Finite Group Databasebeta

← #15892713

Subgroups — #15892713

Order 3328.

1126 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
1C2×#11718659C211yesmaximal, characteristic, complemented
2C2×#11718774C211yesmaximal, characteristic
3C2×#11718663C211yesmaximal, characteristic
4#13117324C211yesmaximal, characteristic
5#13117239C211yesmaximal, characteristic, complemented
6#13116385C211yesmaximal, characteristic, complemented
7#13116381C211yesmaximal, characteristic, complemented
8#13116376C211yesmaximal, characteristic, complemented
9#13116371C211yesmaximal, characteristic, complemented
10#13116366C211yesmaximal, characteristic, complemented
11#13116361C211yesmaximal, characteristic, complemented
12#13116358C211yesmaximal, characteristic, complemented
13#13116354C211yesmaximal, characteristic, complemented
14C2×#11718142C211yesmaximal, characteristic, complemented
15C2×C13×#427C211yesmaximal, characteristic, complemented
16C22×#6127C2211yescharacteristic
17C23×C8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×C13×#117C2211yescharacteristic
20C2×#88180C2211yescharacteristic
21C22×C4×Dic13C2211yescharacteristic
22C23×#1055C2211yescharacteristic
23C13×#427—24no—
24C2×C13×#107C2211yescharacteristic
25C2×C13×#107C2211yescharacteristic
26C2×C13×#106C2211yescharacteristic
27C2×C13×#106C2211yescharacteristic
28#11718774—24no—
29C2×#88061C2211yescharacteristic
30C2×#88061C2211yescharacteristic
31C2×#88056C2211yescharacteristic
32C2×#88056C2211yescharacteristic
33#11718663—24no—
34#11718659—24no—
35#11718649C2211yescharacteristic
36#11718649C2211yescharacteristic
37#11718638C2211yescharacteristic
38#11718638C2211yescharacteristic
39#11718637C2211yescharacteristic
40#11718637C2211yescharacteristic
41#11718635C2211yescharacteristic
42#11718635C2211yescharacteristic
43C2×#88050C2211yescharacteristic
44C2×#88050C2211yescharacteristic
45C2×#88049C2211yescharacteristic
46C2×#88049C2211yescharacteristic
47C2×#88048C2211yescharacteristic
48C2×#88048C2211yescharacteristic
49C2×#88047C2211yescharacteristic
50C2×#88047C2211yescharacteristic
51#11718619—22no—
52#11718603—22no—
53C4×#6103—22no—
54C4×#6101—22no—
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C22×C22⋊D13—21no—
61C23×Dic13D412yes—
62C2×#6127—22no—
63C2×#6127—22no—
64C2×#6127—22no—
65C2×#6127—22no—
66C2×#6127—22no—
67C2×#6127Q814yes—
68C22×D52C2311yescharacteristic
69C22×C4×D13—21no—
70C22×Dic26C2311yescharacteristic
71C22×C8×C13—22no—
72C22×C8×C13—22no—
73C22×C8×C13D414yes—
74C22×C8×C13C2311yescharacteristic
75C22×C8×C13C2311yescharacteristic
76C13×#117—22no—
77C13×#117—22no—
78C13×#117—22no—
79C13×#117—22no—
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C4⋊C4C2311yescharacteristic
82C2×C13×C4⋊C4C2311yescharacteristic
83C2×C13×C4⋊C4C2311yescharacteristic
84C2×C13×C22⋊C4—21no—
85C2×C13×C22⋊C4—21no—
86C2×C42×C13—21no—
87C2×C42×C13—21no—
88C2×C22⋊Dic13—21no—
89C2×#6103—22no—
90C2×#6103—22no—
91#88180—22no—
92#88180—22no—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×#6101—22no—
96C2×#6101—22no—
97C2×C4×Dic13—22no—
98C2×C4×Dic13—22no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13D412yes—
101C2×C4×Dic13C2311yescharacteristic
102C2×C4×Dic13C2311yescharacteristic
103C22×#1055—22no—
104C22×#1055—22no—
105C22×#1055D414yes—
106C22×#1055C2311yescharacteristic
107C22×#1055C2311yescharacteristic
108C13×#107—22no—
109C13×#107—22no—
110C13×#106—22no—
111C13×#106—22no—
112#88071—22no—
113#88071—22no—
114#88071—22no—
115#88071—22no—
116#88061—22no—
117#88061—22no—
118#88056—22no—
119#88056—22no—
120#88050—22no—
121#88050—22no—
122#88049—22no—
123#88049—22no—
124#88048—22no—
125#88048—22no—
126#88047—22no—
127#88047—22no—
128#23730—131nomaximal, Sylow
129C24×C13C2×D411yescharacteristic
130C23×D13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13C2×D412yes—
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D412yes—
138C22×C4×C13C2×Q812yes—
139C22×C4×C13#4112yes—
140C22×C4×C13#4112yes—
141C22×C4×C13#4112yes—
142C22×C4×C13C2411yescharacteristic
143C2×C22⋊D13—24no—
144C2×C22⋊D13—42no—
145C2×C22⋊D13—42no—
146C22×Dic13—24no—
147C22×Dic13—21no—
148C22×Dic13—24no—
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13—42no—
152C22×Dic13C2×D412yes—
153C22×Dic13C2×D412yes—
154#6127—28no—
155#6127—28no—
156#6127—44no—
157#6127—44no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52—21no—
162C2×D52—21no—
163C2×D52C2×Q812yes—
164C2×C4×D13—24no—
165C2×C4×D13—42no—
166C2×C4×D13—42no—
167C2×Dic26—21no—
168C2×Dic26—21no—
169C2×Dic26—21no—
170C2×Dic26—21no—
171C2×Dic26—21no—
172C2×Dic26C2×Q812yes—
173C2×C8×C13—28no—
174C2×C8×C13—21no—
175C2×C8×C13—21no—
176C2×C8×C13—21no—
177C2×C8×C13—21no—
178C2×C8×C13C2×D412yes—
179C2×C8×C13C2×D412yes—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4—21no—
185C13×C4⋊C4—21no—
186C13×C4⋊C4—21no—
187C13×C4⋊C4—21no—
188C13×C22⋊C4—42no—
189C13×C22⋊C4—42no—
190C42×C13—42no—
191C42×C13—42no—
192C22⋊Dic13—42no—
193#6103—44no—
194#6103—44no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199#6101—44no—
200#6101—44no—
201C4×Dic13—24no—
202C4×Dic13—21no—
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—42no—
207C2×#1055—28no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055C2×D412yes—
213C2×#1055C2×D412yes—
214C2×#427—131no—
215C2×#427D1311yescharacteristic, complemented
216C2×#416—131no—
217C2×#416—131no—
218#2297—131no—
219#2285—131no—
220#2097—131no—
221#2097—131no—
222#2096—131no—
223#2095—131no—
224#2094—131no—
225#2093—131no—
226#2092—131no—
227#2092—131no—
228C2×#336—131no—
229C23×C13—22no—
230C23×C13—22no—
231C23×C13C22⋊Q812yes—
232C23×C13#12312yes—
233C23×C13#12712yes—
234C23×C13C22×D411yescharacteristic
235C22×D13—22no—
236C22×D13—41no—
237C22×D13—41no—
238C22×D13—42no—
239C22×D13—42no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—42no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13C22⋊Q814yes—
249C2×C4×C13C22⋊Q814yes—
250C2×C4×C13C22⋊Q812yes—
251C2×C4×C13#12314yes—
252C2×C4×C13#12314yes—
253C2×C4×C13#12312yes—
254C2×C4×C13#12712yes—
255C2×C4×C13C22×D411yescharacteristic
256C2×C4×C13C22×D411yescharacteristic
257C2×C4×C13C22×Q811yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13C2×#4111yescharacteristic
261C22⋊D13—48no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—22no—
266C2×Dic13—22no—
267C2×Dic13—22no—
268C2×Dic13—48no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13#13514yes—
276D52—22no—
277D52—22no—
278D52—41no—
279D52—41no—
280C4×D13—48no—
281Dic26—22no—
282Dic26—22no—
283Dic26—41no—
284Dic26—41no—
285C8×C13—22no—
286C8×C13—22no—
287#1055—22no—
288#1055—22no—
289C22×#41—131no—
290C23×C8—131no—
291C23×C8C2×D1311yescharacteristic
292C2×#117—131no—
293C2×#117C2×D1311yescharacteristic
294C2×#117C2×D1311yescharacteristic
295C22×C42—131no—
296#427—24no—
297#427—264no—
298C2×#107—131no—
299C2×#107—131no—
300C2×#107C2×D1311yescharacteristic
301C2×#107C2×D1311yescharacteristic
302C2×#106—131no—
303C2×#106—131no—
304C2×#106C2×D1311yescharacteristic
305C2×#106C2×D1311yescharacteristic
306#416—264no—
307#416—264no—
308C2×#103—131no—
309C2×#103—131no—
310C2×#103—131no—
311C2×#103—131no—
312C2×#102—131no—
313C2×#102—131no—
314C2×#102—131no—
315C2×#102—131no—
316#391—131no—
317#391—131no—
318#390—131no—
319#390—131no—
320#386—262no—
321#384—131no—
322#384—131no—
323#382—131no—
324#382—131no—
325#380—262no—
326C4×C4⋊C4—262no—
327C4×C22⋊C4—262no—
328#336—131no—
329#336—131no—
330#336—131no—
331#336—131no—
332C22×C13—24no—
333C22×C13—24no—
334C22×C13—22no—
335C22×C13—22no—
336C22×C13#39914yes—
337C22×C13C2×C22⋊Q811yescharacteristic
338C22×C13C2×#12311yescharacteristic
339C22×C13C2×#12711yescharacteristic
340C22×C13C2×#13512yes—
341C22×C13#57512yes—
342C2×D13—44no—
343C2×D13—42no—
344C2×D13—82no—
345C2×D13—81no—
346C2×D13—81no—
347C4×C13—81no—
348C4×C13—81no—
349C4×C13—81no—
350C4×C13—81no—
351C4×C13#39914yes—
352C4×C13C2×C22⋊Q811yescharacteristic
353C4×C13C2×C22⋊Q811yescharacteristic
354C4×C13C2×#12311yescharacteristic
355C4×C13C2×#12311yescharacteristic
356Dic13—24no—
357Dic13—42no—
358Dic13—42no—
359Dic13—81no—
360Dic13—81no—
361C2×#41—134no—
362C2×#41—262no—
363C2×#41—262no—
364C2×#41—262no—
365C2×#41—262no—
366C2×#41—262no—
367C22×Q8—131no—
368C22×D4—131no—
369C22×D4—261no—
370C23×C4—132no—
371C23×C4—261no—
372C23×C4C22×D1311yescharacteristic
373C22×C8—22no—
374C22×C8—22no—
375C22×C8—134no—
376C22×C8—131no—
377C22×C8—131no—
378C22×C8—262no—
379C22×C8—262no—
380C22×C8C22⋊D1314yes—
381C22×C8C22×D1311yescharacteristic
382C22×C8C22×D1311yescharacteristic
383#117—22no—
384#117—22no—
385#117—22no—
386#117—22no—
387#117—262no—
388#117—262no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—262no—
392C2×C4⋊C4—262no—
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C4⋊C4C22×D1311yescharacteristic
396C2×C4⋊C4C22×D1311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—262no—
400C2×C22⋊C4—262no—
401C2×C22⋊C4—261no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—132no—
405C2×C42—131no—
406C2×C42—131no—
407C2×C42—262no—
408C2×C42—262no—
409C2×C42—261no—
410#107—22no—
411#107—22no—
412#107—262no—
413#107—262no—
414#106—22no—
415#106—22no—
416#106—262no—
417#106—262no—
418#103—262no—
419#103—262no—
420#103—262no—
421#103—262no—
422#102—262no—
423#102—262no—
424#102—262no—
425#102—262no—
426#95—262no—
427#95—262no—
428#95—262no—
429#95—262no—
430C2×C13—24no—
431C2×C13C2×#39911yescharacteristic
432C2×C13#307612yes—
433C2×C13#309112yes—
434C2×C13#314711yescharacteristic
435C2×C13#315011yescharacteristic
436D13—82no—
437C24—261no—
438C24D4×D1311yescharacteristic
439#41—268no—
440#41—268no—
441#41—524no—
442#41—524no—
443C2×Q8—132no—
444C2×Q8—261no—
445C2×Q8—261no—
446C2×Q8—261no—
447C2×Q8—261no—
448C2×Q8—261no—
449C2×D4—132no—
450C2×D4—264no—
451C2×D4—261no—
452C2×D4—261no—
453C2×D4—261no—
454C2×D4—261no—
455C2×D4—261no—
456C2×D4—522no—
457C2×D4—522no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—132no—
463C22×C4—132no—
464C22×C4—264no—
465C22×C4—264no—
466C22×C4—261no—
467C22×C4—264no—
468C22×C4—261no—
469C22×C4—261no—
470C22×C4—522no—
471C22×C4—522no—
472C22×C4—522no—
473C22×C4#612712yes—
474C22×C4#612712yes—
475C22×C4D4×D1312yes—
476C22×C4D4×D1311yescharacteristic
477C22×C4Q8×D1312yes—
478C22×C4#613112yes—
479C22×C4C2×C22⋊D1312yes—
480C22×C4C23×D1311yescharacteristic
481C2×C8—28no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—132no—
487C2×C8—132no—
488C2×C8—268no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8C2×C22⋊D1312yes—
494C2×C8C2×C22⋊D1312yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—524no—
508C4⋊C4—524no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—524no—
512C22⋊C4—524no—
513C22⋊C4—522no—
514C42—42no—
515C42—42no—
516C42—264no—
517C42—261no—
518C42—261no—
519C42—261no—
520C42—261no—
521C42—522no—
522C13#2373011yescharacteristic, complemented, Sylow
523C23—22no—
524C23—22no—
525C23—262no—
526C23—521no—
527C23—522no—
528C23—522no—
529C23—521no—
530C23#8814812yes—
531C23#8815012yes—
532C23#8819412yes—
533C23C2×D4×D1311yescharacteristic
534Q8—262no—
535Q8—262no—
536Q8—521no—
537Q8—521no—
538D4—262no—
539D4—262no—
540D4—528no—
541D4—521no—
542D4—521no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—134no—
552C2×C4—262no—
553C2×C4—262no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—528no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—528no—
562C2×C4—522no—
563C2×C4—522no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4#8814814yes—
567C2×C4#8814812yes—
568C2×C4#8815014yes—
569C2×C4#8815012yes—
570C2×C4#8818314yes—
571C2×C4#8819412yes—
572C2×C4#8820014yes—
573C2×C4C2×#612711yescharacteristic
574C2×C4C2×#612711yescharacteristic
575C2×C4C2×D4×D1311yescharacteristic
576C2×C4C2×Q8×D1311yescharacteristic
577C2×C4C2×#613111yescharacteristic
578C2×C4C22×C22⋊D1311yescharacteristic
579C8—22no—
580C8—22no—
581C8—262no—
582C8—262no—
583C22—24no—
584C22—22no—
585C22—24no—
586C22—22no—
587C22—524no—
588C22—522no—
589C22—1041no—
590C22—1042no—
591C22—1041no—
592C22#1171865314yes—
593C22C2×#8814811yescharacteristic
594C22C2×#8815011yescharacteristic
595C22D13×#13512yes—
596C22#1171947312yes—
597C22C2×#8819411yescharacteristic
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—264no—
603C4—522no—
604C4—522no—
605C4—1041no—
606C4—1041no—
607C4#1171865314yes—
608C4C2×#8814811yescharacteristic
609C4C2×#8815011yescharacteristic
610C4C2×#8818311yescharacteristic
611C4C2×#8820011yescharacteristic
612C2—24no—
613C2—1042no—
614C2C2×#1171865311yescharacteristic
615C2#1312462212yes—
616C2#1312475312yes—
617C2#1312526011yescharacteristic
618C2#1312527911yescharacteristic
subgroupG/H
centerC22×C4D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#427C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
hypercenterC2×#427D13
hypocenterC13#23730
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
1. C2×#11718659C2semidirect
5. #13117239C2semidirect
6. #13116385C2semidirect
7. #13116381C2semidirect
8. #13116376C2semidirect
9. #13116371C2semidirect
10. #13116366C2semidirect
11. #13116361C2semidirect
12. #13116358C2semidirect
13. #13116354C2semidirect
14. C2×#11718142C2semidirect
15. C2×C13×#427C2semidirect
215. C2×#427D13semidirect
522. C13#23730semidirect

Maximal subgroups

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

subgroupindex
C2×#117186592
C2×#117187742
C2×#117186632
#131173242
#131172392
#131163852
#131163812
#131163762
#131163712
#131163662
#131163612
#131163582
#131163542
C2×#117181422
C2×C13×#4272
#2373013