Finite Group Databasebeta

← #15789512

Subgroups — #15789512

Order 3328.

2086 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88132C211yesmaximal, characteristic
2C2×#11718603C211yesmaximal, characteristic
3C22×#88071C211yesmaximal, characteristic
4#13113106C212yesmaximal, complemented
5#13113102C212yesmaximal, complemented
6#13113088C212yesmaximal, complemented
7#13113087C212yesmaximal, complemented
8#13113059C211yesmaximal, characteristic
9#13113058C211yesmaximal, characteristic
10#13113058C211yesmaximal, characteristic
11C13×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C24×Dic13C2211yescharacteristic
14C24×Dic13C2211yescharacteristic
15C23×Dic26C2211yescharacteristic
16C22×C13×C22⋊C4C2211yescharacteristic
17C22×C13×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic13C2211yescharacteristic
19C22×C22⋊Dic13C2211yescharacteristic
20C22×C4⋊Dic13C2211yescharacteristic
21C22×#6101C2211yescharacteristic
22C22×#6101C2211yescharacteristic
23C22×#6101C2212yes—
24C2×#88132—24no—
25C2×#88132—24no—
26C2×#88132C418yescomplemented
27C2×C13×#95C2212yes—
28C2×C13×#95C2212yes—
29C2×#88071—24no—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2212yes—
38C2×#88071C2212yes—
39#11718603—24no—
40C25×C13C2311yescharacteristic
41C23×C4×C13—22no—
42C23×C4×C13—22no—
43C23×C4×C13—42no—
44C23×C4×C13—42no—
45C23×C4×C13Q814yescomplemented
46C23×C4×C13C2312yes—
47C23×C4×C13C2311yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2312yes—
50C23×Dic13—22no—
51C23×Dic13—22no—
52C23×Dic13—42no—
53C23×Dic13—42no—
54C23×Dic13—42no—
55C23×Dic13—42no—
56C23×Dic13C2×C414yes—
57C23×Dic13D414yes—
58C23×Dic13D414yes—
59C23×Dic13Q814yescomplemented
60C23×Dic13C2312yes—
61C23×Dic13C2311yescharacteristic
62C23×Dic13C2311yescharacteristic
63C23×Dic13C2312yes—
64C23×Dic13C2311yescharacteristic
65C23×Dic13C2311yescharacteristic
66C22×Dic26—22no—
67C22×Dic26—22no—
68C22×Dic26—42no—
69C22×Dic26—42no—
70C22×Dic26C2×C414yescomplemented
71C2×C13×C22⋊C4—22no—
72C2×C13×C22⋊C4—22no—
73C2×C13×C22⋊C4—22no—
74C2×C13×C22⋊C4—22no—
75C2×C13×C22⋊C4—22no—
76C2×C13×C22⋊C4C2×C414yes—
77C2×C22⋊Dic13—22no—
78C2×C22⋊Dic13—22no—
79C2×C22⋊Dic13—22no—
80C2×C22⋊Dic13—22no—
81C2×C22⋊Dic13—22no—
82C2×C22⋊Dic13C2×C414yes—
83C2×C4⋊Dic13—22no—
84C2×C4⋊Dic13—22no—
85C2×C4⋊Dic13C2×C414yescomplemented
86C2×#6101—22no—
87C2×#6101—22no—
88C2×#6101—22no—
89C2×#6101—22no—
90C2×#6101—24no—
91C2×#6101—42no—
92C2×#6101—42no—
93C2×#6101C2×C414yescomplemented
94C2×#6101C2×C414yescomplemented
95#88132—232no—
96C13×#95—42no—
97C13×#95—42no—
98#88071—24no—
99#88071—24no—
100#88071—24no—
101#88071—42no—
102#88071—42no—
103#88071—42no—
104#88071—42no—
105#88071—42no—
106#88071—42no—
107#14281—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×Q812yes—
116C24×C13C2×Q812yes—
117C24×C13#4112yes—
118C24×C13#4112yes—
119C24×C13C2411yescharacteristic
120C22×C4×C13—28no—
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—22no—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13—22no—
133C22×C4×C13—22no—
134C22×C4×C13—22no—
135C22×C4×C13—22no—
136C22×C4×C13—22no—
137C22×C4×C13—22no—
138C22×C4×C13—22no—
139C22×C4×C13—22no—
140C22×C4×C13—22no—
141C22×C4×C13—22no—
142C22×C4×C13—48no—
143C22×C4×C13—44no—
144C22×C4×C13—44no—
145C22×C4×C13—48no—
146C22×C4×C13C22×C412yes—
147C22×C4×C13C22×C412yes—
148C22×C4×C13C2×Q814yes—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—28no—
154C22×Dic13—22no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
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×Dic13—22no—
170C22×Dic13—22no—
171C22×Dic13—22no—
172C22×Dic13—22no—
173C22×Dic13—22no—
174C22×Dic13—22no—
175C22×Dic13—48no—
176C22×Dic13—48no—
177C22×Dic13—44no—
178C22×Dic13—44no—
179C22×Dic13—48no—
180C22×Dic13—44no—
181C22×Dic13—44no—
182C22×Dic13—48no—
183C22×Dic13C22⋊C4116yes—
184C22×Dic13C22×C412yes—
185C22×Dic13C22×C412yes—
186C22×Dic13C22×C412yes—
187C22×Dic13C22×C412yes—
188C22×Dic13C2×D412yes—
189C22×Dic13C2×D412yes—
190C22×Dic13C2×D412yes—
191C22×Dic13C2×D412yes—
192C22×Dic13C2×Q814yes—
193C2×Dic26—28no—
194C2×Dic26—48no—
195C2×Dic26—44no—
196C2×Dic26—48no—
197C2×Dic26—44no—
198C13×C22⋊C4—28no—
199C13×C22⋊C4—44no—
200C22⋊Dic13—28no—
201C22⋊Dic13—44no—
202C4⋊Dic13—28no—
203#6101—28no—
204#6101—28no—
205#6101—48no—
206C22×C22⋊Q8—131no—
207C2×#380—131no—
208C22×#95—131no—
209#1446—132no—
210#1445—132no—
211#1441—132no—
212#1441—132no—
213C22⋊#95—131no—
214C22⋊#95—131no—
215C22⋊#95—131no—
216C22⋊#95D1311yescharacteristic
217C23×C13—24no—
218C23×C13—24no—
219C23×C13—22no—
220C23×C13—22no—
221C23×C13—24no—
222C23×C13—22no—
223C23×C13—22no—
224C23×C13—24no—
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×C13C2×C22⋊C414yes—
234C23×C13C4×Q814yes—
235C23×C13C4×Q814yes—
236C23×C13C22⋊Q814yes—
237C23×C13C22⋊Q814yes—
238C23×C13#12414yes—
239C23×C13C4⋊Q814yes—
240C23×C13C23×C411yescharacteristic
241C23×C13C22×D411yescharacteristic
242C23×C13C22×D411yescharacteristic
243C23×C13C22×Q811yescharacteristic
244C23×C13C22×Q811yescharacteristic
245C23×C13C2×#4111yescharacteristic
246C23×C13C2×#4111yescharacteristic
247C23×C13H(5,𝔽2)12yes—
248C23×C13H(5,𝔽2)12yes—
249C23×C13H(5,𝔽2)12yes—
250C23×C132−1+412yes—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—24no—
254C2×C4×C13—24no—
255C2×C4×C13—24no—
256C2×C4×C13—24no—
257C2×C4×C13—24no—
258C2×C4×C13—416no—
259C2×C4×C13—41no—
260C2×C4×C13—42no—
261C2×C4×C13—41no—
262C2×C4×C13—42no—
263C2×C4×C13—41no—
264C2×C4×C13—42no—
265C2×C4×C13—41no—
266C2×C4×C13—42no—
267C2×C4×C13—42no—
268C2×C4×C13—42no—
269C2×C4×C13—42no—
270C2×C4×C13—42no—
271C2×C4×C13—42no—
272C2×C4×C13—42no—
273C2×C4×C13—42no—
274C2×C4×C13—42no—
275C2×C4×C13—42no—
276C2×Dic13—24no—
277C2×Dic13—24no—
278C2×Dic13—24no—
279C2×Dic13—22no—
280C2×Dic13—24no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—22no—
285C2×Dic13—22no—
286C2×Dic13—24no—
287C2×Dic13—24no—
288C2×Dic13—24no—
289C2×Dic13—24no—
290C2×Dic13—24no—
291C2×Dic13—416no—
292C2×Dic13—416no—
293C2×Dic13—41no—
294C2×Dic13—42no—
295C2×Dic13—41no—
296C2×Dic13—42no—
297C2×Dic13—41no—
298C2×Dic13—42no—
299C2×Dic13—41no—
300C2×Dic13—42no—
301C2×Dic13—42no—
302C2×Dic13—42no—
303C2×Dic13—42no—
304C2×Dic13—42no—
305C2×Dic13—42no—
306C2×Dic13—42no—
307C2×Dic13—42no—
308C2×Dic13—42no—
309C2×Dic13—42no—
310C2×Dic13C2×C22⋊C414yes—
311C2×Dic13C2×C22⋊C414yes—
312Dic26—416no—
313C23×Q8—131no—
314C24×C4—131no—
315C24×C4—131no—
316C24×C4C2×D1311yescharacteristic
317C2×C22⋊Q8—138no—
318C2×C22⋊Q8—264no—
319C2×C22⋊Q8—264no—
320C22×C4⋊C4—131no—
321C22×C4⋊C4—131no—
322C22×C4⋊C4—131no—
323C22×C4⋊C4—132no—
324C22×C22⋊C4—131no—
325C22×C22⋊C4—131no—
326C22×C22⋊C4C2×D1311yescharacteristic
327C22×C22⋊C4C2×D1311yescharacteristic
328#380—264no—
329C2×#95—132no—
330C2×#95—132no—
331C2×#95—132no—
332C2×#95—132no—
333C2×#95—132no—
334C2×#95—132no—
335C2×#95—132no—
336C2×#95—132no—
337C2×#95—132no—
338C2×#95—264no—
339C2×#95C2×D1312yes—
340C2×#95C2×D1312yes—
341C22×C13—24no—
342C22×C13—24no—
343C22×C13—28no—
344C22×C13—28no—
345C22×C13—24no—
346C22×C13—24no—
347C22×C13—24no—
348C22×C13—24no—
349C22×C13—28no—
350C22×C13—28no—
351C22×C13#39118yes—
352C22×C13C22×C22⋊C411yescharacteristic
353C22×C13C2×C4×Q811yescharacteristic
354C22×C13C2×C4×Q811yescharacteristic
355C22×C13#51812yes—
356C22×C13#52012yes—
357C22×C13C2×C22⋊Q811yescharacteristic
358C22×C13C2×C22⋊Q811yescharacteristic
359C22×C13C2×#12411yescharacteristic
360C22×C13C2×C4⋊Q811yescharacteristic
361C22×C13#53411yescharacteristic
362C22×C13#53412yes—
363C22×C13#53411yescharacteristic
364C22×C13#53511yescharacteristic
365C22×C13#53611yescharacteristic
366C22×C13#53712yes—
367C22×C13#53812yes—
368C22×C13#53812yes—
369C22×C13#53912yes—
370C22×C13#53912yes—
371C22×C13#54312yes—
372C22×C13#54312yes—
373C22×C13#54412yes—
374C22×C13#54412yes—
375C4×C13—42no—
376C4×C13—42no—
377C4×C13—44no—
378C4×C13—82no—
379Dic13—24no—
380Dic13—24no—
381Dic13—42no—
382Dic13—42no—
383Dic13—44no—
384Dic13—82no—
385C25C22×D1311yescharacteristic
386C22×Q8—134no—
387C22×Q8—262no—
388C22×Q8—262no—
389C22×Q8—522no—
390C22×Q8—522no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—42no—
394C23×C4—42no—
395C23×C4—131no—
396C23×C4—131no—
397C23×C4—134no—
398C23×C4—134no—
399C23×C4—134no—
400C23×C4—131no—
401C23×C4—131no—
402C23×C4—132no—
403C23×C4—134no—
404C23×C4—132no—
405C23×C4—262no—
406C23×C4—262no—
407C23×C4—522no—
408C23×C4—522no—
409C23×C4—522no—
410C23×C4—522no—
411C23×C4Dic2614yescomplemented
412C23×C4C22×D1312yes—
413C23×C4C22×D1312yes—
414C23×C4C22×D1311yescharacteristic
415C23×C4C22×D1311yescharacteristic
416C22⋊Q8—2632no—
417C2×C4⋊C4—134no—
418C2×C4⋊C4—134no—
419C2×C4⋊C4—134no—
420C2×C4⋊C4—262no—
421C2×C4⋊C4—262no—
422C2×C4⋊C4—262no—
423C2×C4⋊C4—262no—
424C2×C4⋊C4—262no—
425C2×C4⋊C4—262no—
426C2×C4⋊C4—264no—
427C2×C4⋊C4—522no—
428C2×C4⋊C4—522no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—134no—
435C2×C22⋊C4—262no—
436C2×C22⋊C4—262no—
437C2×C22⋊C4—262no—
438C2×C22⋊C4—262no—
439C2×C22⋊C4—262no—
440C2×C22⋊C4C4×D1314yes—
441#95—42no—
442#95—42no—
443#95—264no—
444#95—264no—
445#95—264no—
446#95—522no—
447#95—522no—
448#95—522no—
449#95—522no—
450#95—522no—
451#95—522no—
452C2×C13—28no—
453C2×C13C2×#39111yescharacteristic
454C2×C13#230811yescharacteristic
455C2×C13#231111yescharacteristic
456C2×C13#232912yes—
457C2×C13#232912yes—
458C2×C13#262611yescharacteristic
459C2×C13#262611yescharacteristic
460C2×C13#262711yescharacteristic
461C2×C13#262711yescharacteristic
462C2×C13#263912yes—
463C2×C13#265912yes—
464C24—22no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24C2×Dic2612yes—
469C24C2×C4×D1312yes—
470C24#612712yes—
471C24D4×D1312yes—
472C24D4×D1312yes—
473C24#612912yes—
474C24Q8×D1312yes—
475C24C23×D1311yescharacteristic
476C2×Q8—268no—
477C2×Q8—528no—
478C2×Q8—524no—
479C2×Q8—528no—
480C2×Q8—524no—
481C22×C4—28no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—48no—
507C22×C4—132no—
508C22×C4—132no—
509C22×C4—132no—
510C22×C4—132no—
511C22×C4—132no—
512C22×C4—132no—
513C22×C4—1316no—
514C22×C4—132no—
515C22×C4—132no—
516C22×C4—134no—
517C22×C4—262no—
518C22×C4—262no—
519C22×C4—262no—
520C22×C4—262no—
521C22×C4—268no—
522C22×C4—262no—
523C22×C4—262no—
524C22×C4—261no—
525C22×C4—261no—
526C22×C4—261no—
527C22×C4—261no—
528C22×C4—262no—
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—262no—
542C22×C4—262no—
543C22×C4—528no—
544C22×C4—524no—
545C22×C4—528no—
546C22×C4—524no—
547C22×C4—528no—
548C22×C4—528no—
549C22×C4—524no—
550C22×C4—524no—
551C22×C4C2×Dic2614yes—
552C22×C4C2×C4×D1312yes—
553C22×C4C2×C4×D1312yes—
554C4⋊C4—268no—
555C4⋊C4—268no—
556C4⋊C4—268no—
557C4⋊C4—528no—
558C22⋊C4—28no—
559C22⋊C4—44no—
560C22⋊C4—268no—
561C22⋊C4—524no—
562C13#1428111yescharacteristic, complemented, Sylow
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23C4×Dic2614yes—
580C23#8813214yes—
581C23C22⋊C4×D1314yes—
582C23#8813914yes—
583C23#8814114yes—
584C23#8814214yes—
585C23#8815014yes—
586C23C22×Dic2611yescharacteristic
587C23C22×C4×D1311yescharacteristic
588C23C2×#612711yescharacteristic
589C23C2×D4×D1311yescharacteristic
590C23C2×D4×D1311yescharacteristic
591C23C2×#612911yescharacteristic
592C23#8825112yes—
593C23#8825112yes—
594C23C2×Q8×D1311yescharacteristic
595C23#8825612yes—
596C23#8825712yes—
597Q8—5216no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—134no—
624C2×C4—134no—
625C2×C4—264no—
626C2×C4—264no—
627C2×C4—262no—
628C2×C4—262no—
629C2×C4—264no—
630C2×C4—264no—
631C2×C4—262no—
632C2×C4—262no—
633C2×C4—262no—
634C2×C4—262no—
635C2×C4—264no—
636C2×C4—264no—
637C2×C4—264no—
638C2×C4—264no—
639C2×C4—264no—
640C2×C4—5216no—
641C2×C4—5216no—
642C2×C4—522no—
643C2×C4—522no—
644C2×C4—521no—
645C2×C4—521no—
646C2×C4—522no—
647C2×C4—522no—
648C2×C4—522no—
649C2×C4—521no—
650C2×C4—522no—
651C2×C4—521no—
652C2×C4—522no—
653C2×C4—522no—
654C2×C4—522no—
655C2×C4—522no—
656C2×C4—522no—
657C2×C4—522no—
658C2×C4—522no—
659C22—28no—
660C22—28no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22—24no—
667C22—24no—
668C22—28no—
669C22#1171830718yes—
670C22C2×C4×Dic2611yescharacteristic
671C22C2×#8813211yescharacteristic
672C22#1171918812yes—
673C22C2×C22⋊C4×D1311yescharacteristic
674C22C2×#8813911yescharacteristic
675C22C2×#8814111yescharacteristic
676C22C2×#8814211yescharacteristic
677C22C2×#8815011yescharacteristic
678C22#1171922412yes—
679C22#1171925012yes—
680C22#1171925112yes—
681C22#1171926512yes—
682C22#1171926912yes—
683C22#1171929411yescharacteristic
684C22#1171929711yescharacteristic
685C22#1171929812yes—
686C22#1171930212yes—
687C22#1171931012yes—
688C22#1171932812yes—
689C22#1171933912yes—
690C22#1171935311yescharacteristic
691C22#1171935811yescharacteristic
692C22#1171937012yes—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—264no—
698C4—264no—
699C4—524no—
700C4—522no—
701C4—522no—
702C4—1042no—
703C2—28no—
704C2C2×#1171830711yescharacteristic
705C2#1311741011yescharacteristic
706C2#1311742311yescharacteristic
707C2#1311756412yes—
708C2#1311756712yes—
709C2#1312064511yescharacteristic
710C2#1312064711yescharacteristic
711C2#1312065311yescharacteristic
712C2#1312065511yescharacteristic
713C2#1312076612yes—
714C2#1312095312yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×C22⋊#95C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC22⋊#95D13
hypocenterC13#14281
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #13113106C2semidirect
5. #13113102C2semidirect
6. #13113088C2semidirect
7. #13113087C2semidirect
26. C2×#88132C4semidirect
45. C23×C4×C13Q8semidirect
59. C23×Dic13Q8semidirect
70. C22×Dic26C2×C4semidirect
85. C2×C4⋊Dic13C2×C4semidirect
93. C2×#6101C2×C4semidirect
94. C2×#6101C2×C4semidirect
411. C23×C4Dic26semidirect
562. C13#14281semidirect

Maximal subgroups

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

subgroupindex
C22×#881322
C2×#117186032
C22×#880712
#131131062
#131131022
#131130882
#131130872
#131130592
#131130582
C13×C22⋊#952
#1428113