Finite Group Databasebeta

← #15869939

Subgroups — #15869939

Order 3328.

924 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718789C211yesmaximal, characteristic
2C4×#88183C211yesmaximal, characteristic
3#13115449C211yesmaximal, characteristic, complemented
4#13115442C211yesmaximal, characteristic, complemented
5#13115400C211yesmaximal, characteristic, complemented
6#13115393C211yesmaximal, characteristic, complemented
7#13115133C211yesmaximal, characteristic
8#13115019C211yesmaximal, characteristic, complemented
9#13114904C211yesmaximal, characteristic, complemented
10#13114712C211yesmaximal, characteristic, complemented
11C8×#6101C211yesmaximal, characteristic, complemented
12#13114621C211yesmaximal, characteristic, complemented
13#13114619C211yesmaximal, characteristic, complemented
14#13114617C211yesmaximal, characteristic, complemented
15C2×C4×C13×M16C211yesmaximal, characteristic, complemented
16C22×C13×M16C2211yescharacteristic
17C22×C13×M16C2211yescharacteristic
18C22×C42×C13C2211yescharacteristic
19C2×#88183C411yescharacteristic
20C2×C4×C22⋊D13C2211yescharacteristic
21C22×#6094C2211yescharacteristic
22C4×C13×M16—24no—
23C2×C13×#97C2211yescharacteristic
24C2×C13×#97C2211yescharacteristic
25C2×C4×C8×C13C2211yescharacteristic
26C2×C4×C8×C13C2211yescharacteristic
27#11718874C411yescharacteristic
28#11718872C411yescharacteristic
29#11718871C411yescharacteristic
30C4×C22⋊Dic13C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33#11718789—24no—
34C2×#88060C2211yescharacteristic
35C2×#88060C2211yescharacteristic
36C2×#88060C2211yescharacteristic
37C2×#88055C2211yescharacteristic
38C2×#88054C2211yescharacteristic
39C2×#88054C2211yescharacteristic
40C2×#88054C2211yescharacteristic
41C2×C8×Dic13C2211yescharacteristic
42C2×C8×Dic13C2211yescharacteristic
43#11718601C411yescharacteristic
44#11718601C411yescharacteristic
45C4×#6103C2211yescharacteristic
46C4×#6103C2211yescharacteristic
47#11718594C411yescharacteristic
48#11718594C411yescharacteristic
49C4×#6101C2211yescharacteristic
50C4×#6101C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C23×C4×C13C2×C411yescharacteristic
62C23×C4×C13C2×C411yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C22×C22⋊D13C2×C411yescharacteristic
65C22×C4×D13C2311yescharacteristic
66C2×C13×M16—22no—
67C2×C13×M16—22no—
68C2×C13×M16—22no—
69C2×C13×M16—22no—
70C2×C13×M16—22no—
71C2×C13×M16D414yes—
72C22×C8×C13C2311yescharacteristic
73C22×C8×C13C2311yescharacteristic
74C22×C8×C13C2311yescharacteristic
75C22×C8×C13C2311yescharacteristic
76C2×C42×C13—22no—
77C2×C42×C13—22no—
78C2×C42×C13C2×C411yescharacteristic
79C2×C42×C13C2×C411yescharacteristic
80C2×C42×C13C2311yescharacteristic
81C2×C42×C13C2311yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C22⋊Dic13C2×C411yescharacteristic
85#88183—24no—
86C4×C22⋊D13—22no—
87C4×C22⋊D13—22no—
88C2×#6103C2×C411yescharacteristic
89C2×#6103C2×C411yescharacteristic
90C2×#6103C2×C411yescharacteristic
91C2×#6103C2×C411yescharacteristic
92C2×#6103C2×C411yescharacteristic
93C2×#6101C2×C411yescharacteristic
94C2×#6101C2×C411yescharacteristic
95C2×#6101C2×C411yescharacteristic
96C2×#6101C2×C411yescharacteristic
97C2×#6101C2×C411yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C2×C4×Dic13C2311yescharacteristic
100C2×C4×Dic13C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C2×#6094—22no—
106C2×#6094—22no—
107C2×#6094—22no—
108C13×#97—22no—
109C13×#97—22no—
110C4×C8×C13—22no—
111C4×C8×C13—22no—
112#88071C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118#88071C2×C411yescharacteristic
119#88071C2×C411yescharacteristic
120#88071C2×C411yescharacteristic
121#88071C2×C411yescharacteristic
122C22⋊#1055—22no—
123C22⋊#1055—22no—
124#88060—22no—
125#88060—22no—
126#88060—22no—
127#88055—22no—
128#88054—22no—
129#88054—22no—
130#88054—22no—
131C8×Dic13—22no—
132C8×Dic13—22no—
133#21565—131nomaximal, Sylow
134C24×C13C22×C411yescharacteristic
135C23×D13C22×C411yescharacteristic
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—22no—
141C22×C4×C13—22no—
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—41no—
145C22×C4×C13—41no—
146C22×C4×C13C22×C411yescharacteristic
147C22×C4×C13C22×C411yescharacteristic
148C22×C4×C13C22×C411yescharacteristic
149C22×C4×C13C22×C411yescharacteristic
150C22×C4×C13C22×C411yescharacteristic
151C22×C4×C13C2×D412yes—
152C22×C4×C13#4112yes—
153C22×C4×C13C2411yescharacteristic
154C2×C22⋊D13—22no—
155C2×C22⋊D13—22no—
156C2×C22⋊D13—41no—
157C22×Dic13C22×C411yescharacteristic
158C22×Dic13C22×C411yescharacteristic
159C22×Dic13C22×C411yescharacteristic
160C22×Dic13C22×C411yescharacteristic
161C22×Dic13C22×C411yescharacteristic
162C22×Dic13C22×C411yescharacteristic
163C22×Dic13C22×C411yescharacteristic
164C2×C4×D13—21no—
165C2×C4×D13—21no—
166C2×C4×D13—21no—
167C2×C4×D13—21no—
168C2×C4×D13—21no—
169C2×C4×D13—21no—
170C13×M16—28no—
171C13×M16—28no—
172C2×C8×C13—21no—
173C2×C8×C13—21no—
174C2×C8×C13—21no—
175C2×C8×C13—21no—
176C2×C8×C13—21no—
177C2×C8×C13—21no—
178C2×C8×C13C2×D412yes—
179C2×C8×C13C2×D412yes—
180C2×C8×C13#4112yes—
181C2×C8×C13#4112yes—
182C2×C8×C13#4112yes—
183C2×C8×C13#4112yes—
184C42×C13—22no—
185C42×C13—22no—
186C42×C13—21no—
187C42×C13—21no—
188C42×C13—21no—
189C42×C13—21no—
190C22⋊Dic13—22no—
191C22⋊Dic13—22no—
192C22⋊Dic13—41no—
193#6103—22no—
194#6103—22no—
195#6103—22no—
196#6103—22no—
197#6103—41no—
198#6101—22no—
199#6101—22no—
200#6101—22no—
201#6101—41no—
202#6101M1614yescomplemented
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—21no—
207C4×Dic13—21no—
208C4×Dic13—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221#6094—44no—
222C2×#436—131no—
223C2×C4×M16D1311yescharacteristic, complemented
224C4×#123—131no—
225#1924—131no—
226#1923—131no—
227#1917—131no—
228#1916—131no—
229#1870—131no—
230#1851—131no—
231#1817—131no—
232#1771—131no—
233C8×C4⋊C4—131no—
234#1753—131no—
235#1753—131no—
236#1753—131no—
237C23×C13—21no—
238C23×C13—21no—
239C23×C13—21no—
240C23×C13—21no—
241C23×C13—21no—
242C23×C13—21no—
243C23×C13C4×D412yes—
244C23×C13C23×C411yescharacteristic
245C22×D13—21no—
246C22×D13—21no—
247C22×D13—21no—
248C22×D13—21no—
249C22×D13—21no—
250C22×D13—21no—
251C22×D13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
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—44no—
269C2×C4×C13—44no—
270C2×C4×C13C4×D412yes—
271C2×C4×C13C2×M1612yes—
272C2×C4×C13#13112yes—
273C2×C4×C13C22×D411yescharacteristic
274C2×C4×C13C2×#4111yescharacteristic
275C2×C4×C13C2×#4111yescharacteristic
276C2×C4×C13C2×#4111yescharacteristic
277C2×C4×C13C2×#4111yescharacteristic
278C2×C4×C13C2×#4111yescharacteristic
279C22⋊D13—44no—
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×Dic13—21no—
291C2×Dic13—21no—
292C2×Dic13—21no—
293C2×Dic13—21no—
294C2×Dic13—21no—
295C2×Dic13—21no—
296C2×Dic13—21no—
297C2×Dic13—21no—
298C2×Dic13—21no—
299C2×Dic13C2×M1612yes—
300C2×Dic13C2×M1612yes—
301C4×D13—41no—
302C4×D13—41no—
303C4×D13—41no—
304C4×D13—41no—
305C8×C13—22no—
306C8×C13—22no—
307C8×C13—22no—
308C8×C13#12314yescomplemented
309#1055—41no—
310#1055—41no—
311#1055—41no—
312#1055—41no—
313C22×M16—131no—
314C22×M16C2×D1311yescharacteristic
315C22×M16C2×D1311yescharacteristic
316C2×#123—131no—
317C2×C4×D4—131no—
318C22×C42C2×D1311yescharacteristic
319#436—264no—
320C2×C4⋊C8—131no—
321C2×C4⋊C8—131no—
322C2×C4⋊C8—131no—
323C2×C22⋊C8—131no—
324C2×C22⋊C8—131no—
325C2×C22⋊C8—131no—
326C2×C22⋊C8—131no—
327C2×C22⋊C8—131no—
328C4×M16—24no—
329C2×#97—131no—
330C2×#97C2×D1311yescharacteristic
331C2×#97C2×D1311yescharacteristic
332C2×C4×C8—131no—
333C2×C4×C8—131no—
334C2×C4×C8C2×D1311yescharacteristic
335C2×C4×C8C2×D1311yescharacteristic
336#388—131no—
337#388—131no—
338#387—131no—
339#387—131no—
340#386—131no—
341#385—131no—
342#381—131no—
343C4×C4⋊C4—131no—
344C4×C4⋊C4—131no—
345C4×C22⋊C4—131no—
346C4×C22⋊C4—131no—
347C4×C22⋊C4—131no—
348#336—131no—
349#336—131no—
350#336—131no—
351#336—131no—
352#336—131no—
353#336—131no—
354#336—131no—
355#336—131no—
356#336—131no—
357#336—131no—
358C22×C13—22no—
359C22×C13—22no—
360C22×C13—22no—
361C22×C13—41no—
362C22×C13—41no—
363C22×C13—41no—
364C22×C13—41no—
365C22×C13C2×C4×D411yescharacteristic
366C22×C13C4×#4111yescharacteristic
367C22×C13C4×#4111yescharacteristic
368C22×C13C22×M1611yescharacteristic
369C22×C13C2×#13111yescharacteristic
370C22×C13#56811yescharacteristic
371C22×C13#56811yescharacteristic
372C2×D13—41no—
373C2×D13—41no—
374C2×D13—41no—
375C2×D13—41no—
376C2×D13—41no—
377C2×D13—41no—
378C2×D13—41no—
379C4×C13—22no—
380C4×C13—22no—
381C4×C13—22no—
382C4×C13—41no—
383C4×C13—41no—
384C4×C13C2×#12311yescharacteristic
385C4×C13#52511yescharacteristic
386C4×C13#52911yescharacteristic
387C4×C13#52911yescharacteristic
388Dic13—22no—
389Dic13—22no—
390Dic13—41no—
391Dic13—41no—
392Dic13—41no—
393Dic13—41no—
394Dic13—41no—
395C22×D4—131no—
396C23×C4—131no—
397C23×C4C4×D1311yescharacteristic
398C23×C4C4×D1311yescharacteristic
399C23×C4C22×D1311yescharacteristic
400C2×M16—22no—
401C2×M16—22no—
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—262no—
406C2×M16—262no—
407C2×M16—262no—
408C2×M16C22⋊D1314yes—
409C22×C8—131no—
410C22×C8—131no—
411C22×C8—131no—
412C22×C8—131no—
413C22×C8C22×D1311yescharacteristic
414C22×C8C22×D1311yescharacteristic
415C22×C8C22×D1311yescharacteristic
416C22×C8C22×D1311yescharacteristic
417#123—264no—
418C4×D4—262no—
419C4×D4—262no—
420C2×C4⋊C4—131no—
421C2×C4⋊C4—131no—
422C2×C4⋊C4—131no—
423C2×C4⋊C4—131no—
424C2×C4⋊C4—131no—
425C2×C22⋊C4—131no—
426C2×C22⋊C4—131no—
427C2×C22⋊C4—131no—
428C2×C22⋊C4—131no—
429C2×C22⋊C4—131no—
430C2×C22⋊C4—131no—
431C2×C22⋊C4—131no—
432C2×C22⋊C4—131no—
433C2×C42—22no—
434C2×C42—22no—
435C2×C42—131no—
436C2×C42—131no—
437C2×C42—131no—
438C2×C42C4×D1311yescharacteristic
439C2×C42C4×D1311yescharacteristic
440C2×C42C22×D1311yescharacteristic
441C2×C42C22×D1311yescharacteristic
442C4⋊C8—262no—
443C4⋊C8—262no—
444C4⋊C8—262no—
445C22⋊C8—262no—
446C22⋊C8—262no—
447C22⋊C8—262no—
448C22⋊C8—262no—
449C22⋊C8—262no—
450#97—22no—
451#97—22no—
452#97—262no—
453C4×C8—22no—
454C4×C8—22no—
455C4×C8—262no—
456C4×C8—262no—
457#95—131no—
458#95—131no—
459#95—131no—
460#95—131no—
461#95—131no—
462#95—131no—
463#95—131no—
464#95—131no—
465#95—131no—
466#95—131no—
467C2×C13—42no—
468C2×C13C4×#12311yescharacteristic
469C2×C13#293311yescharacteristic
470C2×C13D4×M1611yescharacteristic
471C2×C13#296511yescharacteristic
472C2×C13#296711yescharacteristic
473C2×C13#296911yescharacteristic
474C2×C13#297111yescharacteristic
475D13—81no—
476C24—131no—
477C24C2×C4×D1311yescharacteristic
478C2×D4—262no—
479C2×D4—262no—
480C2×D4—521no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—131no—
492C22×C4—131no—
493C22×C4—131no—
494C22×C4—131no—
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×C4C2×C4×D1311yescharacteristic
505C22×C4C2×C4×D1311yescharacteristic
506C22×C4C2×C4×D1311yescharacteristic
507C22×C4C2×C4×D1311yescharacteristic
508C22×C4C2×C4×D1311yescharacteristic
509C22×C4#612712yes—
510C22×C4C2×C22⋊D1312yes—
511C22×C4C23×D1311yescharacteristic
512M16—28no—
513M16—28no—
514M16—524no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—261no—
522C2×C8—261no—
523C2×C8—261no—
524C2×C8—261no—
525C2×C8—261no—
526C2×C8—261no—
527C2×C8—261no—
528C2×C8—261no—
529C2×C8—261no—
530C2×C8—261no—
531C2×C8—261no—
532C2×C8—261no—
533C2×C8#612712yes—
534C2×C8#612712yes—
535C2×C8#612712yes—
536C2×C8#612712yes—
537C2×C8C2×C22⋊D1312yes—
538C2×C8C2×C22⋊D1312yes—
539C4⋊C4—134no—
540C4⋊C4—262no—
541C4⋊C4—262no—
542C4⋊C4—262no—
543C4⋊C4—521no—
544C22⋊C4—262no—
545C22⋊C4—262no—
546C22⋊C4—262no—
547C22⋊C4—262no—
548C22⋊C4—262no—
549C22⋊C4—262no—
550C22⋊C4—521no—
551C22⋊C4—521no—
552C42—22no—
553C42—22no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—261no—
559C42—261no—
560C42—261no—
561C42—261no—
562C42—261no—
563C42—261no—
564C13#2156511yescharacteristic, complemented, Sylow
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—261no—
572C23—261no—
573C23—261no—
574C23—261no—
575C23—261no—
576C23—261no—
577C23—261no—
578C23C4×C22⋊D1312yes—
579C23C22×C4×D1311yescharacteristic
580D4—524no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—132no—
600C2×C4—132no—
601C2×C4—261no—
602C2×C4—261no—
603C2×C4—261no—
604C2×C4—261no—
605C2×C4—261no—
606C2×C4—261no—
607C2×C4—261no—
608C2×C4—261no—
609C2×C4—261no—
610C2×C4—261no—
611C2×C4—261no—
612C2×C4—261no—
613C2×C4—261no—
614C2×C4—261no—
615C2×C4—261no—
616C2×C4—261no—
617C2×C4—261no—
618C2×C4—261no—
619C2×C4—261no—
620C2×C4—521no—
621C2×C4—521no—
622C2×C4—521no—
623C2×C4—521no—
624C2×C4M16×D1312yes—
625C2×C4#8816112yes—
626C2×C4C4×C22⋊D1312yes—
627C2×C4C2×#612711yescharacteristic
628C2×C4C2×#612711yescharacteristic
629C2×C4C2×#612711yescharacteristic
630C2×C4C2×#612711yescharacteristic
631C2×C4C2×#612711yescharacteristic
632C2×C4C22×C22⋊D1311yescharacteristic
633C8—22no—
634C8—22no—
635C8—22no—
636C8—521no—
637C8—521no—
638C8—521no—
639C8—521no—
640C8#8818314yescomplemented
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—521no—
649C22—521no—
650C22—521no—
651C22—521no—
652C22—521no—
653C22—521no—
654C22—521no—
655C22C4×#612711yescharacteristic
656C22C4×#612711yescharacteristic
657C22C2×M16×D1311yescharacteristic
658C22C2×#8816111yescharacteristic
659C22#1171945011yescharacteristic
660C22#1171945011yescharacteristic
661C22C2×C4×C22⋊D1311yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—262no—
668C4—262no—
669C4—521no—
670C4—521no—
671C4—521no—
672C4—521no—
673C4—521no—
674C4#1171918411yescharacteristic
675C4#1171918411yescharacteristic
676C4C2×#8818311yescharacteristic
677C4#1171949611yescharacteristic
678C2—42no—
679C2—1041no—
680C2C4×#8818311yescharacteristic
681C2#1312356111yescharacteristic
682C2M16×C22⋊D1311yescharacteristic
683C2#1312379811yescharacteristic
684C2#1312382111yescharacteristic
685C2#1312384011yescharacteristic
686C2#1312385411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC2×C4×C13×M16C2
prime radicalC22×C22⋊D13C2×C4
cosocleC22×C4×C13C24
hypercenterC2×C4×M16D13
hypocenterC13#21565
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
3. #13115449C2semidirect
4. #13115442C2semidirect
5. #13115400C2semidirect
6. #13115393C2semidirect
8. #13115019C2semidirect
9. #13114904C2semidirect
10. #13114712C2semidirect
11. C8×#6101C2semidirect
12. #13114621C2semidirect
13. #13114619C2semidirect
14. #13114617C2semidirect
15. C2×C4×C13×M16C2semidirect
202. #6101M16semidirect
223. C2×C4×M16D13semidirect
308. C8×C13#123semidirect
564. C13#21565semidirect
640. C8#88183semidirect

Maximal subgroups

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

subgroupindex
C2×#117187892
C4×#881832
#131154492
#131154422
#131154002
#131153932
#131151332
#131150192
#131149042
#131147122
C8×#61012
#131146212
#131146192
#131146172
C2×C4×C13×M162
#2156513