Finite Group Databasebeta

← #15921785

Subgroups — #15921785

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718537C211yesmaximal, characteristic, complemented
2#13118554C211yesmaximal, characteristic
3#13118521C211yesmaximal, characteristic
4#13116321C211yesmaximal, characteristic, complemented
5#13116320C211yesmaximal, characteristic, complemented
6#13116176C211yesmaximal, characteristic, complemented
7#13116162C211yesmaximal, characteristic, complemented
8#13115572C211yesmaximal, characteristic, complemented
9#13115570C211yesmaximal, characteristic, complemented
10#13115035C211yesmaximal, characteristic
11#13115024C211yesmaximal, characteristic
12#13114997C211yesmaximal, characteristic, complemented
13#13114869C211yesmaximal, characteristic
14#13114617C211yesmaximal, characteristic
15C2×C13×#428C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C2×#88144C2211yescharacteristic
20C2×#88142C2211yescharacteristic
21C13×#428—24no—
22C2×C13×#107C2211yescharacteristic
23C2×C13×#107C2211yescharacteristic
24C2×C13×#106C2211yescharacteristic
25C2×C13×#106C2211yescharacteristic
26#11718873—21no—
27#11718873C2211yescharacteristic
28#11718873C2211yescharacteristic
29#11718871—21no—
30C4×C22⋊Dic13C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32C2×#88058C2211yescharacteristic
33C2×#88057C2211yescharacteristic
34C2×#88055C2211yescharacteristic
35C2×C8×Dic13C2211yescharacteristic
36#11718647—21no—
37#11718646C2211yescharacteristic
38#11718646C2211yescharacteristic
39#11718645—21no—
40#11718638C2211yescharacteristic
41#11718638C2211yescharacteristic
42#11718634C2211yescharacteristic
43#11718634C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88047C2211yescharacteristic
46C2×#88047C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48#11718609—21no—
49#11718608—21no—
50#11718594—21no—
51#11718590—21no—
52#11718537—24no—
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61C23×C4×C13—21no—
62C23×C4×C13C2311yescharacteristic
63C22×Dic26—21no—
64C2×C13×M16—22no—
65C2×C13×M16—22no—
66C2×C13×M16Q814yescomplemented
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C13×#117—22no—
70C13×#117—41no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—41no—
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C22⋊C4—21no—
79C2×C42×C13—21no—
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82C2×C22⋊Dic13D411yescharacteristic
83C2×C22⋊Dic13D411yescharacteristic
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×#6101—21no—
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99#88144—22no—
100#88144—41no—
101#88142—22no—
102#88142—41no—
103C13×#107—22no—
104C13×#107D414yes—
105C13×#106—22no—
106C13×#106—22no—
107#88071—21no—
108#88071—21no—
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#88071D411yescharacteristic
118#88071D411yescharacteristic
119C22⋊#1055Q814yescomplemented
120#88058—22no—
121#88057—22no—
122#88055—22no—
123C8×Dic13—22no—
124#88048—22no—
125#88047—22no—
126#88047—41no—
127#88047—41no—
128#26079—131nomaximal, Sylow
129C24×C13C2×D411yescharacteristic
130C22×C4×C13—21no—
131C22×C4×C13—21no—
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—42no—
139C22×C4×C13—42no—
140C22×C4×C13—81no—
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×Q812yes—
143C22×C4×C13C2×Q812yes—
144C22×C4×C13C2411yescharacteristic
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13C2×D411yescharacteristic
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C2×Dic26—42no—
154C2×Dic26—42no—
155C2×Dic26—81no—
156C13×M16—28no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13C2×D412yes—
160C2×C8×C13C2×Q812yes—
161C2×C8×C13C2×Q812yes—
162C2×C8×C13#4112yes—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—42no—
170C13×C4⋊C4C2×D412yes—
171C13×C4⋊C4C2×D412yes—
172C13×C22⋊C4—42no—
173C42×C13—42no—
174C22⋊Dic13—41no—
175C22⋊Dic13—41no—
176C22⋊Dic13—42no—
177C22⋊Dic13—81no—
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—42no—
187C4⋊Dic13—81no—
188C4⋊Dic13—81no—
189#6101—42no—
190#6101—42no—
191#6101—42no—
192#6101—81no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055C2×Q812yes—
202C2×#1055C2×Q812yes—
203C2×#501—131no—
204C2×#428D1311yescharacteristic
205#2442—131no—
206#2439—131no—
207#2085—131no—
208#2085—131no—
209#2059—131no—
210#2057—131no—
211#1943—131no—
212#1942—131no—
213#1854—131no—
214#1852—131no—
215#1846—131no—
216#1809—131no—
217#1753—131no—
218C23×C13—21no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13C4⋊Q812yes—
225C23×C13C22×D411yescharacteristic
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—84no—
245C2×C4×C13C4⋊Q812yes—
246C2×C4×C13C22×D411yescharacteristic
247C2×C4×C13C22×Q811yescharacteristic
248C2×C4×C13C22×Q811yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13H(5,𝔽2)11yescharacteristic
251C2×C4×C132−1+411yescharacteristic
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
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—
276Dic26—84no—
277C8×C13—22no—
278C8×C13C22⋊Q814yes—
279#1055—22no—
280#1055—22no—
281C22×M16C2×D1311yescharacteristic
282C2×C4⋊Q8—131no—
283C2×#125—131no—
284C2×#117C2×D1311yescharacteristic
285C22×C4⋊C4C2×D1311yescharacteristic
286#501—264no—
287#428—24no—
288C2×#107—131no—
289C2×#107—131no—
290C2×#107—131no—
291C2×#107—131no—
292C2×#107C2×D1311yescharacteristic
293C2×#107C2×D1311yescharacteristic
294C2×#106—131no—
295C2×#106—131no—
296C2×#106C2×D1311yescharacteristic
297C2×#106C2×D1311yescharacteristic
298C2×C22⋊C8—131no—
299C2×#97—131no—
300C2×C4×C8—131no—
301#400—261no—
302#398—261no—
303#396—261no—
304#393—261no—
305#391—261no—
306#389—131no—
307#389—131no—
308#389—131no—
309#389—131no—
310#387—261no—
311#385—261no—
312#382—131no—
313#382—131no—
314#380—131no—
315#380—131no—
316#380—261no—
317C4×C22⋊C4—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#336—131no—
325#336—131no—
326C22×C13—22no—
327C22×C13—22no—
328C22×C13—22no—
329C22×C13—41no—
330C22×C13—41no—
331C22×C13—41no—
332C22×C13—41no—
333C22×C13C2×C4⋊Q811yescharacteristic
334C22×C13D4211yescharacteristic
335C22×C13#54711yescharacteristic
336C22×C13#57511yescharacteristic
337C22×C13#57511yescharacteristic
338C22×C13#57611yescharacteristic
339C22×C13#57811yescharacteristic
340C4×C13—22no—
341C4×C13—42no—
342C4×C13—42no—
343C4×C13—81no—
344C4×C13—81no—
345C4×C13C2×C22⋊Q811yescharacteristic
346C4×C13#54411yescharacteristic
347C4×C13D4×Q811yescharacteristic
348C4×C13#55411yescharacteristic
349Dic13—41no—
350Dic13—41no—
351Dic13—41no—
352Dic13—41no—
353Dic13—81no—
354Dic13—81no—
355Dic13—81no—
356Dic13—81no—
357C22×Q8—261no—
358C23×C4—21no—
359C23×C4C22×D1311yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16Dic2614yescomplemented
363C22×C8—131no—
364C22×C8—131no—
365C22×C8C22×D1311yescharacteristic
366C22×C8C22×D1311yescharacteristic
367C4⋊Q8—262no—
368C4⋊Q8—521no—
369#125—262no—
370#125—521no—
371#117—22no—
372#117—41no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4C22×D1311yescharacteristic
388C2×C4⋊C4C22×D1311yescharacteristic
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C4⋊C4C22×D1311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—131no—
393C2×C22⋊C4—131no—
394C2×C22⋊C4—261no—
395C2×C22⋊C4—261no—
396C2×C42—21no—
397C2×C42—131no—
398C2×C42—131no—
399#107—22no—
400#107—262no—
401#107—262no—
402#107—521no—
403#107—521no—
404#107C22⋊D1314yes—
405#106—22no—
406#106—22no—
407#106—262no—
408#106—262no—
409C22⋊C8—134no—
410#97—262no—
411C4×C8—262no—
412#95—131no—
413#95—131no—
414#95—261no—
415#95—261no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424C2×C13—42no—
425C2×C13#244011yescharacteristic
426C2×C13#316411yescharacteristic
427C2×C13#316611yescharacteristic
428C2×C13#327311yescharacteristic
429C2×C13#327711yescharacteristic
430C2×C13#334711yescharacteristic
431C2×C13#334911yescharacteristic
432C24D4×D1311yescharacteristic
433C2×Q8—522no—
434C2×Q8—522no—
435C2×Q8—1041no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4—131no—
448C22×C4—131no—
449C22×C4—131no—
450C22×C4—131no—
451C22×C4—261no—
452C22×C4—261no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4C2×Dic2612yes—
456C22×C4D4×D1311yescharacteristic
457C22×C4Q8×D1312yes—
458C22×C4C23×D1311yescharacteristic
459M16—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—132no—
463C2×C8—132no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—261no—
467C2×C8—261no—
468C2×C8C2×Dic2612yes—
469C2×C8C2×Dic2612yes—
470C2×C8#612712yes—
471C2×C8C2×C22⋊D1312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
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—522no—
488C4⋊C4—522no—
489C4⋊C4—522no—
490C4⋊C4—522no—
491C4⋊C4—1041no—
492C4⋊C4—1041no—
493C4⋊C4—1041no—
494C4⋊C4C2×C22⋊D1312yes—
495C4⋊C4C2×C22⋊D1312yes—
496C22⋊C4—42no—
497C22⋊C4—521no—
498C22⋊C4—521no—
499C22⋊C4—522no—
500C22⋊C4—1041no—
501C42—42no—
502C42—261no—
503C42—261no—
504C42—261no—
505C42—261no—
506C13#2607911yescharacteristic, complemented, Sylow
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#8814212yes—
514C23C2×D4×D1311yescharacteristic
515Q8—1044no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—261no—
536C2×C4—261no—
537C2×C4—261no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—261no—
541C2×C4—261no—
542C2×C4—261no—
543C2×C4—261no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—261no—
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#8814212yes—
560C2×C4C22×Dic2611yescharacteristic
561C2×C4C2×#612711yescharacteristic
562C2×C4#8825111yescharacteristic
563C2×C4C2×Q8×D1311yescharacteristic
564C2×C4#8825411yescharacteristic
565C2×C4C22×C22⋊D1311yescharacteristic
566C8—22no—
567C8—262no—
568C8—262no—
569C8#8817814yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#8814211yescharacteristic
578C22#1171921211yescharacteristic
579C22#1171947811yescharacteristic
580C22#1171947911yescharacteristic
581C22#1171948211yescharacteristic
582C22#1171948411yescharacteristic
583C22D4×C22⋊D1311yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—521no—
590C4—521no—
591C4—521no—
592C4—521no—
593C4—1041no—
594C4—1041no—
595C4—1041no—
596C4—1041no—
597C4#1171920511yescharacteristic
598C4#1171921611yescharacteristic
599C4C2×#8817811yescharacteristic
600C4Q8×C22⋊D1311yescharacteristic
601C2—42no—
602C2#1311853211yescharacteristic
603C2#1312537411yescharacteristic
604C2#1312538611yescharacteristic
605C2#1312644111yescharacteristic
606C2#1312648711yescharacteristic
607C2#1312733711yescharacteristic
608C2#1312736711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#428C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterC2×#428D13
hypocenterC13#26079
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#11718537C2semidirect
4. #13116321C2semidirect
5. #13116320C2semidirect
6. #13116176C2semidirect
7. #13116162C2semidirect
8. #13115572C2semidirect
9. #13115570C2semidirect
12. #13114997C2semidirect
66. C2×C13×M16Q8semidirect
119. C22⋊#1055Q8semidirect
362. C2×M16Dic26semidirect
506. C13#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#117185372
#131185542
#131185212
#131163212
#131163202
#131161762
#131161622
#131155722
#131155702
#131150352
#131150242
#131149972
#131148692
#131146172
C2×C13×#4282
#2607913