Finite Group Databasebeta

← #15926586

Subgroups — #15926586

Order 3328.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117839C211yesmaximal, characteristic
2#13116403C211yesmaximal, characteristic, complemented
3#13116338C211yesmaximal, characteristic, complemented
4#13116214C211yesmaximal, characteristic, complemented
5#13115984C211yesmaximal, characteristic, complemented
6#13115794C211yesmaximal, characteristic, complemented
7#13115418C211yesmaximal, characteristic, complemented
8#13115257C211yesmaximal, characteristic
9#13115143C211yesmaximal, characteristic
10#13115103C211yesmaximal, characteristic, complemented
11#13115028C211yesmaximal, characteristic
12#13115016C211yesmaximal, characteristic
13#13114848C211yesmaximal, characteristic, complemented
14C2×#11718121C211yesmaximal, characteristic
15C13×#2442C211yesmaximal, characteristic
16C2×C13×#125C2211yescharacteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C2×#88178C2211yescharacteristic
20C23×#1055C2211yescharacteristic
21C2×C4×Dic26C2211yescharacteristic
22C13×#400—21no—
23C13×#398—21no—
24C13×#389C2211yescharacteristic
25C13×#389C2211yescharacteristic
26C13×#385—21no—
27C13×#382C2211yescharacteristic
28C13×#380—21no—
29C13×#380C2211yescharacteristic
30C4×C13×C22⋊C4C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32C2×#88050C2211yescharacteristic
33C2×#88050C2211yescharacteristic
34C2×#88050C2211yescharacteristic
35C2×#88050C2211yescharacteristic
36C2×#88048C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38#11718607—21no—
39C22⋊C4×Dic13—21no—
40C4×C4⋊Dic13C2211yescharacteristic
41#11718590C2211yescharacteristic
42#11718589C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44C2×C4⋊#1055C2211yescharacteristic
45#11718316—21no—
46#11718308—21no—
47#11718121C2211yescharacteristic
48#11718121C2211yescharacteristic
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C23×C4×C13—21no—
60C23×C4×C13C2311yescharacteristic
61C23×Dic13—21no—
62C22×Dic26C2311yescharacteristic
63C13×#125—22no—
64C13×#125—41no—
65C2×C13×C4⋊C4—22no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—41no—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4D411yescharacteristic
78C2×C13×C22⋊C4D411yescharacteristic
79C2×C42×C13C2311yescharacteristic
80C2×C42×C13C2311yescharacteristic
81C2×C22⋊Dic13—21no—
82C2×C4⋊Dic13—22no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—41no—
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88#88178—22no—
89#88178Q814yes—
90C2×#6101—21no—
91C2×#6101—21no—
92C2×#6101—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C22×#1055—22no—
96C22×#1055—41no—
97C22×#1055—41no—
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C4×Dic26—22no—
103C4×Dic26—41no—
104C13×#95—21no—
105C13×#95—21no—
106C13×#95—21no—
107C13×#95—21no—
108C13×#95—21no—
109C13×#95D411yescharacteristic
110C13×#95D411yescharacteristic
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115C22⋊#1055—22no—
116#88050—22no—
117#88050—41no—
118#88050—41no—
119#88050—41no—
120#88048—22no—
121#88048—41no—
122C4⋊#1055—41no—
123C4⋊#1055—41no—
124#26416—131nomaximal, Sylow
125C24×C13C2×D411yescharacteristic
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—42no—
135C22×C4×C13—42no—
136C22×C4×C13—81no—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×Q812yes—
143C22×C4×C13#4112yes—
144C22×C4×C13C2411yescharacteristic
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—42no—
150C22×Dic13—42no—
151C22×Dic13—81no—
152C2×Dic26—21no—
153C2×Dic26—21no—
154C2×Dic26—21no—
155C2×Dic26—21no—
156C2×Dic26—21no—
157C2×Dic26C2×Q812yes—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—42no—
166C13×C4⋊C4—42no—
167C13×C4⋊C4—42no—
168C13×C4⋊C4—81no—
169C13×C4⋊C4—81no—
170C13×C4⋊C4#4112yes—
171C13×C22⋊C4—22no—
172C13×C22⋊C4—22no—
173C13×C22⋊C4—42no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—21no—
177C42×C13—21no—
178C22⋊Dic13—24no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—42no—
184C4⋊Dic13C2×Q812yes—
185C4⋊Dic13#4112yes—
186#6101—24no—
187#6101—42no—
188#6101—81no—
189C4×Dic13—42no—
190C4×Dic13—81no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—44no—
204#2442D1311yescharacteristic
205#2360—131no—
206#2101—131no—
207#2088—131no—
208#2066—131no—
209#2026—131no—
210#1989—131no—
211#1920—131no—
212#1892—131no—
213#1872—131no—
214#1865—131no—
215#1853—131no—
216#1850—131no—
217#1805—131no—
218C2×#340—131no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13C22⋊Q812yes—
224C23×C13#13512yes—
225C23×C13#13712yes—
226C23×C13C22×D411yescharacteristic
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—22no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—84no—
258C2×C4×C13C22⋊Q812yes—
259C2×C4×C13C22×Q811yescharacteristic
260C2×C4×C13C2×#4111yescharacteristic
261C2×C4×C13C2×#4111yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13H(5,𝔽2)11yescharacteristic
264C2×C4×C13H(5,𝔽2)11yescharacteristic
265C2×Dic13—22no—
266C2×Dic13—22no—
267C2×Dic13—22no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—84no—
278Dic26—22no—
279Dic26—22no—
280Dic26—41no—
281Dic26—41no—
282#1055—41no—
283#1055—41no—
284#1055—41no—
285#1055—41no—
286C23×C8—131no—
287C2×#125C2×D1311yescharacteristic
288C2×C22⋊Q8—131no—
289C2×C4×Q8—131no—
290C22×C4⋊C4—131no—
291C22×C4⋊C4C2×D1311yescharacteristic
292C2×#106—131no—
293C2×#106—131no—
294C2×C4⋊C8—131no—
295C2×C4⋊C8—131no—
296C2×#103—131no—
297C2×#103—131no—
298C2×#103—131no—
299C2×#103—131no—
300C2×C22⋊C8—131no—
301#400—21no—
302#398—21no—
303#391—131no—
304#389—131no—
305#389—261no—
306#389C2×D1311yescharacteristic
307#389C2×D1311yescharacteristic
308#387—261no—
309#385—21no—
310#385—261no—
311#382C2×D1311yescharacteristic
312#380—21no—
313#380C2×D1311yescharacteristic
314C4×C4⋊C4—131no—
315C4×C22⋊C4—261no—
316C4×C22⋊C4C2×D1311yescharacteristic
317#340—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#336—131no—
328#336—131no—
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—22no—
332C22×C13—22no—
333C22×C13—22no—
334C22×C13—22no—
335C22×C13#47914yes—
336C22×C13C2×C22⋊Q811yescharacteristic
337C22×C13#54611yescharacteristic
338C22×C13#54611yescharacteristic
339C22×C13C2×#13511yescharacteristic
340C22×C13C2×#13711yescharacteristic
341C22×C13#57711yescharacteristic
342C22×C13#57711yescharacteristic
343C4×C13—22no—
344C4×C13—41no—
345C4×C13—41no—
346C4×C13—42no—
347C4×C13—41no—
348C4×C13—41no—
349C4×C13—81no—
350C4×C13—81no—
351C4×C13—81no—
352C4×C13#52911yescharacteristic
353C4×C13#53811yescharacteristic
354C4×C13#55411yescharacteristic
355C4×C13#55411yescharacteristic
356Dic13—42no—
357Dic13—42no—
358Dic13—42no—
359Dic13—81no—
360C22×Q8—131no—
361C23×C4—21no—
362C23×C4—261no—
363C23×C4C22×D1311yescharacteristic
364C22×C8—131no—
365C22×C8—131no—
366C22×C8—131no—
367C22×C8—131no—
368C22×C8—262no—
369C22×C8—521no—
370C22×C8—521no—
371#125—22no—
372#125—41no—
373C22⋊Q8—134no—
374C22⋊Q8—262no—
375C4×Q8—262no—
376C4×Q8—521no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—41no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4—262no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4—521no—
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C4⋊C4C22×D1311yescharacteristic
396C2×C4⋊C4C22×D1311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—261no—
399C2×C22⋊C4C22⋊D1311yescharacteristic
400C2×C22⋊C4C22⋊D1311yescharacteristic
401C2×C42—261no—
402C2×C42—261no—
403C2×C42C22×D1311yescharacteristic
404C2×C42C22×D1311yescharacteristic
405#106—262no—
406#106—521no—
407C4⋊C8—521no—
408C4⋊C8—521no—
409#103—262no—
410#103—521no—
411#103—521no—
412#103—521no—
413C22⋊C8—262no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95C22⋊D1311yescharacteristic
424#95C22⋊D1311yescharacteristic
425C2×C13—24no—
426C2×C13#245011yescharacteristic
427C2×C13C2×#47911yescharacteristic
428C2×C13#308411yescharacteristic
429C2×C13#329611yescharacteristic
430C2×C13#329611yescharacteristic
431C2×C13#331611yescharacteristic
432C2×C13#332011yescharacteristic
433C24C2×C22⋊D1311yescharacteristic
434C2×Q8—132no—
435C2×Q8—261no—
436C2×Q8—261no—
437C2×Q8—261no—
438C2×Q8—261no—
439C2×Q8—261no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—261no—
452C22×C4—261no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—522no—
456C22×C4—522no—
457C22×C4—1041no—
458C22×C4Q8×D1312yes—
459C22×C4#613112yes—
460C22×C4C2×C22⋊D1311yescharacteristic
461C22×C4C2×C22⋊D1311yescharacteristic
462C22×C4C2×C22⋊D1311yescharacteristic
463C22×C4C2×C22⋊D1311yescharacteristic
464C22×C4C2×C22⋊D1311yescharacteristic
465C22×C4C23×D1311yescharacteristic
466C2×C8—261no—
467C2×C8—261no—
468C2×C8—261no—
469C2×C8—261no—
470C2×C8—261no—
471C2×C8—261no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—524no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—132no—
492C4⋊C4—132no—
493C4⋊C4—264no—
494C4⋊C4—261no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—522no—
499C4⋊C4—522no—
500C4⋊C4—1041no—
501C4⋊C4#612912yes—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—42no—
505C22⋊C4—264no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—522no—
511C42—1041no—
512C13#2641611yescharacteristic, complemented, Sylow
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23#8820012yes—
518C23#8820412yes—
519C23#8820512yes—
520C23C22×C22⋊D1311yescharacteristic
521Q8—262no—
522Q8—262no—
523Q8—521no—
524Q8—521no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—262no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—1044no—
569C2×C4#8820012yes—
570C2×C4C2×#612911yescharacteristic
571C2×C4#8825111yescharacteristic
572C2×C4C2×Q8×D1311yescharacteristic
573C2×C4C2×#613111yescharacteristic
574C2×C4#41×D1311yescharacteristic
575C2×C4#8825611yescharacteristic
576C8—521no—
577C8—521no—
578C8—521no—
579C8—521no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22#1171872414yes—
587C22#1171951111yescharacteristic
588C22C2×#8820011yescharacteristic
589C22C2×#8820411yescharacteristic
590C22C2×#8820511yescharacteristic
591C22#1171953411yescharacteristic
592C22#1171954111yescharacteristic
593C22#1171954211yescharacteristic
594C4—22no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—522no—
604C4—522no—
605C4—522no—
606C4—1041no—
607C4#1171932911yescharacteristic
608C4#1171934011yescharacteristic
609C4#1171934111yescharacteristic
610C4#1171934911yescharacteristic
611C2—24no—
612C2#1311865611yescharacteristic
613C2C2×#1171872411yescharacteristic
614C2#1312469411yescharacteristic
615C2#1312666511yescharacteristic
616C2#1312667111yescharacteristic
617C2#1312694711yescharacteristic
618C2#1312700111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2442C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2442D13
hypocenterC13#26416
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. #13116403C2semidirect
3. #13116338C2semidirect
4. #13116214C2semidirect
5. #13115984C2semidirect
6. #13115794C2semidirect
7. #13115418C2semidirect
10. #13115103C2semidirect
13. #13114848C2semidirect
512. C13#26416semidirect

Maximal subgroups

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

subgroupindex
#131178392
#131164032
#131163382
#131162142
#131159842
#131157942
#131154182
#131152572
#131151432
#131151032
#131150282
#131150162
#131148482
C2×#117181212
C13×#24422
#2641613