Finite Group Databasebeta

← #16035657

Subgroups — #16035657

Order 3328.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123142C214yesmaximal, complemented
2#13120808C214yesmaximal, complemented
3#13119335C214yesmaximal, complemented
4#13119333C214yesmaximal, complemented
5#13118982C214yesmaximal, complemented
6#13118974C214yesmaximal, complemented
7#13117448C211yesmaximal, characteristic
8#13117448C211yesmaximal, characteristic
9#13117448C211yesmaximal, characteristic
10C2×#11718611C211yesmaximal, characteristic
11C2×#11718607C211yesmaximal, characteristic
12C2×C22⋊C4×Dic13C211yesmaximal, characteristic
13C13×#2315C211yesmaximal, characteristic
14C24×Dic13C2211yescharacteristic
15C22×C13×C22⋊C4C2211yescharacteristic
16C22×C13×C22⋊C4C2211yescharacteristic
17C22×C13×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic13C2211yescharacteristic
19C22×C22⋊Dic13C2211yescharacteristic
20C22×C22⋊Dic13C2211yescharacteristic
21C22×C4⋊Dic13C2212yes—
22C22×#6101C2212yes—
23C22×C4×Dic13C2212yes—
24C2×#88143C2214yescomplemented
25C13×#385C2214yes—
26C13×#385C2214yes—
27C13×#380C2212yes—
28C13×#380C2212yes—
29#11718871C2212yes—
30#11718871C2214yes—
31#11718871C2212yes—
32#11718871C2214yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36#11718645C2214yescomplemented
37#11718644C2214yescomplemented
38#11718643C2214yescomplemented
39#11718642C2214yescomplemented
40#11718641C2214yescomplemented
41#11718639C2214yescomplemented
42#11718633C2214yescomplemented
43#11718611C2214yes—
44#11718611C2212yes—
45#11718611C2212yes—
46#11718607C2214yes—
47#11718607C2214yes—
48#11718607C2212yes—
49#11718607C2212yes—
50#11718607C2212yes—
51#11718607C2212yes—
52#11718607C2214yes—
53#11718603C2214yes—
54#11718603C2214yes—
55#11718603C2212yes—
56#11718603C2212yes—
57C22⋊C4×Dic13C2214yes—
58C22⋊C4×Dic13C2214yes—
59#11718320C2214yescomplemented
60#11718319C2214yescomplemented
61#11718319C2214yescomplemented
62#11718318C2214yescomplemented
63#11718314C2214yescomplemented
64#11718313C2214yescomplemented
65#11718311C2214yescomplemented
66#11718308C2214yescomplemented
67C25×C13C2311yescharacteristic
68C23×C4×C13C2311yescharacteristic
69C23×C4×C13C2311yescharacteristic
70C23×C4×C13C2312yes—
71C23×C4×C13C2312yes—
72C23×Dic13—24no—
73C23×Dic13—44no—
74C23×Dic13C2311yescharacteristic
75C23×Dic13C2312yes—
76C23×Dic13C2312yes—
77C23×Dic13C2312yes—
78C23×Dic13C2311yescharacteristic
79C23×Dic13C2311yescharacteristic
80C23×Dic13C2311yescharacteristic
81C23×Dic13C2312yes—
82C2×C13×C4⋊C4C2314yes—
83C2×C13×C4⋊C4C2314yes—
84C2×C13×C22⋊C4—24no—
85C2×C13×C22⋊C4—24no—
86C2×C13×C22⋊C4—24no—
87C2×C13×C22⋊C4C2312yes—
88C2×C13×C22⋊C4C2312yes—
89C2×C13×C22⋊C4C2312yes—
90C2×C13×C22⋊C4C2312yes—
91C2×C13×C22⋊C4C2312yes—
92C2×C13×C22⋊C4C2312yes—
93C2×C22⋊Dic13—24no—
94C2×C22⋊Dic13—24no—
95C2×C22⋊Dic13Q818yes—
96C2×C22⋊Dic13C2312yes—
97C2×C22⋊Dic13C2312yes—
98C2×C22⋊Dic13C2312yes—
99C2×C22⋊Dic13C2312yes—
100C2×C22⋊Dic13C2312yes—
101C2×C22⋊Dic13C2312yes—
102C2×C4⋊Dic13—44no—
103C2×C4⋊Dic13C2314yes—
104C2×C4⋊Dic13C2314yes—
105C2×#6101—44no—
106C2×#6101C2314yes—
107C2×#6101C2314yes—
108C2×#6101C2314yes—
109C2×#6101C2314yes—
110C2×#6101C2314yes—
111C2×#6101C2314yes—
112C2×#6101C2314yes—
113C2×#6101C2314yes—
114C2×C4×Dic13—44no—
115C2×C4×Dic13C2314yes—
116C2×C4×Dic13C2314yes—
117#88143—48no—
118C13×#95C2314yes—
119C13×#95C2314yes—
120#88071C2314yes—
121#88071C2314yes—
122#88071C2314yes—
123#88071C2314yes—
124#88071C2314yes—
125#88071C2314yes—
126#88071C2314yes—
127#88071C2314yes—
128#88071C2314yes—
129#88071C2314yes—
130#88071C2314yes—
131#88071C2314yes—
132#34994—131nomaximal, Sylow
133C24×C13—42no—
134C24×C13—42no—
135C24×C13C2×Q814yes—
136C24×C13#4114yes—
137C24×C13#4114yes—
138C24×C13C2411yescharacteristic
139C24×C13C2411yescharacteristic
140C24×C13C2411yescharacteristic
141C22×C4×C13—24no—
142C22×C4×C13—24no—
143C22×C4×C13—22no—
144C22×C4×C13—22no—
145C22×C4×C13—42no—
146C22×C4×C13—42no—
147C22×C4×C13—42no—
148C22×C4×C13—42no—
149C22×C4×C13—42no—
150C22×C4×C13—42no—
151C22×C4×C13C2412yes—
152C22×C4×C13C2414yes—
153C22×C4×C13C2414yes—
154C22×C4×C13C2412yes—
155C22×Dic13—22no—
156C22×Dic13—24no—
157C22×Dic13—24no—
158C22×Dic13—24no—
159C22×Dic13—24no—
160C22×Dic13—22no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—24no—
164C22×Dic13—24no—
165C22×Dic13—42no—
166C22×Dic13—42no—
167C22×Dic13—42no—
168C22×Dic13—42no—
169C22×Dic13—42no—
170C22×Dic13—42no—
171C22×Dic13—42no—
172C22×Dic13—42no—
173C22×Dic13—42no—
174C22×Dic13—416no—
175C22×Dic13C2×Q818yes—
176C22×Dic13C2412yes—
177C22×Dic13C2412yes—
178C22×Dic13C2414yes—
179C22×Dic13C2412yes—
180C22×Dic13C2414yes—
181C22×Dic13C2412yes—
182C13×C4⋊C4—44no—
183C13×C4⋊C4—44no—
184C13×C22⋊C4—24no—
185C13×C22⋊C4—24no—
186C13×C22⋊C4—24no—
187C13×C22⋊C4—24no—
188C13×C22⋊C4—24no—
189C13×C22⋊C4—24no—
190C22⋊Dic13—24no—
191C22⋊Dic13—24no—
192C22⋊Dic13—24no—
193C22⋊Dic13—24no—
194C22⋊Dic13C2×Q818yes—
195C22⋊Dic13C2×Q818yes—
196C4⋊Dic13—28no—
197C4⋊Dic13—44no—
198#6101—28no—
199#6101—44no—
200#6101—44no—
201#6101—44no—
202#6101—44no—
203#6101—44no—
204#6101—44no—
205#6101—44no—
206C4×Dic13—28no—
207C4×Dic13—44no—
208#2851—134no—
209#2646—134no—
210#2505—134no—
211#2505—134no—
212#2473—134no—
213#2473—134no—
214#2315—131no—
215#2315—131no—
216#2315—131no—
217#2315D1311yescharacteristic
218C2×#385—131no—
219C2×#385—131no—
220C2×C4×C22⋊C4—131no—
221C23×C13—22no—
222C23×C13—22no—
223C23×C13—22no—
224C23×C13—22no—
225C23×C13—22no—
226C23×C13—22no—
227C23×C13—44no—
228C23×C13—48no—
229C23×C13—48no—
230C23×C13—44no—
231C23×C13#125116yescomplemented
232C23×C13C22×Q812yes—
233C23×C13C22×Q812yes—
234C23×C13C22×Q812yes—
235C23×C13C2×#4112yes—
236C23×C13C2×#4112yes—
237C23×C13C2×#4112yes—
238C23×C13C2×#4112yes—
239C23×C13C2×#4112yes—
240C23×C13C2×#4112yes—
241C23×C13C2511yescharacteristic
242C2×C4×C13—24no—
243C2×C4×C13—24no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—24no—
247C2×C4×C13—24no—
248C2×C4×C13—24no—
249C2×C4×C13—24no—
250C2×C4×C13—24no—
251C2×C4×C13—24no—
252C2×C4×C13—48no—
253C2×C4×C13—48no—
254C2×C4×C13—82no—
255C2×C4×C13—82no—
256C2×Dic13—28no—
257C2×Dic13—22no—
258C2×Dic13—24no—
259C2×Dic13—22no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—24no—
266C2×Dic13—24no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—24no—
270C2×Dic13—24no—
271C2×Dic13—48no—
272C2×Dic13—48no—
273C2×Dic13—48no—
274C2×Dic13—48no—
275C2×Dic13—44no—
276C2×Dic13—82no—
277C2×Dic13—82no—
278C2×Dic13C22×Q818yes—
279C2×Dic13H(5,𝔽2)14yes—
280C24×C4—131no—
281C2×#125—134no—
282C22×C4⋊C4—132no—
283C22×C4⋊C4—132no—
284C22×C22⋊C4—131no—
285C22×C22⋊C4—131no—
286C22×C22⋊C4—131no—
287C22×C22⋊C4C2×D1311yescharacteristic
288C22×C22⋊C4C2×D1311yescharacteristic
289C22×C22⋊C4C2×D1311yescharacteristic
290C22×C42—132no—
291#400—134no—
292#400—134no—
293#400—134no—
294#400—134no—
295#398—134no—
296#398—134no—
297#398—134no—
298#398—134no—
299#389—134no—
300#389—134no—
301#387—134no—
302#387—134no—
303#387—134no—
304#387—134no—
305#385—134no—
306#385—132no—
307#385—134no—
308#385—132no—
309#385—132no—
310#385—134no—
311#385—134no—
312#385—132no—
313#385—132no—
314#385—132no—
315#385—132no—
316#385—132no—
317#385—134no—
318#385—134no—
319#385C2×D1314yes—
320#385C2×D1314yes—
321#382—134no—
322#380—134no—
323#380—134no—
324#380—132no—
325#380—132no—
326#380C2×D1312yes—
327#380C2×D1312yes—
328C4×C22⋊C4—134no—
329C4×C22⋊C4—134no—
330C2×#95—132no—
331C2×#95—132no—
332C2×#95—132no—
333C22×C13—24no—
334C22×C13—24no—
335C22×C13—24no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—22no—
341C22×C13—22no—
342C22×C13—24no—
343C22×C13—24no—
344C22×C13—24no—
345C22×C13—416no—
346C22×C13C2×#12514yes—
347C22×C13C2×#12514yes—
348C22×C13C2×#12514yes—
349C22×C13C23×Q811yescharacteristic
350C22×C13C22×#4111yescharacteristic
351C22×C13C22×#4111yescharacteristic
352C22×C13C2×H(5,𝔽2)12yes—
353C22×C13#58512yes—
354C4×C13—44no—
355C4×C13—44no—
356C4×C13—42no—
357C4×C13—42no—
358Dic13—24no—
359Dic13—28no—
360Dic13—44no—
361Dic13—42no—
362Dic13—42no—
363Dic13—42no—
364C25C22×D1311yescharacteristic
365C23×C4—132no—
366C23×C4—131no—
367C23×C4—132no—
368C23×C4—132no—
369C23×C4—131no—
370C23×C4—132no—
371C23×C4—131no—
372C23×C4—131no—
373C23×C4—264no—
374C23×C4—524no—
375C23×C4C22×D1311yescharacteristic
376C23×C4C22×D1312yes—
377C23×C4C22×D1311yescharacteristic
378C23×C4C22×D1312yes—
379#125—528no—
380C2×C4⋊C4—134no—
381C2×C4⋊C4—134no—
382C2×C4⋊C4—134no—
383C2×C4⋊C4—134no—
384C2×C4⋊C4—134no—
385C2×C4⋊C4—134no—
386C2×C4⋊C4—134no—
387C2×C4⋊C4—134no—
388C2×C4⋊C4—134no—
389C2×C4⋊C4—134no—
390C2×C4⋊C4—524no—
391C2×C4⋊C4—524no—
392C2×C4⋊C4C22×D1314yes—
393C2×C4⋊C4C22×D1314yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—138no—
398C2×C22⋊C4—132no—
399C2×C22⋊C4—132no—
400C2×C22⋊C4—132no—
401C2×C22⋊C4—132no—
402C2×C22⋊C4—132no—
403C2×C22⋊C4—132no—
404C2×C22⋊C4—264no—
405C2×C22⋊C4—264no—
406C2×C22⋊C4C22×D1312yes—
407C2×C22⋊C4C22×D1312yes—
408C2×C22⋊C4C22×D1312yes—
409C2×C22⋊C4C22×D1312yes—
410C2×C22⋊C4C22×D1312yes—
411C2×C22⋊C4C22×D1312yes—
412C2×C42—134no—
413C2×C42—134no—
414C2×C42—524no—
415#95—134no—
416#95—134no—
417#95—134no—
418#95—134no—
419#95—134no—
420#95—134no—
421#95—134no—
422#95—134no—
423#95—134no—
424#95—134no—
425#95—134no—
426#95—134no—
427#95C22×D1314yes—
428#95C22×D1314yes—
429C2×C13—24no—
430C2×C13—24no—
431C2×C13—24no—
432C2×C13C22×#12511yescharacteristic
433C2×C13#345912yes—
434C2×C13#349011yescharacteristic
435C2×C13#349111yescharacteristic
436C2×C13#350112yes—
437C24—42no—
438C24—42no—
439C24#612714yes—
440C24#612914yes—
441C24Q8×D1314yes—
442C24C23×D1311yescharacteristic
443C24C23×D1311yescharacteristic
444C24C23×D1311yescharacteristic
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—134no—
456C22×C4—132no—
457C22×C4—138no—
458C22×C4—132no—
459C22×C4—132no—
460C22×C4—134no—
461C22×C4—132no—
462C22×C4—264no—
463C22×C4—264no—
464C22×C4—264no—
465C22×C4—262no—
466C22×C4—264no—
467C22×C4—264no—
468C22×C4—262no—
469C22×C4—264no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—522no—
473C22×C4—522no—
474C22×C4—522no—
475C22×C4—522no—
476C22×C4—522no—
477C22×C4—522no—
478C22×C4—522no—
479C22×C4—522no—
480C22×C4—522no—
481C22×C4—5216no—
482C22×C4C23×D1312yes—
483C22×C4C23×D1314yes—
484C22×C4C23×D1314yes—
485C22×C4C23×D1312yes—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—268no—
489C4⋊C4—268no—
490C4⋊C4—524no—
491C4⋊C4—524no—
492C4⋊C4—524no—
493C4⋊C4—524no—
494C4⋊C4—524no—
495C4⋊C4—524no—
496C4⋊C4—524no—
497C4⋊C4—524no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—138no—
505C22⋊C4—138no—
506C22⋊C4—264no—
507C22⋊C4—264no—
508C22⋊C4—264no—
509C22⋊C4—264no—
510C42—268no—
511C42—524no—
512C13#3499411yescharacteristic, complemented, Sylow
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—44no—
521C23—48no—
522C23—48no—
523C23#88143116yescomplemented
524C23C2×#612712yes—
525C23C2×#612712yes—
526C23C2×#612712yes—
527C23C2×#612912yes—
528C23C2×#612912yes—
529C23C2×#612912yes—
530C23C2×Q8×D1312yes—
531C23C2×Q8×D1312yes—
532C23C2×Q8×D1312yes—
533C23C24×D1311yescharacteristic
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—138no—
549C2×C4—134no—
550C2×C4—268no—
551C2×C4—262no—
552C2×C4—264no—
553C2×C4—262no—
554C2×C4—264no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—264no—
559C2×C4—264no—
560C2×C4—264no—
561C2×C4—262no—
562C2×C4—262no—
563C2×C4—264no—
564C2×C4—264no—
565C2×C4—528no—
566C2×C4—528no—
567C2×C4—528no—
568C2×C4—528no—
569C2×C4—524no—
570C2×C4—1042no—
571C2×C4—1042no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—416no—
585C22C2×#8814314yes—
586C22C2×#8814314yes—
587C22C2×#8814314yes—
588C22C22×#612711yescharacteristic
589C22C22×#612911yescharacteristic
590C22C22×Q8×D1311yescharacteristic
591C22D13×H(5,𝔽2)12yes—
592C22#1171965012yes—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—264no—
598C4—268no—
599C4—524no—
600C4—522no—
601C4—522no—
602C4—522no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#8814311yescharacteristic
607C2#1312824212yes—
608C2#1312852111yescharacteristic
609C2#1312852911yescharacteristic
610C2#1312863612yes—
subgroupG/H
center, FrattiniC23C24×D13
commutator, socle, cosocleC23×C13C25
FittingC13×#2315C2
prime radicalC25×C13C23
hypercenter#2315D13
hypocenterC13#34994
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. #13123142C2semidirect
2. #13120808C2semidirect
3. #13119335C2semidirect
4. #13119333C2semidirect
5. #13118982C2semidirect
6. #13118974C2semidirect
24. C2×#88143C22semidirect
36. #11718645C22semidirect
37. #11718644C22semidirect
38. #11718643C22semidirect
39. #11718642C22semidirect
40. #11718641C22semidirect
41. #11718639C22semidirect
42. #11718633C22semidirect
59. #11718320C22semidirect
60. #11718319C22semidirect
61. #11718319C22semidirect
62. #11718318C22semidirect
63. #11718314C22semidirect
64. #11718313C22semidirect
65. #11718311C22semidirect
66. #11718308C22semidirect
231. C23×C13#125semidirect
512. C13#34994semidirect
523. C23#88143semidirect

Maximal subgroups

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

subgroupindex
#131231422
#131208082
#131193352
#131193332
#131189822
#131189742
#131174482
C2×#117186112
C2×#117186072
C2×C22⋊C4×Dic132
C13×#23152
#3499413