Finite Group Databasebeta

← #15830494

Subgroups — #15830494

Order 3328.

882 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123925C211yesmaximal, characteristic, complemented
2#13117487C211yesmaximal, characteristic, complemented
3#13113413C212yesmaximal
4#13113400C212yesmaximal, complemented
5#13113387C212yesmaximal, complemented
6#13113374C212yesmaximal
7#13113361C211yesmaximal, characteristic
8#13113348C211yesmaximal, characteristic, complemented
9#13113335C211yesmaximal, characteristic, complemented
10#13113322C211yesmaximal, characteristic
11C13×#3523C211yesmaximal, characteristic
12C13×#557C411yescharacteristic, complemented
13C13×#557C411yescharacteristic, complemented
14C13×#557C412yescomplemented
15C13×#556C411yescharacteristic, complemented
16C13×#556C412yescomplemented
17C13×#552—22no—
18C13×#552—21no—
19D4×Q8×C13—21no—
20D4×Q8×C13C411yescharacteristic, complemented
21C13×#541—21no—
22C13×#541—22no—
23C13×#533C2211yescharacteristic
24C13×#533C2212yes—
25C13×#529C2211yescharacteristic
26C13×#529C2212yes—
27C13×#519—21no—
28C2×C4×Q8×C13C2211yescharacteristic
29#11718891—21no—
30#11718685—21no—
31#11718685—22no—
32Q8×#1055—21no—
33#11718646—21no—
34#11718642—22no—
35#11718598C2212yes—
36#11718597C2212yes—
37C4×C4⋊Dic13C2211yescharacteristic
38#11718593C2211yescharacteristic
39#11718587C2211yescharacteristic
40#11718587C2212yes—
41#11718586C2212yes—
42#11718586C2211yescharacteristic
43#11718130C2212yes—
44#11718130C2212yes—
45#11718130C2212yes—
46#11718130C2212yes—
47#11718130C2211yescharacteristic
48#11718130C2211yescharacteristic
49#11718130C2212yes—
50#11718129C2212yes—
51#11718129C2211yescharacteristic
52#11718129C2211yescharacteristic
53C22×Q8×C13—21no—
54C22×Q8×C13D411yescharacteristic
55C13×C4⋊Q8—21no—
56C13×C4⋊Q8—22no—
57C13×C4⋊Q8C2×C411yescharacteristic
58C13×C4⋊Q8C2×C412yes—
59C13×C4⋊Q8C2×C411yescharacteristic
60C13×C4⋊Q8C2×C412yes—
61C13×#126—22no—
62C13×#126—22no—
63C13×#126—22no—
64C13×#126—22no—
65C13×#126—21no—
66C13×#125—22no—
67C13×#125—22no—
68C13×#125—22no—
69C13×#125—22no—
70C13×#125—21no—
71C13×#125—22no—
72C13×#125—21no—
73C13×#125C2×C411yescharacteristic
74C13×#125C2×C412yes—
75C13×#124C2×C411yescharacteristic
76C13×#124C2×C412yes—
77C13×#123—21no—
78C13×#123—22no—
79C13×C22⋊Q8—21no—
80C13×C22⋊Q8—22no—
81C13×C22⋊Q8—21no—
82C13×C22⋊Q8—22no—
83C13×C22⋊Q8—22no—
84C13×C22⋊Q8—21no—
85C13×C22⋊Q8C2311yescharacteristic
86C13×C22⋊Q8C2312yes—
87C13×#121C2311yescharacteristic
88C4×Q8×C13—21no—
89C4×Q8×C13—22no—
90C4×Q8×C13—21no—
91C4×Q8×C13—21no—
92C4×Q8×C13—22no—
93C4×Q8×C13C2×C411yescharacteristic
94C4×Q8×C13C2×C412yes—
95C4×Q8×C13C2×C412yes—
96C4×Q8×C13C2×C412yes—
97C4×Q8×C13C2×C412yes—
98C4×Q8×C13C2×C411yescharacteristic
99C4×Q8×C13C2×C411yescharacteristic
100C4×Q8×C13C2×C412yes—
101C4×Q8×C13D411yescharacteristic
102C4×Q8×C13D411yescharacteristic
103C4×Q8×C13D412yes—
104C4×D4×C13—21no—
105C4×D4×C13—22no—
106C4×D4×C13C2×C411yescharacteristic
107C4×D4×C13C2×C412yes—
108C13×#117—21no—
109C13×#117—22no—
110C13×#117—22no—
111C13×#117—21no—
112C13×#117—21no—
113C13×#117—22no—
114C2×C13×C4⋊C4D412yes—
115C2×C13×C4⋊C4D411yescharacteristic
116C2×C42×C13C2311yescharacteristic
117C2×C42×C13C2312yes—
118C2×C4⋊Dic13—21no—
119C2×C4⋊Dic13—22no—
120C2×C4×Dic13—21no—
121C2×C4×Dic13—21no—
122#88071—21no—
123#88071—22no—
124#88071—22no—
125#88071—22no—
126#88071C2311yescharacteristic
127#88071C2311yescharacteristic
128#88071C2312yes—
129C22⋊#1055C2311yescharacteristic
130C22⋊#1055C2312yes—
131C22⋊#1055C2311yescharacteristic
132C4⋊#1055—21no—
133C4⋊#1055—22no—
134C4⋊#1055—22no—
135C4⋊#1055—21no—
136#88043—22no—
137#88043—21no—
138C4×#1055—21no—
139C4×#1055—22no—
140#17601—131nomaximal, Sylow
141C2×Q8×C13—22no—
142C2×Q8×C13—21no—
143C2×Q8×C13—42no—
144C2×Q8×C13—44no—
145C2×Q8×C13C22⋊C411yescharacteristic
146C2×Q8×C13C22⋊C411yescharacteristic
147C2×Q8×C13C22⋊C412yes—
148C2×Q8×C13C22×C411yescharacteristic
149C2×Q8×C13C22×C412yes—
150C2×D4×C13—21no—
151C2×D4×C13C22×C411yescharacteristic
152C22×C4×C13—22no—
153C22×C4×C13—21no—
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D412yes—
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D412yes—
158C22×C4×C13C2411yescharacteristic
159C22×Dic13—21no—
160C22×Dic13—21no—
161C22×Dic13—22no—
162C13×C4⋊C4—22no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—22no—
166C13×C4⋊C4—22no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—22no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—22no—
172C13×C4⋊C4—22no—
173C13×C4⋊C4—22no—
174C13×C4⋊C4—22no—
175C13×C4⋊C4—22no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—22no—
179C13×C4⋊C4C22⋊C412yes—
180C13×C4⋊C4C22⋊C411yescharacteristic
181C13×C4⋊C4C22⋊C412yes—
182C13×C4⋊C4C22⋊C411yescharacteristic
183C13×C4⋊C4C22⋊C412yes—
184C13×C4⋊C4C22⋊C412yes—
185C13×C4⋊C4C22⋊C412yes—
186C13×C4⋊C4C22×C411yescharacteristic
187C13×C4⋊C4C22×C412yes—
188C13×C4⋊C4C22×C411yescharacteristic
189C13×C22⋊C4—22no—
190C13×C22⋊C4—21no—
191C13×C22⋊C4—21no—
192C13×C22⋊C4—22no—
193C13×C22⋊C4—21no—
194C13×C22⋊C4—22no—
195C13×C22⋊C4—22no—
196C13×C22⋊C4—21no—
197C13×C22⋊C4—21no—
198C42×C13—22no—
199C42×C13—22no—
200C42×C13—22no—
201C42×C13—22no—
202C42×C13—21no—
203C42×C13C22×C412yes—
204C42×C13C22×C411yescharacteristic
205C42×C13C22×C411yescharacteristic
206C42×C13C22×C412yes—
207C42×C13C2×D411yescharacteristic
208C42×C13C2×D412yes—
209C42×C13C2×D412yes—
210C42×C13C2×D411yescharacteristic
211C4⋊Dic13—42no—
212C4⋊Dic13—82no—
213C4×Dic13—42no—
214C4×Dic13—81no—
215C2×#1055—21no—
216C2×#1055—22no—
217C2×#1055—21no—
218#3523D1311yescharacteristic
219#2984—131no—
220#2320—131no—
221#1521—132no—
222#1518—132no—
223#1515—132no—
224#1512—132no—
225#1509—131no—
226#1506—131no—
227#1503—131no—
228#1500—131no—
229C23×C13—21no—
230C23×C13C22×D411yescharacteristic
231Q8×C13—22no—
232Q8×C13—22no—
233Q8×C13—42no—
234Q8×C13—44no—
235Q8×C13—41no—
236Q8×C13—42no—
237Q8×C13—42no—
238D4×C13—41no—
239D4×C13—41no—
240D4×C13—41no—
241C2×C4×C13—21no—
242C2×C4×C13—22no—
243C2×C4×C13—21no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—22no—
250C2×C4×C13—21no—
251C2×C4×C13—42no—
252C2×C4×C13—41no—
253C2×C4×C13—44no—
254C2×C4×C13—42no—
255C2×C4×C13C2×C22⋊C411yescharacteristic
256C2×C4×C13C2×C22⋊C412yes—
257C2×C4×C13C2×C22⋊C412yes—
258C2×C4×C13C2×C22⋊C412yes—
259C2×C4×C13C2×C22⋊C411yescharacteristic
260C2×C4×C13C2×C22⋊C411yescharacteristic
261C2×C4×C13C2×C22⋊C412yes—
262C2×C4×C13C2×C22⋊C411yescharacteristic
263C2×C4×C13C23×C411yescharacteristic
264C2×C4×C13C22×D411yescharacteristic
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—42no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274#1055—41no—
275#1055—42no—
276#1055—41no—
277#557Dic1311yescharacteristic, complemented
278#557Dic1311yescharacteristic, complemented
279#557Dic1312yescomplemented
280#556Dic1311yescharacteristic, complemented
281#556Dic1312yescomplemented
282#552—22no—
283#552—21no—
284D4×Q8—21no—
285D4×Q8Dic1311yescharacteristic, complemented
286#541—21no—
287#541—22no—
288#533C2×D1311yescharacteristic
289#533C2×D1312yes—
290#529C2×D1311yescharacteristic
291#529C2×D1312yes—
292#519—21no—
293C2×C4×Q8C2×D1311yescharacteristic
294#446—261no—
295#446—262no—
296C8×Q8—261no—
297#432—131no—
298#432—132no—
299#431—131no—
300#431—132no—
301#391—261no—
302#389—261no—
303#387—262no—
304#383—132no—
305#382—132no—
306C4×C4⋊C4—131no—
307#376—131no—
308#328—131no—
309#328—131no—
310#328—132no—
311#328—132no—
312#328—132no—
313#328—132no—
314#328—132no—
315#327—131no—
316#327—131no—
317#327—132no—
318C22×C13—21no—
319C22×C13—21no—
320C22×C13—41no—
321C22×C13—41no—
322C22×C13—41no—
323C22×C13C22×C22⋊C411yescharacteristic
324C22×C13#57711yescharacteristic
325C22×C13#57811yescharacteristic
326C4×C13—21no—
327C4×C13—21no—
328C4×C13—21no—
329C4×C13—21no—
330C4×C13—22no—
331C4×C13—22no—
332C4×C13—21no—
333C4×C13—22no—
334C4×C13—22no—
335C4×C13—42no—
336C4×C13—41no—
337C4×C13—41no—
338C4×C13—42no—
339C4×C13—41no—
340C4×C13#42112yes—
341Dic13—81no—
342Dic13—82no—
343Dic13—81no—
344C22×Q8—21no—
345C22×Q8C22⋊D1311yescharacteristic
346C4⋊Q8—21no—
347C4⋊Q8—22no—
348C4⋊Q8C2×Dic1311yescharacteristic
349C4⋊Q8C2×Dic1312yes—
350C4⋊Q8C2×Dic1311yescharacteristic
351C4⋊Q8C2×Dic1312yes—
352#126—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#126—21no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125—22no—
361#125—21no—
362#125—22no—
363#125—21no—
364#125C2×Dic1311yescharacteristic
365#125C2×Dic1312yes—
366#124C2×Dic1311yescharacteristic
367#124C2×Dic1312yes—
368#123—21no—
369#123—22no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8C22×D1311yescharacteristic
377C22⋊Q8C22×D1312yes—
378#121C22×D1311yescharacteristic
379C4×Q8—21no—
380C4×Q8—22no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8—22no—
384C4×Q8C2×Dic1311yescharacteristic
385C4×Q8C2×Dic1312yes—
386C4×Q8C2×Dic1312yes—
387C4×Q8C2×Dic1312yes—
388C4×Q8C2×Dic1312yes—
389C4×Q8C2×Dic1311yescharacteristic
390C4×Q8C2×Dic1311yescharacteristic
391C4×Q8C2×Dic1312yes—
392C4×Q8C22⋊D1311yescharacteristic
393C4×Q8C22⋊D1311yescharacteristic
394C4×Q8C22⋊D1312yes—
395C4×D4—21no—
396C4×D4—22no—
397C4×D4C2×Dic1311yescharacteristic
398C4×D4C2×Dic1312yes—
399#117—21no—
400#117—22no—
401#117—22no—
402#117—21no—
403#117—21no—
404#117—22no—
405C2×C4⋊C4—261no—
406C2×C4⋊C4—262no—
407C2×C4⋊C4C22⋊D1312yes—
408C2×C4⋊C4C22⋊D1311yescharacteristic
409C2×C42—261no—
410C2×C42—261no—
411C2×C42C22×D1311yescharacteristic
412C2×C42C22×D1312yes—
413C4⋊C8—262no—
414C4⋊C8—261no—
415C4⋊C8—261no—
416C4⋊C8—262no—
417C22⋊C8—131no—
418C22⋊C8—131no—
419C22⋊C8—132no—
420#97—262no—
421#97—261no—
422C4×C8—261no—
423C4×C8—262no—
424#95—131no—
425#95—131no—
426#95—132no—
427#95—261no—
428#95—262no—
429#95—262no—
430#95—262no—
431C2×C13—21no—
432C2×C13—21no—
433C2×C13—81no—
434C2×C13#288411yescharacteristic
435C2×C13#289811yescharacteristic
436C2×C13C2×#42111yescharacteristic
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8C22⋊Dic1311yescharacteristic
442C2×Q8C22⋊Dic1311yescharacteristic
443C2×Q8C22⋊Dic1312yes—
444C2×Q8C22×Dic1311yescharacteristic
445C2×Q8C22×Dic1312yes—
446C2×D4—21no—
447C2×D4C22×Dic1311yescharacteristic
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—261no—
451C22×C4—261no—
452C22×C4—262no—
453C22×C4C2×C22⋊D1311yescharacteristic
454C22×C4C2×C22⋊D1312yes—
455C22×C4C2×C22⋊D1311yescharacteristic
456C22×C4C2×C22⋊D1312yes—
457C22×C4C23×D1311yescharacteristic
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—262no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—522no—
479C4⋊C4—1042no—
480C4⋊C4C22⋊Dic1312yes—
481C4⋊C4C22⋊Dic1311yescharacteristic
482C4⋊C4C22⋊Dic1312yes—
483C4⋊C4C22⋊Dic1311yescharacteristic
484C4⋊C4C22⋊Dic1312yes—
485C4⋊C4C22⋊Dic1312yes—
486C4⋊C4C22⋊Dic1312yes—
487C4⋊C4C22×Dic1311yescharacteristic
488C4⋊C4C22×Dic1312yes—
489C4⋊C4C22×Dic1311yescharacteristic
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—522no—
505C42—1041no—
506C42C22×Dic1312yes—
507C42C22×Dic1311yescharacteristic
508C42C22×Dic1311yescharacteristic
509C42C22×Dic1312yes—
510C42C2×C22⋊D1311yescharacteristic
511C42C2×C22⋊D1312yes—
512C42C2×C22⋊D1312yes—
513C42C2×C22⋊D1311yescharacteristic
514C13#1760111yescharacteristic, complemented, Sylow
515C23—21no—
516C23C22×C22⋊D1311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—42no—
520Q8—41no—
521Q8—42no—
522Q8—44no—
523Q8—42no—
524D4—41no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—522no—
544C2×C4—521no—
545C2×C4—521no—
546C2×C4—522no—
547C2×C4—522no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4C2×C22⋊Dic1311yescharacteristic
551C2×C4C2×C22⋊Dic1312yes—
552C2×C4C2×C22⋊Dic1312yes—
553C2×C4C2×C22⋊Dic1312yes—
554C2×C4C2×C22⋊Dic1311yescharacteristic
555C2×C4C2×C22⋊Dic1311yescharacteristic
556C2×C4C2×C22⋊Dic1312yes—
557C2×C4C2×C22⋊Dic1311yescharacteristic
558C2×C4C23×Dic1311yescharacteristic
559C2×C4C22×C22⋊D1311yescharacteristic
560C8—521no—
561C8—521no—
562C8—522no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22#1171954211yescharacteristic
569C22#1171954311yescharacteristic
570C22C22×C22⋊Dic1311yescharacteristic
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—1041no—
586C4—1042no—
587C4—1041no—
588C4#1171889712yes—
589C2—21no—
590C2—21no—
591C2—81no—
592C2#1312330011yescharacteristic
593C2#1312336911yescharacteristic
594C2C2×#1171889711yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#3523C2
socleC22×C13C22×C22⋊C4
prime radicalC13×#121C23
cosocleC22×C4×C13C24
hypercenter#3523D13
hypocenterC13#17601
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3523

Complements

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

NH ≅ G/Ntype
1. #13123925C2semidirect
2. #13117487C2semidirect
4. #13113400C2semidirect
5. #13113387C2semidirect
8. #13113348C2semidirect
9. #13113335C2semidirect
12. C13×#557C4semidirect
13. C13×#557C4semidirect
14. C13×#557C4semidirect
15. C13×#556C4semidirect
16. C13×#556C4semidirect
20. D4×Q8×C13C4semidirect
277. #557Dic13semidirect
278. #557Dic13semidirect
279. #557Dic13semidirect
280. #556Dic13semidirect
281. #556Dic13semidirect
285. D4×Q8Dic13semidirect
514. C13#17601semidirect

Maximal subgroups

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

subgroupindex
#131239252
#131174872
#131134132
#131134002
#131133872
#131133742
#131133612
#131133482
#131133352
#131133222
C13×#35232
#1760113