D13—21no—
250C22×D13—21no—
251C22×D13—21no—
252C22×D13—21no—
253C22×D13—21no—
254C22×D13C2×C4212yes—
255C22×D13C2×C4212yes—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—21no—
269C2×C4×C13—21no—
270C2×C4×C13—21no—
271C2×C4×C13—21no—
272C2×C4×C13—21no—
273C2×C4×C13C2×C4212yes—
274C2×C4×C13C2×C4212yes—
275C2×C4×C13C2×C4212yes—
276C2×C4×C13#13112yes—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—21no—
287C2×Dic13—21no—
288C2×Dic13—21no—
289C2×Dic13—21no—
290C2×Dic13C2×C4212yes—
291C2×Dic13#13112yes—
292C2×Dic13C23×C411yescharacteristic
293C2×Dic13C23×C411yescharacteristic
294C2×Dic13H(5,𝔽2)11yescharacteristic
295C2×Dic13H(5,𝔽2)11yescharacteristic
296C2×Dic13H(5,𝔽2)11yescharacteristic
297C2×Dic132−1+411yescharacteristic
298D52—44no—
299C4×D13—22no—
300C4×D13—22no—
301C4×D13—22no—
302C4×D13—22no—
303C4×D13—22no—
304C4×D13—22no—
305C4×D13—22no—
306C4×D13—22no—
307C4×D13—44no—
308C4×D13—44no—
309#1057—22no—
310#1057—22no—
311#1057—22no—
312#1057—22no—
313#1057—22no—
314#1057—22no—
315#1057—22no—
316#1057—22no—
317C22×M16—131no—
318C22×M16—131no—
319C22×M16—131no—
320C23×C8—131no—
321C2×C4×D4—131no—
322C2×#117—131no—
323C22×C42—131no—
324#436—138no—
325#435—138no—
326C2×C4⋊C8—131no—
327C2×C4⋊C8—131no—
328C2×C22⋊C8—131no—
329C2×C22⋊C8—131no—
330C2×C22⋊C8—131no—
331C2×C22⋊C8—131no—
332C2×#97—131no—
333C2×#97—131no—
334C2×#97—131no—
335C2×C4×C8—131no—
336C2×C4×C8—131no—
337C2×C4×C8—131no—
338C2×C4×C8—131no—
339C2×C4×C8—131no—
340#388—131no—
341#388—131no—
342#387—131no—
343#387#26111yescharacteristic
344#386—131no—
345#386—131no—
346#385—131no—
347#385—131no—
348C4×C4⋊C4—131no—
349C4×C22⋊C4—131no—
350C4×C22⋊C4—131no—
351#376—131no—
352#336—131no—
353#336—131no—
354#336—131no—
355#336—131no—
356#336—131no—
357#336—131no—
358#336—131no—
359#336—131no—
360#336—131no—
361#336—131no—
362C22×C13C22×C4211yescharacteristic
363C22×C13#51811yescharacteristic
364C22×C13#52011yescharacteristic
365C22×C13C2×#13111yescharacteristic
366C22×C13C2×#13111yescharacteristic
367C22×C13#56811yescharacteristic
368C22×C13#56811yescharacteristic
369C2×D13—22no—
370C2×D13—22no—
371C2×D13—22no—
372C2×D13—22no—
373C2×D13—22no—
374C2×D13—22no—
375C2×D13—41no—
376C2×D13—41no—
377C2×D13—41no—
378C2×D13—41no—
379C2×D13—41no—
380C2×D13—41no—
381C2×D13—41no—
382C2×D13—41no—
383C4×C13—22no—
384C4×C13—22no—
385C4×C13—22no—
386C4×C13—22no—
387C4×C13—41no—
388C4×C13—41no—
389C4×C13—41no—
390Dic13—22no—
391Dic13—22no—
392Dic13—41no—
393Dic13—41no—
394Dic13—41no—
395Dic13#51811yescharacteristic
396Dic13#51811yescharacteristic
397Dic13#52011yescharacteristic
398Dic13#52011yescharacteristic
399C22×D4—131no—
400C23×C4—131no—
401C23×C4—131no—
402C23×C4—131no—
403C23×C4—131no—
404C2×M16—134no—
405C2×M16—134no—
406C2×M16—134no—
407C2×M16—521no—
408C2×M16—521no—
409C2×M16—521no—
410C2×M16—521no—
411C2×M16—521no—
412C2×M16—521no—
413C22×C8—131no—
414C22×C8—131no—
415C22×C8—131no—
416C22×C8—131no—
417C22×C8—131no—
418C22×C8—131no—
419C22×C8—131no—
420C22×C8—131no—
421C22×C8—134no—
422C22×C8—521no—
423C22×C8—521no—
424C4×D4—134no—
425C4×D4—134no—
426#117—521no—
427#117—521no—
428C2×C4⋊C4—131no—
429C2×C4⋊C4—131no—
430C2×C4⋊C4—131no—
431C2×C4⋊C4C2×#26111yescharacteristic
432C2×C4⋊C4C2×#26111yescharacteristic
433C2×C22⋊C4—131no—
434C2×C22⋊C4—131no—
435C2×C22⋊C4—131no—
436C2×C22⋊C4—131no—
437C2×C22⋊C4—131no—
438C2×C22⋊C4—131no—
439C2×C22⋊C4—131no—
440C2×C22⋊C4—131no—
441C2×C42—131no—
442C2×C42—131no—
443C2×C42—131no—
444C2×C42—131no—
445C2×C42—131no—
446C2×C42—131no—
447C2×C42—521no—
448C2×C42—521no—
449C2×C42C2×#26111yescharacteristic
450C4⋊C8—134no—
451C4⋊C8—134no—
452C22⋊C8—134no—
453C22⋊C8—134no—
454C22⋊C8—134no—
455C22⋊C8—134no—
456#97—134no—
457#97—521no—
458#97—521no—
459C4×C8—134no—
460C4×C8—521no—
461C4×C8—521no—
462C4×C8—521no—
463C4×C8—521no—
464#95—131no—
465#95—131no—
466#95—131no—
467#95—131no—
468#95—131no—
469#95—131no—
470#95C2×#26111yescharacteristic
471#95C2×#26111yescharacteristic
472#95C2×#26111yescharacteristic
473#95C2×#26111yescharacteristic
474C2×C13#227511yescharacteristic
475C2×C13#287311yescharacteristic
476C2×C13C4×#13111yescharacteristic
477C2×C13#297611yescharacteristic
478C2×C13#297611yescharacteristic
479C2×C13#299311yescharacteristic
480C2×C13#299311yescharacteristic
481D13—42no—
482D13—42no—
483C24—131no—
484C24—131no—
485C2×D4—134no—
486C2×D4—521no—
487C2×D4—521no—
488C22×C4—131no—
489C22×C4—131no—
490C22×C4—131no—
491C22×C4—132no—
492C22×C4—132no—
493C22×C4—132no—
494C22×C4—132no—
495C22×C4—131no—
496C22×C4—131no—
497C22×C4—131no—
498C22×C4—261no—
499C22×C4—261no—
500C22×C4—261no—
501C22×C4—261no—
502C22×C4—261no—
503C22×C4—261no—
504C22×C4—261no—
505C22×C4—261no—
506C22×C4—262no—
507C22×C4—262no—
508C22×C4—521no—
509C22×C4—521no—
510C22×C4—521no—
511C22×C4—521no—
512C22×C4C22×#26111yescharacteristic
513C22×C4C22×#26111yescharacteristic
514C22×C4C22×#26111yescharacteristic
515C22×C4C22×#26111yescharacteristic
516C22×C4C22×#26111yescharacteristic
517C22×C4C22×#26111yescharacteristic
518C22×C4C22×#26111yescharacteristic
519M16—524no—
520M16—524no—
521M16—524no—
522C2×C8—132no—
523C2×C8—132no—
524C2×C8—132no—
525C2×C8—132no—
526C2×C8—132no—
527C2×C8—132no—
528C2×C8—132no—
529C2×C8—132no—
530C2×C8—261no—
531C2×C8—261no—
532C2×C8—261no—
533C2×C8—261no—
534C2×C8—261no—
535C2×C8—261no—
536C2×C8—261no—
537C2×C8—261no—
538C2×C8—261no—
539C2×C8—261no—
540C2×C8—261no—
541C2×C8—261no—
542C2×C8—261no—
543C2×C8—261no—
544C2×C8—261no—
545C2×C8—261no—
546C2×C8—524no—
547C4⋊C4—41no—
548C4⋊C4—262no—
549C4⋊C4—262no—
550C4⋊C4—521no—
551C4⋊C4C4×#26114yes—
552C22⋊C4—134no—
553C22⋊C4—134no—
554C22⋊C4—262no—
555C22⋊C4—262no—
556C22⋊C4—521no—
557C22⋊C4—521no—
558C22⋊C4—521no—
559C22⋊C4—521no—
560C42—41no—
561C42—132no—
562C42—132no—
563C42—261no—
564C42—261no—
565C42—261no—
566C42—261no—
567C42—261no—
568C42—261no—
569C42—261no—
570C42—261no—
571C42—521no—
572C13#2056111yescharacteristic, complemented, Sylow
573C23—132no—
574C23—132no—
575C23—261no—
576C23—261no—
577C23—261no—
578C23—261no—
579C23—261no—
580C23—261no—
581C23—261no—
582C23—261no—
583C23—261no—
584C23—261no—
585C23—261no—
586C23—261no—
587C23C23×#26111yescharacteristic
588D4—524no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—131no—
607C2×C4—131no—
608C2×C4—131no—
609C2×C4—131no—
610C2×C4—131no—
611C2×C4—131no—
612C2×C4—132no—
613C2×C4—132no—
614C2×C4—262no—
615C2×C4—262no—
616C2×C4—262no—
617C2×C4—262no—
618C2×C4—262no—
619C2×C4—262no—
620C2×C4—262no—
621C2×C4—262no—
622C2×C4—261no—
623C2×C4—261no—
624C2×C4—261no—
625C2×C4—261no—
626C2×C4—261no—
627C2×C4—261no—
628C2×C4—261no—
629C2×C4—261no—
630C2×C4—261no—
631C2×C4—261no—
632C2×C4—261no—
633C2×C4—261no—
634C2×C4—261no—
635C2×C4—524no—
636C2×C4—524no—
637C2×C4C2×C4×#26112yes—
638C2×C4C2×C4×#26112yes—
639C2×C4C2×C4×#26112yes—
640C2×C4#8824012yes—
641C8—262no—
642C8—262no—
643C8—262no—
644C8—262no—
645C8—262no—
646C8—262no—
647C8—262no—
648C8—262no—
649C22—262no—
650C22—262no—
651C22—262no—
652C22—262no—
653C22—262no—
654C22—262no—
655C22—521no—
656C22—521no—
657C22—521no—
658C22—521no—
659C22—521no—
660C22—521no—
661C22—521no—
662C22—521no—
663C22C22×C4×#26111yescharacteristic
664C22#1171962311yescharacteristic
665C22C2×#8824011yescharacteristic
666C22D13⋊#13111yescharacteristic
667C22#1171962911yescharacteristic
668C22#1171962911yescharacteristic
669C22#1171963111yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—131no—
678C4—131no—
679C4—131no—
680C4—131no—
681C4—262no—
682C4—262no—
683C4—521no—
684C4—521no—
685C4—521no—
686C2—522no—
687C2—522no—
688C2#1313127511yescharacteristic
689C2#1313140111yescharacteristic
690C2C4×#8824011yescharacteristic
691C2#1313170411yescharacteristic
692C2#1313170611yescharacteristic
693C2#1313178011yescharacteristic
694C2#1313178211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutatorC22×C13C22×C42
FittingC13×#387C4
socleC23×C13C23×C4
prime radicalC22×D52C2×C4
cosocleC22×Dic13C24
hypercenter#387#261
hypocenterC13#20561
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #13130635C2semidirect
4. #13130628C2semidirect
5. #13130624C2semidirect
6. #13130577C2semidirect
7. #13130577C2semidirect
8. #13130577C2semidirect
9. #13130570C2semidirect
10. #13130565C2semidirect
11. C4×#88111C2semidirect
12. #13130555C2semidirect
13. #13130503C2semidirect
14. #13130500C2semidirect
37. #11719076C4semidirect
38. #11719075C4semidirect
82. C2×#6118C2×C4semidirect
85. C2×D13⋊C8C2×C4semidirect
112. #88112C2×C4semidirect
115. #88111C2×C4semidirect
116. #88111C2×C4semidirect
119. #88110C2×C4semidirect
120. C4⋊#1057C2×C4semidirect
125. C4×#1057C2×C4semidirect
572. C13#20561semidirect

Maximal subgroups

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

subgroupindex
C2×#117190762
C2×#117190752
#131306352
#131306282
#131306242
#131305772
#131305702
#131305652
C4×#881112
#131305552
#131305032
#131305002
#131172742
#2056113