Finite Group Databasebeta

← #15829010

Subgroups — #15829010

Order 3328.

1386 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88178C211yesmaximal, characteristic
2C2×#11718868C211yesmaximal, characteristic
3#13113201C214yesmaximal, complemented
4#13113199C214yesmaximal, complemented
5#13113165C214yesmaximal, complemented
6C2×C13×#407C211yesmaximal, characteristic
7C24×C4×C13C2211yescharacteristic
8C23×Dic26C411yescharacteristic
9C22×C13×M16—21no—
10C22×C22⋊Dic13C412yes—
11C22×C4⋊Dic13C411yescharacteristic
12C2×#88178—21no—
13C2×#88178—21no—
14C2×#88178—21no—
15C2×#88178—21no—
16C2×#88178—21no—
17C2×#88178—21no—
18C2×#88178—21no—
19C2×#88178—21no—
20C2×#88178C412yes—
21C2×#88178C412yes—
22C2×#88178C2212yes—
23C2×#88178C2212yes—
24C22×#6101C2212yes—
25C22×#6099—21no—
26C13×#407—21no—
27C13×#407—21no—
28C13×#407C2212yes—
29C13×#407C2212yes—
30C2×C13×C22⋊C8—21no—
31C2×C13×C22⋊C8C2212yes—
32#11718868—21no—
33#11718868—21no—
34#11718868C2212yes—
35#11718868C2212yes—
36C2×C22⋊#1055—21no—
37C2×C22⋊#1055C2212yes—
38#11718061C22116yescomplemented
39C25×C13C2×C411yescharacteristic
40C23×C4×C13—44no—
41C23×C4×C13C2×C411yescharacteristic
42C23×C4×C13C2×C411yescharacteristic
43C23×C4×C13C2×C411yescharacteristic
44C23×C4×C13D411yescharacteristic
45C23×C4×C13D411yescharacteristic
46C23×C4×C13D411yescharacteristic
47C23×C4×C13D411yescharacteristic
48C23×C4×C13D411yescharacteristic
49C23×C4×C13D411yescharacteristic
50C23×C4×C13D411yescharacteristic
51C23×C4×C13D411yescharacteristic
52C23×C4×C13C2311yescharacteristic
53C23×C4×C13C2311yescharacteristic
54C23×C4×C13C2311yescharacteristic
55C23×Dic13C2×C412yes—
56C23×Dic13C2×C412yes—
57C22×Dic26—21no—
58C22×Dic26—21no—
59C22×Dic26—21no—
60C22×Dic26—21no—
61C22×Dic26—44no—
62C22×Dic26C2×C412yes—
63C22×Dic26C2×C412yes—
64C2×C13×M16—22no—
65C2×C13×M16—22no—
66C2×C13×M16—44no—
67C22×C8×C13—22no—
68C2×C22⋊Dic13—22no—
69C2×C22⋊Dic13—22no—
70C2×C22⋊Dic13—22no—
71C2×C22⋊Dic13—22no—
72C2×C22⋊Dic13C2×C414yes—
73C2×C22⋊Dic13C2×C414yes—
74C2×C4⋊Dic13—21no—
75C2×C4⋊Dic13—21no—
76C2×C4⋊Dic13—21no—
77C2×C4⋊Dic13—21no—
78C2×C4⋊Dic13C2×C412yes—
79C2×C4⋊Dic13C2×C412yes—
80#88178—22no—
81#88178—22no—
82#88178—22no—
83#88178—22no—
84#88178—22no—
85#88178—22no—
86#88178—22no—
87#88178—22no—
88#88178—22no—
89#88178—22no—
90#88178—22no—
91#88178—22no—
92#88178—22no—
93#88178—22no—
94#88178—22no—
95#88178—22no—
96C2×#6101—22no—
97C2×#6101—22no—
98C2×#6101—22no—
99C2×#6101—22no—
100C2×#6101C2×C414yes—
101C2×#6101C2314yes—
102C2×#6099—22no—
103C2×#6099—22no—
104C2×#6099—44no—
105C22×#1055—22no—
106C13×C22⋊C8—22no—
107C13×C22⋊C8—22no—
108C13×C22⋊C8—22no—
109C13×C22⋊C8C2314yes—
110C22⋊#1055—22no—
111C22⋊#1055—22no—
112C22⋊#1055—22no—
113C22⋊#1055C2314yes—
114#17397—131nomaximal, Sylow
115C24×C13—24no—
116C24×C13—42no—
117C24×C13—42no—
118C24×C13C22⋊C411yescharacteristic
119C24×C13C22⋊C411yescharacteristic
120C24×C13C22⋊C411yescharacteristic
121C24×C13C22⋊C411yescharacteristic
122C24×C13C22×C411yescharacteristic
123C24×C13C22×C411yescharacteristic
124C24×C13C22×C411yescharacteristic
125C22×C4×C13—24no—
126C22×C4×C13—24no—
127C22×C4×C13—24no—
128C22×C4×C13—42no—
129C22×C4×C13—42no—
130C22×C4×C13—42no—
131C22×C4×C13—42no—
132C22×C4×C13—42no—
133C22×C4×C13—42no—
134C22×C4×C13—42no—
135C22×C4×C13—42no—
136C22×C4×C13—42no—
137C22×C4×C13—42no—
138C22×C4×C13—42no—
139C22×C4×C13C22⋊C411yescharacteristic
140C22×C4×C13C22⋊C411yescharacteristic
141C22×C4×C13C22⋊C411yescharacteristic
142C22×C4×C13C22⋊C411yescharacteristic
143C22×C4×C13C22⋊C411yescharacteristic
144C22×C4×C13C22⋊C411yescharacteristic
145C22×C4×C13C22⋊C411yescharacteristic
146C22×C4×C13C22⋊C411yescharacteristic
147C22×C4×C13C22⋊C411yescharacteristic
148C22×C4×C13C22⋊C411yescharacteristic
149C22×C4×C13C22⋊C411yescharacteristic
150C22×C4×C13C22⋊C411yescharacteristic
151C22×C4×C13C22×C411yescharacteristic
152C22×C4×C13C22×C411yescharacteristic
153C22×C4×C13C22×C411yescharacteristic
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D411yescharacteristic
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2×D411yescharacteristic
159C22×C4×C13C2×D411yescharacteristic
160C22×C4×C13C2×D411yescharacteristic
161C22×C4×C13C2×D411yescharacteristic
162C22×C4×C13C2×D411yescharacteristic
163C22×C4×C13C2×D411yescharacteristic
164C22×C4×C13C2×D411yescharacteristic
165C22×C4×C13C2×D411yescharacteristic
166C22×C4×C13C2411yescharacteristic
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—22no—
170C22×Dic13—22no—
171C22×Dic13—44no—
172C22×Dic13—44no—
173C22×Dic13C22×C414yes—
174C22×Dic13C22×C414yes—
175C2×Dic26—22no—
176C2×Dic26—22no—
177C2×Dic26—22no—
178C2×Dic26—22no—
179C2×Dic26—44no—
180C2×Dic26—44no—
181C2×Dic26—44no—
182C2×Dic26—44no—
183C2×Dic26—44no—
184C2×Dic26—44no—
185C13×M16—44no—
186C13×M16—44no—
187C2×C8×C13—24no—
188C2×C8×C13—44no—
189C22⋊Dic13—24no—
190C22⋊Dic13—24no—
191C22⋊Dic13—24no—
192C22⋊Dic13—24no—
193C4⋊Dic13—22no—
194C4⋊Dic13—22no—
195C4⋊Dic13—22no—
196C4⋊Dic13—22no—
197#6101—24no—
198#6101—24no—
199#6101—24no—
200#6101—24no—
201#6099—44no—
202#6099—44no—
203C2×#1055—24no—
204C2×#1055—44no—
205C22×C22⋊Q8—131no—
206C2×#407—131no—
207C2×#407D1311yescharacteristic
208#1472—134no—
209#1472—134no—
210#1463—134no—
211C23×C13—22no—
212C23×C13—22no—
213C23×C13—22no—
214C23×C13—22no—
215C23×C13—22no—
216C23×C13—22no—
217C23×C13—22no—
218C23×C13—22no—
219C23×C13—22no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13—22no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13—41no—
228C23×C13—41no—
229C23×C13—41no—
230C23×C13—41no—
231C23×C13—416no—
232C23×C13#10114yes—
233C23×C13C2×C22⋊C411yescharacteristic
234C23×C13C2×C22⋊C411yescharacteristic
235C23×C13C2×C22⋊C411yescharacteristic
236C23×C13C2×C22⋊C411yescharacteristic
237C23×C13C2×C22⋊C411yescharacteristic
238C23×C13C2×C22⋊C411yescharacteristic
239C23×C13C23×C411yescharacteristic
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13#9914yes—
268C2×C4×C13C2×C22⋊C411yescharacteristic
269C2×C4×C13C2×C22⋊C411yescharacteristic
270C2×C4×C13C2×C22⋊C411yescharacteristic
271C2×C4×C13C2×C22⋊C411yescharacteristic
272C2×C4×C13C2×C22⋊C411yescharacteristic
273C2×C4×C13C2×C22⋊C411yescharacteristic
274C2×C4×C13C22×D411yescharacteristic
275C2×C4×C13C22×D411yescharacteristic
276C2×Dic13—24no—
277C2×Dic13—24no—
278C2×Dic13—28no—
279C2×Dic13—44no—
280C2×Dic13—44no—
281C2×Dic13—44no—
282C2×Dic13—44no—
283C2×Dic13—44no—
284Dic26—44no—
285Dic26—44no—
286Dic26—44no—
287Dic26—44no—
288C8×C13—44no—
289#1055—44no—
290C23×Q8—131no—
291C24×C4C2×D1311yescharacteristic
292C22×M16—21no—
293C22×M16—261no—
294C2×C22⋊Q8—132no—
295C2×C22⋊Q8—132no—
296C2×C22⋊Q8—132no—
297C2×C22⋊Q8—132no—
298C2×C22⋊Q8—261no—
299C2×C22⋊Q8—261no—
300C2×C22⋊Q8—261no—
301C2×C22⋊Q8—261no—
302C2×C22⋊Q8—261no—
303C2×C22⋊Q8—261no—
304C2×C22⋊Q8—261no—
305C2×C22⋊Q8—261no—
306C22×C4⋊C4—131no—
307C22×C4⋊C4—132no—
308C22×C22⋊C4—132no—
309#407—21no—
310#407—21no—
311#407—132no—
312#407—132no—
313#407—261no—
314#407—261no—
315#407C2×D1312yes—
316#407C2×D1312yes—
317C2×C22⋊C8—21no—
318C2×C22⋊C8—132no—
319C2×C22⋊C8—261no—
320C2×C22⋊C8C2×D1312yes—
321#324—1316no—
322C22×C13—28no—
323C22×C13—21no—
324C22×C13—21no—
325C22×C13—21no—
326C22×C13—21no—
327C22×C13—21no—
328C22×C13—21no—
329C22×C13—21no—
330C22×C13—21no—
331C22×C13—21no—
332C22×C13—21no—
333C22×C13—21no—
334C22×C13—21no—
335C22×C13—21no—
336C22×C13—21no—
337C22×C13—21no—
338C22×C13—21no—
339C22×C13—44no—
340C22×C13—44no—
341C22×C13—44no—
342C22×C13—44no—
343C22×C13—44no—
344C22×C13—44no—
345C22×C13C2×#9912yes—
346C22×C13C2×#10112yes—
347C22×C13C2×#10112yes—
348C22×C13C2×#10112yes—
349C22×C13C22×C22⋊C411yescharacteristic
350C22×C13#56812yes—
351C4×C13—21no—
352C4×C13—21no—
353C4×C13—21no—
354C4×C13—21no—
355C4×C13—21no—
356C4×C13—21no—
357C4×C13C2×#9912yes—
358C4×C13C2×#9912yes—
359Dic13—44no—
360Dic13—44no—
361C25C4×D1311yescharacteristic
362C22×Q8—132no—
363C22×Q8—132no—
364C22×Q8—261no—
365C22×Q8—261no—
366C22×Q8—261no—
367C22×Q8—261no—
368C22×Q8—524no—
369C23×C4—44no—
370C23×C4—132no—
371C23×C4—132no—
372C23×C4C4×D1311yescharacteristic
373C23×C4C4×D1311yescharacteristic
374C23×C4C4×D1311yescharacteristic
375C23×C4D5211yescharacteristic
376C23×C4D5211yescharacteristic
377C23×C4D5211yescharacteristic
378C23×C4D5211yescharacteristic
379C23×C4C22⋊D1311yescharacteristic
380C23×C4C22⋊D1311yescharacteristic
381C23×C4C22⋊D1311yescharacteristic
382C23×C4C22⋊D1311yescharacteristic
383C23×C4C22×D1311yescharacteristic
384C23×C4C22×D1311yescharacteristic
385C23×C4C22×D1311yescharacteristic
386C2×M16—22no—
387C2×M16—22no—
388C2×M16—44no—
389C2×M16—262no—
390C2×M16—262no—
391C2×M16—524no—
392C22×C8—22no—
393C22×C8—262no—
394C22⋊Q8—262no—
395C22⋊Q8—262no—
396C22⋊Q8—262no—
397C22⋊Q8—262no—
398C22⋊Q8—262no—
399C22⋊Q8—262no—
400C22⋊Q8—262no—
401C22⋊Q8—262no—
402C22⋊Q8—262no—
403C22⋊Q8—262no—
404C22⋊Q8—262no—
405C22⋊Q8—262no—
406C22⋊Q8—262no—
407C22⋊Q8—262no—
408C22⋊Q8—262no—
409C22⋊Q8—262no—
410C2×C4⋊C4—134no—
411C2×C4⋊C4—132no—
412C2×C4⋊C4—132no—
413C2×C4⋊C4—134no—
414C2×C4⋊C4—261no—
415C2×C4⋊C4—261no—
416C2×C4⋊C4—261no—
417C2×C4⋊C4—261no—
418C2×C4⋊C4—262no—
419C2×C4⋊C4—262no—
420C2×C4⋊C4—262no—
421C2×C4⋊C4—262no—
422C2×C22⋊C4—134no—
423C2×C22⋊C4—134no—
424C2×C22⋊C4—262no—
425C2×C22⋊C4—262no—
426C2×C22⋊C4—262no—
427C2×C22⋊C4—262no—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—22no—
431C22⋊C8—134no—
432C22⋊C8—262no—
433C22⋊C8—262no—
434C22⋊C8—262no—
435C22⋊C8C22×D1314yes—
436C2×C13—22no—
437C2×C13—22no—
438C2×C13—22no—
439C2×C13—22no—
440C2×C13—22no—
441C2×C13—41no—
442C2×C13—41no—
443C2×C13#287911yescharacteristic
444C2×C13C22×#9911yescharacteristic
445C2×C13C22×#10111yescharacteristic
446C24—24no—
447C24—42no—
448C24—42no—
449C24#610311yescharacteristic
450C24#610311yescharacteristic
451C24#610311yescharacteristic
452C24#610311yescharacteristic
453C24C2×C4×D1311yescharacteristic
454C24C2×C4×D1311yescharacteristic
455C24C2×C4×D1311yescharacteristic
456C2×Q8—262no—
457C2×Q8—262no—
458C2×Q8—262no—
459C2×Q8—262no—
460C2×Q8—524no—
461C2×Q8—524no—
462C2×Q8—524no—
463C2×Q8—524no—
464C2×Q8—524no—
465C2×Q8—524no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—134no—
481C22×C4—134no—
482C22×C4—262no—
483C22×C4—262no—
484C22×C4—262no—
485C22×C4—262no—
486C22×C4—524no—
487C22×C4—524no—
488C22×C4#610311yescharacteristic
489C22×C4#610311yescharacteristic
490C22×C4#610311yescharacteristic
491C22×C4#610311yescharacteristic
492C22×C4#610311yescharacteristic
493C22×C4#610311yescharacteristic
494C22×C4#610311yescharacteristic
495C22×C4#610311yescharacteristic
496C22×C4#610311yescharacteristic
497C22×C4#610311yescharacteristic
498C22×C4#610311yescharacteristic
499C22×C4#610311yescharacteristic
500C22×C4C2×C4×D1311yescharacteristic
501C22×C4C2×C4×D1311yescharacteristic
502C22×C4C2×C4×D1311yescharacteristic
503C22×C4C2×D5211yescharacteristic
504C22×C4C2×D5211yescharacteristic
505C22×C4C2×D5211yescharacteristic
506C22×C4C2×D5211yescharacteristic
507C22×C4C2×D5211yescharacteristic
508C22×C4C2×D5211yescharacteristic
509C22×C4C2×C22⋊D1311yescharacteristic
510C22×C4C2×C22⋊D1311yescharacteristic
511C22×C4C2×C22⋊D1311yescharacteristic
512C22×C4C2×C22⋊D1311yescharacteristic
513C22×C4C2×C22⋊D1311yescharacteristic
514C22×C4C2×C22⋊D1311yescharacteristic
515C22×C4C23×D1311yescharacteristic
516M16—44no—
517M16—44no—
518M16—524no—
519M16—524no—
520C2×C8—24no—
521C2×C8—44no—
522C2×C8—264no—
523C2×C8—524no—
524C4⋊C4—264no—
525C4⋊C4—264no—
526C4⋊C4—262no—
527C4⋊C4—262no—
528C4⋊C4—262no—
529C4⋊C4—262no—
530C4⋊C4—264no—
531C4⋊C4—264no—
532C22⋊C4—264no—
533C22⋊C4—264no—
534C22⋊C4—264no—
535C22⋊C4—264no—
536C13#1739711yescharacteristic, complemented, Sylow
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—416no—
558C23#8806414yes—
559C23C2×#610311yescharacteristic
560C23C2×#610311yescharacteristic
561C23C2×#610311yescharacteristic
562C23C2×#610311yescharacteristic
563C23C2×#610311yescharacteristic
564C23C2×#610311yescharacteristic
565C23C22×C4×D1311yescharacteristic
566Q8—524no—
567Q8—524no—
568Q8—524no—
569Q8—524no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—268no—
598C2×C4—264no—
599C2×C4—264no—
600C2×C4—524no—
601C2×C4—524no—
602C2×C4—524no—
603C2×C4—524no—
604C2×C4—524no—
605C2×C4#8804614yes—
606C2×C4C2×#610311yescharacteristic
607C2×C4C2×#610311yescharacteristic
608C2×C4C2×#610311yescharacteristic
609C2×C4C2×#610311yescharacteristic
610C2×C4C2×#610311yescharacteristic
611C2×C4C2×#610311yescharacteristic
612C2×C4C22×D5211yescharacteristic
613C2×C4C22×C22⋊D1311yescharacteristic
614C8—44no—
615C8—524no—
616C22—28no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22C2×#8804612yes—
640C22C2×#8806412yes—
641C22C2×#8806412yes—
642C22C2×#8806412yes—
643C22#1171945012yes—
644C22C22×#610311yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—524no—
652C4—524no—
653C4C2×#8804612yes—
654C4C2×#8804612yes—
655C2—22no—
656C2—22no—
657C2—22no—
658C2—22no—
659C2—22no—
660C2—41no—
661C2—41no—
662C2#1312328011yescharacteristic
663C2C22×#8804611yescharacteristic
664C2C22×#8806411yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC23×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#407C2
socleC22×C13C22×C22⋊C4
prime radicalC25×C13C2×C4
cosocleC22×C4×C13C24
hypercenterC2×#407D13
hypocenterC13#17397
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
3. #13113201C2semidirect
4. #13113199C2semidirect
5. #13113165C2semidirect
38. #11718061C22semidirect
536. C13#17397semidirect

Maximal subgroups

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

subgroupindex
C22×#881782
C2×#117188682
#131132012
#131131992
#131131652
C2×C13×#4072
#1739713