Finite Group Databasebeta

← #15926858

Subgroups — #15926858

Order 3328.

748 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13118549C211yesmaximal, characteristic
2#13117839C211yesmaximal, characteristic
3#13116400C211yesmaximal, characteristic, complemented
4#13116334C211yesmaximal, characteristic, complemented
5#13116216C211yesmaximal, characteristic, complemented
6#13115988C211yesmaximal, characteristic, complemented
7#13115791C211yesmaximal, characteristic, complemented
8#13115422C211yesmaximal, characteristic, complemented
9#13115269C211yesmaximal, characteristic
10#13115146C211yesmaximal, characteristic
11#13115107C211yesmaximal, characteristic, complemented
12#13115031C211yesmaximal, characteristic
13#13114842C211yesmaximal, characteristic, complemented
14#13114556C211yesmaximal, characteristic
15C13×#1851C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C2×#88178C2211yescharacteristic
20C2×#88124C2211yescharacteristic
21C2×C4×Dic26C2211yescharacteristic
22C2×C13×C4⋊C8C2211yescharacteristic
23C2×C13×C4⋊C8C2211yescharacteristic
24C2×C13×C22⋊C8C2211yescharacteristic
25C4×C13×C22⋊C4C2211yescharacteristic
26#11718873C2211yescharacteristic
27C2×#88057C2211yescharacteristic
28C2×#88057C2211yescharacteristic
29C2×#88056C2211yescharacteristic
30C2×#88056C2211yescharacteristic
31C2×#88056C2211yescharacteristic
32C2×#88056C2211yescharacteristic
33#11718610—21no—
34#11718609—21no—
35#11718607—21no—
36C22⋊C4×Dic13—21no—
37#11718596C2211yescharacteristic
38C4×C4⋊Dic13C2211yescharacteristic
39#11718590C2211yescharacteristic
40#11718589C2211yescharacteristic
41#11718589C2211yescharacteristic
42#11718589C2211yescharacteristic
43#11718316—21no—
44#11718315—21no—
45#11718308—21no—
46#11718308—21no—
47C13×#336C2211yescharacteristic
48C13×#336C2211yescharacteristic
49#11718143C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718143C2211yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C23×Dic13—21no—
61C22×Dic26C2311yescharacteristic
62C2×C13×M16—22no—
63C2×C13×M16—41no—
64C2×C13×M16—41no—
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C2×C13×C22⋊C4D411yescharacteristic
70C2×C13×C22⋊C4D411yescharacteristic
71C2×C42×C13C2311yescharacteristic
72C2×C42×C13C2311yescharacteristic
73C2×C22⋊Dic13—21no—
74C2×C22⋊Dic13—21no—
75C2×C22⋊Dic13—21no—
76#88180—22no—
77#88180—41no—
78C2×C4⋊Dic13—22no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13—21no—
82C2×C4⋊Dic13—41no—
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90#88178—22no—
91#88178Q814yes—
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C2×#6101—21no—
98C2×#6101—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13—21no—
102#88124—22no—
103#88124—41no—
104C4×Dic26—22no—
105C4×Dic26—41no—
106C13×C4⋊C8—41no—
107C13×C4⋊C8—41no—
108C13×C22⋊C8—22no—
109C13×#95D411yescharacteristic
110C13×#95D411yescharacteristic
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88057—22no—
119#88057—41no—
120#88056—22no—
121#88056—41no—
122#88056—41no—
123#88056—41no—
124#26434—131nomaximal, Sylow
125C24×C13C2×D411yescharacteristic
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13C2×D411yescharacteristic
131C22×C4×C13C2×D411yescharacteristic
132C22×C4×C13C2×D411yescharacteristic
133C22×C4×C13C2×D411yescharacteristic
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×Q812yes—
136C22×C4×C13#4112yes—
137C22×C4×C13C2411yescharacteristic
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—42no—
147C22×Dic13—42no—
148C22×Dic13—81no—
149C2×Dic26—21no—
150C2×Dic26—21no—
151C2×Dic26—21no—
152C2×Dic26—21no—
153C2×Dic26—21no—
154C2×Dic26C2×Q812yes—
155C13×M16—44no—
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C13×C22⋊C4—22no—
169C13×C22⋊C4—41no—
170C42×C13—21no—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—21no—
174C22⋊Dic13—24no—
175C22⋊Dic13—42no—
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—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—42no—
189C4⋊Dic13—42no—
190C4⋊Dic13—81no—
191C4⋊Dic13C2×Q812yes—
192C4⋊Dic13#4112yes—
193C4⋊Dic13#4112yes—
194#6101—24no—
195#6101—42no—
196#6101—42no—
197#6101—42no—
198#6101—42no—
199#6101—81no—
200#6101—81no—
201C4×Dic13—42no—
202C4×Dic13—42no—
203C4×Dic13—81no—
204#2441—131no—
205#2360—131no—
206#2100—131no—
207#2088—131no—
208#2066—131no—
209#2026—131no—
210#1989—131no—
211#1920—131no—
212#1894—131no—
213#1873—131no—
214#1865—131no—
215#1853—131no—
216#1851D1311yescharacteristic
217#1803—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
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
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×C13C22⋊Q812yes—
245C2×C4×C13#13712yes—
246C2×C4×C13C22×Q811yescharacteristic
247C2×C4×C13C2×#4111yescharacteristic
248C2×C4×C13C2×#4111yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13H(5,𝔽2)11yescharacteristic
251C2×C4×C13H(5,𝔽2)11yescharacteristic
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—22no—
255C2×Dic13—22no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
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—84no—
277Dic26—22no—
278Dic26—22no—
279Dic26—41no—
280Dic26—41no—
281C8×C13—41no—
282C8×C13—41no—
283C8×C13—41no—
284C8×C13—41no—
285C22×M16C2×D1311yescharacteristic
286C2×#125—131no—
287C2×C22⋊Q8—131no—
288C2×C4×Q8—131no—
289C2×#117—131no—
290C22×C4⋊C4—131no—
291C2×#106—131no—
292C2×#106—131no—
293C2×C4⋊C8C2×D1311yescharacteristic
294C2×C4⋊C8C2×D1311yescharacteristic
295C2×#103—131no—
296C2×#103—131no—
297C2×#103—131no—
298C2×#103—131no—
299C2×C22⋊C8C2×D1311yescharacteristic
300#399—261no—
301#396—261no—
302#391—131no—
303#389—131no—
304#389—131no—
305#389—131no—
306#389—261no—
307#389—261no—
308#387—261no—
309#387—261no—
310#385—261no—
311#384—131no—
312#380—131no—
313C4×C4⋊C4—131no—
314C4×C22⋊C4—261no—
315C4×C22⋊C4C2×D1311yescharacteristic
316#340—131no—
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#336C2×D1311yescharacteristic
327#336C2×D1311yescharacteristic
328C22×C13—22no—
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—41no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13C2×C22⋊Q811yescharacteristic
336C22×C13#54611yescharacteristic
337C22×C13#54611yescharacteristic
338C22×C13C2×#13711yescharacteristic
339C22×C13#57511yescharacteristic
340C22×C13#57711yescharacteristic
341C22×C13#57711yescharacteristic
342C4×C13—22no—
343C4×C13—22no—
344C4×C13—41no—
345C4×C13—41no—
346C4×C13—41no—
347C4×C13#52911yescharacteristic
348C4×C13#53811yescharacteristic
349C4×C13#55411yescharacteristic
350C4×C13#55411yescharacteristic
351Dic13—42no—
352Dic13—42no—
353Dic13—42no—
354Dic13—42no—
355Dic13—81no—
356Dic13—81no—
357Dic13—81no—
358Dic13—81no—
359C22×Q8—131no—
360C23×C4—261no—
361C23×C4C22×D1311yescharacteristic
362C2×M16—22no—
363C2×M16—41no—
364C2×M16—41no—
365C22×C8C22×D1311yescharacteristic
366C22×C8C22×D1311yescharacteristic
367C22×C8C22×D1311yescharacteristic
368C22×C8C22×D1311yescharacteristic
369#125—262no—
370#125—521no—
371C22⋊Q8—134no—
372C22⋊Q8—262no—
373C4×Q8—262no—
374C4×Q8—521no—
375#117—262no—
376#117—521no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
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—261no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4—261no—
395C2×C4⋊C4—521no—
396C2×C22⋊C4—261no—
397C2×C22⋊C4—261no—
398C2×C22⋊C4—261no—
399C2×C22⋊C4D5211yescharacteristic
400C2×C22⋊C4D5211yescharacteristic
401C2×C42—261no—
402C2×C42—261no—
403C2×C42—261no—
404C2×C42C22×D1311yescharacteristic
405C2×C42C22×D1311yescharacteristic
406#106—262no—
407#106—521no—
408C4⋊C8—41no—
409C4⋊C8—41no—
410#103—262no—
411#103—521no—
412#103—521no—
413#103—521no—
414C22⋊C8—22no—
415#95—261no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95D5211yescharacteristic
423#95D5211yescharacteristic
424C2×C13—42no—
425C2×C13#245011yescharacteristic
426C2×C13#307811yescharacteristic
427C2×C13#308411yescharacteristic
428C2×C13#331711yescharacteristic
429C2×C13#332011yescharacteristic
430C2×C13#332311yescharacteristic
431C2×C13#333111yescharacteristic
432C24C2×D5211yescharacteristic
433C2×Q8—132no—
434C2×Q8—261no—
435C2×Q8—261no—
436C2×Q8—261no—
437C2×Q8—261no—
438C2×Q8—261no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—261no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—522no—
452C22×C4—522no—
453C22×C4—1041no—
454C22×C4C2×D5211yescharacteristic
455C22×C4C2×D5211yescharacteristic
456C22×C4C2×D5211yescharacteristic
457C22×C4C2×D5211yescharacteristic
458C22×C4C2×D5211yescharacteristic
459C22×C4#612912yes—
460C22×C4Q8×D1312yes—
461C22×C4C23×D1311yescharacteristic
462M16—44no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C4⋊C4—132no—
476C4⋊C4—132no—
477C4⋊C4—132no—
478C4⋊C4—264no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—261no—
490C4⋊C4—522no—
491C4⋊C4—522no—
492C4⋊C4—522no—
493C4⋊C4—522no—
494C4⋊C4—522no—
495C4⋊C4—522no—
496C4⋊C4—1041no—
497C4⋊C4—1041no—
498C4⋊C4—1041no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—264no—
502C22⋊C4—522no—
503C22⋊C4—1041no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—522no—
509C42—522no—
510C42—1041no—
511C13#2643411yescharacteristic, complemented, Sylow
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23#8815112yes—
519C23C22×D5211yescharacteristic
520Q8—262no—
521Q8—262no—
522Q8—521no—
523Q8—521no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—262no—
542C2×C4—262no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
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—1044no—
566C2×C4#8815112yes—
567C2×C4#8816312yes—
568C2×C4C2×#612911yescharacteristic
569C2×C4#8825111yescharacteristic
570C2×C4#8825111yescharacteristic
571C2×C4C2×Q8×D1311yescharacteristic
572C2×C4#41×D1311yescharacteristic
573C2×C4#41×D1311yescharacteristic
574C8—41no—
575C8—41no—
576C8—41no—
577C8—41no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×#8815111yescharacteristic
586C22#1171925911yescharacteristic
587C22#1171925911yescharacteristic
588C22C2×#8816311yescharacteristic
589C22#1171945311yescharacteristic
590C22#1171945611yescharacteristic
591C22#1171945611yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—522no—
598C4—522no—
599C4—522no—
600C4—522no—
601C4—1041no—
602C4—1041no—
603C4—1041no—
604C4—1041no—
605C4#1171929811yescharacteristic
606C4#1171934111yescharacteristic
607C4#1171934111yescharacteristic
608C4#1171935411yescharacteristic
609C2—42no—
610C2#1311865811yescharacteristic
611C2#1312464311yescharacteristic
612C2#1312470011yescharacteristic
613C2#1312696411yescharacteristic
614C2#1312700911yescharacteristic
615C2#1312705411yescharacteristic
616C2#1312715311yescharacteristic
subgroupG/H
centerC23C22×D52
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1851C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1851D13
hypocenterC13#26434
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
3. #13116400C2semidirect
4. #13116334C2semidirect
5. #13116216C2semidirect
6. #13115988C2semidirect
7. #13115791C2semidirect
8. #13115422C2semidirect
11. #13115107C2semidirect
13. #13114842C2semidirect
511. C13#26434semidirect

Maximal subgroups

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

subgroupindex
#131185492
#131178392
#131164002
#131163342
#131162162
#131159882
#131157912
#131154222
#131152692
#131151462
#131151072
#131150312
#131148422
#131145562
C13×#18512
#2643413