Finite Group Databasebeta

← #15869979

Subgroups — #15869979

Order 3328.

878 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×#11718672C211yesmaximal, characteristic
2C2×#11718762C211yesmaximal, characteristic, complemented
3#13118095C211yesmaximal, characteristic
4#13115452C211yesmaximal, characteristic, complemented
5#13115445C211yesmaximal, characteristic, complemented
6#13115403C211yesmaximal, characteristic, complemented
7#13115396C211yesmaximal, characteristic, complemented
8#13115014C211yesmaximal, characteristic, complemented
9#13114905C211yesmaximal, characteristic, complemented
10#13114709C211yesmaximal, characteristic, complemented
11#13114702C211yesmaximal, characteristic, complemented
12#13114622C211yesmaximal, characteristic, complemented
13#13114615C211yesmaximal, characteristic, complemented
14#13114610C211yesmaximal, characteristic, complemented
15C13×#1869C211yesmaximal, characteristic
16C23×C8×C13C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×#88180C2211yescharacteristic
19C22×#6099C2211yescharacteristic
20C23×#1055C2211yescharacteristic
21C2×C13×C22⋊C8C2211yescharacteristic
22C2×C13×C22⋊C8C2211yescharacteristic
23C2×C13×C22⋊C8C2211yescharacteristic
24#11718883C411yescharacteristic
25#11718883C411yescharacteristic
26C4×C22⋊Dic13C2211yescharacteristic
27C4×C22⋊Dic13C2211yescharacteristic
28C4×C22⋊Dic13C2211yescharacteristic
29C2×C22⋊#1055C2211yescharacteristic
30C2×C22⋊#1055C2211yescharacteristic
31#11718762—24no—
32C2×#88055C2211yescharacteristic
33C2×#88055C2211yescharacteristic
34C2×#88055C2211yescharacteristic
35C2×#88054C2211yescharacteristic
36C2×#88054C2211yescharacteristic
37C2×C8×Dic13C2211yescharacteristic
38C2×C8×Dic13C2211yescharacteristic
39C2×C8×Dic13C2211yescharacteristic
40#11718672—24no—
41#11718647C411yescharacteristic
42#11718644C411yescharacteristic
43#11718614C411yescharacteristic
44#11718613C411yescharacteristic
45#11718613C411yescharacteristic
46#11718610C411yescharacteristic
47C4×#6101C2211yescharacteristic
48C4×#6101C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×#88043C2211yescharacteristic
51C13×#336C2211yescharacteristic
52C13×#336C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718142C2211yescharacteristic
60#11718142C2211yescharacteristic
61C22×D4×C13C2×C411yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C22×C8×C13—22no—
65C22×C8×C13—22no—
66C22×C8×C13—22no—
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C22×C8×C13C2311yescharacteristic
71C4×D4×C13—22no—
72C4×D4×C13—22no—
73C2×C13×C4⋊C4C2×C411yescharacteristic
74C2×C13×C22⋊C4C2×C411yescharacteristic
75C2×C13×C22⋊C4C2×C411yescharacteristic
76C2×C42×C13C2311yescharacteristic
77C2×C22⋊Dic13C2×C411yescharacteristic
78C2×C22⋊Dic13C2×C411yescharacteristic
79C2×C22⋊Dic13C2×C411yescharacteristic
80C2×C22⋊Dic13C2×C411yescharacteristic
81C2×C22⋊Dic13C2×C411yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C22⋊Dic13C2×C411yescharacteristic
85#88180—22no—
86#88180—22no—
87C2×C4⋊Dic13C2×C411yescharacteristic
88C2×C4⋊Dic13C2×C411yescharacteristic
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×#6099—22no—
98C2×#6099—22no—
99C2×#6099D414yes—
100C22×#1055—22no—
101C22×#1055—22no—
102C22×#1055—22no—
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C22×#1055C2311yescharacteristic
107C13×C22⋊C8—22no—
108C13×C22⋊C8—22no—
109C13×C22⋊C8—22no—
110#88071C2×C411yescharacteristic
111#88071C2×C411yescharacteristic
112#88071C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118#88071C2×C411yescharacteristic
119#88071C2×C411yescharacteristic
120C22⋊#1055—22no—
121C22⋊#1055—22no—
122#88055—22no—
123#88055—22no—
124#88055—22no—
125#88054—22no—
126#88054—22no—
127C8×Dic13—22no—
128C8×Dic13—22no—
129C8×Dic13—22no—
130C4⋊#1055—22no—
131#88043—22no—
132#21568—131nomaximal, Sylow
133C24×C13C22×C411yescharacteristic
134C24×C13C22×C411yescharacteristic
135C2×D4×C13—22no—
136C2×D4×C13—22no—
137C2×D4×C13—41no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13—21no—
147C22×C4×C13—21no—
148C22×C4×C13C22×C411yescharacteristic
149C22×C4×C13C22×C411yescharacteristic
150C22×C4×C13C22×C411yescharacteristic
151C22×C4×C13C22×C411yescharacteristic
152C22×C4×C13C2×D412yes—
153C22×C4×C13#4112yes—
154C22×C4×C13C2411yescharacteristic
155C22×Dic13C22×C411yescharacteristic
156C22×Dic13C22×C411yescharacteristic
157C22×Dic13C22×C411yescharacteristic
158C22×Dic13C22×C411yescharacteristic
159C22×Dic13C22×C411yescharacteristic
160C22×Dic13C22×C411yescharacteristic
161C22×Dic13C22×C411yescharacteristic
162C22×Dic13C22×C411yescharacteristic
163C2×C8×C13—28no—
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×C13—21no—
174C2×C8×C13#4112yes—
175C2×C8×C13#4112yes—
176C13×C4⋊C4—22no—
177C13×C22⋊C4—22no—
178C13×C22⋊C4—22no—
179C42×C13—21no—
180C42×C13—21no—
181C22⋊Dic13—22no—
182C22⋊Dic13—22no—
183C22⋊Dic13—22no—
184C22⋊Dic13—22no—
185C22⋊Dic13—22no—
186C22⋊Dic13—22no—
187C22⋊Dic13—41no—
188C22⋊Dic13—41no—
189C4⋊Dic13—22no—
190C4⋊Dic13—22no—
191#6101—22no—
192#6101—41no—
193#6101—41no—
194#6101—41no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203#6099—28no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—44no—
213C2×#1055C2×D412yes—
214C2×#1055C2×D412yes—
215C2×#1055#4112yes—
216C2×#1055#4112yes—
217C2×#435—131no—
218C2×#405—131no—
219#2395—131no—
220#1924—131no—
221#1923—131no—
222#1917—131no—
223#1916—131no—
224#1869D1311yescharacteristic
225#1850—131no—
226#1818—131no—
227#1771—131no—
228#1770—131no—
229#1753—131no—
230#1752—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—21no—
239C23×C13—21no—
240C23×C13—21no—
241C23×C13—21no—
242C23×C13—21no—
243C23×C13—21no—
244C23×C13C4×D412yes—
245C23×C13#13112yes—
246C23×C13C23×C411yescharacteristic
247D4×C13—44no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
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—21no—
262C2×C4×C13—21no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13C4×D412yes—
268C2×C4×C13C22×D411yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×C4×C13C2×#4111yescharacteristic
273C2×C4×C13C2×#4111yescharacteristic
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×Dic13—21no—
292C2×Dic13—21no—
293C2×Dic13—21no—
294C2×Dic13—21no—
295C2×Dic13—21no—
296C2×Dic13—21no—
297C2×Dic13#13112yes—
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×C8C2×D1311yescharacteristic
309C2×C4×D4C2×D1311yescharacteristic
310C2×#117—131no—
311#435—264no—
312C2×C4⋊C8—131no—
313C2×C4⋊C8—131no—
314C2×C4⋊C8—131no—
315C2×C22⋊C8—131no—
316C2×C22⋊C8—131no—
317C2×C22⋊C8C2×D1311yescharacteristic
318C2×C22⋊C8C2×D1311yescharacteristic
319C2×C22⋊C8C2×D1311yescharacteristic
320#405—264no—
321C2×#97—131no—
322C2×#97—131no—
323C2×#97—131no—
324C2×#97—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⋊C4—131no—
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—22no—
355C22×C13—22no—
356C22×C13—22no—
357C22×C13—41no—
358C22×C13—41no—
359C22×C13—41no—
360C22×C13—41no—
361C22×C13—41no—
362C22×C13—41no—
363C22×C13—41no—
364C22×C13—41no—
365C22×C13C2×C4×D411yescharacteristic
366C22×C13C4×#4111yescharacteristic
367C22×C13C4×#4111yescharacteristic
368C22×C13C2×#13111yescharacteristic
369C22×C13C2×#13111yescharacteristic
370C22×C13#56811yescharacteristic
371C22×C13#56811yescharacteristic
372C4×C13—22no—
373C4×C13—41no—
374C4×C13—41no—
375C4×C13—41no—
376C4×C13C2×#12311yescharacteristic
377C4×C13#52511yescharacteristic
378C4×C13#52911yescharacteristic
379C4×C13#52911yescharacteristic
380Dic13—22no—
381Dic13—41no—
382Dic13—41no—
383Dic13—41no—
384Dic13—41no—
385Dic13—41no—
386Dic13—41no—
387Dic13—41no—
388C22×D4C4×D1311yescharacteristic
389C23×C4C22×D1311yescharacteristic
390C23×C4C22×D1311yescharacteristic
391C2×M16—134no—
392C2×M16—262no—
393C2×M16—262no—
394C22×C8—22no—
395C22×C8—22no—
396C22×C8—22no—
397C22×C8—131no—
398C22×C8—131no—
399C22×C8—131no—
400C22×C8—131no—
401C22×C8—262no—
402C22×C8—262no—
403C22×C8—262no—
404C22×C8C22×D1311yescharacteristic
405C22×C8C22×D1311yescharacteristic
406C22×C8C22×D1311yescharacteristic
407C22×C8C22×D1311yescharacteristic
408C4×D4—22no—
409C4×D4—22no—
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⋊C4C4×D1311yescharacteristic
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×C42—131no—
433C2×C42C22×D1311yescharacteristic
434C4⋊C8—262no—
435C4⋊C8—262no—
436C4⋊C8—262no—
437C22⋊C8—22no—
438C22⋊C8—22no—
439C22⋊C8—22no—
440C22⋊C8—262no—
441C22⋊C8—262no—
442#97—262no—
443#97—262no—
444#97—262no—
445#97—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#95—131no—
458#95—131no—
459C2×C13—42no—
460C2×C13—42no—
461C2×C13C4×#12311yescharacteristic
462C2×C13#293311yescharacteristic
463C2×C13#293511yescharacteristic
464C2×C13#296611yescharacteristic
465C2×C13#296711yescharacteristic
466C2×C13#296811yescharacteristic
467C2×C13#296811yescharacteristic
468C24C2×C4×D1311yescharacteristic
469C24C2×C4×D1311yescharacteristic
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—41no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—131no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—131no—
487C22×C4—131no—
488C22×C4—131no—
489C22×C4—131no—
490C22×C4—131no—
491C22×C4C2×C4×D1311yescharacteristic
492C22×C4C2×C4×D1311yescharacteristic
493C22×C4C2×C4×D1311yescharacteristic
494C22×C4C2×C4×D1311yescharacteristic
495C22×C4D4×D1312yes—
496C22×C4#612912yes—
497C22×C4C23×D1311yescharacteristic
498M16—268no—
499C2×C8—28no—
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—21no—
510C2×C8—132no—
511C2×C8—132no—
512C2×C8—132no—
513C2×C8—132no—
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×C8#612912yes—
524C2×C8#612912yes—
525C4⋊C4—22no—
526C4⋊C4—262no—
527C4⋊C4—262no—
528C4⋊C4—262no—
529C4⋊C4—521no—
530C4⋊C4—521no—
531C4⋊C4—521no—
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—521no—
541C22⋊C4—521no—
542C42—21no—
543C42—21no—
544C42—261no—
545C42—261no—
546C42—261no—
547C42—261no—
548C42—261no—
549C42—261no—
550C42—261no—
551C42—261no—
552C13#2156811yescharacteristic, complemented, Sylow
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#8813512yes—
566C23#8815512yes—
567C23C22×C4×D1311yescharacteristic
568D4—44no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
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—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—132no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—261no—
597C2×C4—261no—
598C2×C4—261no—
599C2×C4—261no—
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#8813512yes—
613C2×C4C2×D4×D1311yescharacteristic
614C2×C4C2×#612911yescharacteristic
615C2×C4C2×#612911yescharacteristic
616C2×C4C2×#612911yescharacteristic
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—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#8813511yescharacteristic
642C22C4×#612911yescharacteristic
643C22C4×#612911yescharacteristic
644C22C2×#8815511yescharacteristic
645C22#1171945011yescharacteristic
646C22D13×#13111yescharacteristic
647C22#1171945511yescharacteristic
648C4—22no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—262no—
653C4—521no—
654C4—521no—
655C4—521no—
656C4—521no—
657C4—521no—
658C4—521no—
659C4—521no—
660C4#1171929111yescharacteristic
661C4#1171930511yescharacteristic
662C4#1171930511yescharacteristic
663C4D13×#12311yescharacteristic
664C2—42no—
665C2—42no—
666C2#1311736111yescharacteristic
667C2#1312355711yescharacteristic
668C2#1312357611yescharacteristic
669C2#1312380411yescharacteristic
670C2#1312381811yescharacteristic
671C2#1312382411yescharacteristic
672C2#1312382711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC13×#1869C2
prime radicalC22×D4×C13C2×C4
cosocleC22×C4×C13C24
hypercenter#1869D13
hypocenterC13#21568
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C2×#11718762C2semidirect
4. #13115452C2semidirect
5. #13115445C2semidirect
6. #13115403C2semidirect
7. #13115396C2semidirect
8. #13115014C2semidirect
9. #13114905C2semidirect
10. #13114709C2semidirect
11. #13114702C2semidirect
12. #13114622C2semidirect
13. #13114615C2semidirect
14. #13114610C2semidirect
552. C13#21568semidirect

Maximal subgroups

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

subgroupindex
C2×#117186722
C2×#117187622
#131180952
#131154522
#131154452
#131154032
#131153962
#131150142
#131149052
#131147092
#131147022
#131146222
#131146152
#131146102
C13×#18692
#2156813