Finite Group Databasebeta

← #15791415

Subgroups — #15791415

Order 3328.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718642C212yesmaximal, complemented
2#13113107C212yesmaximal, complemented
3#13113104C212yesmaximal, complemented
4#13113081C212yesmaximal
5#13113073C212yesmaximal
6#13113065C212yesmaximal, complemented
7#13113059C211yesmaximal, characteristic
8#13113059C211yesmaximal, characteristic
9C22×C13×C22⋊Q8C211yesmaximal, characteristic
10C23×Q8×C13C2211yescharacteristic
11C24×C4×C13C2211yescharacteristic
12C2×C13×C22⋊Q8—28no—
13C2×C13×C22⋊Q8C418yescomplemented
14C22×C13×C4⋊C4C2212yes—
15C22×C13×C4⋊C4C2211yescharacteristic
16C22×C13×C22⋊C4C2212yes—
17C22×C22⋊Dic13C2212yes—
18C22×C22⋊Dic13C2211yescharacteristic
19C22×C22⋊Dic13C2211yescharacteristic
20C22×C4⋊Dic13C2212yes—
21C22×C4×Dic13C2212yes—
22C2×#88071C2212yes—
23C2×#88071C2212yes—
24C2×#88071C2212yes—
25C2×#88071C2212yes—
26C2×#88071C2211yescharacteristic
27C2×#88071C2212yes—
28C2×#88071C2212yes—
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2211yescharacteristic
33#11718642—28no—
34C25×C13C2311yescharacteristic
35C22×Q8×C13—24no—
36C22×Q8×C13—41no—
37C22×Q8×C13—41no—
38C22×Q8×C13—42no—
39C22×Q8×C13C2×C414yes—
40C23×C4×C13—24no—
41C23×C4×C13—41no—
42C23×C4×C13—41no—
43C23×C4×C13—42no—
44C23×C4×C13C2×C414yes—
45C23×C4×C13C2312yes—
46C23×C4×C13C2311yescharacteristic
47C23×C4×C13C2312yes—
48C23×C4×C13C2311yescharacteristic
49C23×Dic13C2312yes—
50C23×Dic13C2312yes—
51C23×Dic13C2312yes—
52C23×Dic13C2312yes—
53C13×C22⋊Q8—232no—
54C2×C13×C4⋊C4—24no—
55C2×C13×C4⋊C4—24no—
56C2×C13×C4⋊C4C2×C418yes—
57C2×C13×C4⋊C4C2×C414yes—
58C2×C13×C4⋊C4D418yes—
59C2×C13×C22⋊C4—24no—
60C2×C13×C22⋊C4—24no—
61C2×C13×C22⋊C4C2×C418yes—
62C2×C22⋊Dic13—24no—
63C2×C22⋊Dic13—22no—
64C2×C22⋊Dic13—24no—
65C2×C22⋊Dic13—41no—
66C2×C22⋊Dic13—41no—
67C2×C22⋊Dic13—42no—
68C2×C22⋊Dic13—41no—
69C2×C22⋊Dic13—41no—
70C2×C22⋊Dic13Q814yescomplemented
71C2×C4⋊Dic13—24no—
72C2×C4⋊Dic13—24no—
73C2×C4⋊Dic13—42no—
74C2×C4×Dic13—24no—
75C2×C4×Dic13—24no—
76C2×C4×Dic13—42no—
77#88071—24no—
78#88071—24no—
79#88071—24no—
80#88071—24no—
81#88071—41no—
82#88071—42no—
83#88071—42no—
84#88071—42no—
85#88071—42no—
86#88071—41no—
87#88071—42no—
88#14448—131nomaximal, Sylow
89C24×C13—22no—
90C24×C13—22no—
91C24×C13—21no—
92C24×C13—21no—
93C24×C13—21no—
94C24×C13—22no—
95C24×C13—22no—
96C24×C13—21no—
97C24×C13C22×C412yes—
98C24×C13C2×Q812yes—
99C24×C13#4112yes—
100C24×C13C2411yescharacteristic
101C2×Q8×C13—28no—
102C2×Q8×C13—44no—
103C2×Q8×C13—44no—
104C2×Q8×C13—48no—
105C2×Q8×C13—48no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—21no—
110C22×C4×C13—21no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—21no—
117C22×C4×C13—21no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—28no—
124C22×C4×C13—44no—
125C22×C4×C13—44no—
126C22×C4×C13—48no—
127C22×C4×C13—48no—
128C22×C4×C13C22×C414yes—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C22×C414yes—
131C22×C4×C13C22×C412yes—
132C22×C4×C13C2×D414yes—
133C22×C4×C13C2×D414yes—
134C22×C4×C13C2×D414yes—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—22no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—22no—
160C22×Dic13—22no—
161C22×Dic13C2×Q814yes—
162C22×Dic13#4114yes—
163C13×C4⋊C4—216no—
164C13×C4⋊C4C22⋊C4116yescomplemented
165C13×C22⋊C4—216no—
166C22⋊Dic13—44no—
167C22⋊Dic13—48no—
168C22⋊Dic13—44no—
169C4⋊Dic13—48no—
170C4×Dic13—48no—
171C22×C22⋊Q8D1311yescharacteristic
172C2×#387—132no—
173#1446—132no—
174#1445—132no—
175#1439—132no—
176#1437—132no—
177C4⋊#95—132no—
178C22⋊#95—131no—
179C22⋊#95—131no—
180C23×C13—22no—
181C23×C13—22no—
182C23×C13—22no—
183C23×C13—22no—
184C23×C13—22no—
185C23×C13—22no—
186C23×C13—22no—
187C23×C13—22no—
188C23×C13—22no—
189C23×C13—22no—
190C23×C13—24no—
191C23×C13—24no—
192C23×C13—24no—
193C23×C13—24no—
194C23×C13—42no—
195C23×C13—42no—
196C23×C13—42no—
197C23×C13—42no—
198C23×C13—42no—
199C23×C13—42no—
200C23×C13—42no—
201C23×C13—42no—
202C23×C13C4×Q814yes—
203C23×C13C23×C411yescharacteristic
204C23×C13C22×D412yes—
205C23×C13C22×Q811yescharacteristic
206C23×C13C2×#4111yescharacteristic
207C23×C13C2×#4112yes—
208C23×C13H(5,𝔽2)11yescharacteristic
209C23×C13H(5,𝔽2)11yescharacteristic
210C23×C13H(5,𝔽2)12yes—
211C23×C13H(5,𝔽2)11yescharacteristic
212C23×C132−1+412yes—
213C23×C132−1+411yescharacteristic
214Q8×C13—416no—
215C2×C4×C13—24no—
216C2×C4×C13—22no—
217C2×C4×C13—22no—
218C2×C4×C13—24no—
219C2×C4×C13—24no—
220C2×C4×C13—24no—
221C2×C4×C13—24no—
222C2×C4×C13—24no—
223C2×C4×C13—24no—
224C2×C4×C13—24no—
225C2×C4×C13—24no—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—24no—
230C2×C4×C13—24no—
231C2×C4×C13—416no—
232C2×C4×C13—42no—
233C2×C4×C13—42no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13C2×C22⋊C418yes—
239C2×C4×C13C2×C22⋊C414yes—
240C2×Dic13—24no—
241C2×Dic13—24no—
242C2×Dic13—24no—
243C2×Dic13—24no—
244C2×Dic13—24no—
245C2×Dic13—24no—
246C2×Dic13—42no—
247C2×Dic13—42no—
248C2×Dic13—42no—
249C2×Dic13—42no—
250C2×Dic13—42no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—42no—
254C2×Dic13—42no—
255C2×Dic13—42no—
256C2×Dic13—42no—
257C2×Dic13—42no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268C23×Q8C2×D1311yescharacteristic
269C24×C4C2×D1311yescharacteristic
270C2×C22⋊Q8—28no—
271C2×C22⋊Q8Dic1318yescomplemented
272C22×C4⋊C4—132no—
273C22×C4⋊C4C2×D1312yes—
274C22×C4⋊C4C2×D1311yescharacteristic
275C22×C22⋊C4—131no—
276C22×C22⋊C4—132no—
277C22×C22⋊C4—131no—
278C22×C22⋊C4C2×D1312yes—
279C22×C42—132no—
280#387—268no—
281C2×#95—132no—
282C2×#95—132no—
283C2×#95—132no—
284C2×#95—132no—
285C2×#95—131no—
286C2×#95—132no—
287C2×#95—131no—
288C2×#95—132no—
289C2×#95—132no—
290C2×#95—132no—
291C2×#95—132no—
292C22×C13—24no—
293C22×C13—24no—
294C22×C13—24no—
295C22×C13—42no—
296C22×C13—42no—
297C22×C13—42no—
298C22×C13—42no—
299C22×C13—44no—
300C22×C13—44no—
301C22×C13—44no—
302C22×C13—44no—
303C22×C13C22×C22⋊C411yescharacteristic
304C22×C13C2×C4×Q811yescharacteristic
305C22×C13C4×#4111yescharacteristic
306C22×C13#51811yescharacteristic
307C22×C13#51811yescharacteristic
308C22×C13#51911yescharacteristic
309C22×C13#52011yescharacteristic
310C22×C13#53512yes—
311C22×C13#53512yes—
312C22×C13#53811yescharacteristic
313C22×C13#53912yes—
314C22×C13#53911yescharacteristic
315C22×C13#54212yes—
316C22×C13#54311yescharacteristic
317C22×C13#54411yescharacteristic
318C22×C13#54612yes—
319C22×C13#54712yes—
320C22×C13#54812yes—
321C22×C13D4×Q812yes—
322C22×C13#55112yes—
323C22×C13#55412yes—
324C22×C13#56212yes—
325C22×C13#56312yes—
326C4×C13—24no—
327C4×C13—28no—
328C4×C13—42no—
329C4×C13—44no—
330Dic13—44no—
331Dic13—44no—
332Dic13—82no—
333Dic13—82no—
334C25C22×D1311yescharacteristic
335C22×Q8—24no—
336C22×Q8—41no—
337C22×Q8—41no—
338C22×Q8—42no—
339C22×Q8C2×Dic1314yes—
340C23×C4—24no—
341C23×C4—41no—
342C23×C4—41no—
343C23×C4—42no—
344C23×C4—132no—
345C23×C4—132no—
346C23×C4—132no—
347C23×C4—132no—
348C23×C4C2×Dic1314yes—
349C23×C4C22×D1312yes—
350C23×C4C22×D1311yescharacteristic
351C23×C4C22×D1312yes—
352C23×C4C22×D1311yescharacteristic
353C22⋊Q8—232no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—264no—
357C2×C4⋊C4—264no—
358C2×C4⋊C4—522no—
359C2×C4⋊C4C2×Dic1318yes—
360C2×C4⋊C4C2×Dic1314yes—
361C2×C4⋊C4C22⋊D1318yes—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—134no—
365C2×C22⋊C4—264no—
366C2×C22⋊C4—264no—
367C2×C22⋊C4—262no—
368C2×C22⋊C4—521no—
369C2×C22⋊C4—521no—
370C2×C22⋊C4—522no—
371C2×C22⋊C4—521no—
372C2×C22⋊C4—521no—
373C2×C22⋊C4C2×Dic1318yes—
374C2×C42—264no—
375C2×C42—264no—
376C2×C42—522no—
377#95—264no—
378#95—264no—
379#95—264no—
380#95—264no—
381#95—521no—
382#95—522no—
383#95—521no—
384#95—522no—
385#95—522no—
386#95—522no—
387#95—522no—
388C2×C13—44no—
389C2×C13#230911yescharacteristic
390C2×C13#232911yescharacteristic
391C2×C13#233211yescharacteristic
392C2×C13#233911yescharacteristic
393C2×C13Q8×C22⋊C411yescharacteristic
394C2×C13#236811yescharacteristic
395C2×C13#237811yescharacteristic
396C2×C13#269212yes—
397C2×C13#269612yes—
398C2×C13#272012yes—
399C2×C13#272312yes—
400C24—21no—
401C24—21no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—21no—
406C24—21no—
407C24—22no—
408C24Q8×D1312yes—
409C24#613112yes—
410C24C22×Dic1312yes—
411C24C23×D1311yescharacteristic
412C2×Q8—28no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—48no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—134no—
440C22×C4—134no—
441C22×C4—262no—
442C22×C4—262no—
443C22×C4—262no—
444C22×C4—262no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4—262no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—262no—
453C22×C4—262no—
454C22×C4—262no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—262no—
465C22×C4—262no—
466C22×C4—262no—
467C22×C4C22×Dic1314yes—
468C22×C4C22×Dic1312yes—
469C22×C4C22×Dic1314yes—
470C22×C4C22×Dic1312yes—
471C22×C4C2×C22⋊D1314yes—
472C22×C4C2×C22⋊D1314yes—
473C22×C4C2×C22⋊D1314yes—
474C4⋊C4—216no—
475C4⋊C4—528no—
476C4⋊C4C22⋊Dic13116yescomplemented
477C22⋊C4—216no—
478C22⋊C4—524no—
479C22⋊C4—528no—
480C22⋊C4—524no—
481C42—528no—
482C13#1444811yescharacteristic, complemented, Sylow
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23Q8×Dic1314yes—
506C23#8825111yescharacteristic
507C23#8825111yescharacteristic
508C23#8825112yes—
509C23C2×Q8×D1311yescharacteristic
510C23C2×#613111yescharacteristic
511C23#8825412yes—
512C23#41×D1312yes—
513C23#8825611yescharacteristic
514C23#8825711yescharacteristic
515C23C23×Dic1311yescharacteristic
516C23C22×C22⋊D1312yes—
517Q8—416no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—264no—
542C2×C4—264no—
543C2×C4—264no—
544C2×C4—264no—
545C2×C4—264no—
546C2×C4—264no—
547C2×C4—522no—
548C2×C4—522no—
549C2×C4—522no—
550C2×C4—522no—
551C2×C4—522no—
552C2×C4—522no—
553C2×C4—522no—
554C2×C4—522no—
555C2×C4—522no—
556C2×C4—522no—
557C2×C4—522no—
558C2×C4—522no—
559C2×C4—522no—
560C2×C4—522no—
561C2×C4—522no—
562C2×C4—522no—
563C2×C4—522no—
564C2×C4—522no—
565C2×C4—522no—
566C2×C4—522no—
567C2×C4—522no—
568C2×C4—522no—
569C2×C4C2×C22⋊Dic1318yes—
570C2×C4C2×C22⋊Dic1314yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22#1171932811yescharacteristic
583C22#1171933911yescharacteristic
584C22#1171934112yes—
585C22#1171934212yes—
586C22#1171934312yes—
587C22#1171934512yes—
588C22#1171934911yescharacteristic
589C22#1171935011yescharacteristic
590C22#1171935112yes—
591C22#1171935212yes—
592C22#1171950311yescharacteristic
593C22#1171950311yescharacteristic
594C22#1171951012yes—
595C22#1171951012yes—
596C22C2×Q8×Dic1311yescharacteristic
597C22#1171951711yescharacteristic
598C22Q8×C22⋊D1312yes—
599C22#1171952212yes—
600C22#1171952912yes—
601C22#41×Dic1311yescharacteristic
602C22#1171953211yescharacteristic
603C22#1171953412yes—
604C22C22×C22⋊Dic1311yescharacteristic
605C4—28no—
606C4—24no—
607C4—42no—
608C4—44no—
609C4—524no—
610C4—524no—
611C4—1042no—
612C4—1042no—
613C2—44no—
614C2#1311741811yescharacteristic
615C2#1311757111yescharacteristic
616C2#1311759411yescharacteristic
617C2#1311766611yescharacteristic
618C2Q8×C22⋊Dic1311yescharacteristic
619C2#1311792111yescharacteristic
620C2#1311800211yescharacteristic
621C2#1312132612yes—
622C2#1312138012yes—
623C2#1312171012yes—
624C2#1312174912yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC22×C13×C22⋊Q8C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC22×C22⋊Q8D13
hypocenterC13#14448
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11718642C2semidirect
2. #13113107C2semidirect
3. #13113104C2semidirect
6. #13113065C2semidirect
13. C2×C13×C22⋊Q8C4semidirect
70. C2×C22⋊Dic13Q8semidirect
164. C13×C4⋊C4C22⋊C4semidirect
271. C2×C22⋊Q8Dic13semidirect
476. C4⋊C4C22⋊Dic13semidirect
482. C13#14448semidirect

Maximal subgroups

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

subgroupindex
C2×#117186422
#131131072
#131131042
#131130812
#131130732
#131130652
#131130592
C22×C13×C22⋊Q82
#1444813