Finite Group Databasebeta

← #15793588

Subgroups — #15793588

Order 3328.

1586 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718652C211yesmaximal, characteristic
2C2×#11718600C211yesmaximal, characteristic
3#13113100C211yesmaximal, characteristic, complemented
4#13113089C211yesmaximal, characteristic, complemented
5#13113084C212yesmaximal, complemented
6#13113080C212yesmaximal, complemented
7#13113080C212yesmaximal, complemented
8#13113078C212yesmaximal, complemented
9#13113076C211yesmaximal, characteristic
10#13113063C211yesmaximal, characteristic, complemented
11C2×C13×#387C211yesmaximal, characteristic, complemented
12C23×D52C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C22×C42×C13C2211yescharacteristic
16C22×#6103C2212yes—
17C22×#6103C2212yes—
18C22×#6103C2212yes—
19C22×#6103C2212yes—
20C22×#6103C2212yes—
21C22×#6103C2212yes—
22C22×C4⋊Dic13C2211yescharacteristic
23C22×#6101C2212yes—
24C13×#387—24no—
25C2×C13×#95C2211yescharacteristic
26C2×C13×#95C2211yescharacteristic
27C2×C13×#95C2211yescharacteristic
28C2×C13×#95C2211yescharacteristic
29C2×#88071C2211yescharacteristic
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2211yescharacteristic
36#11718652C418yescomplemented
37#11718600—24no—
38C24×D13C2312yes—
39C23×C4×C13C2311yescharacteristic
40C23×C4×C13C2311yescharacteristic
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2311yescharacteristic
43C23×C4×C13C2311yescharacteristic
44C23×C4×C13C2311yescharacteristic
45C23×C4×C13C2311yescharacteristic
46C23×Dic13C2312yes—
47C23×Dic13C2312yes—
48C23×Dic13C2312yes—
49C22×D52—22no—
50C22×D52—22no—
51C22×D52—42no—
52C22×D52—42no—
53C22×D52C2×C414yes—
54C2×C13×C4⋊C4—22no—
55C2×C13×C4⋊C4—22no—
56C2×C13×C4⋊C4—22no—
57C2×C13×C4⋊C4—42no—
58C2×C13×C4⋊C4C2×C414yes—
59C2×C42×C13—42no—
60C2×C42×C13D414yes—
61C2×#6103—24no—
62C2×#6103—24no—
63C2×#6103—42no—
64C2×#6103—42no—
65C2×#6103—42no—
66C2×#6103—42no—
67C2×#6103—42no—
68C2×#6103—42no—
69C2×#6103—42no—
70C2×#6103—42no—
71C2×#6103—42no—
72C2×#6103—42no—
73C2×#6103—42no—
74C2×#6103—42no—
75C2×#6103—42no—
76C2×#6103—42no—
77C2×#6103C2×C418yes—
78C2×#6103C2×C418yes—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13—42no—
81C2×#6101—42no—
82C2×#6101—42no—
83C2×#6101—42no—
84C13×#95—22no—
85C13×#95—22no—
86C13×#95—22no—
87C13×#95—22no—
88#88071—22no—
89#88071—42no—
90#88071—42no—
91#88071—42no—
92#88071—42no—
93#88071—42no—
94#88071C2×C414yes—
95#14624—131nomaximal, Sylow
96C24×C13C2411yescharacteristic
97C23×D13—22no—
98C23×D13—22no—
99C23×D13—22no—
100C23×D13—22no—
101C23×D13—22no—
102C23×D13—22no—
103C23×D13—22no—
104C23×D13—22no—
105C23×D13—22no—
106C23×D13—22no—
107C23×D13—22no—
108C23×D13—22no—
109C23×D13—22no—
110C23×D13—22no—
111C23×D13C22×C414yes—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—21no—
115C22×C4×C13—21no—
116C22×C4×C13—21no—
117C22×C4×C13—21no—
118C22×C4×C13—21no—
119C22×C4×C13—21no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—22no—
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—22no—
128C22×C4×C13—22no—
129C22×C4×C13—22no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—22no—
135C22×C4×C13—21no—
136C22×C4×C13—22no—
137C22×C4×C13—22no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—22no—
141C22×C4×C13—21no—
142C22×C4×C13C22×C412yes—
143C22×C4×C13C22×C412yes—
144C22×C4×C13C22×C412yes—
145C22×C4×C13C2×D412yes—
146C22×C4×C13C2×D412yes—
147C22×C4×C13C2×D412yes—
148C22×C4×C13#4112yes—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—22no—
160C22×Dic13—22no—
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×Dic13C22×C414yes—
170C2×D52—28no—
171C2×D52—48no—
172C2×D52—44no—
173C2×D52—44no—
174C2×D52—48no—
175C13×C4⋊C4—28no—
176C13×C4⋊C4—44no—
177C42×C13—44no—
178#6103—48no—
179#6103—48no—
180#6103—48no—
181#6103—48no—
182#6103—48no—
183#6103—48no—
184C4⋊Dic13—44no—
185#6101—48no—
186C2×#394—131no—
187C2×#390—131no—
188C2×#387D1311yescharacteristic, complemented
189#1444—131no—
190#1441—131no—
191#1440—132no—
192#1439—132no—
193#1439—132no—
194#1439—132no—
195#1438—131no—
196#1434—131no—
197C23×C13C23×C411yescharacteristic
198C23×C13C22×D411yescharacteristic
199C23×C13C2×#4111yescharacteristic
200C23×C13H(5,𝔽2)12yes—
201C23×C13H(5,𝔽2)12yes—
202C23×C13H(5,𝔽2)11yescharacteristic
203C23×C13H(5,𝔽2)11yescharacteristic
204C23×C13H(5,𝔽2)11yescharacteristic
205C23×C13H(5,𝔽2)12yes—
206C23×C132−1+412yes—
207C23×C132−1+411yescharacteristic
208C22×D13—24no—
209C22×D13—24no—
210C22×D13—24no—
211C22×D13—24no—
212C22×D13—24no—
213C22×D13—24no—
214C22×D13—24no—
215C22×D13—42no—
216C22×D13—42no—
217C22×D13—42no—
218C22×D13—42no—
219C22×D13—42no—
220C22×D13—42no—
221C22×D13—42no—
222C22×D13—42no—
223C22×D13—42no—
224C22×D13—42no—
225C22×D13—42no—
226C22×D13—42no—
227C22×D13—42no—
228C22×D13—42no—
229C22×D13—42no—
230C22×D13—42no—
231C22×D13—42no—
232C22×D13—42no—
233C22×D13—42no—
234C22×D13—42no—
235C22×D13—42no—
236C22×D13—42no—
237C22×D13—42no—
238C22×D13—42no—
239C22×D13—42no—
240C22×D13—42no—
241C22×D13—42no—
242C22×D13—42no—
243C2×C4×C13—24no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—24no—
247C2×C4×C13—22no—
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—22no—
257C2×C4×C13—22no—
258C2×C4×C13—41no—
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—42no—
266C2×C4×C13—42no—
267C2×C4×C13—42no—
268C2×C4×C13—42no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—41no—
275C2×C4×C13—42no—
276C2×C4×C13—42no—
277C2×C4×C13—42no—
278C2×C4×C13—42no—
279C2×C4×C13C4×D414yes—
280C2×Dic13—24no—
281C2×Dic13—24no—
282C2×Dic13—24no—
283C2×Dic13—42no—
284C2×Dic13—42no—
285C2×Dic13—42no—
286C2×Dic13—42no—
287C2×Dic13—42no—
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—
301D52—416no—
302C23×D4—131no—
303C22×C4⋊C4—131no—
304C22×C4⋊C4—132no—
305C22×C4⋊C4C2×D1311yescharacteristic
306C22×C4⋊C4C2×D1311yescharacteristic
307C22×C22⋊C4—132no—
308C22×C22⋊C4—132no—
309C22×C22⋊C4—132no—
310C22×C22⋊C4—132no—
311C22×C22⋊C4—132no—
312C22×C22⋊C4—132no—
313C22×C42C2×D1311yescharacteristic
314#394—138no—
315#390—264no—
316#387—24no—
317C2×#95—131no—
318C2×#95—131no—
319C2×#95—132no—
320C2×#95—132no—
321C2×#95—132no—
322C2×#95—132no—
323C2×#95—132no—
324C2×#95C2×D1311yescharacteristic
325C2×#95C2×D1311yescharacteristic
326C2×#95C2×D1311yescharacteristic
327C2×#95C2×D1311yescharacteristic
328C22×C13C2×C4×D411yescharacteristic
329C22×C13#51811yescharacteristic
330C22×C13#51812yes—
331C22×C13#52011yescharacteristic
332C22×C13#52012yes—
333C22×C13#53411yescharacteristic
334C22×C13#53511yescharacteristic
335C22×C13#53511yescharacteristic
336C22×C13#53511yescharacteristic
337C22×C13#53611yescharacteristic
338C22×C13#53711yescharacteristic
339C22×C13#53812yes—
340C22×C13#54011yescharacteristic
341C22×C13#54211yescharacteristic
342C22×C13#54212yes—
343C22×C13#56012yes—
344C22×C13#56012yes—
345C22×C13#56012yes—
346C22×C13#56212yes—
347C22×C13#56212yes—
348C22×C13#56212yes—
349C22×C13#56212yes—
350C22×C13#56312yes—
351C2×D13—44no—
352C2×D13—44no—
353C2×D13—44no—
354C2×D13—44no—
355C2×D13—44no—
356C2×D13—44no—
357C2×D13—44no—
358C2×D13—82no—
359C2×D13—82no—
360C2×D13—82no—
361C2×D13—82no—
362C2×D13—82no—
363C2×D13—82no—
364C2×D13—82no—
365C2×D13—82no—
366C4×C13—24no—
367C4×C13—42no—
368C4×C13—42no—
369C4×C13—42no—
370C4×C13—42no—
371C4×C13—81no—
372C4×C13—81no—
373Dic13—44no—
374Dic13—82no—
375Dic13—82no—
376C25—132no—
377C22×D4—134no—
378C22×D4—262no—
379C22×D4—262no—
380C22×D4—522no—
381C22×D4—522no—
382C23×C4—132no—
383C23×C4—132no—
384C23×C4—132no—
385C23×C4C22×D1311yescharacteristic
386C23×C4C22×D1311yescharacteristic
387C23×C4C22×D1311yescharacteristic
388C23×C4C22×D1311yescharacteristic
389C23×C4C22×D1311yescharacteristic
390C23×C4C22×D1311yescharacteristic
391C23×C4C22×D1311yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—42no—
396C2×C4⋊C4—262no—
397C2×C4⋊C4—522no—
398C2×C4⋊C4—522no—
399C2×C4⋊C4—522no—
400C2×C4⋊C4—522no—
401C2×C4⋊C4C4×D1314yes—
402C2×C22⋊C4—138no—
403C2×C22⋊C4—138no—
404C2×C22⋊C4—264no—
405C2×C22⋊C4—264no—
406C2×C22⋊C4—522no—
407C2×C22⋊C4—522no—
408C2×C22⋊C4—522no—
409C2×C22⋊C4—522no—
410C2×C22⋊C4—522no—
411C2×C22⋊C4—522no—
412C2×C22⋊C4—522no—
413C2×C22⋊C4—522no—
414C2×C22⋊C4—522no—
415C2×C22⋊C4—522no—
416C2×C22⋊C4—522no—
417C2×C22⋊C4—522no—
418C2×C22⋊C4—522no—
419C2×C22⋊C4—522no—
420C2×C42—42no—
421C2×C42C22⋊D1314yes—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—134no—
427#95—262no—
428#95—522no—
429#95—522no—
430#95—522no—
431#95—522no—
432#95—522no—
433C2×C13#232111yescharacteristic
434C2×C13#232411yescharacteristic
435C2×C13#232611yescharacteristic
436C2×C13#237612yes—
437C2×C13#237712yes—
438C2×C13#281311yescharacteristic
439C2×C13#281611yescharacteristic
440C2×C13#281811yescharacteristic
441C2×C13#282011yescharacteristic
442C2×C13#282912yes—
443C2×C13#283012yes—
444D13—84no—
445C24—134no—
446C24—262no—
447C24—262no—
448C24—262no—
449C24—262no—
450C24—262no—
451C24—262no—
452C24—262no—
453C24—262no—
454C24—262no—
455C24—262no—
456C24—262no—
457C24—262no—
458C24—262no—
459C24—262no—
460C24C23×D1311yescharacteristic
461C2×D4—268no—
462C2×D4—528no—
463C2×D4—524no—
464C2×D4—528no—
465C2×D4—524no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—134no—
497C22×C4—262no—
498C22×C4—262no—
499C22×C4—262no—
500C22×C4—262no—
501C22×C4—262no—
502C22×C4—262no—
503C22×C4—262no—
504C22×C4—262no—
505C22×C4—262no—
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×C4C2×C4×D1312yes—
518C22×C4C2×C4×D1312yes—
519C22×C4C2×C4×D1312yes—
520C22×C4#612712yes—
521C22×C4C2×C22⋊D1312yes—
522C22×C4C2×C22⋊D1312yes—
523C22×C4C2×C22⋊D1312yes—
524C4⋊C4—28no—
525C4⋊C4—44no—
526C4⋊C4—524no—
527C4⋊C4—528no—
528C22⋊C4—528no—
529C22⋊C4—528no—
530C22⋊C4—528no—
531C22⋊C4—528no—
532C22⋊C4—528no—
533C22⋊C4—528no—
534C42—44no—
535C13#1462411yescharacteristic, complemented, Sylow
536C23—264no—
537C23—264no—
538C23—264no—
539C23—264no—
540C23—264no—
541C23—264no—
542C23—264no—
543C23—522no—
544C23—522no—
545C23—522no—
546C23—522no—
547C23—522no—
548C23—522no—
549C23—522no—
550C23—522no—
551C23—522no—
552C23—522no—
553C23—522no—
554C23—522no—
555C23—522no—
556C23—522no—
557C23—522no—
558C23—522no—
559C23—522no—
560C23—522no—
561C23—522no—
562C23—522no—
563C23—522no—
564C23—522no—
565C23—522no—
566C23—522no—
567C23—522no—
568C23—522no—
569C23—522no—
570C23—522no—
571C23C22×C4×D1311yescharacteristic
572C23C2×#612711yescharacteristic
573C23#8825112yes—
574C23#8825412yes—
575C23#8825611yescharacteristic
576C23#8825611yescharacteristic
577C23#8825611yescharacteristic
578C23#8825612yes—
579C23#8825612yes—
580C23#8825711yescharacteristic
581C23C22×C22⋊D1311yescharacteristic
582D4—5216no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—264no—
620C2×C4—264no—
621C2×C4—264no—
622C2×C4—522no—
623C2×C4—522no—
624C2×C4—522no—
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×C4C4×C22⋊D1314yes—
641C22—524no—
642C22—524no—
643C22—524no—
644C22—524no—
645C22—524no—
646C22—524no—
647C22—524no—
648C22—1042no—
649C22—1042no—
650C22—1042no—
651C22—1042no—
652C22—1042no—
653C22—1042no—
654C22—1042no—
655C22—1042no—
656C22#1171922611yescharacteristic
657C22#1171922811yescharacteristic
658C22#1171923911yescharacteristic
659C22#1171924011yescharacteristic
660C22#1171925012yes—
661C22#1171926512yes—
662C22#1171927912yes—
663C22#1171928712yes—
664C22#1171935212yes—
665C22#1171935212yes—
666C22#1171937112yes—
667C22#1171937212yes—
668C22#1171940412yes—
669C22#1171940512yes—
670C22#1171941612yes—
671C22#1171941612yes—
672C22C2×C4×C22⋊D1311yescharacteristic
673C22#1171950911yescharacteristic
674C22#1171952111yescharacteristic
675C22#1171953511yescharacteristic
676C22#1171953511yescharacteristic
677C22#1171953511yescharacteristic
678C22#1171953811yescharacteristic
679C4—24no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—524no—
687C4—1042no—
688C4—1042no—
689C2—1044no—
690C2#1311749011yescharacteristic
691C2#1311752911yescharacteristic
692C2#1311754311yescharacteristic
693C2#1311798312yes—
694C2#1311799112yes—
695C2#1312283811yescharacteristic
696C2#1312286311yescharacteristic
697C2#1312287611yescharacteristic
698C2#1312288611yescharacteristic
699C2#1312296612yes—
700C2#1312297712yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#387C2
socle, cosocleC24×C13C24
prime radicalC23×D52C22
hypercenterC2×#387D13
hypocenterC13#14624
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
3. #13113100C2semidirect
4. #13113089C2semidirect
5. #13113084C2semidirect
6. #13113080C2semidirect
7. #13113080C2semidirect
8. #13113078C2semidirect
10. #13113063C2semidirect
11. C2×C13×#387C2semidirect
36. #11718652C4semidirect
188. C2×#387D13semidirect
535. C13#14624semidirect

Maximal subgroups

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

subgroupindex
C2×#117186522
C2×#117186002
#131131002
#131130892
#131130842
#131130802
#131130782
#131130762
#131130632
C2×C13×#3872
#1462413