Finite Group Databasebeta

← #15785652

Subgroups — #15785652

Order 3328.

2222 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88131C211yesmaximal, characteristic
2C2×#11718611C211yesmaximal, characteristic
3C2×#11718604C211yesmaximal, characteristic
4C22×#88071C211yesmaximal, characteristic
5#13113090C212yesmaximal, complemented
6#13113086C212yesmaximal, complemented
7#13113065C212yesmaximal, complemented
8#13113060C212yesmaximal, complemented
9#13113059C211yesmaximal, characteristic
10#13113058C211yesmaximal, characteristic
11C2×C13×#385C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C24×Dic13C2211yescharacteristic
14C24×Dic13C2211yescharacteristic
15C22×C13×C4⋊C4C2212yes—
16C22×C13×C22⋊C4C2211yescharacteristic
17C22×C13×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic13C2211yescharacteristic
19C22×C22⋊Dic13C2211yescharacteristic
20C22×C4⋊Dic13C2212yes—
21C22×#6101C2212yes—
22C22×C4×Dic13C2212yes—
23C2×#88131—24no—
24C2×#88131—24no—
25C2×#88131C418yescomplemented
26C13×#385—24no—
27C2×C13×#95C2212yes—
28C2×#88071—24no—
29C2×#88071C2211yescharacteristic
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2211yescharacteristic
33C2×#88071C2211yescharacteristic
34C2×#88071C2211yescharacteristic
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2212yes—
38C2×#88071C2212yes—
39C2×#88071C2212yes—
40#11718611—24no—
41#11718604—24no—
42C25×C13C2311yescharacteristic
43C23×C4×C13—22no—
44C23×C4×C13—24no—
45C23×C4×C13—22no—
46C23×C4×C13—22no—
47C23×C4×C13—42no—
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2311yescharacteristic
50C23×C4×C13C2312yes—
51C23×C4×C13C2312yes—
52C23×Dic13—22no—
53C23×Dic13—22no—
54C23×Dic13—22no—
55C23×Dic13—22no—
56C23×Dic13—22no—
57C23×Dic13—24no—
58C23×Dic13—22no—
59C23×Dic13—22no—
60C23×Dic13—42no—
61C23×Dic13C2×C414yes—
62C23×Dic13D414yes—
63C23×Dic13Q814yes—
64C23×Dic13C2311yescharacteristic
65C23×Dic13C2311yescharacteristic
66C23×Dic13C2312yes—
67C23×Dic13C2312yes—
68C23×Dic13C2311yescharacteristic
69C23×Dic13C2311yescharacteristic
70C2×C13×C4⋊C4—24no—
71C2×C13×C4⋊C4—24no—
72C2×C13×C4⋊C4—42no—
73C2×C13×C22⋊C4—22no—
74C2×C13×C22⋊C4—22no—
75C2×C13×C22⋊C4—22no—
76C2×C13×C22⋊C4C2×C414yes—
77C2×C13×C22⋊C4D414yes—
78C2×C13×C22⋊C4D414yes—
79C2×C22⋊Dic13—22no—
80C2×C22⋊Dic13—22no—
81C2×C22⋊Dic13—22no—
82C2×C22⋊Dic13—22no—
83C2×C22⋊Dic13—22no—
84C2×C22⋊Dic13C2×C414yes—
85C2×C4⋊Dic13—24no—
86C2×C4⋊Dic13—24no—
87C2×C4⋊Dic13—42no—
88C2×#6101—24no—
89C2×#6101—24no—
90C2×#6101C2×C418yescomplemented
91C2×C4×Dic13—24no—
92C2×C4×Dic13—24no—
93C2×C4×Dic13C2×C418yescomplemented
94#88131—232no—
95C13×#95—24no—
96#88071—24no—
97#88071—24no—
98#88071—22no—
99#88071—22no—
100#88071—22no—
101#88071—22no—
102#88071—24no—
103#88071—24no—
104#88071—24no—
105#88071—42no—
106#88071—42no—
107#13962—131nomaximal, Sylow
108C24×C13—22no—
109C24×C13—22no—
110C24×C13—22no—
111C24×C13—22no—
112C24×C13C22×C412yes—
113C24×C13C2×D412yes—
114C24×C13C2×D412yes—
115C24×C13C2×D412yes—
116C24×C13C2×Q812yes—
117C24×C13#4112yes—
118C24×C13#4112yes—
119C24×C13C2411yescharacteristic
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—28no—
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—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—22no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—22no—
139C22×C4×C13—22no—
140C22×C4×C13—22no—
141C22×C4×C13—48no—
142C22×C4×C13—48no—
143C22×C4×C13—44no—
144C22×C4×C13—44no—
145C22×C4×C13C22×C414yes—
146C22×C4×C13C2×D414yes—
147C22×C4×C13C2×D414yes—
148C22×C4×C13#4114yes—
149C22×Dic13—28no—
150C22×Dic13—28no—
151C22×Dic13—28no—
152C22×Dic13—28no—
153C22×Dic13—28no—
154C22×Dic13—28no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C22×Dic13—21no—
160C22×Dic13—21no—
161C22×Dic13—21no—
162C22×Dic13—21no—
163C22×Dic13—28no—
164C22×Dic13—22no—
165C22×Dic13—22no—
166C22×Dic13—22no—
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—22no—
170C22×Dic13—22no—
171C22×Dic13—22no—
172C22×Dic13—22no—
173C22×Dic13—22no—
174C22×Dic13—22no—
175C22×Dic13—22no—
176C22×Dic13—21no—
177C22×Dic13—21no—
178C22×Dic13—21no—
179C22×Dic13—21no—
180C22×Dic13—22no—
181C22×Dic13—22no—
182C22×Dic13—21no—
183C22×Dic13—21no—
184C22×Dic13—22no—
185C22×Dic13—22no—
186C22×Dic13—48no—
187C22×Dic13—48no—
188C22×Dic13—44no—
189C22×Dic13—44no—
190C22×Dic13C4⋊C4116yescomplemented
191C22×Dic13C22×C414yes—
192C22×Dic13C22×C412yes—
193C22×Dic13C22×C412yes—
194C22×Dic13C2×D412yes—
195C22×Dic13C2×D412yes—
196C22×Dic13C2×Q812yes—
197C22×Dic13C2×Q812yes—
198C22×Dic13#4114yes—
199C13×C4⋊C4—48no—
200C13×C22⋊C4—44no—
201C13×C22⋊C4C22⋊C4116yes—
202C22⋊Dic13—28no—
203C22⋊Dic13—44no—
204C4⋊Dic13—48no—
205#6101—216no—
206C4×Dic13—216no—
207C22×#117—131no—
208C2×#385—131no—
209C2×#385D1311yescharacteristic
210C2×#381—131no—
211C22×#95—131no—
212#1441—132no—
213#1441—132no—
214C4⋊#95—132no—
215#1434—132no—
216C22⋊#95—131no—
217C22⋊#95—131no—
218C23×C13—24no—
219C23×C13—24no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13—24no—
223C23×C13—22no—
224C23×C13—22no—
225C23×C13—24no—
226C23×C13—24no—
227C23×C13—24no—
228C23×C13—24no—
229C23×C13—24no—
230C23×C13—24no—
231C23×C13—24no—
232C23×C13—24no—
233C23×C13—24no—
234C23×C13C2×C22⋊C414yes—
235C23×C13C2×C4⋊C414yes—
236C23×C13#11714yes—
237C23×C13#12114yes—
238C23×C13#12114yes—
239C23×C13C22⋊Q814yes—
240C23×C13C22⋊Q814yes—
241C23×C13C23×C411yescharacteristic
242C23×C13C22×D411yescharacteristic
243C23×C13C22×D411yescharacteristic
244C23×C13C22×D411yescharacteristic
245C23×C13C22×Q811yescharacteristic
246C23×C13C2×#4111yescharacteristic
247C23×C13C2×#4111yescharacteristic
248C23×C13C2×#4112yes—
249C23×C13C2×#4112yes—
250C23×C13H(5,𝔽2)12yes—
251C23×C132−1+412yes—
252C2×C4×C13—24no—
253C2×C4×C13—24no—
254C2×C4×C13—24no—
255C2×C4×C13—24no—
256C2×C4×C13—24no—
257C2×C4×C13—24no—
258C2×C4×C13—24no—
259C2×C4×C13—24no—
260C2×C4×C13—24no—
261C2×C4×C13—416no—
262C2×C4×C13—42no—
263C2×C4×C13—42no—
264C2×C4×C13—42no—
265C2×C4×C13—42no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—42no—
273C2×C4×C13—42no—
274C2×C4×C13—42no—
275C2×C4×C13—42no—
276C2×C4×C13C2×C22⋊C418yes—
277C2×Dic13—22no—
278C2×Dic13—22no—
279C2×Dic13—22no—
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—22no—
285C2×Dic13—22no—
286C2×Dic13—22no—
287C2×Dic13—22no—
288C2×Dic13—22no—
289C2×Dic13—232no—
290C2×Dic13—24no—
291C2×Dic13—24no—
292C2×Dic13—24no—
293C2×Dic13—24no—
294C2×Dic13—24no—
295C2×Dic13—24no—
296C2×Dic13—416no—
297C2×Dic13—42no—
298C2×Dic13—42no—
299C2×Dic13—42no—
300C2×Dic13—42no—
301C2×Dic13—42no—
302C2×Dic13—42no—
303C2×Dic13—42no—
304C2×Dic13—42no—
305C2×Dic13—41no—
306C2×Dic13—41no—
307C2×Dic13—41no—
308C2×Dic13—41no—
309C2×Dic13—41no—
310C2×Dic13—41no—
311C2×Dic13—42no—
312C2×Dic13—42no—
313C2×Dic13—42no—
314C2×Dic13—42no—
315C2×Dic13C2×C4⋊C414yes—
316C2×Dic13C2×C4⋊C414yes—
317C24×C4—131no—
318C24×C4—131no—
319C24×C4C2×D1311yescharacteristic
320C2×#117—138no—
321C2×#117—264no—
322C2×#117—264no—
323C22×C4⋊C4—132no—
324C22×C4⋊C4—132no—
325C22×C4⋊C4C2×D1312yes—
326C22×C22⋊C4—131no—
327C22×C22⋊C4—131no—
328C22×C22⋊C4C2×D1311yescharacteristic
329C22×C22⋊C4C2×D1311yescharacteristic
330C22×C42—132no—
331#385—24no—
332#385—264no—
333#381—264no—
334C2×#95—132no—
335C2×#95—132no—
336C2×#95—131no—
337C2×#95—132no—
338C2×#95—132no—
339C2×#95—131no—
340C2×#95—132no—
341C2×#95—131no—
342C2×#95—131no—
343C2×#95—132no—
344C2×#95—132no—
345C2×#95—264no—
346C2×#95C2×D1312yes—
347C22×C13—24no—
348C22×C13—28no—
349C22×C13—28no—
350C22×C13—24no—
351C22×C13—24no—
352C22×C13—24no—
353C22×C13—24no—
354C22×C13—28no—
355C22×C13—28no—
356C22×C13—24no—
357C22×C13#38018yes—
358C22×C13C22×C22⋊C411yescharacteristic
359C22×C13C22×C4⋊C411yescharacteristic
360C22×C13C2×#11711yescharacteristic
361C22×C13C4×#4112yes—
362C22×C13#52012yes—
363C22×C13C2×#12111yescharacteristic
364C22×C13C2×#12111yescharacteristic
365C22×C13C2×C22⋊Q811yescharacteristic
366C22×C13C2×C22⋊Q811yescharacteristic
367C22×C13#52511yescharacteristic
368C22×C13#52511yescharacteristic
369C22×C13#53312yes—
370C22×C13#53511yescharacteristic
371C22×C13#53611yescharacteristic
372C22×C13#54412yes—
373C22×C13#54612yes—
374C22×C13#54612yes—
375C22×C13#54712yes—
376C22×C13#54712yes—
377C22×C13#55112yes—
378C22×C13#55112yes—
379C22×C13#55212yes—
380C22×C13#55212yes—
381C4×C13—28no—
382C4×C13—44no—
383C4×C13—81no—
384C4×C13—81no—
385Dic13—24no—
386Dic13—24no—
387Dic13—44no—
388Dic13—44no—
389Dic13—81no—
390Dic13—81no—
391C25C22×D1311yescharacteristic
392C23×C4—22no—
393C23×C4—24no—
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—42no—
397C23×C4—134no—
398C23×C4—134no—
399C23×C4—134no—
400C23×C4—131no—
401C23×C4—131no—
402C23×C4—132no—
403C23×C4—132no—
404C23×C4—131no—
405C23×C4—131no—
406C23×C4—264no—
407C23×C4—262no—
408C23×C4—262no—
409C23×C4—262no—
410C23×C4—262no—
411C23×C4—262no—
412C23×C4—262no—
413C23×C4—262no—
414C23×C4—522no—
415C23×C4C22×D1311yescharacteristic
416C23×C4C22×D1311yescharacteristic
417C23×C4C22×D1312yes—
418C23×C4C22×D1312yes—
419#117—2632no—
420C2×C4⋊C4—24no—
421C2×C4⋊C4—24no—
422C2×C4⋊C4—42no—
423C2×C4⋊C4—138no—
424C2×C4⋊C4—264no—
425C2×C4⋊C4—264no—
426C2×C4⋊C4—264no—
427C2×C4⋊C4—264no—
428C2×C4⋊C4—522no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—134no—
433C2×C22⋊C4—262no—
434C2×C22⋊C4—262no—
435C2×C22⋊C4—262no—
436C2×C22⋊C4—262no—
437C2×C22⋊C4—262no—
438C2×C22⋊C4C4×D1314yes—
439C2×C22⋊C4D5214yes—
440C2×C22⋊C4C22⋊D1314yes—
441C2×C42—138no—
442C2×C42—264no—
443C2×C42—264no—
444#95—24no—
445#95—264no—
446#95—264no—
447#95—264no—
448#95—262no—
449#95—262no—
450#95—262no—
451#95—262no—
452#95—264no—
453#95—264no—
454#95—522no—
455#95—522no—
456C2×C13—28no—
457C2×C13C2×#38011yescharacteristic
458C2×C13#229611yescharacteristic
459C2×C13#231711yescharacteristic
460C2×C13#233912yes—
461C2×C13#234312yes—
462C2×C13#241111yescharacteristic
463C2×C13#241111yescharacteristic
464C2×C13#241311yescharacteristic
465C2×C13#241311yescharacteristic
466C2×C13#243812yes—
467C2×C13#243812yes—
468C24—22no—
469C24—22no—
470C24—22no—
471C24—22no—
472C24C2×C4×D1312yes—
473C24C2×D5212yes—
474C24D4×D1312yes—
475C24#612912yes—
476C24Q8×D1312yes—
477C24#613112yes—
478C24C2×C22⋊D1312yes—
479C24C23×D1311yescharacteristic
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—28no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—1316no—
506C22×C4—132no—
507C22×C4—132no—
508C22×C4—132no—
509C22×C4—132no—
510C22×C4—132no—
511C22×C4—132no—
512C22×C4—134no—
513C22×C4—134no—
514C22×C4—268no—
515C22×C4—268no—
516C22×C4—268no—
517C22×C4—268no—
518C22×C4—268no—
519C22×C4—268no—
520C22×C4—261no—
521C22×C4—261no—
522C22×C4—261no—
523C22×C4—261no—
524C22×C4—261no—
525C22×C4—261no—
526C22×C4—261no—
527C22×C4—261no—
528C22×C4—268no—
529C22×C4—262no—
530C22×C4—262no—
531C22×C4—262no—
532C22×C4—262no—
533C22×C4—262no—
534C22×C4—262no—
535C22×C4—262no—
536C22×C4—262no—
537C22×C4—262no—
538C22×C4—262no—
539C22×C4—262no—
540C22×C4—262no—
541C22×C4—261no—
542C22×C4—261no—
543C22×C4—261no—
544C22×C4—261no—
545C22×C4—262no—
546C22×C4—262no—
547C22×C4—261no—
548C22×C4—261no—
549C22×C4—262no—
550C22×C4—262no—
551C22×C4—528no—
552C22×C4—528no—
553C22×C4—524no—
554C22×C4—524no—
555C22×C4C2×C4×D1314yes—
556C22×C4C2×D5214yes—
557C22×C4#612914yes—
558C22×C4C2×C22⋊D1314yes—
559C4⋊C4—48no—
560C4⋊C4—2616no—
561C4⋊C4—528no—
562C22⋊C4—44no—
563C22⋊C4—268no—
564C22⋊C4—524no—
565C22⋊C4#6103116yes—
566C42—2616no—
567C13#1396211yescharacteristic, complemented, Sylow
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—22no—
575C23—24no—
576C23—22no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23C4⋊C4×D1314yes—
585C23#8814614yes—
586C23#8814914yes—
587C23#8815114yes—
588C23C2×#610314yes—
589C23#8819214yes—
590C23#8820014yes—
591C23C22×C4×D1311yescharacteristic
592C23C22×D5211yescharacteristic
593C23C2×D4×D1311yescharacteristic
594C23C2×#612911yescharacteristic
595C23C2×#612912yes—
596C23#8825112yes—
597C23C2×Q8×D1311yescharacteristic
598C23C2×#613111yescharacteristic
599C23#41×D1312yes—
600C23#8825712yes—
601C23C22×C22⋊D1311yescharacteristic
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—134no—
627C2×C4—134no—
628C2×C4—262no—
629C2×C4—262no—
630C2×C4—262no—
631C2×C4—262no—
632C2×C4—262no—
633C2×C4—262no—
634C2×C4—262no—
635C2×C4—262no—
636C2×C4—262no—
637C2×C4—262no—
638C2×C4—262no—
639C2×C4—262no—
640C2×C4—2632no—
641C2×C4—264no—
642C2×C4—264no—
643C2×C4—264no—
644C2×C4—264no—
645C2×C4—264no—
646C2×C4—264no—
647C2×C4—5216no—
648C2×C4—522no—
649C2×C4—522no—
650C2×C4—522no—
651C2×C4—522no—
652C2×C4—522no—
653C2×C4—522no—
654C2×C4—522no—
655C2×C4—522no—
656C2×C4—521no—
657C2×C4—521no—
658C2×C4—522no—
659C2×C4—522no—
660C2×C4—521no—
661C2×C4—521no—
662C2×C4—522no—
663C2×C4—522no—
664C2×C4—521no—
665C2×C4—521no—
666C2×C4C2×#610318yes—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—28no—
673C22—28no—
674C22—24no—
675C22—28no—
676C22—28no—
677C22#1171864818yes—
678C22C2×C4⋊C4×D1311yescharacteristic
679C22C2×#8814611yescharacteristic
680C22C2×#8814911yescharacteristic
681C22C2×#8815111yescharacteristic
682C22C4×#612912yes—
683C22#1171925112yes—
684C22#1171925912yes—
685C22#1171926012yes—
686C22#1171929111yescharacteristic
687C22#1171929212yes—
688C22#1171929711yescharacteristic
689C22#1171932712yes—
690C22#1171932812yes—
691C22#1171934212yes—
692C22#1171934712yes—
693C22#1171935311yescharacteristic
694C22#1171935512yes—
695C22#1171936111yescharacteristic
696C22C22×#610311yescharacteristic
697C22C2×#8819211yescharacteristic
698C22#1171951112yes—
699C22C2×#8820011yescharacteristic
700C22#1171952912yes—
701C4—28no—
702C4—44no—
703C4—81no—
704C4—81no—
705C4—264no—
706C4—264no—
707C4—524no—
708C4—524no—
709C4—1041no—
710C4—1041no—
711C2—28no—
712C2C2×#1171864811yescharacteristic
713C2#1311731511yescharacteristic
714C2#1311745511yescharacteristic
715C2#1311766112yes—
716C2#1311769412yes—
717C2#1311821711yescharacteristic
718C2#1311821911yescharacteristic
719C2#1311823911yescharacteristic
720C2#1311824111yescharacteristic
721C2#1311850812yes—
722C2#1311851012yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#385C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC2×#385D13
hypocenterC13#13962
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #13113090C2semidirect
6. #13113086C2semidirect
7. #13113065C2semidirect
8. #13113060C2semidirect
25. C2×#88131C4semidirect
90. C2×#6101C2×C4semidirect
93. C2×C4×Dic13C2×C4semidirect
190. C22×Dic13C4⋊C4semidirect
567. C13#13962semidirect

Maximal subgroups

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

subgroupindex
C22×#881312
C2×#117186112
C2×#117186042
C22×#880712
#131130902
#131130862
#131130652
#131130602
#131130592
#131130582
C2×C13×#3852
#1396213