Finite Group Databasebeta

← #15869957

Subgroups — #15869957

Order 3328.

906 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718386C211yesmaximal, characteristic
2C2×#11718369C211yesmaximal, characteristic, complemented
3#13118097C211yesmaximal, characteristic
4#13115446C211yesmaximal, characteristic, complemented
5#13115401C211yesmaximal, characteristic, complemented
6#13115400C211yesmaximal, characteristic, complemented
7#13115130C211yesmaximal, characteristic
8#13115017C211yesmaximal, characteristic, complemented
9#13114906C211yesmaximal, characteristic, complemented
10#13114701C211yesmaximal, characteristic, complemented
11#13114697C211yesmaximal, characteristic, complemented
12#13114610C211yesmaximal, characteristic, complemented
13C22⋊C4×#1055C211yesmaximal, characteristic, complemented
14#13114606C211yesmaximal, characteristic, complemented
15C13×#1924C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊D13C2211yescharacteristic
17C23×#1055C2211yescharacteristic
18C22×#6094C2211yescharacteristic
19C22×C8×D13C2211yescharacteristic
20C2×#88126C2211yescharacteristic
21C2×C13×C4⋊C8C2211yescharacteristic
22C2×C13×C22⋊C8C2211yescharacteristic
23C2×C13×C22⋊C8C2211yescharacteristic
24C2×C13×#97C2211yescharacteristic
25C4×C13×C22⋊C4C2211yescharacteristic
26C4×C22⋊Dic13C2211yescharacteristic
27C2×C22⋊#1055C2211yescharacteristic
28C2×#88060C2211yescharacteristic
29C2×#88060C2211yescharacteristic
30C2×#88055C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×C8×Dic13C2211yescharacteristic
34C2×C8×Dic13C2211yescharacteristic
35C2×C8×Dic13C2211yescharacteristic
36#11718622C411yescharacteristic
37#11718620C411yescharacteristic
38#11718613C411yescharacteristic
39#11718610C411yescharacteristic
40C4×#6103C2211yescharacteristic
41C4×C4⋊Dic13C2211yescharacteristic
42C4×#6101C2211yescharacteristic
43C2×C4×#1055C2211yescharacteristic
44C2×C4×#1055C2211yescharacteristic
45#11718386—24no—
46#11718369—24no—
47#11718336C411yescharacteristic
48#11718335C411yescharacteristic
49#11718319C411yescharacteristic
50#11718318C411yescharacteristic
51C13×#336C2211yescharacteristic
52C13×#336C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C22×C22⋊D13C2×C411yescharacteristic
63C22×C4×D13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C2×C13×C22⋊C4C2×C411yescharacteristic
69C2×C13×C22⋊C4C2×C411yescharacteristic
70C2×C42×C13C2311yescharacteristic
71C2×C42×C13C2311yescharacteristic
72C2×C22⋊Dic13C2×C411yescharacteristic
73C2×C22⋊Dic13C2×C411yescharacteristic
74C2×C22⋊Dic13C2×C411yescharacteristic
75C4×C22⋊D13—22no—
76C4×C22⋊D13—22no—
77C2×#6103C2×C411yescharacteristic
78C2×#6103C2×C411yescharacteristic
79C2×#6103C2×C411yescharacteristic
80C2×#6103C2×C411yescharacteristic
81C2×#6103C2×C411yescharacteristic
82C2×C4⋊Dic13C2×C411yescharacteristic
83C2×C4⋊Dic13C2×C411yescharacteristic
84C2×#6101C2×C411yescharacteristic
85C2×#6101C2×C411yescharacteristic
86C2×#6101C2×C411yescharacteristic
87C2×#6101C2×C411yescharacteristic
88C2×#6101C2×C411yescharacteristic
89C2×C4×Dic13C2311yescharacteristic
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C22×#1055—22no—
93C22×#1055—22no—
94C22×#1055—22no—
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C2×#6094—22no—
100C2×#6094—22no—
101C2×#6094—22no—
102C2×C8×D13—22no—
103C2×C8×D13—22no—
104C2×C8×D13D414yes—
105#88126—22no—
106#88126—22no—
107C13×C4⋊C8—22no—
108C13×C22⋊C8—22no—
109C13×C22⋊C8—22no—
110C13×#97—22no—
111C13×#95C2×C411yescharacteristic
112C13×#95C2×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
121C22⋊#1055—22no—
122#88060—22no—
123#88060—22no—
124#88055—22no—
125#88054—22no—
126#88054—22no—
127C8×Dic13—22no—
128C8×Dic13—22no—
129C8×Dic13—22no—
130C4×#1055—22no—
131C4×#1055—22no—
132#21566—131nomaximal, Sylow
133C24×C13C22×C411yescharacteristic
134C23×D13C22×C411yescharacteristic
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13C22×C411yescharacteristic
142C22×C4×C13C22×C411yescharacteristic
143C22×C4×C13C22×C411yescharacteristic
144C22×C4×C13C22×C411yescharacteristic
145C22×C4×C13C22×C411yescharacteristic
146C22×C4×C13C2411yescharacteristic
147C2×C22⋊D13—22no—
148C2×C22⋊D13—22no—
149C2×C22⋊D13—22no—
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C22×Dic13C22×C411yescharacteristic
153C22×Dic13C22×C411yescharacteristic
154C22×Dic13C22×C411yescharacteristic
155C22×Dic13C22×C411yescharacteristic
156C22×Dic13C22×C411yescharacteristic
157C2×C4×D13—21no—
158C2×C4×D13—21no—
159C2×C4×D13—21no—
160C2×C4×D13—21no—
161C2×C4×D13C2×D412yes—
162C2×C4×D13#4112yes—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13—21no—
172C2×C8×C13—21no—
173C2×C8×C13C2×D412yes—
174C2×C8×C13#4112yes—
175C13×C22⋊C4—22no—
176C13×C22⋊C4—22no—
177C42×C13—21no—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13—21no—
181C22⋊Dic13—22no—
182C22⋊Dic13—22no—
183C22⋊Dic13—22no—
184#6103—22no—
185#6103—22no—
186#6103—22no—
187#6103—22no—
188#6103—22no—
189C4⋊Dic13—22no—
190C4⋊Dic13—22no—
191#6101—22no—
192#6101—22no—
193#6101—22no—
194#6101—22no—
195#6101—22no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—44no—
211C2×#1055C2×D412yes—
212C2×#1055#4112yes—
213C2×#1055#4112yes—
214C2×#1055#4112yes—
215#6094—28no—
216C8×D13—28no—
217C2×C8×D4—131no—
218C2×#405—131no—
219#2395—131no—
220#1924—131no—
221#1924D1311yescharacteristic, complemented
222#1917—131no—
223#1917—131no—
224#1870—131no—
225#1851—131no—
226#1818—131no—
227C8×C4⋊C4—131no—
228C8×C4⋊C4—131no—
229C8×C22⋊C4—131no—
230C8×C22⋊C4—131no—
231C8×C22⋊C4—131no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—21no—
238C23×C13#13112yes—
239C23×C13C23×C411yescharacteristic
240C22×D13—21no—
241C22×D13—21no—
242C22×D13—21no—
243C22×D13—21no—
244C22×D13—21no—
245C22×D13—21no—
246C22×D13C4×D412yes—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—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—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13#13112yes—
266C2×C4×C13C22×D411yescharacteristic
267C2×C4×C13C2×#4111yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C22⋊D13—28no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
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×Dic13—21no—
291C2×Dic13C4×D412yes—
292C2×Dic13#13112yes—
293C2×Dic13#13112yes—
294C4×D13—22no—
295C4×D13—22no—
296C4×D13—22no—
297C4×D13—22no—
298C8×C13—22no—
299C8×C13—22no—
300C8×C13—41no—
301C8×C13—41no—
302#1055—22no—
303#1055—41no—
304#1055—41no—
305#1055#12314yes—
306C22×M16—131no—
307C23×C8—131no—
308C23×C8—131no—
309C2×C4×D4—131no—
310C2×#117—131no—
311C8×D4—264no—
312C2×C4⋊C8—131no—
313C2×C4⋊C8—131no—
314C2×C4⋊C8C2×D1311yescharacteristic
315C2×C22⋊C8—131no—
316C2×C22⋊C8—131no—
317C2×C22⋊C8—131no—
318C2×C22⋊C8C2×D1311yescharacteristic
319C2×C22⋊C8C2×D1311yescharacteristic
320#405—264no—
321C2×#97—131no—
322C2×#97C2×D1311yescharacteristic
323C2×C4×C8—131no—
324C2×C4×C8—131no—
325C2×C4×C8—131no—
326C2×C4×C8—131no—
327C2×C4×C8—131no—
328#400—131no—
329#399—131no—
330#398—131no—
331#397—131no—
332#397—131no—
333#397—131no—
334#394—131no—
335#394—131no—
336C4×C4⋊C4—131no—
337C4×C4⋊C4—131no—
338C4×C22⋊C4—131no—
339C4×C22⋊C4—131no—
340C4×C22⋊C4C2×D1311yescharacteristic
341#336—131no—
342#336—131no—
343#336—131no—
344#336—131no—
345#336—131no—
346#336—131no—
347#336—131no—
348#336—131no—
349#336C2×D1311yescharacteristic
350#336C2×D1311yescharacteristic
351C22×C13—22no—
352C22×C13—22no—
353C22×C13—22no—
354C22×C13—41no—
355C22×C13—41no—
356C22×C13—41no—
357C22×C13—41no—
358C22×C13C2×C4×D411yescharacteristic
359C22×C13C4×#4111yescharacteristic
360C22×C13C4×#4111yescharacteristic
361C22×C13C2×#13111yescharacteristic
362C22×C13C2×#13111yescharacteristic
363C22×C13C2×#13111yescharacteristic
364C22×C13C2×#13111yescharacteristic
365C2×D13—22no—
366C2×D13—22no—
367C2×D13—22no—
368C2×D13—41no—
369C2×D13—41no—
370C2×D13—41no—
371C2×D13—41no—
372C4×C13—22no—
373C4×C13—41no—
374C4×C13—41no—
375C4×C13—41no—
376C4×C13—41no—
377C4×C13C2×#12311yescharacteristic
378C4×C13#52511yescharacteristic
379C4×C13#52911yescharacteristic
380C4×C13#52911yescharacteristic
381Dic13—22no—
382Dic13—22no—
383Dic13—22no—
384Dic13—41no—
385Dic13—41no—
386Dic13—41no—
387Dic13—41no—
388C22×D4—131no—
389C23×C4—131no—
390C23×C4C22×D1311yescharacteristic
391C2×M16—262no—
392C2×M16—262no—
393C2×M16—262no—
394C22×C8—131no—
395C22×C8—131no—
396C22×C8—134no—
397C22×C8—131no—
398C22×C8—131no—
399C22×C8—262no—
400C22×C8—262no—
401C22×C8—262no—
402C22×C8—262no—
403C22×C8—262no—
404C22×C8C22×D1311yescharacteristic
405C22×C8C22×D1311yescharacteristic
406C22×C8C22×D1311yescharacteristic
407C22×C8C22×D1311yescharacteristic
408C4×D4—262no—
409C4×D4—262no—
410#117—262no—
411#117—262no—
412C2×C4⋊C4—131no—
413C2×C4⋊C4—131no—
414C2×C4⋊C4—131no—
415C2×C4⋊C4—131no—
416C2×C4⋊C4—131no—
417C2×C4⋊C4—131no—
418C2×C4⋊C4—131no—
419C2×C22⋊C4—131no—
420C2×C22⋊C4—131no—
421C2×C22⋊C4—131no—
422C2×C22⋊C4—131no—
423C2×C22⋊C4—131no—
424C2×C22⋊C4—131no—
425C2×C22⋊C4—131no—
426C2×C22⋊C4—131no—
427C2×C22⋊C4C4×D1311yescharacteristic
428C2×C22⋊C4C4×D1311yescharacteristic
429C2×C42—131no—
430C2×C42—131no—
431C2×C42—131no—
432C2×C42C22×D1311yescharacteristic
433C2×C42C22×D1311yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—262no—
436C4⋊C8—262no—
437C22⋊C8—22no—
438C22⋊C8—22no—
439C22⋊C8—262no—
440C22⋊C8—262no—
441C22⋊C8—262no—
442#97—22no—
443#97—262no—
444C4×C8—262no—
445C4×C8—262no—
446C4×C8—262no—
447C4×C8—262no—
448C4×C8—262no—
449#95—131no—
450#95—131no—
451#95—131no—
452#95—131no—
453#95—131no—
454#95—131no—
455#95—131no—
456#95—131no—
457#95C4×D1311yescharacteristic
458#95C4×D1311yescharacteristic
459C2×C13—42no—
460C2×C13C4×#12311yescharacteristic
461C2×C13#292911yescharacteristic
462C2×C13#293111yescharacteristic
463C2×C13#296611yescharacteristic
464C2×C13#296611yescharacteristic
465C2×C13#296611yescharacteristic
466C2×C13#296611yescharacteristic
467D13—42no—
468C24—131no—
469C24C2×C4×D1311yescharacteristic
470C2×D4—262no—
471C2×D4—262no—
472C2×D4—262no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—131no—
480C22×C4—131no—
481C22×C4—131no—
482C22×C4—132no—
483C22×C4—132no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—131no—
487C22×C4—131no—
488C22×C4—261no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—261no—
492C22×C4C2×C4×D1311yescharacteristic
493C22×C4C2×C4×D1311yescharacteristic
494C22×C4C2×C4×D1311yescharacteristic
495C22×C4C2×C4×D1311yescharacteristic
496C22×C4C2×C4×D1311yescharacteristic
497C22×C4C23×D1311yescharacteristic
498M16—268no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—132no—
510C2×C8—132no—
511C2×C8—132no—
512C2×C8—132no—
513C2×C8—268no—
514C2×C8—261no—
515C2×C8—261no—
516C2×C8—261no—
517C2×C8—261no—
518C2×C8—261no—
519C2×C8—261no—
520C2×C8—261no—
521C2×C8—261no—
522C2×C8—524no—
523C2×C8D4×D1312yes—
524C2×C8#612912yes—
525C4⋊C4—262no—
526C4⋊C4—262no—
527C4⋊C4—262no—
528C4⋊C4—262no—
529C4⋊C4—262no—
530C4⋊C4—262no—
531C4⋊C4—262no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—262no—
535C22⋊C4—262no—
536C22⋊C4—262no—
537C22⋊C4—262no—
538C22⋊C4—262no—
539C22⋊C4—262no—
540C22⋊C4—262no—
541C22⋊C4—262no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—261no—
547C42—261no—
548C42—261no—
549C42—261no—
550C42—261no—
551C42—261no—
552C13#2156611yescharacteristic, complemented, Sylow
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—132no—
560C23—261no—
561C23—261no—
562C23—261no—
563C23—261no—
564C23—261no—
565C23—261no—
566C23#8816112yes—
567C23C22×C4×D1311yescharacteristic
568D4—268no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—132no—
588C2×C4—132no—
589C2×C4—132no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—262no—
597C2×C4—262no—
598C2×C4—262no—
599C2×C4—262no—
600C2×C4—261no—
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#8816112yes—
613C2×C4C2×D4×D1311yescharacteristic
614C2×C4C2×#612911yescharacteristic
615C2×C4#41×D1311yescharacteristic
616C2×C4#41×D1311yescharacteristic
617C2×C4#41×D1311yescharacteristic
618C2×C4#41×D1311yescharacteristic
619C8—22no—
620C8—22no—
621C8—41no—
622C8—41no—
623C8—134no—
624C8—262no—
625C8—521no—
626C8—521no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—262no—
635C22—262no—
636C22—262no—
637C22—521no—
638C22—521no—
639C22—521no—
640C22—521no—
641C22#1171922711yescharacteristic
642C22C4×#612911yescharacteristic
643C22C4×D4×D1311yescharacteristic
644C22C2×#8816111yescharacteristic
645C22C2×#8816111yescharacteristic
646C22D13×#13111yescharacteristic
647C22D13×#13111yescharacteristic
648C4—22no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—262no—
654C4—262no—
655C4—262no—
656C4—521no—
657C4—521no—
658C4—521no—
659C4—521no—
660C4#1171931111yescharacteristic
661C4D13×#12311yescharacteristic
662C4#1171938511yescharacteristic
663C4#1171941111yescharacteristic
664C2—42no—
665C2—522no—
666C2#1311736311yescharacteristic
667C2#1312352711yescharacteristic
668C2#1312354511yescharacteristic
669C2#1312380511yescharacteristic
670C2#1312380611yescharacteristic
671C2#1312380611yescharacteristic
672C2#1312380911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC13×#1924C2
prime radicalC22×C22⋊D13C2×C4
cosocleC22×C4×C13C24
hypercenter#1924D13
hypocenterC13#21566
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
2. C2×#11718369C2semidirect
4. #13115446C2semidirect
5. #13115401C2semidirect
6. #13115400C2semidirect
8. #13115017C2semidirect
9. #13114906C2semidirect
10. #13114701C2semidirect
11. #13114697C2semidirect
12. #13114610C2semidirect
13. C22⋊C4×#1055C2semidirect
14. #13114606C2semidirect
15. C13×#1924C2semidirect
221. #1924D13semidirect
552. C13#21566semidirect

Maximal subgroups

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

subgroupindex
C2×#117183862
C2×#117183692
#131180972
#131154462
#131154012
#131154002
#131151302
#131150172
#131149062
#131147012
#131146972
#131146102
C22⋊C4×#10552
#131146062
C13×#19242
#2156613