Finite Group Databasebeta

← #15864623

Subgroups — #15864623

Order 3328.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719496C211yesmaximal, characteristic
2C2×#11718794C212yesmaximal, complemented
3C2×#11718873C211yesmaximal, characteristic, complemented
4#13115274C212yesmaximal, complemented
5#13115269C212yesmaximal, complemented
6#13115264C212yesmaximal, complemented
7#13115259C212yesmaximal, complemented
8#13114556C212yesmaximal, complemented
9C23×C13×M16C211yesmaximal, characteristic, complemented
10C24×C4×C13C2211yescharacteristic
11C22×#6127C2211yescharacteristic
12C22×C13×M16—24no—
13C22×C13×M16—28no—
14C22×C13×M16C2212yes—
15C22×C13×M16C2212yes—
16C23×C8×C13C2212yes—
17C2×#88207—21no—
18C22×C22⋊Dic13—21no—
19#11719496—24no—
20#11719496C418yescomplemented
21C2×#88184C2211yescharacteristic
22C2×#88184C2211yescharacteristic
23C2×#88183—21no—
24C2×C4×C22⋊D13—21no—
25C2×C4×C22⋊D13—21no—
26C2×#88180C2211yescharacteristic
27C22×C4⋊Dic13C2212yes—
28C2×#88178C2211yescharacteristic
29C2×#88178C2211yescharacteristic
30#11718873—21no—
31#11718873—21no—
32#11718873C2212yes—
33#11718873C2212yes—
34C2×#88071—21no—
35#11718794—28no—
36C2×#88061C2214yescomplemented
37C2×#88056C2214yescomplemented
38#11718143C2214yescomplemented
39#11718143C2214yescomplemented
40C25×C13D411yescharacteristic
41C23×C4×C13—24no—
42C23×C4×C13—24no—
43C23×C4×C13C2×C414yes—
44C23×C4×C13D411yescharacteristic
45C23×C4×C13D411yescharacteristic
46C23×C4×C13D411yescharacteristic
47C23×C4×C13D414yes—
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2311yescharacteristic
50C23×C4×C13C2311yescharacteristic
51C22×C22⋊D13—21no—
52C22×C22⋊D13—21no—
53C22×C22⋊D13—21no—
54C23×Dic13—22no—
55C2×#6127—22no—
56C2×#6127—24no—
57C2×#6127—24no—
58C2×#6127C2×C414yes—
59C22×D52C2311yescharacteristic
60C22×C4×D13—21no—
61C22×Dic26C2311yescharacteristic
62C2×C13×M16—232no—
63C2×C13×M16—24no—
64C2×C13×M16—24no—
65C2×C13×M16—24no—
66C2×C13×M16—28no—
67C2×C13×M16D418yescomplemented
68C22×C8×C13—24no—
69C22×C8×C13—28no—
70C22×C8×C13C2314yes—
71#88207—28no—
72C2×C22⋊Dic13—21no—
73C2×C22⋊Dic13—21no—
74C2×C22⋊Dic13—21no—
75C2×C22⋊Dic13—22no—
76C2×C22⋊Dic13—22no—
77C2×C22⋊Dic13—44no—
78#88184—22no—
79#88184—22no—
80#88184C2×C414yescomplemented
81#88184C2×C414yescomplemented
82#88183—28no—
83C4×C22⋊D13—28no—
84C4×C22⋊D13—28no—
85C2×#6103—21no—
86C2×#6103—21no—
87#88180—22no—
88#88180C2×C414yes—
89C2×C4⋊Dic13—24no—
90C2×C4⋊Dic13—24no—
91C2×C4⋊Dic13—22no—
92C2×C4⋊Dic13C2314yes—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95#88178—22no—
96#88178—22no—
97#88178C2×C414yescomplemented
98#88178C2×C414yescomplemented
99C2×#6101—21no—
100C2×#6101—21no—
101C2×C4×Dic13—21no—
102#88071—22no—
103#88071—22no—
104#88061—28no—
105#88056—28no—
106#20971—131nomaximal, Sylow
107C24×C13—24no—
108C24×C13—22no—
109C24×C13—22no—
110C24×C13—42no—
111C24×C13C22⋊C414yes—
112C24×C13C2×D411yescharacteristic
113C24×C13C2×D411yescharacteristic
114C24×C13C2×D411yescharacteristic
115C23×D13—21no—
116C22×C4×C13—24no—
117C22×C4×C13—216no—
118C22×C4×C13—28no—
119C22×C4×C13—24no—
120C22×C4×C13—24no—
121C22×C4×C13—28no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13C22⋊C414yes—
127C22×C4×C13C22⋊C414yes—
128C22×C4×C13C22⋊C414yes—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C22×C412yes—
131C22×C4×C13C22×C412yes—
132C22×C4×C13C2×D411yescharacteristic
133C22×C4×C13C2×D411yescharacteristic
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D412yes—
136C22×C4×C13C2×D412yes—
137C22×C4×C13C2×D412yes—
138C22×C4×C13C2×D414yes—
139C22×C4×C13#4114yes—
140C22×C4×C13C2411yescharacteristic
141C2×C22⋊D13—24no—
142C2×C22⋊D13—24no—
143C2×C22⋊D13—24no—
144C2×C22⋊D13—44no—
145C2×C22⋊D13—44no—
146C2×C22⋊D13—42no—
147C2×C22⋊D13—42no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13—24no—
152C22×Dic13—44no—
153C22×Dic13—44no—
154C22×Dic13—44no—
155#6127—216no—
156#6127—44no—
157#6127—44no—
158C2×D52—22no—
159C2×D52—22no—
160C2×D52—21no—
161C2×D52C22×C412yes—
162C2×C4×D13—24no—
163C2×C4×D13—44no—
164C2×Dic26—22no—
165C2×Dic26—22no—
166C2×Dic26—21no—
167C2×Dic26C22×C412yes—
168C13×M16—216no—
169C13×M16—216no—
170C2×C8×C13—216no—
171C2×C8×C13—24no—
172C2×C8×C13—24no—
173C2×C8×C13C2×D418yes—
174C22⋊Dic13—24no—
175C22⋊Dic13—24no—
176C22⋊Dic13—24no—
177C22⋊Dic13—44no—
178C22⋊Dic13—44no—
179#6103—24no—
180#6103—24no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—24no—
184C4⋊Dic13—24no—
185C4⋊Dic13—44no—
186C4⋊Dic13C22×C412yes—
187C4⋊Dic13C22×C412yes—
188#6101—24no—
189#6101—24no—
190C4×Dic13—24no—
191C23×M16D1311yescharacteristic, complemented
192C2×#525—131no—
193C2×#417—132no—
194C2×#380—131no—
195#1895—132no—
196#1894—132no—
197#1893—132no—
198#1892—132no—
199#1736—132no—
200C23×C13—22no—
201C23×C13—21no—
202C23×C13—21no—
203C23×C13—22no—
204C23×C13—22no—
205C23×C13—22no—
206C23×C13—21no—
207C23×C13—21no—
208C23×C13—22no—
209C23×C13—28no—
210C23×C13—28no—
211C23×C13—44no—
212C23×C13—48no—
213C23×C13—48no—
214C23×C13C2×C22⋊C412yes—
215C23×C13C2×C22⋊C412yes—
216C23×C13C2×C22⋊C412yes—
217C23×C13#12114yes—
218C23×C13AGL(1,ℤ/8)12yes—
219C23×C13#13712yes—
220C23×C13C22×D411yescharacteristic
221C22×D13—22no—
222C22×D13—42no—
223C22×D13—41no—
224C22×D13—41no—
225C22×D13—42no—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—28no—
229C2×C4×C13—24no—
230C2×C4×C13—24no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—28no—
236C2×C4×C13—28no—
237C2×C4×C13C2×C22⋊C412yes—
238C2×C4×C13C2×C22⋊C412yes—
239C2×C4×C13C2×C22⋊C412yes—
240C2×C4×C13C4×D418yes—
241C2×C4×C13#12014yes—
242C2×C4×C13#12114yes—
243C2×C4×C13#12314yes—
244C2×C4×C13C23×C411yescharacteristic
245C2×C4×C13C22×D412yes—
246C2×C4×C13C22×D411yescharacteristic
247C2×C4×C13C2×#4112yes—
248C22⋊D13—48no—
249C22⋊D13—48no—
250C22⋊D13—48no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—42no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—42no—
258C2×Dic13—44no—
259C2×Dic13—44no—
260C2×Dic13—44no—
261C2×Dic13—88no—
262D52—24no—
263D52—41no—
264D52—41no—
265C4×D13—48no—
266Dic26—24no—
267Dic26—41no—
268Dic26—41no—
269C8×C13—28no—
270C22×#41—131no—
271C24×C4C2×D1311yescharacteristic
272C22×M16—24no—
273C22×M16—28no—
274C22×M16C2×D1312yes—
275C22×M16C2×D1312yes—
276C23×C8C2×D1312yes—
277#525—138no—
278#525—264no—
279C2×#123—261no—
280C2×C22⋊Q8—131no—
281C2×C22⋊Q8—131no—
282C2×#121—131no—
283C2×#121—131no—
284C2×#120—261no—
285C2×C4×D4—261no—
286C2×C4×D4—261no—
287C2×#117—131no—
288C22×C4⋊C4—132no—
289C22×C22⋊C4—261no—
290#417—268no—
291C2×#103—134no—
292C2×#102—134no—
293#380—132no—
294#380—132no—
295#380—261no—
296#380—261no—
297C2×#95—261no—
298#340—134no—
299#340—134no—
300C22×C13—22no—
301C22×C13—22no—
302C22×C13—22no—
303C22×C13—22no—
304C22×C13—24no—
305C22×C13—24no—
306C22×C13—24no—
307C22×C13—24no—
308C22×C13—22no—
309C22×C13—22no—
310C22×C13—22no—
311C22×C13—416no—
312C22×C13—42no—
313C22×C13—42no—
314C22×C13#38614yes—
315C22×C13#41714yescomplemented
316C22×C13C22×C22⋊C411yescharacteristic
317C22×C13C2×#12112yes—
318C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C13C2×#13711yescharacteristic
320C22×C13#57512yes—
321C2×D13—42no—
322C2×D13—42no—
323C2×D13—42no—
324C2×D13—82no—
325C2×D13—82no—
326C4×C13—22no—
327C4×C13—22no—
328C4×C13#38614yes—
329C4×C13C2×C4×D412yes—
330C4×C13C2×#12011yescharacteristic
331C4×C13C2×#12311yescharacteristic
332Dic13—42no—
333Dic13—42no—
334Dic13—42no—
335Dic13—84no—
336C25D5211yescharacteristic
337C2×#41—134no—
338C2×#41—262no—
339C2×#41—264no—
340C2×#41—264no—
341C22×Q8—131no—
342C22×D4—131no—
343C22×D4—261no—
344C22×D4—261no—
345C22×D4—261no—
346C23×C4—24no—
347C23×C4—24no—
348C23×C4—261no—
349C23×C4—262no—
350C23×C4C4×D1314yes—
351C23×C4D5211yescharacteristic
352C23×C4D5211yescharacteristic
353C23×C4D5211yescharacteristic
354C23×C4C22⋊D1314yes—
355C23×C4C22×D1311yescharacteristic
356C23×C4C22×D1311yescharacteristic
357C23×C4C22×D1311yescharacteristic
358C2×M16—232no—
359C2×M16—24no—
360C2×M16—24no—
361C2×M16—24no—
362C2×M16—28no—
363C2×M16C22⋊D1318yescomplemented
364C22×C8—24no—
365C22×C8—28no—
366C22×C8C22×D1314yes—
367#123—268no—
368C22⋊Q8—134no—
369C22⋊Q8—134no—
370C22⋊Q8—262no—
371C22⋊Q8—262no—
372#121—134no—
373#121—134no—
374#121—262no—
375#121—262no—
376#120—268no—
377C4×D4—268no—
378C4×D4—268no—
379#117—134no—
380#117—262no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—134no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—264no—
387C2×C4⋊C4—264no—
388C2×C4⋊C4—262no—
389C2×C22⋊C4—261no—
390C2×C22⋊C4—261no—
391C2×C22⋊C4—261no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4—261no—
394C2×C22⋊C4—262no—
395C2×C22⋊C4—262no—
396C2×C22⋊C4—524no—
397C2×C42—261no—
398#103—268no—
399#102—268no—
400#95—262no—
401#95—262no—
402C2×C13—24no—
403C2×C13—24no—
404C2×C13—42no—
405C2×C13C2×#38611yescharacteristic
406C2×C13C2×#41711yescharacteristic
407C2×C13#289611yescharacteristic
408C2×C13#305512yes—
409C2×C13#305612yes—
410D13—82no—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—42no—
415C24—261no—
416C24#610314yes—
417C24C2×D5211yescharacteristic
418C24C2×D5211yescharacteristic
419C24C2×D5211yescharacteristic
420#41—2616no—
421#41—524no—
422#41—524no—
423C2×Q8—132no—
424C2×Q8—261no—
425C2×Q8—262no—
426C2×Q8—262no—
427C2×D4—132no—
428C2×D4—264no—
429C2×D4—264no—
430C2×D4—264no—
431C2×D4—262no—
432C2×D4—262no—
433C2×D4—261no—
434C2×D4—524no—
435C2×D4—522no—
436C2×D4—522no—
437C2×D4—524no—
438C22×C4—24no—
439C22×C4—216no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—264no—
452C22×C4—264no—
453C22×C4—524no—
454C22×C4—524no—
455C22×C4—524no—
456C22×C4—524no—
457C22×C4#610314yes—
458C22×C4#610314yes—
459C22×C4#610314yes—
460C22×C4C2×C4×D1312yes—
461C22×C4C2×C4×D1312yes—
462C22×C4C2×C4×D1312yes—
463C22×C4C2×D5211yescharacteristic
464C22×C4C2×D5211yescharacteristic
465C22×C4C2×D5211yescharacteristic
466C22×C4#612714yes—
467C22×C4C2×C22⋊D1312yes—
468C22×C4C2×C22⋊D1312yes—
469C22×C4C2×C22⋊D1312yes—
470C22×C4C2×C22⋊D1314yes—
471C22×C4C23×D1311yescharacteristic
472M16—216no—
473M16—216no—
474C2×C8—216no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8C2×C22⋊D1318yes—
478C4⋊C4—132no—
479C4⋊C4—132no—
480C4⋊C4—264no—
481C4⋊C4—264no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—264no—
485C4⋊C4—264no—
486C4⋊C4—524no—
487C22⋊C4—264no—
488C22⋊C4—264no—
489C22⋊C4—264no—
490C22⋊C4—264no—
491C22⋊C4—264no—
492C22⋊C4—524no—
493C22⋊C4—524no—
494C42—264no—
495C13#2097111yescharacteristic, complemented, Sylow
496C23—21no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—28no—
506C23—28no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—262no—
511C23—521no—
512C23—521no—
513C23—522no—
514C23—522no—
515C23#8816212yes—
516C23#8816312yes—
517C23C2×#610312yes—
518C23C2×#610312yes—
519C23C2×#610312yes—
520C23#8818414yes—
521C23C22×D5211yescharacteristic
522Q8—264no—
523Q8—521no—
524Q8—521no—
525D4—264no—
526D4—528no—
527D4—528no—
528D4—528no—
529D4—521no—
530D4—521no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—262no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4—522no—
548C2×C4—522no—
549C2×C4—528no—
550C2×C4—524no—
551C2×C4—524no—
552C2×C4—524no—
553C2×C4—1048no—
554C2×C4C2×#610312yes—
555C2×C4C2×#610312yes—
556C2×C4C2×#610312yes—
557C2×C4C4×C22⋊D1318yes—
558C2×C4#8818314yes—
559C2×C4#8818414yes—
560C2×C4#8820714yes—
561C2×C4C22×C4×D1311yescharacteristic
562C2×C4C2×#612712yes—
563C2×C4C22×C22⋊D1312yes—
564C2×C4C22×C22⋊D1311yescharacteristic
565C8—28no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—522no—
581C22—522no—
582C22—522no—
583C22—1042no—
584C22—1042no—
585C22#1171879414yescomplemented
586C22#1171887414yes—
587C22C2×#8816211yescharacteristic
588C22C2×#8816311yescharacteristic
589C22#1171945312yes—
590C22C22×#610311yescharacteristic
591C22C2×#8818412yes—
592C4—22no—
593C4—22no—
594C4—522no—
595C4—522no—
596C4—522no—
597C4—1044no—
598C4#1171887414yes—
599C4C2×C4×C22⋊D1312yes—
600C4C2×#8818311yescharacteristic
601C4C2×#8820711yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—1042no—
606C2C2×#1171887411yescharacteristic
607C2C2×#1171879411yescharacteristic
608C2#1312335611yescharacteristic
609C2#1312445512yes—
610C2#1312446612yes—
subgroupG/H
centerC23C22×D52
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC23×C13×M16C2
socleC23×C13C22×D4
prime radicalC2×#11719496C2
cosocleC22×C4×C13C24
hypercenterC23×M16D13
hypocenterC13#20971
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C2×#11718794C2semidirect
3. C2×#11718873C2semidirect
4. #13115274C2semidirect
5. #13115269C2semidirect
6. #13115264C2semidirect
7. #13115259C2semidirect
8. #13114556C2semidirect
9. C23×C13×M16C2semidirect
20. #11719496C4semidirect
36. C2×#88061C22semidirect
37. C2×#88056C22semidirect
38. #11718143C22semidirect
39. #11718143C22semidirect
67. C2×C13×M16D4semidirect
80. #88184C2×C4semidirect
81. #88184C2×C4semidirect
97. #88178C2×C4semidirect
98. #88178C2×C4semidirect
191. C23×M16D13semidirect
315. C22×C13#417semidirect
363. C2×M16C22⋊D13semidirect
495. C13#20971semidirect
585. C22#11718794semidirect

Maximal subgroups

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

subgroupindex
C2×#117194962
C2×#117187942
C2×#117188732
#131152742
#131152692
#131152642
#131152592
#131145562
C23×C13×M162
#2097113