Finite Group Databasebeta

← #15862737

Subgroups — #15862737

Order 3328.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719228C211yesmaximal, characteristic
2C2×#11718790C211yesmaximal, characteristic
3#13117429C211yesmaximal, characteristic
4#13114930C212yesmaximal, complemented
5#13114929C212yesmaximal, complemented
6#13114925C212yesmaximal, complemented
7#13114921C212yesmaximal, complemented
8#13114556C211yesmaximal, characteristic, complemented
9#13114552C211yesmaximal, characteristic, complemented
10C2×#11718121C211yesmaximal, characteristic, complemented
11C13×#1736C211yesmaximal, characteristic, complemented
12C22×#6127C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C2×C13×#117C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16C2×#88180C2211yescharacteristic
17C22×C4⋊Dic13C2211yescharacteristic
18C23×#1055C2211yescharacteristic
19#11719228—24no—
20#11719228C418yescomplemented
21C2×#88146C2212yes—
22C2×C4⋊C4×D13C2212yes—
23C2×#88135—22no—
24C2×C4×D52—21no—
25C2×C4×Dic26—21no—
26#11718874—22no—
27#11718790—24no—
28C2×#88060C2212yes—
29C2×#88060C2212yes—
30#11718649—21no—
31#11718648C2212yes—
32#11718648C2212yes—
33#11718635—21no—
34C13×#340C2212yes—
35C13×#340C2212yes—
36#11718143C2212yes—
37#11718143C2212yes—
38#11718121C2211yescharacteristic
39#11718121C2212yes—
40#11718121C2211yescharacteristic
41#11718121C2212yes—
42#11718121C2211yescharacteristic
43#11718121C2211yescharacteristic
44C23×C4×C13—21no—
45C23×C4×C13C2311yescharacteristic
46C22×C22⋊D13D412yes—
47C23×Dic13—21no—
48C2×#6127—22no—
49C2×#6127—22no—
50C2×#6127—22no—
51C2×#6127—22no—
52C2×#6127C2×C414yes—
53C2×#6127D414yes—
54C22×D52D411yescharacteristic
55C22×C4×D13—22no—
56C22×C4×D13C2312yes—
57C22×Dic26D411yescharacteristic
58C2×C13×M16—22no—
59C2×C13×M16—22no—
60C2×C13×M16—22no—
61C22×C8×C13C2312yes—
62C13×#117—22no—
63C13×#117C2×C414yes—
64C2×C13×C4⋊C4—22no—
65C2×C13×C4⋊C4—22no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4C2312yes—
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C22⋊C4—21no—
71C2×C42×C13—21no—
72C2×C22⋊Dic13—21no—
73C2×#6103—22no—
74C2×#6103—22no—
75C2×#6103—22no—
76#88180—22no—
77#88180—22no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13C2×C414yes—
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2312yes—
84C2×#6101—22no—
85C2×C4×Dic13—22no—
86C2×C4×Dic13—21no—
87C22×#1055—22no—
88C22×#1055—22no—
89C22×#1055—22no—
90C22×#1055C2311yescharacteristic
91C22×#1055C2311yescharacteristic
92#88146—24no—
93#88146C2×C418yescomplemented
94C4⋊C4×D13—24no—
95C4⋊C4×D13C2×C418yescomplemented
96#88135—216no—
97C4×D52—28no—
98C4×Dic26—28no—
99#88071—22no—
100#88071—22no—
101#88060—24no—
102#88060—24no—
103#20703—131nomaximal, Sylow
104C24×C13C2×D411yescharacteristic
105C23×D13C2×D412yes—
106C22×C4×C13—21no—
107C22×C4×C13—21no—
108C22×C4×C13—22no—
109C22×C4×C13—42no—
110C22×C4×C13—81no—
111C22×C4×C13—81no—
112C22×C4×C13C22×C412yes—
113C22×C4×C13C2×D411yescharacteristic
114C22×C4×C13C2×D412yes—
115C22×C4×C13#4112yes—
116C22×C4×C13#4112yes—
117C22×C4×C13#4112yes—
118C22×C4×C13#4112yes—
119C22×C4×C13C2411yescharacteristic
120C2×C22⋊D13—24no—
121C2×C22⋊D13—24no—
122C2×C22⋊D13C22⋊C418yes—
123C22×Dic13—22no—
124C22×Dic13—24no—
125C22×Dic13—21no—
126C22×Dic13—21no—
127C22×Dic13—81no—
128C22×Dic13—81no—
129C22×Dic13C2×D412yes—
130#6127—28no—
131#6127—28no—
132#6127—28no—
133#6127—28no—
134C2×D52—22no—
135C2×D52—22no—
136C2×D52C22⋊C414yes—
137C2×C4×D13—22no—
138C2×C4×D13—22no—
139C2×C4×D13—22no—
140C2×C4×D13—22no—
141C2×C4×D13—28no—
142C2×C4×D13—82no—
143C2×C4×D13—82no—
144C2×C4×D13C22×C414yes—
145C2×C4×D13C2×D414yes—
146C2×Dic26—22no—
147C2×Dic26—22no—
148C2×Dic26C22⋊C414yes—
149C13×M16—44no—
150C2×C8×C13—22no—
151C2×C8×C13—22no—
152C2×C8×C13—22no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—22no—
156C13×C4⋊C4—22no—
157C13×C4⋊C4—42no—
158C13×C4⋊C4C22×C412yes—
159C13×C4⋊C4C22×C412yes—
160C13×C22⋊C4—24no—
161C42×C13—24no—
162C22⋊Dic13—42no—
163#6103—28no—
164#6103—82no—
165#6103—82no—
166C4⋊Dic13—24no—
167C4⋊Dic13—22no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13C22×C414yes—
173#6101—28no—
174C4×Dic13—28no—
175C4×Dic13—42no—
176C2×#1055—21no—
177C2×#1055—21no—
178C2×#1055—21no—
179C2×#1055—21no—
180C2×#1055—21no—
181C2×#1055—21no—
182C2×#1055—44no—
183C2×#520—131no—
184C2×#408—131no—
185#2311—131no—
186#1825—132no—
187#1825—132no—
188#1824—132no—
189#1823—132no—
190#1736—131no—
191#1736D1311yescharacteristic, complemented
192#1735—131no—
193C2×#340—131no—
194C23×C13—21no—
195C23×C13—21no—
196C23×C13—21no—
197C23×C13—21no—
198C23×C13C2×C22⋊C412yes—
199C23×C13#12312yes—
200C23×C13#12312yes—
201C23×C13C22×D411yescharacteristic
202C22×D13—22no—
203C22×D13—22no—
204C22×D13—22no—
205C22×D13—22no—
206C22×D13—22no—
207C22×D13—22no—
208C22×D13C2×C22⋊C414yes—
209C2×C4×C13—22no—
210C2×C4×C13—22no—
211C2×C4×C13—41no—
212C2×C4×C13—41no—
213C2×C4×C13—41no—
214C2×C4×C13—41no—
215C2×C4×C13—42no—
216C2×C4×C13—42no—
217C2×C4×C13—42no—
218C2×C4×C13—84no—
219C2×C4×C13C2×C22⋊C412yes—
220C2×C4×C13#11714yes—
221C2×C4×C13#11714yes—
222C2×C4×C13#12314yes—
223C2×C4×C13#12314yes—
224C2×C4×C13#12312yes—
225C2×C4×C13#12312yes—
226C2×C4×C13C23×C411yescharacteristic
227C2×C4×C13C22×D411yescharacteristic
228C2×C4×C13C2×#4111yescharacteristic
229C2×C4×C13C2×#4111yescharacteristic
230C2×C4×C13C2×#4111yescharacteristic
231C2×C4×C13C2×#4111yescharacteristic
232C22⋊D13—216no—
233C2×Dic13—22no—
234C2×Dic13—22no—
235C2×Dic13—24no—
236C2×Dic13—42no—
237C2×Dic13—42no—
238C2×Dic13—41no—
239C2×Dic13—41no—
240C2×Dic13—41no—
241C2×Dic13—41no—
242C2×Dic13—41no—
243C2×Dic13—41no—
244C2×Dic13—84no—
245C2×Dic13C2×C22⋊C414yes—
246D52—28no—
247C4×D13—24no—
248C4×D13—24no—
249C4×D13—24no—
250C4×D13—24no—
251C4×D13—88no—
252Dic26—28no—
253C8×C13—42no—
254#1055—41no—
255#1055—41no—
256C22×#41—131no—
257C22×M16C2×D1311yescharacteristic
258C23×C8—131no—
259#520—138no—
260#520—264no—
261C2×C4×Q8—261no—
262C2×C4×D4—261no—
263C2×C4×D4—262no—
264C2×#117—131no—
265C2×#117—132no—
266C2×#117C2×D1311yescharacteristic
267C22×C4⋊C4—131no—
268C22×C4⋊C4—132no—
269C22×C4⋊C4C2×D1311yescharacteristic
270#408—264no—
271C2×C22⋊C8—132no—
272C2×C22⋊C8—132no—
273#391—261no—
274#390—261no—
275#386—262no—
276#380—132no—
277#380—132no—
278#340—132no—
279#340—132no—
280#340—131no—
281#340—132no—
282#340—131no—
283#340—132no—
284#340—131no—
285#340—131no—
286#340C2×D1312yes—
287#340C2×D1312yes—
288C22×C13—22no—
289C22×C13—22no—
290C22×C13—22no—
291C22×C13—22no—
292C22×C13—22no—
293C22×C13—22no—
294C22×C13#38114yes—
295C22×C13C22×C22⋊C411yescharacteristic
296C22×C13C2×#12311yescharacteristic
297C22×C13C2×#12311yescharacteristic
298C22×C13#57611yescharacteristic
299C22×C13#57711yescharacteristic
300C22×C13#57711yescharacteristic
301C22×C13#57811yescharacteristic
302C2×D13—24no—
303C2×D13—24no—
304C2×D13—24no—
305C2×D13—42no—
306C2×D13—42no—
307C2×D13—42no—
308C2×D13—42no—
309C4×C13—42no—
310C4×C13—42no—
311C4×C13—82no—
312C4×C13#38114yes—
313C4×C13C2×#11711yescharacteristic
314C4×C13C2×#11711yescharacteristic
315C4×C13C2×#12311yescharacteristic
316C4×C13C2×#12311yescharacteristic
317Dic13—24no—
318Dic13—44no—
319Dic13—81no—
320Dic13—81no—
321C2×#41—134no—
322C2×#41—134no—
323C2×#41—262no—
324C2×#41—262no—
325C2×#41—262no—
326C2×#41—262no—
327C22×Q8—131no—
328C22×D4—132no—
329C22×D4—131no—
330C23×C4—21no—
331C23×C4—132no—
332C23×C4—262no—
333C23×C4—261no—
334C23×C4C22×D1311yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C22×C8—131no—
339C22×C8—131no—
340C22×C8—262no—
341C22×C8—262no—
342C22×C8—262no—
343C22×C8C22×D1312yes—
344C4×Q8—268no—
345C4×D4—268no—
346C4×D4—2616no—
347#117—22no—
348#117—138no—
349#117—264no—
350#117—262no—
351#117—262no—
352#117C4×D1314yes—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—134no—
357C2×C4⋊C4—132no—
358C2×C4⋊C4—138no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—131no—
361C2×C4⋊C4—264no—
362C2×C4⋊C4—262no—
363C2×C4⋊C4—261no—
364C2×C4⋊C4—262no—
365C2×C4⋊C4C22×D1312yes—
366C2×C4⋊C4C22×D1311yescharacteristic
367C2×C4⋊C4C22×D1311yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—262no—
370C2×C22⋊C4—261no—
371C2×C22⋊C4—262no—
372C2×C22⋊C4—262no—
373C2×C42—21no—
374C2×C42—262no—
375C2×C42—261no—
376C22⋊C8—264no—
377C22⋊C8—264no—
378#95—262no—
379#95—262no—
380C2×C13—24no—
381C2×C13C2×#38111yescharacteristic
382C2×C13#289711yescharacteristic
383C2×C13#289811yescharacteristic
384C2×C13#309611yescharacteristic
385C2×C13#309711yescharacteristic
386C2×C13#309811yescharacteristic
387C2×C13#309911yescharacteristic
388D13—44no—
389C24—132no—
390C24D4×D1311yescharacteristic
391#41—268no—
392#41—268no—
393#41—268no—
394#41—268no—
395C2×Q8—134no—
396C2×Q8—262no—
397C2×Q8—262no—
398C2×D4—138no—
399C2×D4—134no—
400C2×D4—264no—
401C2×D4—264no—
402C2×D4—262no—
403C2×D4—262no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—132no—
411C22×C4—134no—
412C22×C4—134no—
413C22×C4—268no—
414C22×C4—262no—
415C22×C4—262no—
416C22×C4—262no—
417C22×C4—262no—
418C22×C4—262no—
419C22×C4—264no—
420C22×C4—261no—
421C22×C4—261no—
422C22×C4—1042no—
423C22×C4—1042no—
424C22×C4—1041no—
425C22×C4—1041no—
426C22×C4C2×C4×D1312yes—
427C22×C4#612712yes—
428C22×C4#612712yes—
429C22×C4D4×D1311yescharacteristic
430C22×C4D4×D1312yes—
431C22×C4#612912yes—
432C22×C4#613112yes—
433C22×C4C23×D1311yescharacteristic
434M16—44no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—261no—
439C2×C8—261no—
440C2×C8—261no—
441C2×C8—261no—
442C2×C8—261no—
443C2×C8—261no—
444C2×C8—524no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—134no—
451C4⋊C4—268no—
452C4⋊C4—264no—
453C4⋊C4—262no—
454C4⋊C4—261no—
455C4⋊C4—261no—
456C4⋊C4—261no—
457C4⋊C4—261no—
458C4⋊C4C2×C4×D1312yes—
459C4⋊C4C2×C4×D1312yes—
460C22⋊C4—24no—
461C22⋊C4—268no—
462C22⋊C4—522no—
463C22⋊C4—1042no—
464C22⋊C4—1042no—
465C42—24no—
466C42—268no—
467C42—522no—
468C13#2070311yescharacteristic, complemented, Sylow
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—134no—
474C23—262no—
475C23—262no—
476C23—262no—
477C23—262no—
478C23—262no—
479C23—262no—
480C23C22⋊C4×D1312yes—
481C23#8813712yes—
482C23#8814812yes—
483C23C2×D4×D1311yescharacteristic
484Q8—268no—
485D4—2616no—
486D4—268no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—134no—
498C2×C4—262no—
499C2×C4—262no—
500C2×C4—264no—
501C2×C4—264no—
502C2×C4—264no—
503C2×C4—264no—
504C2×C4—264no—
505C2×C4—522no—
506C2×C4—522no—
507C2×C4—521no—
508C2×C4—521no—
509C2×C4—521no—
510C2×C4—521no—
511C2×C4—521no—
512C2×C4—521no—
513C2×C4—1048no—
514C2×C4—1044no—
515C2×C4#8812614yes—
516C2×C4C22⋊C4×D1312yes—
517C2×C4#8813714yes—
518C2×C4#8813712yes—
519C2×C4#8814614yes—
520C2×C4#8814814yes—
521C2×C4#8814812yes—
522C2×C4C22×C4×D1311yescharacteristic
523C2×C4C2×#612711yescharacteristic
524C2×C4C2×#612711yescharacteristic
525C2×C4C2×D4×D1311yescharacteristic
526C2×C4C2×#612911yescharacteristic
527C2×C4C2×#613111yescharacteristic
528C8—42no—
529C8—521no—
530C8—521no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—264no—
538C22—264no—
539C22—264no—
540C22—522no—
541C22—522no—
542C22—522no—
543C22—522no—
544C22#1171832514yes—
545C22C2×C22⋊C4×D1311yescharacteristic
546C22C2×#8813711yescharacteristic
547C22C2×#8814811yescharacteristic
548C22#1171947911yescharacteristic
549C22#1171948011yescharacteristic
550C22#1171948311yescharacteristic
551C22#1171948411yescharacteristic
552C4—42no—
553C4—42no—
554C4—82no—
555C4—264no—
556C4—524no—
557C4—1041no—
558C4—1041no—
559C4#1171832514yes—
560C4C2×#8812611yescharacteristic
561C4C2×#8813711yescharacteristic
562C4C2×#8814611yescharacteristic
563C4C2×#8814811yescharacteristic
564C2—24no—
565C2—524no—
566C2C2×#1171832511yescharacteristic
567C2#1312336011yescharacteristic
568C2#1312336711yescharacteristic
569C2#1312479211yescharacteristic
570C2#1312480311yescharacteristic
571C2#1312481811yescharacteristic
572C2#1312482911yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1736C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenter#1736D13
hypocenterC13#20703
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #13114930C2semidirect
5. #13114929C2semidirect
6. #13114925C2semidirect
7. #13114921C2semidirect
8. #13114556C2semidirect
9. #13114552C2semidirect
10. C2×#11718121C2semidirect
11. C13×#1736C2semidirect
20. #11719228C4semidirect
93. #88146C2×C4semidirect
95. C4⋊C4×D13C2×C4semidirect
191. #1736D13semidirect
468. C13#20703semidirect

Maximal subgroups

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

subgroupindex
C2×#117192282
C2×#117187902
#131174292
#131149302
#131149292
#131149252
#131149212
#131145562
#131145522
C2×#117181212
C13×#17362
#2070313