Finite Group Databasebeta

← #15786573

Subgroups — #15786573

Order 3328.

2222 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#6101C211yesmaximal, characteristic
2C2×#11718594C212yesmaximal, complemented
3C2×#11718607C211yesmaximal, characteristic
4#13113094C212yesmaximal, complemented
5#13113087C212yesmaximal, complemented
6#13113069C212yesmaximal, complemented
7#13113059C211yesmaximal, characteristic
8#13113058C211yesmaximal, characteristic
9#13113058C211yesmaximal, characteristic
10#13113058C211yesmaximal, characteristic
11C2×C4×C13×C22⋊C4C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C24×Dic13C2211yescharacteristic
14C24×Dic13C2211yescharacteristic
15C22×C13×C22⋊C4C2211yescharacteristic
16C22×C13×C22⋊C4C2211yescharacteristic
17C22×C42×C13C2211yescharacteristic
18C22×C42×C13C2211yescharacteristic
19C22×C22⋊Dic13C2211yescharacteristic
20C22×C22⋊Dic13C2211yescharacteristic
21C22×#6101—24no—
22C22×#6101—28no—
23C22×#6101C2212yes—
24C22×#6101C2212yes—
25C22×#6101C2212yes—
26C4×C13×C22⋊C4—24no—
27C2×C13×#95C2211yescharacteristic
28C2×C13×#95C2211yescharacteristic
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2212yes—
38#11718607C418yescomplemented
39#11718594—28no—
40C25×C13C2311yescharacteristic
41C23×C4×C13—22no—
42C23×C4×C13—22no—
43C23×C4×C13—24no—
44C23×C4×C13—42no—
45C23×C4×C13D414yes—
46C23×C4×C13C2311yescharacteristic
47C23×C4×C13C2311yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2311yescharacteristic
50C23×C4×C13C2311yescharacteristic
51C23×C4×C13C2311yescharacteristic
52C23×Dic13—22no—
53C23×Dic13—22no—
54C23×Dic13—24no—
55C23×Dic13—24no—
56C23×Dic13—22no—
57C23×Dic13—24no—
58C23×Dic13—22no—
59C23×Dic13—22no—
60C23×Dic13—42no—
61C23×Dic13C2×C414yes—
62C23×Dic13C2312yes—
63C23×Dic13C2312yes—
64C23×Dic13C2312yes—
65C23×Dic13C2312yes—
66C2×C13×C22⋊C4—22no—
67C2×C13×C22⋊C4—22no—
68C2×C13×C22⋊C4—22no—
69C2×C13×C22⋊C4—22no—
70C2×C13×C22⋊C4C2×C414yes—
71C2×C13×C22⋊C4Q814yes—
72C2×C42×C13—22no—
73C2×C42×C13—22no—
74C2×C42×C13—24no—
75C2×C42×C13—42no—
76C2×C22⋊Dic13—22no—
77C2×C22⋊Dic13—22no—
78C2×C22⋊Dic13—22no—
79C2×C22⋊Dic13—22no—
80C2×C22⋊Dic13—22no—
81C2×C22⋊Dic13C2×C414yes—
82C2×#6101—232no—
83C2×#6101—24no—
84C2×#6101—24no—
85C2×#6101—24no—
86C2×#6101—24no—
87C2×#6101—24no—
88C2×#6101—24no—
89C2×#6101—24no—
90C2×#6101C2×C418yescomplemented
91C2×#6101D418yes—
92C13×#95—22no—
93C13×#95—22no—
94#88071—24no—
95#88071—24no—
96#88071—24no—
97#88071—24no—
98#88071—24no—
99#88071—42no—
100#88071—42no—
101#88071—42no—
102#88071C2×C418yescomplemented
103#14039—131nomaximal, Sylow
104C24×C13—22no—
105C24×C13—22no—
106C24×C13—22no—
107C24×C13—22no—
108C24×C13C22×C412yes—
109C24×C13C2×D412yes—
110C24×C13C2×Q812yes—
111C24×C13#4112yes—
112C24×C13#4112yes—
113C24×C13#4112yes—
114C24×C13#4112yes—
115C24×C13C2411yescharacteristic
116C22×C4×C13—21no—
117C22×C4×C13—21no—
118C22×C4×C13—22no—
119C22×C4×C13—28no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—22no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—22no—
136C22×C4×C13—22no—
137C22×C4×C13—22no—
138C22×C4×C13—22no—
139C22×C4×C13—22no—
140C22×C4×C13—28no—
141C22×C4×C13—44no—
142C22×C4×C13—48no—
143C22×C4×C13—48no—
144C22×C4×C13C22×C412yes—
145C22×C4×C13C22×C412yes—
146C22×C4×C13C2×D414yes—
147C22×C4×C13C2×D412yes—
148C22×C4×C13C2×D412yes—
149C22×C4×C13C2×Q812yes—
150C22×C4×C13C2×Q812yes—
151C22×Dic13—28no—
152C22×Dic13—216no—
153C22×Dic13—28no—
154C22×Dic13—28no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—216no—
160C22×Dic13—28no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—22no—
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—22no—
177C22×Dic13—22no—
178C22×Dic13—22no—
179C22×Dic13—22no—
180C22×Dic13—48no—
181C22×Dic13—48no—
182C22×Dic13—44no—
183C22×Dic13—44no—
184C22×Dic13C22×C414yes—
185C22×Dic13C22×C414yes—
186C22×Dic13C2×D414yes—
187C22×Dic13C2×D414yes—
188C22×Dic13#4114yes—
189C13×C22⋊C4—28no—
190C13×C22⋊C4—28no—
191C42×C13—44no—
192C42×C13—44no—
193C22⋊Dic13—28no—
194C22⋊Dic13—44no—
195#6101—216no—
196#6101—216no—
197#6101—216no—
198C23×C4⋊C4—131no—
199C2×#387—132no—
200C2×#385—131no—
201C2×C4×C22⋊C4D1311yescharacteristic
202#1442—132no—
203#1441—132no—
204#1436—132no—
205C22⋊#95—131no—
206C22⋊#95—131no—
207C22⋊#95—131no—
208C22⋊#95—131no—
209C23×C13—24no—
210C23×C13—24no—
211C23×C13—22no—
212C23×C13—22no—
213C23×C13—24no—
214C23×C13—22no—
215C23×C13—22no—
216C23×C13—24no—
217C23×C13—24no—
218C23×C13—24no—
219C23×C13—24no—
220C23×C13—24no—
221C23×C13—24no—
222C23×C13—24no—
223C23×C13—24no—
224C23×C13—24no—
225C23×C13#11714yes—
226C23×C13C4×D414yes—
227C23×C13C4×Q814yes—
228C23×C13C22⋊Q814yes—
229C23×C13#12314yes—
230C23×C13#12514yes—
231C23×C13#12614yes—
232C23×C13C23×C411yescharacteristic
233C23×C13C22×D411yescharacteristic
234C23×C13C22×D412yes—
235C23×C13C22×Q811yescharacteristic
236C23×C13C2×#4111yescharacteristic
237C23×C13C2×#4111yescharacteristic
238C23×C13C2×#4111yescharacteristic
239C23×C13C2×#4112yes—
240C23×C13C2×#4111yescharacteristic
241C23×C13H(5,𝔽2)12yes—
242C23×C132−1+412yes—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—24no—
246C2×C4×C13—24no—
247C2×C4×C13—24no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—24no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—24no—
257C2×C4×C13—24no—
258C2×C4×C13—416no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—42no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—42no—
269C2×C4×C13—42no—
270C2×C4×C13—42no—
271C2×C4×C13—42no—
272C2×C4×C13—42no—
273C2×C4×C13#12014yes—
274C2×Dic13—232no—
275C2×Dic13—24no—
276C2×Dic13—24no—
277C2×Dic13—24no—
278C2×Dic13—24no—
279C2×Dic13—24no—
280C2×Dic13—24no—
281C2×Dic13—24no—
282C2×Dic13—24no—
283C2×Dic13—24no—
284C2×Dic13—24no—
285C2×Dic13—24no—
286C2×Dic13—24no—
287C2×Dic13—416no—
288C2×Dic13—42no—
289C2×Dic13—42no—
290C2×Dic13—42no—
291C2×Dic13—42no—
292C2×Dic13—42no—
293C2×Dic13—42no—
294C2×Dic13—42no—
295C2×Dic13—42no—
296C2×Dic13—42no—
297C2×Dic13—42no—
298C2×Dic13—42no—
299C2×Dic13—42no—
300C2×Dic13—42no—
301C2×Dic13—42no—
302C2×Dic13—42no—
303C2×Dic13C4×D418yes—
304C24×C4—131no—
305C24×C4—131no—
306C24×C4C2×D1311yescharacteristic
307C22×C4⋊C4—132no—
308C22×C4⋊C4—132no—
309C22×C4⋊C4—132no—
310C22×C4⋊C4—268no—
311C22×C4⋊C4—264no—
312C22×C22⋊C4—131no—
313C22×C22⋊C4—131no—
314C22×C22⋊C4C2×D1311yescharacteristic
315C22×C22⋊C4C2×D1311yescharacteristic
316C22×C42C2×D1311yescharacteristic
317C22×C42C2×D1311yescharacteristic
318#387—268no—
319#385—138no—
320C4×C22⋊C4—24no—
321C2×#95—132no—
322C2×#95—132no—
323C2×#95—132no—
324C2×#95—132no—
325C2×#95—132no—
326C2×#95—132no—
327C2×#95—132no—
328C2×#95—132no—
329C2×#95—132no—
330C2×#95C2×D1311yescharacteristic
331C2×#95C2×D1311yescharacteristic
332C22×C13—24no—
333C22×C13—24no—
334C22×C13—28no—
335C22×C13—28no—
336C22×C13—24no—
337C22×C13—24no—
338C22×C13—28no—
339C22×C13—28no—
340C22×C13—24no—
341C22×C13—24no—
342C22×C13#38718yescomplemented
343C22×C13C2×#11711yescharacteristic
344C22×C13C2×C4×D411yescharacteristic
345C22×C13C2×C4×D412yes—
346C22×C13C2×C4×Q811yescharacteristic
347C22×C13#52012yes—
348C22×C13C2×#12011yescharacteristic
349C22×C13C2×C22⋊Q811yescharacteristic
350C22×C13C2×#12311yescharacteristic
351C22×C13#52512yes—
352C22×C13#52511yescharacteristic
353C22×C13C2×#12511yescharacteristic
354C22×C13C2×#12611yescharacteristic
355C22×C13#53411yescharacteristic
356C22×C13#53611yescharacteristic
357C22×C13#53912yes—
358C22×C13#54612yes—
359C22×C13#54612yes—
360C22×C13#54812yes—
361C22×C13D4×Q812yes—
362C22×C13#55212yes—
363C22×C13#55212yes—
364C22×C13#55312yes—
365C22×C13#55412yes—
366C4×C13—24no—
367C4×C13—42no—
368C4×C13—42no—
369C4×C13—42no—
370C4×C13—42no—
371C4×C13—42no—
372Dic13—28no—
373Dic13—44no—
374Dic13—44no—
375Dic13—82no—
376C25C22×D1311yescharacteristic
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—42no—
381C23×C4—132no—
382C23×C4—134no—
383C23×C4—132no—
384C23×C4—132no—
385C23×C4—132no—
386C23×C4—264no—
387C23×C4—264no—
388C23×C4—264no—
389C23×C4—262no—
390C23×C4—262no—
391C23×C4—262no—
392C23×C4—262no—
393C23×C4—262no—
394C23×C4—522no—
395C23×C4C22⋊D1314yes—
396C23×C4C22×D1311yescharacteristic
397C23×C4C22×D1311yescharacteristic
398C23×C4C22×D1311yescharacteristic
399C23×C4C22×D1311yescharacteristic
400C23×C4C22×D1311yescharacteristic
401C23×C4C22×D1311yescharacteristic
402C2×C4⋊C4—138no—
403C2×C4⋊C4—138no—
404C2×C4⋊C4—264no—
405C2×C4⋊C4—264no—
406C2×C4⋊C4—264no—
407C2×C4⋊C4—264no—
408C2×C4⋊C4—2632no—
409C2×C4⋊C4—264no—
410C2×C4⋊C4—264no—
411C2×C4⋊C4—264no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—134no—
417C2×C22⋊C4—262no—
418C2×C22⋊C4—262no—
419C2×C22⋊C4—262no—
420C2×C22⋊C4—262no—
421C2×C22⋊C4—262no—
422C2×C22⋊C4Dic2614yes—
423C2×C22⋊C4C4×D1314yes—
424C2×C42—24no—
425C2×C42—22no—
426C2×C42—22no—
427C2×C42—42no—
428#95—22no—
429#95—22no—
430#95—138no—
431#95—264no—
432#95—264no—
433#95—264no—
434#95—264no—
435#95—264no—
436#95—522no—
437#95—522no—
438#95—522no—
439C2×C13—28no—
440C2×C13C2×#38711yescharacteristic
441C2×C13C4×#12011yescharacteristic
442C2×C13#232211yescharacteristic
443C2×C13#235212yes—
444C2×C13#236012yes—
445C2×C13#240511yescharacteristic
446C2×C13#240911yescharacteristic
447C2×C13#247312yes—
448C2×C13#248112yes—
449C2×C13#248411yescharacteristic
450C2×C13#257911yescharacteristic
451C24—22no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24C2×Dic2612yes—
456C24C2×C4×D1312yes—
457C24#612712yes—
458C24#612712yes—
459C24#612712yes—
460C24#612712yes—
461C24C2×C22⋊D1312yes—
462C24C23×D1311yescharacteristic
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—28no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—134no—
492C22×C4—134no—
493C22×C4—134no—
494C22×C4—134no—
495C22×C4—134no—
496C22×C4—2616no—
497C22×C4—268no—
498C22×C4—2616no—
499C22×C4—268no—
500C22×C4—268no—
501C22×C4—262no—
502C22×C4—262no—
503C22×C4—262no—
504C22×C4—262no—
505C22×C4—268no—
506C22×C4—262no—
507C22×C4—262no—
508C22×C4—262no—
509C22×C4—262no—
510C22×C4—262no—
511C22×C4—262no—
512C22×C4—262no—
513C22×C4—262no—
514C22×C4—262no—
515C22×C4—262no—
516C22×C4—262no—
517C22×C4—262no—
518C22×C4—262no—
519C22×C4—262no—
520C22×C4—262no—
521C22×C4—262no—
522C22×C4—262no—
523C22×C4—262no—
524C22×C4—262no—
525C22×C4—528no—
526C22×C4—528no—
527C22×C4—524no—
528C22×C4—524no—
529C22×C4C2×Dic2612yes—
530C22×C4C2×Dic2612yes—
531C22×C4C2×C4×D1312yes—
532C22×C4C2×C4×D1312yes—
533C22×C4D4×D1314yes—
534C22×C4C2×C22⋊D1312yes—
535C22×C4C2×C22⋊D1312yes—
536C4⋊C4—2616no—
537C4⋊C4—2616no—
538C4⋊C4—2616no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—268no—
542C22⋊C4—524no—
543C42—44no—
544C42—44no—
545C13#1403911yescharacteristic, complemented, Sylow
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23C4×Dic2614yes—
563C23#8812414yes—
564C23#8812614yes—
565C23#8813014yes—
566C23#8817814yes—
567C23C4×C22⋊D1314yes—
568C23#8818314yes—
569C23C22×Dic2611yescharacteristic
570C23C22×C4×D1311yescharacteristic
571C23C2×#612711yescharacteristic
572C23C2×#612711yescharacteristic
573C23C2×#612711yescharacteristic
574C23C2×#612711yescharacteristic
575C23C2×D4×D1312yes—
576C23#8825112yes—
577C23#41×D1312yes—
578C23#8825712yes—
579C23C22×C22⋊D1311yescharacteristic
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—138no—
611C2×C4—2632no—
612C2×C4—264no—
613C2×C4—264no—
614C2×C4—264no—
615C2×C4—264no—
616C2×C4—264no—
617C2×C4—264no—
618C2×C4—264no—
619C2×C4—264no—
620C2×C4—264no—
621C2×C4—264no—
622C2×C4—264no—
623C2×C4—264no—
624C2×C4—5216no—
625C2×C4—522no—
626C2×C4—522no—
627C2×C4—522no—
628C2×C4—522no—
629C2×C4—522no—
630C2×C4—522no—
631C2×C4—522no—
632C2×C4—522no—
633C2×C4—522no—
634C2×C4—522no—
635C2×C4—522no—
636C2×C4—522no—
637C2×C4—522no—
638C2×C4—522no—
639C2×C4—522no—
640C2×C4#8819114yes—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22#1171859418yescomplemented
652C22C2×C4×Dic2611yescharacteristic
653C22C2×#8812411yescharacteristic
654C22C2×#8812611yescharacteristic
655C22C2×#8813011yescharacteristic
656C22#1171919512yes—
657C22#1171919512yes—
658C22#1171923612yes—
659C22#1171923612yes—
660C22D4×Dic2612yes—
661C22#1171924412yes—
662C22#1171924512yes—
663C22C4×D4×D1312yes—
664C22#1171925112yes—
665C22#1171925212yes—
666C22#1171925712yes—
667C22#1171926912yes—
668C22C2×#8817811yescharacteristic
669C22C2×C4×C22⋊D1311yescharacteristic
670C22C2×#8818311yescharacteristic
671C22C2×#8819111yescharacteristic
672C22#1171950911yescharacteristic
673C22#1171953011yescharacteristic
674C22#1171953311yescharacteristic
675C4—24no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—268no—
682C4—524no—
683C4—524no—
684C4—1042no—
685C2—28no—
686C2C2×#1171859411yescharacteristic
687C2C4×#8819111yescharacteristic
688C2#1311750111yescharacteristic
689C2#1311775912yes—
690C2#1311784212yes—
691C2#1311816311yescharacteristic
692C2#1311820111yescharacteristic
693C2#1311897712yes—
694C2#1311909712yes—
695C2#1311911911yescharacteristic
696C2#1312017911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C4×C13×C22⋊C4C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC2×C4×C22⋊C4D13
hypocenterC13#14039
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11718594C2semidirect
4. #13113094C2semidirect
5. #13113087C2semidirect
6. #13113069C2semidirect
38. #11718607C4semidirect
90. C2×#6101C2×C4semidirect
102. #88071C2×C4semidirect
342. C22×C13#387semidirect
545. C13#14039semidirect
651. C22#11718594semidirect

Maximal subgroups

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

subgroupindex
C23×#61012
C2×#117185942
C2×#117186072
#131130942
#131130872
#131130692
#131130592
#131130582
C2×C4×C13×C22⋊C42
#1403913