Finite Group Databasebeta

← #15921791

Subgroups — #15921791

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×#11718363C211yesmaximal, characteristic, complemented
2C2×#11718782C211yesmaximal, characteristic
3#13118560C211yesmaximal, characteristic
4#13118527C211yesmaximal, characteristic
5#13116319C211yesmaximal, characteristic, complemented
6#13116319C211yesmaximal, characteristic, complemented
7#13116175C211yesmaximal, characteristic, complemented
8#13116165C211yesmaximal, characteristic, complemented
9#13115574C211yesmaximal, characteristic, complemented
10#13115569C211yesmaximal, characteristic, complemented
11#13115037C211yesmaximal, characteristic
12#13115026C211yesmaximal, characteristic
13#13115000C211yesmaximal, characteristic, complemented
14#13114869C211yesmaximal, characteristic
15C13×#1753C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C2×#88124C2211yescharacteristic
20C2×#88123C2211yescharacteristic
21C2×C13×C22⋊C8C2211yescharacteristic
22C2×C13×#97C2211yescharacteristic
23C2×C4×C8×C13C2211yescharacteristic
24C4×C13×C22⋊C4C2211yescharacteristic
25#11718873C2211yescharacteristic
26#11718873C2211yescharacteristic
27#11718782—24no—
28C2×#88058C2211yescharacteristic
29C2×#88058C2211yescharacteristic
30C2×#88058C2211yescharacteristic
31C2×#88058C2211yescharacteristic
32C2×#88058C2211yescharacteristic
33C2×#88058C2211yescharacteristic
34C2×#88057C2211yescharacteristic
35C2×#88057C2211yescharacteristic
36C2×#88057C2211yescharacteristic
37C2×#88057C2211yescharacteristic
38#11718609—21no—
39#11718608—21no—
40#11718607—21no—
41#11718603—21no—
42#11718597C2211yescharacteristic
43#11718597C2211yescharacteristic
44#11718589C2211yescharacteristic
45#11718589C2211yescharacteristic
46#11718589C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718363—24no—
49#11718321—21no—
50#11718318—21no—
51#11718316—21no—
52#11718307—21no—
53C13×#336C2211yescharacteristic
54C13×#336C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718143C2211yescharacteristic
59#11718143C2211yescharacteristic
60#11718143C2211yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C23×Dic13—21no—
63C22×Dic26—21no—
64C2×C13×M16—22no—
65C2×C13×M16—22no—
66C2×C13×M16Q814yescomplemented
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C22×C8×C13C2311yescharacteristic
71C2×C13×C22⋊C4D411yescharacteristic
72C2×C13×C22⋊C4D411yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×C42×C13C2311yescharacteristic
75C2×C22⋊Dic13—21no—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13—21no—
78#88180—22no—
79#88180—41no—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13—21no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—41no—
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C2×#6101—21no—
98C2×C4×Dic13—21no—
99#88124—22no—
100#88124—41no—
101#88123—22no—
102#88123—41no—
103C13×C22⋊C8Q814yescomplemented
104C13×#97—22no—
105C4×C8×C13—22no—
106C13×#95D411yescharacteristic
107C13×#95D411yescharacteristic
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#88071—21no—
118#88058—22no—
119#88058—22no—
120#88058—22no—
121#88058—41no—
122#88058—41no—
123#88058D414yes—
124#88057—22no—
125#88057—22no—
126#88057—22no—
127#88057—22no—
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×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×Q812yes—
140C22×C4×C13C2×Q812yes—
141C22×C4×C13C2411yescharacteristic
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13—42no—
151C22×Dic13—42no—
152C22×Dic13—81no—
153C2×Dic26—42no—
154C2×Dic26—42no—
155C2×Dic26—81no—
156C13×M16—28no—
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×C13C2×D412yes—
164C2×C8×C13C2×Q812yes—
165C2×C8×C13C2×Q812yes—
166C2×C8×C13C2×Q812yes—
167C2×C8×C13C2×Q812yes—
168C2×C8×C13#4112yes—
169C13×C22⋊C4—41no—
170C13×C22⋊C4—41no—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C22⋊Dic13—42no—
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—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—42no—
193C4⋊Dic13—42no—
194C4⋊Dic13C2×D412yes—
195C4⋊Dic13C2×D412yes—
196#6101—42no—
197#6101—42no—
198#6101—42no—
199#6101—81no—
200#6101—81no—
201#6101—81no—
202C4×Dic13—42no—
203C2×#501—131no—
204C2×#428—131no—
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#1753D1311yescharacteristic
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—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
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×C13C4⋊Q812yes—
245C2×C4×C13C22×D411yescharacteristic
246C2×C4×C13C22×Q811yescharacteristic
247C2×C4×C13C22×Q811yescharacteristic
248C2×C4×C13C2×#4111yescharacteristic
249C2×C4×C13H(5,𝔽2)11yescharacteristic
250C2×C4×C132−1+411yescharacteristic
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
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—84no—
276Dic26—84no—
277C8×C13—22no—
278C8×C13—22no—
279C8×C13—22no—
280C8×C13C22⋊Q814yes—
281C22×M16C2×D1311yescharacteristic
282C2×C4⋊Q8—131no—
283C2×#125—131no—
284C2×#117—131no—
285C22×C4⋊C4—131no—
286#501—264no—
287#428—264no—
288C2×#107—131no—
289C2×#107—131no—
290C2×#107—131no—
291C2×#107—131no—
292C2×#107—131no—
293C2×#107—131no—
294C2×#106—131no—
295C2×#106—131no—
296C2×#106—131no—
297C2×#106—131no—
298C2×C22⋊C8C2×D1311yescharacteristic
299C2×#97C2×D1311yescharacteristic
300C2×C4×C8C2×D1311yescharacteristic
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⋊C4C2×D1311yescharacteristic
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#336C2×D1311yescharacteristic
325#336C2×D1311yescharacteristic
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—41no—
342C4×C13—41no—
343C4×C13—41no—
344C4×C13—41no—
345C4×C13C2×C22⋊Q811yescharacteristic
346C4×C13#54411yescharacteristic
347C4×C13D4×Q811yescharacteristic
348C4×C13#55411yescharacteristic
349Dic13—42no—
350Dic13—42no—
351Dic13—81no—
352Dic13—81no—
353Dic13—81no—
354Dic13—81no—
355Dic13—81no—
356Dic13—81no—
357C22×Q8—261no—
358C23×C4—261no—
359C23×C4C22×D1311yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16Dic2614yescomplemented
363C22×C8C22×D1311yescharacteristic
364C22×C8C22×D1311yescharacteristic
365C22×C8C22×D1311yescharacteristic
366C22×C8C22×D1311yescharacteristic
367C4⋊Q8—262no—
368C4⋊Q8—521no—
369#125—262no—
370#125—521no—
371#117—262no—
372#117—521no—
373C2×C4⋊C4—131no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—262no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—521no—
391C2×C22⋊C4—261no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4—261no—
394C2×C22⋊C4D5211yescharacteristic
395C2×C22⋊C4D5211yescharacteristic
396C2×C42—261no—
397C2×C42C22×D1311yescharacteristic
398C2×C42C22×D1311yescharacteristic
399#107—134no—
400#107—262no—
401#107—262no—
402#107—262no—
403#107—521no—
404#107—521no—
405#106—262no—
406#106—262no—
407#106—262no—
408#106—262no—
409C22⋊C8Dic2614yescomplemented
410#97—22no—
411C4×C8—22no—
412#95—261no—
413#95—261no—
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#95D5211yescharacteristic
423#95D5211yescharacteristic
424C2×C13—42no—
425C2×C13#244011yescharacteristic
426C2×C13#316411yescharacteristic
427C2×C13#316611yescharacteristic
428C2×C13#327311yescharacteristic
429C2×C13#327711yescharacteristic
430C2×C13#334711yescharacteristic
431C2×C13#334911yescharacteristic
432C24C2×D5211yescharacteristic
433C2×Q8—522no—
434C2×Q8—522no—
435C2×Q8—1041no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—261no—
441C22×C4—261no—
442C22×C4—261no—
443C22×C4—261no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4—522no—
449C22×C4—522no—
450C22×C4—1041no—
451C22×C4C2×Dic2612yes—
452C22×C4C2×Dic2612yes—
453C22×C4C2×D5211yescharacteristic
454C22×C4C2×D5211yescharacteristic
455C22×C4C2×D5211yescharacteristic
456C22×C4C2×D5211yescharacteristic
457C22×C4C2×D5211yescharacteristic
458C22×C4C23×D1311yescharacteristic
459M16—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8C2×Dic2612yes—
467C2×C8C2×Dic2612yes—
468C2×C8C2×Dic2612yes—
469C2×C8C2×Dic2612yes—
470C2×C8D4×D1312yes—
471C2×C8#612912yes—
472C4⋊C4—132no—
473C4⋊C4—132no—
474C4⋊C4—261no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—261no—
478C4⋊C4—261no—
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—522no—
489C4⋊C4—522no—
490C4⋊C4—522no—
491C4⋊C4—522no—
492C4⋊C4—522no—
493C4⋊C4—1041no—
494C4⋊C4—1041no—
495C4⋊C4—1041no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—522no—
499C22⋊C4—522no—
500C22⋊C4—1041no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—522no—
506C13#2607911yescharacteristic, complemented, Sylow
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#8812312yes—
514C23C22×D5211yescharacteristic
515Q8—1044no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—262no—
535C2×C4—262no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
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—1044no—
559C2×C4#8812312yes—
560C2×C4C22×Dic2611yescharacteristic
561C2×C4C22×Dic2611yescharacteristic
562C2×C4C2×D4×D1311yescharacteristic
563C2×C4C2×#612911yescharacteristic
564C2×C4#8825611yescharacteristic
565C2×C4#8825711yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#8813214yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#8812311yescharacteristic
578C22D4×D5211yescharacteristic
579C22#1171926011yescharacteristic
580C22#1171945311yescharacteristic
581C22#1171945311yescharacteristic
582C22#1171945711yescharacteristic
583C22#1171945811yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—522no—
590C4—522no—
591C4—1041no—
592C4—1041no—
593C4—1041no—
594C4—1041no—
595C4—1041no—
596C4—1041no—
597C4C2×#8813211yescharacteristic
598C4#1171922411yescharacteristic
599C4D4×Dic2611yescharacteristic
600C4#1171924811yescharacteristic
601C2—42no—
602C2#1311853811yescharacteristic
603C2#1312537511yescharacteristic
604C2#1312538811yescharacteristic
605C2#1312644811yescharacteristic
606C2#1312649411yescharacteristic
607C2#1312733811yescharacteristic
608C2#1312736811yescharacteristic
subgroupG/H
centerC23C22×D52
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1753C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1753D13
hypocenterC13#26079
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#11718363C2semidirect
5. #13116319C2semidirect
6. #13116319C2semidirect
7. #13116175C2semidirect
8. #13116165C2semidirect
9. #13115574C2semidirect
10. #13115569C2semidirect
13. #13115000C2semidirect
66. C2×C13×M16Q8semidirect
103. C13×C22⋊C8Q8semidirect
362. C2×M16Dic26semidirect
409. C22⋊C8Dic26semidirect
506. C13#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#117183632
C2×#117187822
#131185602
#131185272
#131163192
#131161752
#131161652
#131155742
#131155692
#131150372
#131150262
#131150002
#131148692
C13×#17532
#2607913