Finite Group Databasebeta

← #15830615

Subgroups — #15830615

Order 3328.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123494C211yesmaximal, characteristic, complemented
2#13117729C211yesmaximal, characteristic, complemented
3#13113413C212yesmaximal, complemented
4#13113408C211yesmaximal, characteristic, complemented
5#13113387C212yesmaximal, complemented
6#13113379C211yesmaximal, characteristic
7#13113361C212yesmaximal, complemented
8#13113356C211yesmaximal, characteristic, complemented
9#13113335C212yesmaximal, complemented
10#13113327C211yesmaximal, characteristic
11C13×#3495C211yesmaximal, characteristic
12C13×#556—21no—
13C13×#554—21no—
14C13×#552—22no—
15C13×#552—21no—
16C13×#552—21no—
17C13×#551—22no—
18C13×#544C411yescharacteristic, complemented
19C13×#542C412yescomplemented
20C13×#541C412yescomplemented
21C13×#536C411yescharacteristic, complemented
22C13×#533C2212yes—
23C13×#529C2212yes—
24C13×#525C2211yescharacteristic
25C2×C13×C22⋊Q8C2211yescharacteristic
26C13×#520C411yescharacteristic, complemented
27C13×#519C411yescharacteristic, complemented
28C2×C13×#117C2211yescharacteristic
29#11718873C2211yescharacteristic
30C4×C22⋊Dic13C2211yescharacteristic
31#11718868C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33#11718661—22no—
34#11718660—21no—
35#11718658—21no—
36#11718646—21no—
37C4⋊C4×Dic13—21no—
38#11718612—22no—
39#11718598C2212yes—
40C4×C4⋊Dic13C2212yes—
41#11718587C2212yes—
42#11718586C2212yes—
43#11718130C2212yes—
44#11718130C2212yes—
45#11718130C2212yes—
46#11718130C2212yes—
47#11718130C2211yescharacteristic
48#11718130C2212yes—
49#11718130C2211yescharacteristic
50#11718129C2211yescharacteristic
51#11718129C2212yes—
52#11718129C2211yescharacteristic
53C2×C13×#41C2×C411yescharacteristic
54C22×Q8×C13C2×C411yescharacteristic
55C23×C4×C13C2311yescharacteristic
56C13×C4⋊Q8—21no—
57C13×C4⋊Q8C2×C412yes—
58C13×C4⋊Q8C2×C412yes—
59C13×#126—22no—
60C13×#126—22no—
61C13×#126—22no—
62C13×#126—22no—
63C13×#125—22no—
64C13×#125—22no—
65C13×#125—21no—
66C13×#125—22no—
67C13×#125C2×C412yes—
68C13×#124—22no—
69C13×#124C2×C412yes—
70C13×#123—21no—
71C13×#123—22no—
72C13×#123—22no—
73C13×#123—22no—
74C13×C22⋊Q8—21no—
75C13×C22⋊Q8—21no—
76C13×C22⋊Q8—22no—
77C13×C22⋊Q8—21no—
78C13×C22⋊Q8—21no—
79C13×C22⋊Q8—22no—
80C13×C22⋊Q8—22no—
81C13×C22⋊Q8C2×C412yes—
82C13×C22⋊Q8C2×C411yescharacteristic
83C13×C22⋊Q8C2312yes—
84C13×C22⋊Q8C2311yescharacteristic
85C13×#121C2×C411yescharacteristic
86C13×#121C2311yescharacteristic
87C13×#120—21no—
88C4×Q8×C13—21no—
89C4×Q8×C13—21no—
90C4×Q8×C13—22no—
91C4×Q8×C13C2×C412yes—
92C4×Q8×C13C2×C412yes—
93C4×Q8×C13C2×C412yes—
94C4×D4×C13—21no—
95C4×D4×C13—21no—
96C4×D4×C13—22no—
97C4×D4×C13—22no—
98C4×D4×C13C2×C412yes—
99C13×#117—22no—
100C13×#117—21no—
101C13×#117—22no—
102C13×#117—22no—
103C13×#117C2×C411yescharacteristic
104C13×#117C2×C411yescharacteristic
105C13×#117C2×C411yescharacteristic
106C13×#117C2×C412yes—
107C13×#117D411yescharacteristic
108C13×#117D411yescharacteristic
109C13×#117D412yes—
110C2×C13×C4⋊C4—21no—
111C2×C13×C4⋊C4—21no—
112C2×C13×C4⋊C4C2×C411yescharacteristic
113C2×C13×C4⋊C4D411yescharacteristic
114C2×C13×C4⋊C4D411yescharacteristic
115C2×C13×C22⋊C4—21no—
116C2×C13×C22⋊C4D412yes—
117C2×C42×C13C2312yes—
118C2×C22⋊Dic13—21no—
119C2×C22⋊Dic13—21no—
120C2×C4⋊Dic13—22no—
121C2×C4⋊Dic13—21no—
122C2×C4×Dic13—21no—
123C2×C4×Dic13—22no—
124C2×#6099—21no—
125C22×#1055—21no—
126#88071—22no—
127#88071—22no—
128#88071C2311yescharacteristic
129#88071C2312yes—
130#88071C2311yescharacteristic
131C22⋊#1055—21no—
132C22⋊#1055—21no—
133C22⋊#1055C2311yescharacteristic
134C22⋊#1055C2312yes—
135C22⋊#1055C2311yescharacteristic
136C4⋊#1055—22no—
137C4⋊#1055—22no—
138#88043—22no—
139C4×#1055—22no—
140#17612—131nomaximal, Sylow
141C24×C13C2×D411yescharacteristic
142C13×#41—42no—
143C2×Q8×C13—21no—
144C2×Q8×C13—22no—
145C2×Q8×C13—42no—
146C2×Q8×C13C22×C412yes—
147C2×Q8×C13C22×C411yescharacteristic
148C2×D4×C13—21no—
149C2×D4×C13—21no—
150C2×D4×C13—21no—
151C2×D4×C13C22×C411yescharacteristic
152C22×C4×C13—21no—
153C22×C4×C13—21no—
154C22×C4×C13—21no—
155C22×C4×C13—22no—
156C22×C4×C13—42no—
157C22×C4×C13C22×C411yescharacteristic
158C22×C4×C13C22×C411yescharacteristic
159C22×C4×C13C2×D412yes—
160C22×C4×C13C2×D411yescharacteristic
161C22×C4×C13C2×D411yescharacteristic
162C22×C4×C13C2×D412yes—
163C22×C4×C13C2×D411yescharacteristic
164C22×C4×C13C2411yescharacteristic
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—22no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—22no—
170C13×C4⋊C4—22no—
171C13×C4⋊C4—22no—
172C13×C4⋊C4—22no—
173C13×C4⋊C4—22no—
174C13×C4⋊C4—22no—
175C13×C4⋊C4—22no—
176C13×C4⋊C4—22no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—22no—
179C13×C4⋊C4—22no—
180C13×C4⋊C4—21no—
181C13×C4⋊C4C22⋊C411yescharacteristic
182C13×C4⋊C4C22⋊C411yescharacteristic
183C13×C4⋊C4C22⋊C412yes—
184C13×C4⋊C4C22⋊C411yescharacteristic
185C13×C4⋊C4C22⋊C411yescharacteristic
186C13×C4⋊C4C22⋊C412yes—
187C13×C4⋊C4C22×C412yes—
188C13×C4⋊C4C22×C411yescharacteristic
189C13×C4⋊C4C22×C411yescharacteristic
190C13×C22⋊C4—22no—
191C13×C22⋊C4—21no—
192C13×C22⋊C4—21no—
193C13×C22⋊C4—22no—
194C13×C22⋊C4—21no—
195C13×C22⋊C4—22no—
196C13×C22⋊C4—22no—
197C13×C22⋊C4—21no—
198C13×C22⋊C4—22no—
199C13×C22⋊C4—21no—
200C13×C22⋊C4C22⋊C412yes—
201C13×C22⋊C4C22⋊C412yes—
202C13×C22⋊C4C22⋊C412yes—
203C13×C22⋊C4C22⋊C412yes—
204C42×C13—22no—
205C42×C13—22no—
206C42×C13—22no—
207C42×C13—21no—
208C42×C13C22×C412yes—
209C42×C13C22×C412yes—
210C42×C13C2×D412yes—
211C42×C13C2×D412yes—
212C22⋊Dic13—42no—
213C22⋊Dic13—81no—
214C4⋊Dic13—44no—
215C4⋊Dic13—42no—
216C4×Dic13—42no—
217C4×Dic13—82no—
218#6099—42no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055—22no—
222C2×#1055—42no—
223#3495D1311yescharacteristic
224#2924—131no—
225#2347—131no—
226#1521—132no—
227#1520—131no—
228#1515—132no—
229#1513—131no—
230#1509—132no—
231#1508—131no—
232#1503—132no—
233#1501—131no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—41no—
238C23×C13—41no—
239C23×C13C2×C22⋊C411yescharacteristic
240C23×C13C2×C22⋊C411yescharacteristic
241C23×C13C22×D411yescharacteristic
242Q8×C13—42no—
243Q8×C13—41no—
244Q8×C13—42no—
245Q8×C13—82no—
246D4×C13—41no—
247D4×C13—42no—
248D4×C13—41no—
249D4×C13—41no—
250D4×C13—81no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—22no—
259C2×C4×C13—21no—
260C2×C4×C13—22no—
261C2×C4×C13—42no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—42no—
266C2×C4×C13—42no—
267C2×C4×C13—82no—
268C2×C4×C13—81no—
269C2×C4×C13C2×C22⋊C412yes—
270C2×C4×C13C2×C22⋊C412yes—
271C2×C4×C13C2×C22⋊C412yes—
272C2×C4×C13C2×C22⋊C412yes—
273C2×C4×C13C2×C22⋊C411yescharacteristic
274C2×C4×C13C2×C22⋊C411yescharacteristic
275C2×C4×C13C23×C411yescharacteristic
276C2×C4×C13C22×D411yescharacteristic
277C2×Dic13—22no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13—42no—
282C2×Dic13—42no—
283C2×Dic13—41no—
284C2×Dic13—42no—
285C2×Dic13—41no—
286#1055—41no—
287#1055—41no—
288#1055—42no—
289#556—21no—
290#554—21no—
291#552—22no—
292#552—21no—
293#552—21no—
294#551—22no—
295#544Dic1311yescharacteristic, complemented
296#542Dic1312yescomplemented
297#541Dic1312yescomplemented
298#536Dic1311yescharacteristic, complemented
299#533C2×D1312yes—
300#529C2×D1312yes—
301#525C2×D1311yescharacteristic
302C2×C22⋊Q8C2×D1311yescharacteristic
303#520Dic1311yescharacteristic, complemented
304#519Dic1311yescharacteristic, complemented
305C2×#117C2×D1311yescharacteristic
306#433—262no—
307#432—132no—
308#431—132no—
309#424—261no—
310#407—131no—
311C2×C22⋊C8—131no—
312#405—261no—
313#389—261no—
314#388—262no—
315#383—132no—
316#380—131no—
317C4×C4⋊C4—132no—
318C4×C4⋊C4—261no—
319C4×C22⋊C4—131no—
320#328—131no—
321#328—131no—
322#328—132no—
323#328—132no—
324#328—132no—
325#328—132no—
326#328—132no—
327#327—131no—
328#327—131no—
329#327—132no—
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—41no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13—81no—
342C22×C13—81no—
343C22×C13C22×C22⋊C411yescharacteristic
344C22×C13#57711yescharacteristic
345C22×C13#57811yescharacteristic
346C4×C13—21no—
347C4×C13—21no—
348C4×C13—21no—
349C4×C13—22no—
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—22no—
353C4×C13—41no—
354C4×C13—42no—
355C4×C13—41no—
356C4×C13—42no—
357C4×C13—41no—
358C4×C13#41012yes—
359Dic13—42no—
360Dic13—82no—
361Dic13—81no—
362C2×#41C2×Dic1311yescharacteristic
363C22×Q8C2×Dic1311yescharacteristic
364C23×C4C22×D1311yescharacteristic
365C2×M16—261no—
366C22×C8—261no—
367C4⋊Q8—21no—
368C4⋊Q8C2×Dic1312yes—
369C4⋊Q8C2×Dic1312yes—
370#126—22no—
371#126—22no—
372#126—22no—
373#126—22no—
374#125—22no—
375#125—22no—
376#125—21no—
377#125—22no—
378#125C2×Dic1312yes—
379#124—22no—
380#124C2×Dic1312yes—
381#123—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8C2×Dic1312yes—
393C22⋊Q8C2×Dic1311yescharacteristic
394C22⋊Q8C22×D1312yes—
395C22⋊Q8C22×D1311yescharacteristic
396#121C2×Dic1311yescharacteristic
397#121C22×D1311yescharacteristic
398#120—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—22no—
402C4×Q8C2×Dic1312yes—
403C4×Q8C2×Dic1312yes—
404C4×Q8C2×Dic1312yes—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4C2×Dic1312yes—
410#117—22no—
411#117—21no—
412#117—22no—
413#117—22no—
414#117C2×Dic1311yescharacteristic
415#117C2×Dic1311yescharacteristic
416#117C2×Dic1311yescharacteristic
417#117C2×Dic1312yes—
418#117C22⋊D1311yescharacteristic
419#117C22⋊D1312yes—
420#117C22⋊D1311yescharacteristic
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—262no—
424C2×C4⋊C4—261no—
425C2×C4⋊C4C2×Dic1311yescharacteristic
426C2×C4⋊C4C22⋊D1311yescharacteristic
427C2×C4⋊C4C22⋊D1311yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—261no—
430C2×C22⋊C4—261no—
431C2×C22⋊C4C22⋊D1312yes—
432C2×C42—261no—
433C2×C42—262no—
434C2×C42C22×D1312yes—
435C4⋊C8—262no—
436C4⋊C8—262no—
437C22⋊C8—131no—
438C22⋊C8—131no—
439C22⋊C8—132no—
440C22⋊C8—261no—
441C22⋊C8—261no—
442#97—262no—
443C4×C8—262no—
444#95—131no—
445#95—131no—
446#95—132no—
447#95—262no—
448#95—262no—
449C2×C13—21no—
450C2×C13—21no—
451C2×C13—41no—
452C2×C13—41no—
453C2×C13—81no—
454C2×C13C2×#41011yescharacteristic
455C2×C13#289811yescharacteristic
456C2×C13#290111yescharacteristic
457C24C2×C22⋊D1311yescharacteristic
458#41—42no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—42no—
462C2×Q8C22×Dic1312yes—
463C2×Q8C22×Dic1311yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4C22×Dic1311yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—262no—
476C22×C4C22×Dic1311yescharacteristic
477C22×C4C22×Dic1311yescharacteristic
478C22×C4C2×C22⋊D1312yes—
479C22×C4C2×C22⋊D1311yescharacteristic
480C22×C4C2×C22⋊D1311yescharacteristic
481C22×C4C2×C22⋊D1312yes—
482C22×C4C2×C22⋊D1311yescharacteristic
483C22×C4C23×D1311yescharacteristic
484M16—522no—
485C2×C8—261no—
486C2×C8—262no—
487C2×C8—261no—
488C2×C8—522no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—522no—
503C4⋊C4—524no—
504C4⋊C4C22⋊Dic1311yescharacteristic
505C4⋊C4C22⋊Dic1311yescharacteristic
506C4⋊C4C22⋊Dic1312yes—
507C4⋊C4C22⋊Dic1311yescharacteristic
508C4⋊C4C22⋊Dic1311yescharacteristic
509C4⋊C4C22⋊Dic1312yes—
510C4⋊C4C22×Dic1312yes—
511C4⋊C4C22×Dic1311yescharacteristic
512C4⋊C4C22×Dic1311yescharacteristic
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—522no—
524C22⋊C4—1041no—
525C22⋊C4C22⋊Dic1312yes—
526C22⋊C4C22⋊Dic1312yes—
527C22⋊C4C22⋊Dic1312yes—
528C22⋊C4C22⋊Dic1312yes—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—522no—
534C42—1042no—
535C42C22×Dic1312yes—
536C42C22×Dic1312yes—
537C42C2×C22⋊D1312yes—
538C42C2×C22⋊D1312yes—
539C13#1761211yescharacteristic, complemented, Sylow
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23C2×C22⋊Dic1311yescharacteristic
546C23C2×C22⋊Dic1311yescharacteristic
547C23C22×C22⋊D1311yescharacteristic
548Q8—42no—
549Q8—41no—
550Q8—42no—
551Q8—82no—
552D4—41no—
553D4—42no—
554D4—41no—
555D4—41no—
556D4—81no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—262no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—521no—
579C2×C4—521no—
580C2×C4—522no—
581C2×C4—522no—
582C2×C4—522no—
583C2×C4—521no—
584C2×C4C2×C22⋊Dic1312yes—
585C2×C4C2×C22⋊Dic1312yes—
586C2×C4C2×C22⋊Dic1312yes—
587C2×C4C2×C22⋊Dic1312yes—
588C2×C4C2×C22⋊Dic1311yescharacteristic
589C2×C4C2×C22⋊Dic1311yescharacteristic
590C2×C4C23×Dic1311yescharacteristic
591C2×C4C22×C22⋊D1311yescharacteristic
592C8—521no—
593C8—521no—
594C8—522no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—81no—
608C22#1171954211yescharacteristic
609C22#1171954311yescharacteristic
610C22C22×C22⋊Dic1311yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—522no—
624C4—1042no—
625C4—1041no—
626C4#1171890112yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—81no—
632C2C2×#1171890111yescharacteristic
633C2#1312336911yescharacteristic
634C2#1312338311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#3495C2
socleC22×C13C22×C22⋊C4
prime radicalC13×#525C22
cosocleC22×C4×C13C24
hypercenter#3495D13
hypocenterC13#17612
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×#117 ◁ #3495

Complements

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

NH ≅ G/Ntype
1. #13123494C2semidirect
2. #13117729C2semidirect
3. #13113413C2semidirect
4. #13113408C2semidirect
5. #13113387C2semidirect
7. #13113361C2semidirect
8. #13113356C2semidirect
9. #13113335C2semidirect
18. C13×#544C4semidirect
19. C13×#542C4semidirect
20. C13×#541C4semidirect
21. C13×#536C4semidirect
26. C13×#520C4semidirect
27. C13×#519C4semidirect
295. #544Dic13semidirect
296. #542Dic13semidirect
297. #541Dic13semidirect
298. #536Dic13semidirect
303. #520Dic13semidirect
304. #519Dic13semidirect
539. C13#17612semidirect

Maximal subgroups

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

subgroupindex
#131234942
#131177292
#131134132
#131134082
#131133872
#131133792
#131133612
#131133562
#131133352
#131133272
C13×#34952
#1761213