Finite Group Databasebeta

← #15799566

Subgroups — #15799566

Order 3328.

1814 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718610C212yesmaximal
2C2×#11718308C212yesmaximal, complemented
3C2×#11718637C211yesmaximal, characteristic, complemented
4C2×#11718603C212yesmaximal
5C2×C22⋊C4×Dic13C211yesmaximal, characteristic
6#13113102C212yesmaximal, complemented
7#13113101C211yesmaximal, characteristic, complemented
8#13113099C211yesmaximal, characteristic, complemented
9#13113081C211yesmaximal, characteristic
10#13113064C211yesmaximal, characteristic, complemented
11C13×#1439C211yesmaximal, characteristic
12C24×Dic13C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C13×C22⋊C4C2212yes—
15C22×C13×C22⋊C4C2211yescharacteristic
16C22×C22⋊Dic13C2212yes—
17C22×C22⋊Dic13C2211yescharacteristic
18C22×C4⋊Dic13C2212yes—
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6101C2212yes—
21C22×#6101C2212yes—
22C22×#6101C2212yes—
23C22×#6101C2212yes—
24C22×C4×Dic13C2211yescharacteristic
25C22×C4×Dic13C2211yescharacteristic
26C2×C13×#95C2211yescharacteristic
27C2×C13×#95C2212yes—
28C2×#88071C2211yescharacteristic
29C2×#88071C2212yes—
30C2×#88071C2211yescharacteristic
31C2×#88071C2212yes—
32C2×#88071C2211yescharacteristic
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35#11718637—24no—
36#11718610—28no—
37#11718603—28no—
38C22⋊C4×Dic13—24no—
39#11718308—28no—
40C25×C13C2311yescharacteristic
41C23×C4×C13C2312yes—
42C23×C4×C13C2312yes—
43C23×C4×C13C2311yescharacteristic
44C23×C4×C13C2311yescharacteristic
45C23×Dic13—22no—
46C23×Dic13—24no—
47C23×Dic13—24no—
48C23×Dic13—22no—
49C23×Dic13Q814yes—
50C23×Dic13C2312yes—
51C23×Dic13C2311yescharacteristic
52C23×Dic13C2312yes—
53C23×Dic13C2311yescharacteristic
54C23×Dic13C2311yescharacteristic
55C23×Dic13C2311yescharacteristic
56C2×C13×C4⋊C4—22no—
57C2×C13×C4⋊C4—42no—
58C2×C13×C22⋊C4—22no—
59C2×C13×C22⋊C4—22no—
60C2×C13×C22⋊C4—24no—
61C2×C13×C22⋊C4—24no—
62C2×C13×C22⋊C4—42no—
63C2×C13×C22⋊C4Q814yes—
64C2×C22⋊Dic13—22no—
65C2×C22⋊Dic13—22no—
66C2×C22⋊Dic13—24no—
67C2×C22⋊Dic13—24no—
68C2×C22⋊Dic13—42no—
69C2×C22⋊Dic13D414yes—
70C2×C4⋊Dic13—24no—
71C2×C4⋊Dic13—22no—
72C2×C4⋊Dic13—24no—
73C2×C4⋊Dic13—42no—
74C2×C4⋊Dic13—42no—
75C2×#6101—24no—
76C2×#6101—24no—
77C2×#6101—24no—
78C2×#6101—24no—
79C2×#6101—24no—
80C2×#6101—24no—
81C2×#6101—24no—
82C2×#6101—24no—
83C2×#6101—24no—
84C2×#6101—24no—
85C2×#6101—42no—
86C2×#6101D418yes—
87C2×C4×Dic13—24no—
88C2×C4×Dic13—24no—
89C2×C4×Dic13—22no—
90C2×C4×Dic13D414yes—
91C13×#95—24no—
92C13×#95—22no—
93#88071—22no—
94#88071—24no—
95#88071—22no—
96#88071—24no—
97#88071—22no—
98#88071—24no—
99#88071—24no—
100#15135—131nomaximal, Sylow
101C24×C13—21no—
102C24×C13—22no—
103C24×C13—21no—
104C24×C13—21no—
105C24×C13—21no—
106C24×C13—22no—
107C24×C13—22no—
108C24×C13—22no—
109C24×C13C2×D412yes—
110C24×C13C2×Q812yes—
111C24×C13C2×Q812yes—
112C24×C13C2411yescharacteristic
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—21no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—22no—
129C22×C4×C13—22no—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13C2×D414yes—
133C22×C4×C13C2×D412yes—
134C22×C4×C13C2×Q812yes—
135C22×C4×C13C2×Q812yes—
136C22×C4×C13#4112yes—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—21no—
140C22×Dic13—22no—
141C22×Dic13—21no—
142C22×Dic13—22no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—28no—
154C22×Dic13—216no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—22no—
160C22×Dic13—22no—
161C22×Dic13—22no—
162C22×Dic13—44no—
163C22×Dic13—48no—
164C22×Dic13—44no—
165C22×Dic13C2×D414yes—
166C22×Dic13C2×D412yes—
167C22×Dic13C2×D412yes—
168C22×Dic13C2×D412yes—
169C22×Dic13C2×D412yes—
170C22×Dic13C2×D414yes—
171C22×Dic13C2×Q812yes—
172C22×Dic13C2×Q812yes—
173C22×Dic13#4112yes—
174C22×Dic13#4114yes—
175C22×Dic13#4112yes—
176C13×C4⋊C4—44no—
177C13×C22⋊C4—28no—
178C13×C22⋊C4—48no—
179C22⋊Dic13—28no—
180C22⋊Dic13—48no—
181C4⋊Dic13—48no—
182C4⋊Dic13—44no—
183#6101—216no—
184#6101—48no—
185#6101—48no—
186#6101—48no—
187C4×Dic13—28no—
188C4×Dic13—44no—
189C2×#399—132no—
190C2×#389—132no—
191C2×#384—131no—
192C2×#380—132no—
193C2×C4×C22⋊C4—131no—
194#1445—132no—
195#1444—131no—
196#1444—131no—
197#1439—131no—
198#1439D1311yescharacteristic
199C4⋊#95—131no—
200C23×C13—22no—
201C23×C13—22no—
202C23×C13—22no—
203C23×C13—24no—
204C23×C13—22no—
205C23×C13—22no—
206C23×C13—22no—
207C23×C13—24no—
208C23×C13—24no—
209C23×C13—24no—
210C23×C13—24no—
211C23×C13—24no—
212C23×C13—42no—
213C23×C13—41no—
214C23×C13—41no—
215C23×C13—41no—
216C23×C13—41no—
217C23×C13—42no—
218C23×C13—42no—
219C23×C13—42no—
220C23×C13—42no—
221C23×C13—42no—
222C23×C13C4⋊Q814yes—
223C23×C13C22×D411yescharacteristic
224C23×C13C22×D411yescharacteristic
225C23×C13C22×D412yes—
226C23×C13C22×Q811yescharacteristic
227C23×C13C22×Q811yescharacteristic
228C23×C13C2×#4111yescharacteristic
229C23×C13C2×#4111yescharacteristic
230C23×C13C2×#4112yes—
231C23×C13C2×#4111yescharacteristic
232C23×C13H(5,𝔽2)12yes—
233C23×C132−1+412yes—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—24no—
237C2×C4×C13—24no—
238C2×C4×C13—22no—
239C2×C4×C13—24no—
240C2×C4×C13—24no—
241C2×C4×C13—24no—
242C2×C4×C13—24no—
243C2×C4×C13—41no—
244C2×C4×C13—42no—
245C2×C4×C13—41no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13—42no—
249C2×C4×C13—42no—
250C2×C4×C13—42no—
251C2×C4×C13—42no—
252C2×C4×C13—42no—
253C2×C4×C13—42no—
254C2×C4×C13—42no—
255C2×C4×C13—42no—
256C2×C4×C13—42no—
257C2×C4×C13C22⋊Q814yes—
258C2×Dic13—22no—
259C2×Dic13—22no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—24no—
265C2×Dic13—24no—
266C2×Dic13—24no—
267C2×Dic13—24no—
268C2×Dic13—24no—
269C2×Dic13—42no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—42no—
274C2×Dic13—41no—
275C2×Dic13—42no—
276C2×Dic13—42no—
277C2×Dic13—42no—
278C2×Dic13—42no—
279C2×Dic13—416no—
280C2×Dic13—42no—
281C2×Dic13—42no—
282C2×Dic13—42no—
283C2×Dic13—42no—
284C2×Dic13—42no—
285C2×Dic13—42no—
286C2×Dic13—42no—
287C2×Dic13#12114yes—
288C2×Dic13C22⋊Q814yes—
289C2×Dic13C22⋊Q818yes—
290C2×Dic13#12314yes—
291C2×Dic13#12318yes—
292C2×Dic13#12714yes—
293C24×C4—131no—
294C22×C4⋊C4—132no—
295C22×C4⋊C4—132no—
296C22×C4⋊C4—132no—
297C22×C4⋊C4—132no—
298C22×C4⋊C4—132no—
299C22×C4⋊C4—131no—
300C22×C4⋊C4C2×D1311yescharacteristic
301C22×C22⋊C4—131no—
302C22×C22⋊C4—132no—
303C22×C22⋊C4C2×D1312yes—
304C22×C22⋊C4C2×D1311yescharacteristic
305C22×C42—131no—
306C22×C42—131no—
307#399—268no—
308#389—268no—
309#384—264no—
310#380—268no—
311C4×C22⋊C4—264no—
312C2×#95—132no—
313C2×#95—131no—
314C2×#95—132no—
315C2×#95—131no—
316C2×#95—132no—
317C2×#95—132no—
318C2×#95—131no—
319C2×#95C2×D1311yescharacteristic
320C2×#95C2×D1312yes—
321C22×C13—24no—
322C22×C13—28no—
323C22×C13—42no—
324C22×C13—42no—
325C22×C13—44no—
326C22×C13—44no—
327C22×C13—44no—
328C22×C13—42no—
329C22×C13—42no—
330C22×C13—42no—
331C22×C13—42no—
332C22×C13C2×#12111yescharacteristic
333C22×C13C2×C22⋊Q811yescharacteristic
334C22×C13C2×C22⋊Q811yescharacteristic
335C22×C13C2×C22⋊Q812yes—
336C22×C13C2×#12312yes—
337C22×C13C2×#12311yescharacteristic
338C22×C13C2×#12711yescharacteristic
339C22×C13C2×C4⋊Q811yescharacteristic
340C22×C13#53211yescharacteristic
341C22×C13#53311yescharacteristic
342C22×C13#53311yescharacteristic
343C22×C13#53511yescharacteristic
344C22×C13#53611yescharacteristic
345C22×C13#54212yes—
346C22×C13#54412yes—
347C22×C13D4212yes—
348C22×C13#54712yes—
349C22×C13#54712yes—
350C22×C13D4×Q812yes—
351C22×C13#55012yes—
352C22×C13#55212yes—
353C22×C13#55312yes—
354C22×C13#55412yes—
355C4×C13—24no—
356C4×C13—44no—
357C4×C13—42no—
358C4×C13—82no—
359Dic13—24no—
360Dic13—42no—
361Dic13—44no—
362Dic13—42no—
363Dic13—82no—
364Dic13#39918yescomplemented
365C25C22×D1311yescharacteristic
366C23×C4—131no—
367C23×C4—131no—
368C23×C4—134no—
369C23×C4—131no—
370C23×C4—132no—
371C23×C4—131no—
372C23×C4—132no—
373C23×C4—262no—
374C23×C4—264no—
375C23×C4—264no—
376C23×C4—262no—
377C23×C4C22×D1312yes—
378C23×C4C22×D1311yescharacteristic
379C23×C4C22×D1312yes—
380C23×C4C22×D1311yescharacteristic
381C2×C4⋊C4—22no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—138no—
384C2×C4⋊C4—264no—
385C2×C4⋊C4—264no—
386C2×C4⋊C4—264no—
387C2×C4⋊C4—264no—
388C2×C4⋊C4—264no—
389C2×C4⋊C4—264no—
390C2×C4⋊C4—264no—
391C2×C4⋊C4—264no—
392C2×C4⋊C4—264no—
393C2×C4⋊C4—264no—
394C2×C4⋊C4—264no—
395C2×C4⋊C4—262no—
396C2×C4⋊C4—264no—
397C2×C4⋊C4—522no—
398C2×C4⋊C4—522no—
399C2×C4⋊C4—522no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—134no—
406C2×C22⋊C4—262no—
407C2×C22⋊C4—262no—
408C2×C22⋊C4—264no—
409C2×C22⋊C4—264no—
410C2×C22⋊C4—522no—
411C2×C22⋊C4Dic2614yes—
412C2×C42—134no—
413C2×C42—264no—
414C2×C42—262no—
415C2×C42—264no—
416#95—24no—
417#95—22no—
418#95—262no—
419#95—264no—
420#95—264no—
421#95—264no—
422#95—262no—
423#95—264no—
424#95—262no—
425C2×C13—44no—
426C2×C13C2×#39911yescharacteristic
427C2×C13#242212yes—
428C2×C13#242312yes—
429C2×C13#244012yes—
430C2×C13#244112yes—
431C2×C13#248611yescharacteristic
432C2×C13#248711yescharacteristic
433C2×C13#254111yescharacteristic
434C2×C13#254211yescharacteristic
435C2×C13#262811yescharacteristic
436C2×C13#285711yescharacteristic
437C24—21no—
438C24—22no—
439C24—21no—
440C24—22no—
441C24—21no—
442C24—21no—
443C24—22no—
444C24—22no—
445C24C2×Dic2612yes—
446C24D4×D1312yes—
447C24Q8×D1312yes—
448C24C23×D1311yescharacteristic
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—132no—
469C22×C4—134no—
470C22×C4—132no—
471C22×C4—134no—
472C22×C4—132no—
473C22×C4—132no—
474C22×C4—132no—
475C22×C4—132no—
476C22×C4—134no—
477C22×C4—132no—
478C22×C4—132no—
479C22×C4—268no—
480C22×C4—2616no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—261no—
484C22×C4—262no—
485C22×C4—261no—
486C22×C4—262no—
487C22×C4—262no—
488C22×C4—262no—
489C22×C4—262no—
490C22×C4—262no—
491C22×C4—261no—
492C22×C4—262no—
493C22×C4—262no—
494C22×C4—261no—
495C22×C4—262no—
496C22×C4—262no—
497C22×C4—262no—
498C22×C4—262no—
499C22×C4—262no—
500C22×C4—262no—
501C22×C4—262no—
502C22×C4—262no—
503C22×C4—262no—
504C22×C4—524no—
505C22×C4—528no—
506C22×C4—524no—
507C22×C4C2×Dic2612yes—
508C22×C4C2×Dic2612yes—
509C22×C4D4×D1314yes—
510C22×C4D4×D1312yes—
511C22×C4#612912yes—
512C4⋊C4—44no—
513C4⋊C4—2616no—
514C4⋊C4—528no—
515C4⋊C4—524no—
516C4⋊C4—528no—
517C4⋊C4—528no—
518C4⋊C4—528no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—268no—
522C22⋊C4—528no—
523C42—268no—
524C42—524no—
525C13#1513511yescharacteristic, complemented, Sylow
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—24no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23#8814214yes—
549C23C22×Dic2611yescharacteristic
550C23C2×D4×D1311yescharacteristic
551C23C2×D4×D1311yescharacteristic
552C23C2×D4×D1312yes—
553C23C2×#612911yescharacteristic
554C23#8825112yes—
555C23C2×Q8×D1311yescharacteristic
556C23#41×D1311yescharacteristic
557C23#41×D1311yescharacteristic
558C23#41×D1312yes—
559C23#8825712yes—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—138no—
584C2×C4—134no—
585C2×C4—134no—
586C2×C4—138no—
587C2×C4—134no—
588C2×C4—134no—
589C2×C4—262no—
590C2×C4—264no—
591C2×C4—262no—
592C2×C4—264no—
593C2×C4—264no—
594C2×C4—264no—
595C2×C4—262no—
596C2×C4—264no—
597C2×C4—264no—
598C2×C4—262no—
599C2×C4—264no—
600C2×C4—5216no—
601C2×C4—521no—
602C2×C4—522no—
603C2×C4—522no—
604C2×C4—522no—
605C2×C4—522no—
606C2×C4—521no—
607C2×C4—522no—
608C2×C4—522no—
609C2×C4—521no—
610C2×C4—521no—
611C2×C4—522no—
612C2×C4—522no—
613C2×C4—522no—
614C2×C4—522no—
615C2×C4—522no—
616C2×C4—522no—
617C2×C4—522no—
618C2×C4#8813214yes—
619C22—28no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22C2×#8813211yescharacteristic
631C22C2×#8814211yescharacteristic
632C22#1171922211yescharacteristic
633C22D4×Dic2612yes—
634C22#1171924412yes—
635C22#1171929512yes—
636C22#1171929711yescharacteristic
637C22#1171930111yescharacteristic
638C22#1171930412yes—
639C22#1171930612yes—
640C22D13×#12111yescharacteristic
641C22#1171931612yes—
642C22#1171932711yescharacteristic
643C22#1171932812yes—
644C22D13×C22⋊Q812yes—
645C22D13×C22⋊Q811yescharacteristic
646C22D13×#12312yes—
647C22D13×#12311yescharacteristic
648C22#1171936012yes—
649C22#1171937412yes—
650C22#1171938011yescharacteristic
651C22#1171941312yes—
652C22D13×#12711yescharacteristic
653C4—24no—
654C4—42no—
655C4—44no—
656C4—82no—
657C4—138no—
658C4—264no—
659C4—522no—
660C4—522no—
661C4—524no—
662C4—1042no—
663C2—44no—
664C2D13×#39911yescharacteristic
665C2#1311834312yes—
666C2#1311835812yes—
667C2#1311853712yes—
668C2#1311854412yes—
669C2#1311913711yescharacteristic
670C2#1311915011yescharacteristic
671C2#1311977711yescharacteristic
672C2#1311978711yescharacteristic
673C2#1312065611yescharacteristic
674C2#1312318611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, cosocleC24×C13C24
FittingC13×#1439C2
prime radicalC25×C13C23
hypercenter#1439D13
hypocenterC13#15135
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#11718308C2semidirect
3. C2×#11718637C2semidirect
6. #13113102C2semidirect
7. #13113101C2semidirect
8. #13113099C2semidirect
10. #13113064C2semidirect
364. Dic13#399semidirect
525. C13#15135semidirect

Maximal subgroups

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

subgroupindex
C2×#117186102
C2×#117183082
C2×#117186372
C2×#117186032
C2×C22⋊C4×Dic132
#131131022
#131131012
#131130992
#131130812
#131130642
C13×#14392
#1513513