Finite Group Databasebeta

← #15892715

Subgroups — #15892715

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×#11718767C211yesmaximal, characteristic, complemented
2C2×#11718659C211yesmaximal, characteristic, complemented
3C2×#11718894C211yesmaximal, characteristic
4#13117327C211yesmaximal, characteristic
5#13117239C211yesmaximal, characteristic, complemented
6#13116386C211yesmaximal, characteristic, complemented
7#13116382C211yesmaximal, characteristic, complemented
8#13116377C211yesmaximal, characteristic, complemented
9#13116372C211yesmaximal, characteristic, complemented
10#13116367C211yesmaximal, characteristic, complemented
11#13116362C211yesmaximal, characteristic, complemented
12#13116359C211yesmaximal, characteristic, complemented
13#13116355C211yesmaximal, characteristic, complemented
14C2×#11718142C211yesmaximal, characteristic, complemented
15C2×C13×#416C211yesmaximal, characteristic
16C22×C13×#41C2211yescharacteristic
17C23×C8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C2×#88180C2211yescharacteristic
21C22×C4×Dic13C2211yescharacteristic
22C23×#1055C2211yescharacteristic
23C13×#416—24no—
24C2×C13×#103C2211yescharacteristic
25C2×C13×#103C2211yescharacteristic
26C2×C13×#102C2211yescharacteristic
27C2×C13×#102C2211yescharacteristic
28#11718894—24no—
29#11718891C2211yescharacteristic
30#11718891C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33#11718882C2211yescharacteristic
34#11718882C2211yescharacteristic
35#11718881—22no—
36C2×#88072C2211yescharacteristic
37C2×#88072C2211yescharacteristic
38C4×C22⋊Dic13—22no—
39#11718767—24no—
40C2×#88058C2211yescharacteristic
41C2×#88058C2211yescharacteristic
42C2×#88057C2211yescharacteristic
43C2×#88057C2211yescharacteristic
44#11718659—24no—
45#11718638C2211yescharacteristic
46#11718638C2211yescharacteristic
47#11718637C2211yescharacteristic
48#11718637C2211yescharacteristic
49C2×#88048C2211yescharacteristic
50C2×#88048C2211yescharacteristic
51C2×#88047C2211yescharacteristic
52C2×#88047C2211yescharacteristic
53#11718603—22no—
54C4×#6101—22no—
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59C2×C13×#41—22no—
60C2×C13×#41—22no—
61C2×C13×#41—22no—
62C2×C13×#41—22no—
63C2×C13×#41—22no—
64C2×C13×#41Q814yes—
65C22×Q8×C13C2311yescharacteristic
66C22×D4×C13—21no—
67C22×D4×C13C2311yescharacteristic
68C23×C4×C13—21no—
69C23×C4×C13C2311yescharacteristic
70C23×Dic13D412yes—
71C22×C8×C13—22no—
72C22×C8×C13—22no—
73C22×C8×C13D414yes—
74C22×C8×C13C2311yescharacteristic
75C22×C8×C13C2311yescharacteristic
76C13×#117—22no—
77C13×#117—22no—
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C22⋊C4—21no—
81C2×C42×C13—21no—
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—22no—
85C2×C22⋊Dic13—22no—
86#88180—22no—
87#88180—22no—
88#88180—22no—
89#88180—22no—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×#6101—22no—
95C2×#6101—22no—
96C2×C4×Dic13—22no—
97C2×C4×Dic13—22no—
98C2×C4×Dic13—21no—
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×#103—22no—
109C13×#103—22no—
110C13×#102—22no—
111C13×#102—22no—
112#88075—22no—
113#88075—22no—
114#88072—22no—
115#88072—22no—
116#88071—22no—
117#88071—22no—
118#88071—22no—
119#88071—22no—
120#88058—22no—
121#88058—22no—
122#88057—22no—
123#88057—22no—
124#88048—22no—
125#88048—22no—
126#88047—22no—
127#88047—22no—
128#23730—131nomaximal, Sylow
129C24×C13—21no—
130C24×C13C2×D411yescharacteristic
131C13×#41—28no—
132C13×#41—28no—
133C13×#41—44no—
134C13×#41—44no—
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13C2×Q812yes—
141C2×D4×C13—24no—
142C2×D4×C13—21no—
143C2×D4×C13—21no—
144C2×D4×C13—21no—
145C2×D4×C13—21no—
146C2×D4×C13—21no—
147C2×D4×C13—42no—
148C2×D4×C13—42no—
149C2×D4×C13C2×Q812yes—
150C22×C4×C13—21no—
151C22×C4×C13—24no—
152C22×C4×C13—21no—
153C22×C4×C13—21no—
154C22×C4×C13—42no—
155C22×C4×C13—42no—
156C22×C4×C13C2×D412yes—
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2×D412yes—
159C22×C4×C13C2×Q812yes—
160C22×C4×C13#4112yes—
161C22×C4×C13#4112yes—
162C22×C4×C13#4112yes—
163C22×C4×C13C2411yescharacteristic
164C22×Dic13—24no—
165C22×Dic13—24no—
166C22×Dic13—21no—
167C22×Dic13—21no—
168C22×Dic13—21no—
169C22×Dic13—21no—
170C22×Dic13—42no—
171C22×Dic13C2×D412yes—
172C22×Dic13C2×D412yes—
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×C22⋊C4—42no—
185C42×C13—42no—
186C22⋊Dic13—44no—
187C22⋊Dic13—42no—
188C22⋊Dic13—42no—
189C22⋊Dic13—44no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198#6101—44no—
199#6101—44no—
200C4×Dic13—24no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—42no—
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×#427—131no—
216C2×#416—131no—
217C2×#416D1311yescharacteristic
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×C13—22no—
232C23×C13—41no—
233C23×C13—41no—
234C23×C13—42no—
235C23×C13—42no—
236C23×C13C22⋊Q812yes—
237C23×C13#12312yes—
238C23×C13#12712yes—
239C23×C13C22×D411yescharacteristic
240Q8×C13—22no—
241Q8×C13—22no—
242Q8×C13—41no—
243Q8×C13—41no—
244D4×C13—22no—
245D4×C13—22no—
246D4×C13—48no—
247D4×C13—41no—
248D4×C13—41no—
249C2×C4×C13—22no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—48no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—42no—
256C2×C4×C13—42no—
257C2×C4×C13C22⋊Q814yes—
258C2×C4×C13C22⋊Q814yes—
259C2×C4×C13C22⋊Q812yes—
260C2×C4×C13#12314yes—
261C2×C4×C13#12314yes—
262C2×C4×C13#12312yes—
263C2×C4×C13#12712yes—
264C2×C4×C13C22×D411yescharacteristic
265C2×C4×C13C22×D411yescharacteristic
266C2×C4×C13C22×Q811yescharacteristic
267C2×C4×C13C2×#4111yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×Dic13—22no—
271C2×Dic13—22no—
272C2×Dic13—22no—
273C2×Dic13—22no—
274C2×Dic13—22no—
275C2×Dic13—48no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—42no—
281C2×Dic13—42no—
282C2×Dic13—42no—
283C2×Dic13—42no—
284C2×Dic13#13514yes—
285C8×C13—22no—
286C8×C13—22no—
287#1055—22no—
288#1055—22no—
289C22×#41C2×D1311yescharacteristic
290C23×C8—131no—
291C23×C8C2×D1311yescharacteristic
292C2×#117—131no—
293C2×#117—131no—
294C2×#117C2×D1311yescharacteristic
295C22×C42—131no—
296#427—264no—
297#427—264no—
298C2×#107—131no—
299C2×#107—131no—
300C2×#107—131no—
301C2×#107—131no—
302C2×#106—131no—
303C2×#106—131no—
304C2×#106—131no—
305C2×#106—131no—
306#416—24no—
307#416—264no—
308C2×#103—131no—
309C2×#103—131no—
310C2×#103C2×D1311yescharacteristic
311C2×#103C2×D1311yescharacteristic
312C2×#102—131no—
313C2×#102—131no—
314C2×#102C2×D1311yescharacteristic
315C2×#102C2×D1311yescharacteristic
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—44no—
337C22×C13—42no—
338C22×C13—82no—
339C22×C13—81no—
340C22×C13—81no—
341C22×C13#39914yes—
342C22×C13C2×C22⋊Q811yescharacteristic
343C22×C13C2×#12311yescharacteristic
344C22×C13C2×#12711yescharacteristic
345C22×C13C2×#13512yes—
346C22×C13#57512yes—
347C4×C13—42no—
348C4×C13—81no—
349C4×C13—81no—
350C4×C13#39914yes—
351C4×C13C2×C22⋊Q811yescharacteristic
352C4×C13C2×C22⋊Q811yescharacteristic
353C4×C13C2×#12311yescharacteristic
354C4×C13C2×#12311yescharacteristic
355Dic13—24no—
356Dic13—42no—
357Dic13—81no—
358Dic13—81no—
359Dic13—81no—
360Dic13—81no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41—22no—
365C2×#41—22no—
366C2×#41Dic2614yes—
367C22×Q8C22×D1311yescharacteristic
368C22×D4—21no—
369C22×D4C22×D1311yescharacteristic
370C23×C4—21no—
371C23×C4—132no—
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—262no—
386#117—262no—
387#117—262no—
388#117—262no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—262no—
394C2×C4⋊C4—262no—
395C2×C4⋊C4C22×D1311yescharacteristic
396C2×C4⋊C4C22×D1311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—262no—
399C2×C22⋊C4—262no—
400C2×C22⋊C4—261no—
401C2×C22⋊C4—261no—
402C2×C42—21no—
403C2×C42—132no—
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—262no—
407C2×C42—262no—
408C2×C42—261no—
409C2×C42—261no—
410#107—262no—
411#107—262no—
412#107—262no—
413#107—262no—
414#106—262no—
415#106—262no—
416#106—262no—
417#106—262no—
418#103—22no—
419#103—22no—
420#103—262no—
421#103—262no—
422#102—22no—
423#102—22no—
424#102—262no—
425#102—262no—
426#95—262no—
427#95—262no—
428#95—262no—
429#95—262no—
430C2×C13—24no—
431C2×C13—82no—
432C2×C13C2×#39911yescharacteristic
433C2×C13#307612yes—
434C2×C13#309112yes—
435C2×C13#314711yescharacteristic
436C2×C13#315011yescharacteristic
437C24—21no—
438C24D4×D1311yescharacteristic
439#41—28no—
440#41—28no—
441#41—44no—
442#41—44no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×Dic2612yes—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4C2×Dic2612yes—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—132no—
465C22×C4—132no—
466C22×C4—264no—
467C22×C4—264no—
468C22×C4—261no—
469C22×C4—261no—
470C22×C4—261no—
471C22×C4—261no—
472C22×C4—522no—
473C22×C4C2×Dic2612yes—
474C22×C4#612712yes—
475C22×C4D4×D1312yes—
476C22×C4D4×D1311yescharacteristic
477C22×C4#612912yes—
478C22×C4#612912yes—
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—261no—
500C4⋊C4—261no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—524no—
508C4⋊C4—524no—
509C22⋊C4—42no—
510C22⋊C4—524no—
511C22⋊C4—524no—
512C22⋊C4—522no—
513C22⋊C4—522no—
514C42—42no—
515C42—264no—
516C42—261no—
517C42—261no—
518C42—261no—
519C42—261no—
520C42—522no—
521C42—522no—
522C13#2373011yescharacteristic, complemented, Sylow
523C23—22no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23#8813212yes—
531C23#8813712yes—
532C23#8819412yes—
533C23C2×D4×D1311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—48no—
541D4—41no—
542D4—41no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
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—528no—
558C2×C4—522no—
559C2×C4—522no—
560C2×C4—522no—
561C2×C4—522no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4#8813214yes—
567C2×C4#8813212yes—
568C2×C4#8813714yes—
569C2×C4#8813712yes—
570C2×C4#8817814yes—
571C2×C4#8818914yes—
572C2×C4#8819412yes—
573C2×C4C22×Dic2611yescharacteristic
574C2×C4C2×#612711yescharacteristic
575C2×C4C2×D4×D1311yescharacteristic
576C2×C4C2×#612911yescharacteristic
577C2×C4C2×#612911yescharacteristic
578C2×C4C22×C22⋊D1311yescharacteristic
579C8—22no—
580C8—22no—
581C8—262no—
582C8—262no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—42no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22#1171861014yes—
593C22C2×#8813211yescharacteristic
594C22C2×#8813711yescharacteristic
595C22D13×#13512yes—
596C22#1171947312yes—
597C22C2×#8819411yescharacteristic
598C4—42no—
599C4—81no—
600C4—81no—
601C4—264no—
602C4—522no—
603C4—1041no—
604C4—1041no—
605C4—1041no—
606C4—1041no—
607C4#1171861014yes—
608C4C2×#8813211yescharacteristic
609C4C2×#8813711yescharacteristic
610C4C2×#8817811yescharacteristic
611C4C2×#8818911yescharacteristic
612C2—24no—
613C2—82no—
614C2C2×#1171861011yescharacteristic
615C2#1312462012yes—
616C2#1312474712yes—
617C2#1312526011yescharacteristic
618C2#1312527911yescharacteristic
subgroupG/H
centerC22×C4D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#416C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
hypercenterC2×#416D13
hypocenterC13#23730
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C2×#11718767C2semidirect
2. C2×#11718659C2semidirect
5. #13117239C2semidirect
6. #13116386C2semidirect
7. #13116382C2semidirect
8. #13116377C2semidirect
9. #13116372C2semidirect
10. #13116367C2semidirect
11. #13116362C2semidirect
12. #13116359C2semidirect
13. #13116355C2semidirect
14. C2×#11718142C2semidirect
522. C13#23730semidirect

Maximal subgroups

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

subgroupindex
C2×#117187672
C2×#117186592
C2×#117188942
#131173272
#131172392
#131163862
#131163822
#131163772
#131163722
#131163672
#131163622
#131163592
#131163552
C2×#117181422
C2×C13×#4162
#2373013