Finite Group Databasebeta

← #15870466

Subgroups — #15870466

Order 3328.

834 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×#11718362C211yesmaximal, characteristic, complemented
2#13118104C211yesmaximal, characteristic
3#13115772C211yesmaximal, characteristic, complemented
4#13115554C211yesmaximal, characteristic, complemented
5#13115400C211yesmaximal, characteristic, complemented
6#13115400C211yesmaximal, characteristic, complemented
7#13115014C211yesmaximal, characteristic
8#13115014C211yesmaximal, characteristic
9#13114820C211yesmaximal, characteristic
10#13114710C211yesmaximal, characteristic, complemented
11#13114708C211yesmaximal, characteristic, complemented
12C8×#6101C211yesmaximal, characteristic, complemented
13#13114617C211yesmaximal, characteristic
14#13114617C211yesmaximal, characteristic
15C2×C13×#405C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C23×C8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×C4×Dic26C2211yescharacteristic
20C13×#405—24no—
21C2×C13×#97C2211yescharacteristic
22C2×C13×#97C2211yescharacteristic
23C2×C4×C8×C13C2211yescharacteristic
24C2×C4×C8×C13C2211yescharacteristic
25C4×C22⋊Dic13C2211yescharacteristic
26C4×C22⋊Dic13C2211yescharacteristic
27C2×C22⋊#1055C2211yescharacteristic
28C2×C22⋊#1055C2211yescharacteristic
29C2×#88055C2211yescharacteristic
30C2×#88055C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34C2×#88054C2211yescharacteristic
35C2×C8×Dic13C2211yescharacteristic
36C2×C8×Dic13C2211yescharacteristic
37#11718647C411yescharacteristic
38#11718645C411yescharacteristic
39#11718645C411yescharacteristic
40#11718640C411yescharacteristic
41#11718610C411yescharacteristic
42#11718609C411yescharacteristic
43#11718609C411yescharacteristic
44#11718608C411yescharacteristic
45C4×C4⋊Dic13C2211yescharacteristic
46C4×#6101C2211yescharacteristic
47C4×#6101C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50#11718362—24no—
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C22×Dic26C2×C411yescharacteristic
63C2×C13×M16—22no—
64C2×C13×M16—22no—
65C2×C13×M16Q814yescomplemented
66C22×C8×C13—22no—
67C22×C8×C13—22no—
68C22×C8×C13—22no—
69C22×C8×C13C2311yescharacteristic
70C22×C8×C13C2311yescharacteristic
71C22×C8×C13C2311yescharacteristic
72C22×C8×C13C2311yescharacteristic
73C13×#117—22no—
74C13×#117—22no—
75C2×C13×C4⋊C4C2×C411yescharacteristic
76C2×C13×C4⋊C4C2×C411yescharacteristic
77C2×C13×C22⋊C4C2×C411yescharacteristic
78C2×C13×C22⋊C4C2×C411yescharacteristic
79C2×C42×C13C2311yescharacteristic
80C2×C42×C13C2311yescharacteristic
81C2×C22⋊Dic13C2×C411yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C22⋊Dic13C2×C411yescharacteristic
85C2×C4⋊Dic13C2×C411yescharacteristic
86C2×C4⋊Dic13C2×C411yescharacteristic
87C2×C4⋊Dic13C2×C411yescharacteristic
88C2×#6101C2×C411yescharacteristic
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×#6101C2×C411yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C4×Dic26—22no—
103C4×Dic26—22no—
104C13×#97—22no—
105C13×#97D414yes—
106C4×C8×C13—22no—
107C4×C8×C13—22no—
108#88071C2×C411yescharacteristic
109#88071C2×C411yescharacteristic
110#88071C2×C411yescharacteristic
111#88071C2×C411yescharacteristic
112#88071C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118C22⋊#1055—22no—
119C22⋊#1055—22no—
120#88055—22no—
121#88055—22no—
122#88054—22no—
123#88054—22no—
124#88054—22no—
125#88054—22no—
126C8×Dic13—22no—
127C8×Dic13—22no—
128C4⋊#1055—22no—
129C4⋊#1055—22no—
130#21606—131nomaximal, Sylow
131C24×C13C22×C411yescharacteristic
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13C22×C411yescharacteristic
137C22×C4×C13C22×C411yescharacteristic
138C22×C4×C13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×C4×C13C2×Q812yes—
142C22×C4×C13#4112yes—
143C22×C4×C13C2411yescharacteristic
144C22×Dic13C22×C411yescharacteristic
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C2×Dic26—22no—
153C2×Dic26—22no—
154C2×Dic26—41no—
155C13×M16—28no—
156C2×C8×C13—28no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13C2×D412yes—
164C2×C8×C13C2×D412yes—
165C2×C8×C13C2×Q812yes—
166C2×C8×C13C2×Q812yes—
167C2×C8×C13#4112yes—
168C2×C8×C13#4112yes—
169C13×C4⋊C4—22no—
170C13×C4⋊C4—22no—
171C13×C22⋊C4—22no—
172C13×C22⋊C4—22no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13C2×D412yes—
176C42×C13#4112yes—
177C22⋊Dic13—22no—
178C22⋊Dic13—22no—
179C22⋊Dic13—41no—
180C22⋊Dic13—41no—
181C4⋊Dic13—22no—
182C4⋊Dic13—22no—
183C4⋊Dic13—41no—
184#6101—22no—
185#6101—22no—
186#6101—22no—
187#6101—22no—
188#6101—41no—
189#6101—41no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—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×#446—131no—
211C2×#405D1311yescharacteristic
212#2396—131no—
213#1984—131no—
214#1939—131no—
215#1917—131no—
216#1917—131no—
217#1850—131no—
218#1850—131no—
219#1798—131no—
220#1771—131no—
221#1771—131no—
222C8×C4⋊C4—131no—
223#1753—131no—
224#1753—131no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13C4×Q812yes—
232C23×C13C23×C411yescharacteristic
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13C4×D412yes—
250C2×C4×C13C4×D412yes—
251C2×C4×C13C4×Q812yes—
252C2×C4×C13C22×D411yescharacteristic
253C2×C4×C13C22×Q811yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13#13112yes—
281C2×Dic13#13112yes—
282Dic26—44no—
283C8×C13—22no—
284C8×C13—22no—
285C8×C13—22no—
286C8×C13C22⋊Q814yes—
287#1055—41no—
288#1055—41no—
289#1055—41no—
290#1055—41no—
291C22×M16C2×D1311yescharacteristic
292C23×C8C2×D1311yescharacteristic
293C2×C4×Q8—131no—
294C2×#117C2×D1311yescharacteristic
295#446—264no—
296C2×C4⋊C8—131no—
297C2×C4⋊C8—131no—
298C2×C4⋊C8—131no—
299C2×C4⋊C8—131no—
300C2×C4⋊C8—131no—
301C2×C4⋊C8—131no—
302C2×C22⋊C8—131no—
303C2×C22⋊C8—131no—
304#405—24no—
305C2×#97—131no—
306C2×#97—131no—
307C2×#97C2×D1311yescharacteristic
308C2×#97C2×D1311yescharacteristic
309C2×C4×C8—131no—
310C2×C4×C8—131no—
311C2×C4×C8C2×D1311yescharacteristic
312C2×C4×C8C2×D1311yescharacteristic
313#400—131no—
314#399—131no—
315#398—131no—
316#398—131no—
317#396—131no—
318#396—131no—
319#395—131no—
320#393—131no—
321C4×C4⋊C4—131no—
322C4×C4⋊C4—131no—
323C4×C4⋊C4—131no—
324C4×C22⋊C4—131no—
325C4×C22⋊C4—131no—
326#336—131no—
327#336—131no—
328#336—131no—
329#336—131no—
330#336—131no—
331#336—131no—
332#336—131no—
333#336—131no—
334#336—131no—
335#336—131no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13—41no—
342C22×C13—41no—
343C22×C13C2×C4×D411yescharacteristic
344C22×C13C2×C4×Q811yescharacteristic
345C22×C13C4×#4111yescharacteristic
346C22×C13C2×#13111yescharacteristic
347C22×C13C2×#13111yescharacteristic
348C22×C13#56811yescharacteristic
349C22×C13#56811yescharacteristic
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—22no—
353C4×C13—41no—
354C4×C13—41no—
355C4×C13C2×C22⋊Q811yescharacteristic
356C4×C13#52511yescharacteristic
357C4×C13#52911yescharacteristic
358C4×C13#53311yescharacteristic
359Dic13—22no—
360Dic13—22no—
361Dic13—41no—
362Dic13—41no—
363Dic13—41no—
364Dic13—41no—
365Dic13—41no—
366Dic13—41no—
367C22×Q8—131no—
368C23×C4C22×D1311yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16Dic2614yescomplemented
372C22×C8—22no—
373C22×C8—22no—
374C22×C8—22no—
375C22×C8—131no—
376C22×C8—131no—
377C22×C8—131no—
378C22×C8—131no—
379C22×C8C22×D1311yescharacteristic
380C22×C8C22×D1311yescharacteristic
381C22×C8C22×D1311yescharacteristic
382C22×C8C22×D1311yescharacteristic
383C4×Q8—262no—
384C4×Q8—262no—
385#117—22no—
386#117—22no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4C4×D1311yescharacteristic
397C2×C4⋊C4C4×D1311yescharacteristic
398C2×C22⋊C4—131no—
399C2×C22⋊C4—131no—
400C2×C22⋊C4—131no—
401C2×C22⋊C4—131no—
402C2×C22⋊C4C4×D1311yescharacteristic
403C2×C22⋊C4C4×D1311yescharacteristic
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—131no—
407C2×C42—131no—
408C2×C42C22×D1311yescharacteristic
409C2×C42C22×D1311yescharacteristic
410C4⋊C8—262no—
411C4⋊C8—262no—
412C4⋊C8—262no—
413C4⋊C8—262no—
414C4⋊C8—262no—
415C4⋊C8—262no—
416C22⋊C8—262no—
417C22⋊C8—262no—
418#97—22no—
419#97—262no—
420#97—262no—
421#97C22⋊D1314yes—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—262no—
425C4×C8—262no—
426#95—131no—
427#95—131no—
428#95—131no—
429#95—131no—
430#95—131no—
431#95—131no—
432#95—131no—
433#95—131no—
434#95—131no—
435#95—131no—
436C2×C13—42no—
437C2×C13C4×C22⋊Q811yescharacteristic
438C2×C13#293311yescharacteristic
439C2×C13#293511yescharacteristic
440C2×C13#296211yescharacteristic
441C2×C13#296211yescharacteristic
442C2×C13#296611yescharacteristic
443C2×C13#296711yescharacteristic
444C24C2×C4×D1311yescharacteristic
445C2×Q8—262no—
446C2×Q8—262no—
447C2×Q8—521no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—131no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—131no—
459C22×C4—131no—
460C22×C4C2×Dic2612yes—
461C22×C4C2×C4×D1311yescharacteristic
462C22×C4C2×C4×D1311yescharacteristic
463C22×C4C2×C4×D1311yescharacteristic
464C22×C4C2×C4×D1311yescharacteristic
465C22×C4C2×C4×D1311yescharacteristic
466C22×C4#612712yes—
467C22×C4C23×D1311yescharacteristic
468M16—28no—
469C2×C8—28no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8C2×Dic2612yes—
489C2×C8C2×Dic2612yes—
490C2×C8#612712yes—
491C2×C8#612712yes—
492C2×C8C2×C22⋊D1312yes—
493C2×C8C2×C22⋊D1312yes—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—262no—
497C4⋊C4—262no—
498C4⋊C4—262no—
499C4⋊C4—262no—
500C4⋊C4—262no—
501C4⋊C4—262no—
502C4⋊C4—521no—
503C4⋊C4—521no—
504C4⋊C4—521no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—262no—
508C22⋊C4—262no—
509C22⋊C4—521no—
510C22⋊C4—521no—
511C42—21no—
512C42—21no—
513C42—261no—
514C42—261no—
515C42—261no—
516C42—261no—
517C42—261no—
518C42—261no—
519C42—261no—
520C42—261no—
521C42#612712yes—
522C42C2×C22⋊D1312yes—
523C13#2160611yescharacteristic, complemented, Sylow
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C4×Dic2612yes—
531C23C22×C4×D1311yescharacteristic
532Q8—524no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—132no—
550C2×C4—132no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—261no—
569C2×C4—261no—
570C2×C4—261no—
571C2×C4—261no—
572C2×C4—261no—
573C2×C4C4×Dic2612yes—
574C2×C4C4×C22⋊D1312yes—
575C2×C4C4×C22⋊D1312yes—
576C2×C4C22×Dic2611yescharacteristic
577C2×C4C2×#612711yescharacteristic
578C2×C4C2×#612711yescharacteristic
579C2×C4C2×#612711yescharacteristic
580C2×C4C2×#612711yescharacteristic
581C2×C4C22×C22⋊D1311yescharacteristic
582C8—22no—
583C8—22no—
584C8—22no—
585C8—521no—
586C8—521no—
587C8—521no—
588C8—521no—
589C8#8817814yes—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×C4×Dic2611yescharacteristic
598C22C4×#612711yescharacteristic
599C22D13×#13111yescharacteristic
600C22D13×#13111yescharacteristic
601C22#1171945511yescharacteristic
602C22#1171945511yescharacteristic
603C22C2×C4×C22⋊D1311yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—41no—
608C4—41no—
609C4—262no—
610C4—262no—
611C4—521no—
612C4—521no—
613C4—521no—
614C4—521no—
615C4—521no—
616C4—521no—
617C4#1171917511yescharacteristic
618C4#1171918411yescharacteristic
619C4C2×#8817811yescharacteristic
620C4#1171949611yescharacteristic
621C2—42no—
622C2C4×#8817811yescharacteristic
623C2#1312355811yescharacteristic
624C2#1312357811yescharacteristic
625C2#1312378011yescharacteristic
626C2#1312378011yescharacteristic
627C2#1312380711yescharacteristic
628C2#1312381711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC2×C13×#405C2
prime radicalC24×C13C22×C4
cosocleC22×C4×C13C24
hypercenterC2×#405D13
hypocenterC13#21606
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×#11718362C2semidirect
3. #13115772C2semidirect
4. #13115554C2semidirect
5. #13115400C2semidirect
6. #13115400C2semidirect
10. #13114710C2semidirect
11. #13114708C2semidirect
12. C8×#6101C2semidirect
65. C2×C13×M16Q8semidirect
371. C2×M16Dic26semidirect
523. C13#21606semidirect

Maximal subgroups

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

subgroupindex
C2×#117183622
#131181042
#131157722
#131155542
#131154002
#131150142
#131148202
#131147102
#131147082
C8×#61012
#131146172
C2×C13×#4052
#2160613