Finite Group Databasebeta

← #15926774

Subgroups — #15926774

Order 3328.

756 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#13118521C211yesmaximal, characteristic
2#13117845C211yesmaximal, characteristic
3#13116407C211yesmaximal, characteristic, complemented
4#13116326C211yesmaximal, characteristic, complemented
5#13116229C211yesmaximal, characteristic, complemented
6#13115975C211yesmaximal, characteristic, complemented
7#13115786C211yesmaximal, characteristic, complemented
8#13115427C211yesmaximal, characteristic, complemented
9#13115270C211yesmaximal, characteristic
10#13115092C211yesmaximal, characteristic, complemented
11#13115026C211yesmaximal, characteristic
12#13115018C211yesmaximal, characteristic
13#13114849C211yesmaximal, characteristic, complemented
14#13114557C211yesmaximal, characteristic
15C13×#1873C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×#88142C2211yescharacteristic
22C2×C13×#103C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×C22⋊C8C2211yescharacteristic
25#11718891C2211yescharacteristic
26C2×#88075C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28#11718873C2211yescharacteristic
29C4×C22⋊Dic13C2211yescharacteristic
30C2×#88058C2211yescharacteristic
31C2×#88058C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34#11718646—21no—
35#11718646C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718642—21no—
38#11718638C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40#11718634C2211yescharacteristic
41#11718611—21no—
42#11718609—21no—
43#11718608—21no—
44C22⋊C4×Dic13—21no—
45#11718594—21no—
46#11718590—21no—
47C13×#340C2211yescharacteristic
48C13×#340C2211yescharacteristic
49#11718143C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59C22×Q8×C13C2311yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C23×Dic13—21no—
62C22×Dic26—21no—
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C13×C22⋊Q8—22no—
66C13×C22⋊Q8Q814yescomplemented
67C13×#117—22no—
68C13×#117—41no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C22⋊C4—21no—
74C2×C13×C22⋊C4—21no—
75C2×C42×C13—21no—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13D411yescharacteristic
78C2×C22⋊Dic13D411yescharacteristic
79Q8×Dic13—22no—
80Q8×Dic13—41no—
81C2×C4⋊Dic13—22no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—41no—
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×#6101—21no—
92C2×#6101—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×#6099—22no—
98C2×#6099—41no—
99C2×#6099—41no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102#88142—22no—
103#88142—41no—
104C13×#103—22no—
105C13×#103—41no—
106C13×C22⋊C8—22no—
107#88075—41no—
108#88075—41no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071D411yescharacteristic
119#88071D411yescharacteristic
120#88058—22no—
121#88058—41no—
122#88054—41no—
123#88054—41no—
124#26428—131nomaximal, Sylow
125C24×C13C2×D411yescharacteristic
126C2×Q8×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13C2×Q812yes—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×Q812yes—
142C22×C4×C13#4112yes—
143C22×C4×C13C2411yescharacteristic
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—42no—
149C22×Dic13—42no—
150C22×Dic13—81no—
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C2×Dic26—42no—
156C2×Dic26—42no—
157C2×Dic26—81no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C13×C4⋊C4—24no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4C2×Q812yes—
170C13×C4⋊C4#4112yes—
171C13×C22⋊C4—24no—
172C13×C22⋊C4—42no—
173C42×C13—42no—
174C22⋊Dic13—22no—
175C22⋊Dic13—41no—
176C22⋊Dic13—81no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—42no—
185C4⋊Dic13—42no—
186C4⋊Dic13—81no—
187C4⋊Dic13—81no—
188C4⋊Dic13#4112yes—
189#6101—42no—
190#6101—42no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—42no—
196C4×Dic13—81no—
197#6099—44no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204#2439—131no—
205#2360—131no—
206#2101—131no—
207#2086—131no—
208#2068—131no—
209#2024—131no—
210#1988—131no—
211#1921—131no—
212#1894—131no—
213#1873D1311yescharacteristic
214#1863—131no—
215#1852—131no—
216#1851—131no—
217#1805—131no—
218#1736—131no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13C22⋊Q812yes—
226C23×C13C22×D411yescharacteristic
227Q8×C13—22no—
228Q8×C13—22no—
229Q8×C13—41no—
230Q8×C13—41no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13C22⋊Q812yes—
250C2×C4×C13C22×Q811yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13H(5,𝔽2)11yescharacteristic
255C2×C4×C13H(5,𝔽2)11yescharacteristic
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—22no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
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—41no—
278C2×Dic13—41no—
279C2×Dic13—84no—
280C2×Dic13#13712yes—
281Dic26—84no—
282C8×C13—41no—
283C8×C13—41no—
284#1055—41no—
285#1055—41no—
286C22×M16—131no—
287C2×C4⋊Q8—131no—
288C2×C22⋊Q8C2×D1311yescharacteristic
289C2×C4×Q8—131no—
290C2×#117C2×D1311yescharacteristic
291C22×C4⋊C4—131no—
292C2×#107—131no—
293C2×#107—131no—
294C2×C4⋊C8—131no—
295C2×C4⋊C8—131no—
296C2×#103—131no—
297C2×#103—131no—
298C2×#103C2×D1311yescharacteristic
299C2×#103C2×D1311yescharacteristic
300C2×C22⋊C8C2×D1311yescharacteristic
301#396—261no—
302#393—261no—
303#391—131no—
304#391—261no—
305#389—131no—
306#389—131no—
307#389—131no—
308#389—261no—
309#387—261no—
310#387—261no—
311#385—261no—
312#382—131no—
313#380—131no—
314C4×C4⋊C4—131no—
315C4×C22⋊C4—131no—
316C4×C22⋊C4—261no—
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#340C2×D1311yescharacteristic
326#340C2×D1311yescharacteristic
327#336—131no—
328#336—131no—
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—22no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13—41no—
336C22×C13C2×C22⋊Q811yescharacteristic
337C22×C13#54611yescharacteristic
338C22×C13#54611yescharacteristic
339C22×C13C2×#13711yescharacteristic
340C22×C13#57511yescharacteristic
341C22×C13#57611yescharacteristic
342C22×C13#57811yescharacteristic
343C4×C13—22no—
344C4×C13—42no—
345C4×C13—42no—
346C4×C13—42no—
347C4×C13—81no—
348C4×C13#52911yescharacteristic
349C4×C13#53811yescharacteristic
350C4×C13#55411yescharacteristic
351C4×C13#55411yescharacteristic
352Dic13—22no—
353Dic13—41no—
354Dic13—41no—
355Dic13—42no—
356Dic13—41no—
357Dic13—81no—
358Dic13—81no—
359Dic13—81no—
360C22×Q8—261no—
361C22×Q8C22×D1311yescharacteristic
362C23×C4—261no—
363C23×C4C22×D1311yescharacteristic
364C2×M16—262no—
365C2×M16—521no—
366C2×M16—521no—
367C22×C8—131no—
368C22×C8—131no—
369C22×C8C22×D1311yescharacteristic
370C22×C8C22×D1311yescharacteristic
371C4⋊Q8—262no—
372C4⋊Q8—521no—
373C22⋊Q8—22no—
374C22⋊Q8Dic2614yescomplemented
375C4×Q8—262no—
376C4×Q8—521no—
377#117—22no—
378#117—41no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—262no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—521no—
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—131no—
398C2×C22⋊C4—131no—
399C2×C22⋊C4—261no—
400C2×C42—21no—
401C2×C42—131no—
402C2×C42—131no—
403C2×C42—261no—
404C2×C42—261no—
405#107—262no—
406#107—521no—
407C4⋊C8—521no—
408C4⋊C8—521no—
409#103—22no—
410#103—41no—
411#103—521no—
412#103—521no—
413C22⋊C8—22no—
414#95—131no—
415#95—131no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424#95—261no—
425C2×C13—42no—
426C2×C13#245011yescharacteristic
427C2×C13#307811yescharacteristic
428C2×C13#308311yescharacteristic
429C2×C13#331411yescharacteristic
430C2×C13#331911yescharacteristic
431C2×C13#332711yescharacteristic
432C2×C13#332811yescharacteristic
433C24D4×D1311yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—522no—
440C2×Q8—522no—
441C2×Q8—1041no—
442C2×Q8C2×Dic2612yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—131no—
452C22×C4—131no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—261no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—522no—
460C22×C4—522no—
461C22×C4—1041no—
462C22×C4C2×Dic2612yes—
463C22×C4D4×D1311yescharacteristic
464C22×C4#612912yes—
465C22×C4C23×D1311yescharacteristic
466M16—524no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—261no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—132no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—522no—
493C4⋊C4—522no—
494C4⋊C4—522no—
495C4⋊C4—522no—
496C4⋊C4—1041no—
497C4⋊C4—1041no—
498C4⋊C4C2×Dic2612yes—
499C4⋊C4#612912yes—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—262no—
503C22⋊C4—521no—
504C22⋊C4—1041no—
505C42—42no—
506C42—261no—
507C42—261no—
508C42—261no—
509C42—261no—
510C42—522no—
511C42—1041no—
512C13#2642811yescharacteristic, complemented, Sylow
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#8813212yes—
520C23C2×D4×D1311yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—1044no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—132no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—262no—
557C2×C4—521no—
558C2×C4—521no—
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#8813212yes—
570C2×C4C22×Dic2611yescharacteristic
571C2×C4C2×#612911yescharacteristic
572C2×C4C2×#612911yescharacteristic
573C2×C4#8825111yescharacteristic
574C2×C4#41×D1311yescharacteristic
575C2×C4#8825611yescharacteristic
576C8—41no—
577C8—41no—
578C8—521no—
579C8—521no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#8813211yescharacteristic
588C22#1171929611yescharacteristic
589C22#1171935911yescharacteristic
590C22#1171947411yescharacteristic
591C22#1171947611yescharacteristic
592C22D13×#13711yescharacteristic
593C22#1171948211yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—262no—
600C4—521no—
601C4—522no—
602C4—521no—
603C4—521no—
604C4—1041no—
605C4—1041no—
606C4—1041no—
607C4#1171924411yescharacteristic
608C4#1171924811yescharacteristic
609C4#1171930511yescharacteristic
610C4#1171932211yescharacteristic
611C2—42no—
612C2#1311866411yescharacteristic
613C2#1312463811yescharacteristic
614C2#1312468511yescharacteristic
615C2#1312691011yescharacteristic
616C2#1312699511yescharacteristic
617C2#1312709711yescharacteristic
618C2#1312711711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1873C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1873D13
hypocenterC13#26428
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
3. #13116407C2semidirect
4. #13116326C2semidirect
5. #13116229C2semidirect
6. #13115975C2semidirect
7. #13115786C2semidirect
8. #13115427C2semidirect
10. #13115092C2semidirect
13. #13114849C2semidirect
66. C13×C22⋊Q8Q8semidirect
374. C22⋊Q8Dic26semidirect
512. C13#26428semidirect

Maximal subgroups

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

subgroupindex
#131185212
#131178452
#131164072
#131163262
#131162292
#131159752
#131157862
#131154272
#131152702
#131150922
#131150262
#131150182
#131148492
#131145572
C13×#18732
#2642813