Finite Group Databasebeta

← #15784901

Subgroups — #15784901

Order 3328.

2510 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
1C22×C4⋊C4×D13C211yesmaximal, characteristic
2C2×#11718650C214yesmaximal
3C2×#11718637C211yesmaximal, characteristic, complemented
4C2×#11718648C211yesmaximal, characteristic
5C2×C4×#6103C211yesmaximal, characteristic
6#13113099C214yesmaximal, complemented
7#13113066C211yesmaximal, characteristic, complemented
8#13113065C211yesmaximal, characteristic, complemented
9C2×C13×#384C211yesmaximal, characteristic, complemented
10C23×C4×D13C2212yes—
11C23×C4×D13C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C4⋊C4C2214yes—
14C22×C13×C4⋊C4C2211yescharacteristic
15C22×C42×C13C2211yescharacteristic
16C22×#6103C2212yes—
17C22×#6103C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6101C2212yes—
21C22×#6101C2214yes—
22C22×C4×Dic13C2211yescharacteristic
23C2×C4⋊C4×D13—24no—
24C2×C4⋊C4×D13—24no—
25C2×C4⋊C4×D13—24no—
26C13×#384—24no—
27C2×#88071C2212yes—
28C2×#88071C2214yes—
29C2×#88071C2214yes—
30C2×#88071C2212yes—
31#11718650—216no—
32#11718648—24no—
33#11718637—24no—
34C4×#6103C418yescomplemented
35C24×D13C2311yescharacteristic
36C23×C4×C13C2312yes—
37C23×C4×C13C2312yes—
38C23×C4×C13C2312yes—
39C23×C4×C13C2311yescharacteristic
40C23×Dic13C2312yes—
41C23×Dic13C2312yes—
42C23×Dic13C2312yes—
43C23×Dic13C2311yescharacteristic
44C22×C4×D13—22no—
45C22×C4×D13—22no—
46C22×C4×D13—22no—
47C22×C4×D13—22no—
48C22×C4×D13—24no—
49C22×C4×D13—24no—
50C22×C4×D13—24no—
51C22×C4×D13—28no—
52C22×C4×D13—44no—
53C22×C4×D13C2×C414yes—
54C22×C4×D13D414yes—
55C22×C4×D13Q814yes—
56C2×C13×C4⋊C4—22no—
57C2×C13×C4⋊C4—22no—
58C2×C13×C4⋊C4—28no—
59C2×C13×C4⋊C4—22no—
60C2×C13×C4⋊C4—22no—
61C2×C13×C4⋊C4—22no—
62C2×C13×C4⋊C4—28no—
63C2×C13×C4⋊C4—44no—
64C2×C13×C4⋊C4D414yes—
65C2×C42×C13—22no—
66C2×C42×C13—22no—
67C2×C42×C13C2×C414yescomplemented
68C2×#6103—24no—
69C2×#6103—28no—
70C2×#6103C2×C418yes—
71C2×#6103D418yes—
72C2×#6103Q818yes—
73C2×C4⋊Dic13—22no—
74C2×C4⋊Dic13—22no—
75C2×C4⋊Dic13—22no—
76C2×C4⋊Dic13—22no—
77C2×C4⋊Dic13—22no—
78C2×C4⋊Dic13D414yes—
79C2×#6101—24no—
80C2×#6101—24no—
81C2×#6101—24no—
82C2×#6101—28no—
83C2×#6101—28no—
84C2×#6101—44no—
85C2×C4×Dic13—22no—
86C2×C4×Dic13—22no—
87C2×C4×Dic13C2×C414yescomplemented
88C4⋊C4×D13—232no—
89#88071—28no—
90#88071—24no—
91#88071—28no—
92#88071C2×C418yescomplemented
93#13880—131nomaximal, Sylow
94C24×C13C2411yescharacteristic
95C23×D13—21no—
96C23×D13—21no—
97C23×D13—21no—
98C23×D13—21no—
99C23×D13—22no—
100C23×D13—22no—
101C23×D13C22×C412yes—
102C23×D13C2×D414yes—
103C23×D13C2×D412yes—
104C23×D13C2×Q814yes—
105C23×D13C2×Q812yes—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—24no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—24no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—24no—
117C22×C4×C13—22no—
118C22×C4×C13—24no—
119C22×C4×C13C22×C414yes—
120C22×C4×C13C22×C412yes—
121C22×C4×C13C2×D414yes—
122C22×C4×C13C2×D412yes—
123C22×C4×C13C2×D412yes—
124C22×C4×C13C2×D414yes—
125C22×C4×C13C2×D412yes—
126C22×C4×C13C2×Q814yes—
127C22×C4×C13C2×Q812yes—
128C22×C4×C13#4112yes—
129C22×C4×C13#4112yes—
130C22×Dic13—22no—
131C22×Dic13—21no—
132C22×Dic13—21no—
133C22×Dic13—22no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—24no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—24no—
142C22×Dic13—24no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—24no—
147C22×Dic13C22×C412yes—
148C22×Dic13C22×C414yes—
149C22×Dic13C2×D414yes—
150C22×Dic13C2×D412yes—
151C22×Dic13C2×D414yes—
152C22×Dic13C2×Q812yes—
153C22×Dic13C2×Q814yes—
154C2×C4×D13—28no—
155C2×C4×D13—28no—
156C2×C4×D13—28no—
157C2×C4×D13—28no—
158C2×C4×D13—28no—
159C2×C4×D13—28no—
160C2×C4×D13—216no—
161C2×C4×D13—416no—
162C2×C4×D13—416no—
163C2×C4×D13—48no—
164C2×C4×D13—48no—
165C2×C4×D13C4⋊C4116yescomplemented
166C13×C4⋊C4—28no—
167C13×C4⋊C4—416no—
168C13×C4⋊C4—44no—
169C42×C13—28no—
170#6103—48no—
171#6103C4⋊C4132yescomplemented
172C4⋊Dic13—28no—
173C4⋊Dic13—44no—
174#6101—216no—
175#6101—416no—
176C4×Dic13—28no—
177C23×C4⋊C4—131no—
178C2×#385—134no—
179C2×#384—131no—
180C2×#384D1311yescharacteristic, complemented
181C2×#380—131no—
182C2×C4×C22⋊C4—131no—
183#1444—134no—
184C4⋊#95—131no—
185C4⋊#95—131no—
186C23×C13C23×C411yescharacteristic
187C23×C13C22×D411yescharacteristic
188C23×C13C22×D412yes—
189C23×C13C22×D411yescharacteristic
190C23×C13C22×D411yescharacteristic
191C23×C13C22×Q811yescharacteristic
192C23×C13C22×Q812yes—
193C23×C13C2×#4111yescharacteristic
194C23×C13C2×#4111yescharacteristic
195C23×C13H(5,𝔽2)12yes—
196C23×C132−1+412yes—
197C22×D13—22no—
198C22×D13—22no—
199C22×D13—24no—
200C22×D13—22no—
201C22×D13—22no—
202C22×D13—22no—
203C22×D13—22no—
204C22×D13—22no—
205C22×D13—22no—
206C22×D13—24no—
207C22×D13—24no—
208C22×D13—24no—
209C22×D13—24no—
210C22×D13—24no—
211C22×D13—24no—
212C22×D13—24no—
213C22×D13—24no—
214C22×D13—24no—
215C22×D13C2×C4⋊C418yes—
216C22×D13C2×C4⋊C414yes—
217C22×D13#12714yes—
218C22×D13C4⋊Q814yes—
219C22×D13C4⋊Q818yes—
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—24no—
227C2×C4×C13—28no—
228C2×C4×C13—42no—
229C2×C4×C13—42no—
230C2×C4×C13—44no—
231C2×C4×C13—42no—
232C2×C4×C13—42no—
233C2×C4×C13—42no—
234C2×C4×C13—44no—
235C2×C4×C13—44no—
236C2×C4×C13C2×C4⋊C414yes—
237C2×C4×C13C2×C4⋊C418yescomplemented
238C2×C4×C13C4×D414yes—
239C2×C4×C13C4×D414yes—
240C2×C4×C13#12014yes—
241C2×C4×C13C22⋊Q814yes—
242C2×C4×C13C22⋊Q814yes—
243C2×C4×C13#12314yes—
244C2×Dic13—24no—
245C2×Dic13—22no—
246C2×Dic13—22no—
247C2×Dic13—24no—
248C2×Dic13—22no—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—24no—
253C2×Dic13—24no—
254C2×Dic13—24no—
255C2×Dic13—24no—
256C2×Dic13—24no—
257C2×Dic13—28no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—44no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—44no—
265C2×Dic13—44no—
266C2×Dic13C2×C4⋊C414yes—
267C2×Dic13C2×C4⋊C418yescomplemented
268C4×D13—232no—
269C4×D13—432no—
270C24×C4—131no—
271C24×C4—132no—
272C22×C4⋊C4—131no—
273C22×C4⋊C4—132no—
274C22×C4⋊C4—131no—
275C22×C4⋊C4—134no—
276C22×C4⋊C4—264no—
277C22×C4⋊C4—264no—
278C22×C4⋊C4—264no—
279C22×C4⋊C4C2×D1311yescharacteristic
280C22×C4⋊C4C2×D1314yes—
281C22×C4⋊C4C2×D1311yescharacteristic
282C22×C22⋊C4—132no—
283C22×C22⋊C4—132no—
284C22×C42—131no—
285C22×C42C2×D1311yescharacteristic
286#385—2616no—
287#384—24no—
288#384—264no—
289#380—264no—
290C4×C22⋊C4—138no—
291C2×#95—132no—
292C2×#95—134no—
293C2×#95—134no—
294C2×#95—132no—
295C22×C13C22×C4⋊C411yescharacteristic
296C22×C13C22×C4⋊C412yes—
297C22×C13C2×C4×D411yescharacteristic
298C22×C13C2×C4×D411yescharacteristic
299C22×C13#52012yes—
300C22×C13C2×#12011yescharacteristic
301C22×C13C2×C22⋊Q811yescharacteristic
302C22×C13C2×C22⋊Q811yescharacteristic
303C22×C13C2×#12311yescharacteristic
304C22×C13C2×#12711yescharacteristic
305C22×C13C2×C4⋊Q811yescharacteristic
306C22×C13C2×C4⋊Q812yes—
307C22×C13#53511yescharacteristic
308C22×C13#53611yescharacteristic
309C22×C13#54412yes—
310C22×C13D4212yes—
311C22×C13D4212yes—
312C22×C13#54712yes—
313C22×C13#54712yes—
314C22×C13D4×Q812yes—
315C22×C13D4×Q812yes—
316C22×C13#55412yes—
317C22×C13#55412yes—
318C2×D13—24no—
319C2×D13—28no—
320C2×D13—28no—
321C2×D13—24no—
322C2×D13—24no—
323C2×D13—28no—
324C2×D13—28no—
325C2×D13—24no—
326C2×D13—24no—
327C2×D13—24no—
328C2×D13#38418yescomplemented
329C4×C13—28no—
330C4×C13—44no—
331C4×C13—82no—
332C4×C13#38518yescomplemented
333Dic13—28no—
334Dic13—24no—
335Dic13—44no—
336Dic13—82no—
337C25—131no—
338C23×C4—132no—
339C23×C4—131no—
340C23×C4—134no—
341C23×C4—134no—
342C23×C4—134no—
343C23×C4—132no—
344C23×C4—132no—
345C23×C4—264no—
346C23×C4—264no—
347C23×C4—268no—
348C23×C4—262no—
349C23×C4—262no—
350C23×C4—262no—
351C23×C4—262no—
352C23×C4—264no—
353C23×C4—524no—
354C23×C4C22×D1312yes—
355C23×C4C22×D1312yes—
356C23×C4C22×D1312yes—
357C23×C4C22×D1311yescharacteristic
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4—134no—
367C2×C4⋊C4—2632no—
368C2×C4⋊C4—264no—
369C2×C4⋊C4—264no—
370C2×C4⋊C4—262no—
371C2×C4⋊C4—264no—
372C2×C4⋊C4—262no—
373C2×C4⋊C4—268no—
374C2×C4⋊C4—268no—
375C2×C4⋊C4—262no—
376C2×C4⋊C4—262no—
377C2×C4⋊C4—262no—
378C2×C4⋊C4—524no—
379C2×C4⋊C4C22⋊D1314yes—
380C2×C22⋊C4—138no—
381C2×C22⋊C4—138no—
382C2×C22⋊C4—138no—
383C2×C22⋊C4—264no—
384C2×C22⋊C4—268no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—134no—
388C2×C42—262no—
389C2×C42—262no—
390C2×C42C4×D1314yescomplemented
391#95—138no—
392#95—264no—
393#95—268no—
394#95—268no—
395C2×C13C2×#38411yescharacteristic
396C2×C13C2×#38511yescharacteristic
397C2×C13#231711yescharacteristic
398C2×C13D4×C4⋊C412yes—
399C2×C13D4×C4⋊C412yes—
400C2×C13#243311yescharacteristic
401C2×C13#243411yescharacteristic
402C2×C13#244012yes—
403C2×C13#244012yes—
404C2×C13#250111yescharacteristic
405C2×C13#250211yescharacteristic
406D13—28no—
407C24—132no—
408C24—134no—
409C24—132no—
410C24—134no—
411C24—132no—
412C24—261no—
413C24—261no—
414C24—262no—
415C24—261no—
416C24—262no—
417C24—261no—
418C24C23×D1311yescharacteristic
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—132no—
433C22×C4—134no—
434C22×C4—134no—
435C22×C4—132no—
436C22×C4—134no—
437C22×C4—132no—
438C22×C4—1316no—
439C22×C4—134no—
440C22×C4—2616no—
441C22×C4—261no—
442C22×C4—262no—
443C22×C4—261no—
444C22×C4—262no—
445C22×C4—262no—
446C22×C4—261no—
447C22×C4—262no—
448C22×C4—261no—
449C22×C4—268no—
450C22×C4—268no—
451C22×C4—268no—
452C22×C4—268no—
453C22×C4—268no—
454C22×C4—268no—
455C22×C4—264no—
456C22×C4—262no—
457C22×C4—264no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—264no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—264no—
464C22×C4—5216no—
465C22×C4—5216no—
466C22×C4—528no—
467C22×C4—528no—
468C22×C4C2×C4×D1314yes—
469C22×C4C2×C4×D1312yes—
470C22×C4#612712yes—
471C22×C4D4×D1314yes—
472C22×C4D4×D1312yes—
473C22×C4D4×D1312yes—
474C22×C4Q8×D1314yes—
475C22×C4Q8×D1312yes—
476C22×C4#613112yes—
477C22×C4C2×C22⋊D1314yes—
478C22×C4C2×C22⋊D1312yes—
479C4⋊C4—28no—
480C4⋊C4—416no—
481C4⋊C4—44no—
482C4⋊C4—2616no—
483C4⋊C4—268no—
484C4⋊C4—5216no—
485C4⋊C4—524no—
486C22⋊C4—1332no—
487C22⋊C4—528no—
488C42—28no—
489C42—268no—
490C13#1388011yescharacteristic, complemented, Sylow
491C23—134no—
492C23—138no—
493C23—138no—
494C23—134no—
495C23—134no—
496C23—262no—
497C23—264no—
498C23—264no—
499C23—264no—
500C23—264no—
501C23—262no—
502C23—264no—
503C23—262no—
504C23—264no—
505C23—262no—
506C23—264no—
507C23—262no—
508C23—262no—
509C23—264no—
510C23—262no—
511C23—264no—
512C23—262no—
513C23—264no—
514C23C22×C4×D1311yescharacteristic
515C23C2×#612711yescharacteristic
516C23C2×D4×D1311yescharacteristic
517C23C2×D4×D1312yes—
518C23C2×D4×D1311yescharacteristic
519C23#8825112yes—
520C23C2×Q8×D1312yes—
521C23C2×Q8×D1311yescharacteristic
522C23C2×#613111yescharacteristic
523C23#8825412yes—
524C23C22×C22⋊D1311yescharacteristic
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—138no—
542C2×C4—134no—
543C2×C4—264no—
544C2×C4—264no—
545C2×C4—264no—
546C2×C4—264no—
547C2×C4—262no—
548C2×C4—264no—
549C2×C4—262no—
550C2×C4—262no—
551C2×C4—262no—
552C2×C4—262no—
553C2×C4—262no—
554C2×C4—264no—
555C2×C4—2632no—
556C2×C4—264no—
557C2×C4—268no—
558C2×C4—5232no—
559C2×C4—524no—
560C2×C4—522no—
561C2×C4—522no—
562C2×C4—522no—
563C2×C4—524no—
564C2×C4—522no—
565C2×C4—524no—
566C2×C4—522no—
567C2×C4C4⋊C4×D1314yes—
568C2×C4C4⋊C4×D1318yescomplemented
569C2×C4#8814714yes—
570C2×C4#8814814yes—
571C2×C4#8815014yes—
572C2×C4C4×C22⋊D1314yes—
573C2×C4#8819114yes—
574C2×C4#8820014yes—
575C22—138no—
576C22—264no—
577C22—264no—
578C22—264no—
579C22—268no—
580C22—268no—
581C22—264no—
582C22—268no—
583C22—264no—
584C22—268no—
585C22—264no—
586C22C2×C4⋊C4×D1311yescharacteristic
587C22C2×C4⋊C4×D1312yes—
588C22C2×#8814711yescharacteristic
589C22#1171920912yes—
590C22C2×#8814811yescharacteristic
591C22#1171921212yes—
592C22C2×#8815011yescharacteristic
593C22#1171921612yes—
594C22D13×#12711yescharacteristic
595C22#1171942011yescharacteristic
596C22#1171942212yes—
597C22D13×C4⋊Q811yescharacteristic
598C22D13×C4⋊Q812yes—
599C22#1171942911yescharacteristic
600C22#1171943112yes—
601C22#1171943212yes—
602C22#1171943412yes—
603C22#1171943512yes—
604C22C2×C4×C22⋊D1311yescharacteristic
605C22C2×#8819111yescharacteristic
606C22D4×C22⋊D1312yes—
607C22C2×#8820011yescharacteristic
608C22Q8×C22⋊D1312yes—
609C4—28no—
610C4—44no—
611C4—82no—
612C4—268no—
613C4—264no—
614C4—524no—
615C4—1042no—
616C4#1171865018yescomplemented
617C2—268no—
618C2D13×#38411yescharacteristic
619C2C2×#1171865011yescharacteristic
620C2#1311745711yescharacteristic
621C2C4⋊C4×C22⋊D1312yes—
622C2#1311773612yes—
623C2#1311847011yescharacteristic
624C2#1311847711yescharacteristic
625C2#1311853412yes—
626C2#1311854112yes—
627C2#1311930411yescharacteristic
628C2#1311931111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#384C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC2×#384D13
hypocenterC13#13880
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
3. C2×#11718637C2semidirect
6. #13113099C2semidirect
7. #13113066C2semidirect
8. #13113065C2semidirect
9. C2×C13×#384C2semidirect
34. C4×#6103C4semidirect
67. C2×C42×C13C2×C4semidirect
87. C2×C4×Dic13C2×C4semidirect
92. #88071C2×C4semidirect
165. C2×C4×D13C4⋊C4semidirect
171. #6103C4⋊C4semidirect
180. C2×#384D13semidirect
237. C2×C4×C13C2×C4⋊C4semidirect
267. C2×Dic13C2×C4⋊C4semidirect
328. C2×D13#384semidirect
332. C4×C13#385semidirect
390. C2×C42C4×D13semidirect
490. C13#13880semidirect
568. C2×C4C4⋊C4×D13semidirect
616. C4#11718650semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D132
C2×#117186502
C2×#117186372
C2×#117186482
C2×C4×#61032
#131130992
#131130662
#131130652
C2×C13×#3842
#1388013