Finite Group Databasebeta

← #15862609

Subgroups — #15862609

Order 3328.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719450C211yesmaximal, characteristic
2C2×#11718782C211yesmaximal, characteristic, complemented
3#13123356C211yesmaximal, characteristic
4C2×#11718794C211yesmaximal, characteristic
5#13114897C214yesmaximal, complemented
6#13114892C214yesmaximal, complemented
7#13114574C211yesmaximal, characteristic, complemented
8C2×#11718146C211yesmaximal, characteristic, complemented
9C2×C13×#430C211yesmaximal, characteristic, complemented
10C22×#6127C2211yescharacteristic
11C22×C13×M16C2211yescharacteristic
12C22×C13×M16C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C2×#88180C2211yescharacteristic
15C22×C4⋊Dic13C2211yescharacteristic
16C22×#6099C2211yescharacteristic
17#11719450—24no—
18#11719450C418yescomplemented
19C2×#88161—22no—
20C2×M16×D13—22no—
21C2×#88155C2214yes—
22C13×#430—24no—
23C2×C13×#108C2214yescomplemented
24#11718794—22no—
25#11718794C2212yes—
26#11718794C2214yes—
27#11718794C2212yes—
28#11718792—21no—
29#11718782—22no—
30#11718782—22no—
31#11718781—21no—
32#11718774—22no—
33C2×#88061—21no—
34C2×#88058C2212yescomplemented
35C2×#88057C2212yescomplemented
36C2×#88056—21no—
37#11718146C2212yescomplemented
38#11718146C2214yescomplemented
39#11718146C2212yescomplemented
40C23×C4×C13C2311yescharacteristic
41C22×C22⋊D13—21no—
42C23×Dic13—21no—
43C2×#6127—21no—
44C2×#6127—22no—
45C2×#6127—22no—
46C2×#6127—21no—
47C2×#6127C2×C414yes—
48C2×#6127D412yes—
49C2×#6127Q814yes—
50C2×#6127C2312yes—
51C22×D52D411yescharacteristic
52C22×C4×D13—21no—
53C22×Dic26D411yescharacteristic
54C2×C13×M16—21no—
55C2×C13×M16—21no—
56C2×C13×M16—22no—
57C2×C13×M16—22no—
58C2×C13×M16—22no—
59C2×C13×M16—22no—
60C2×C13×M16C2×C414yes—
61C2×C13×M16D412yes—
62C2×C13×M16D412yes—
63C2×C13×M16D414yes—
64C2×C13×M16C2312yes—
65C2×C13×M16C2312yes—
66C22×C8×C13—21no—
67C22×C8×C13—21no—
68C22×C8×C13C2312yes—
69C2×C22⋊Dic13—21no—
70#88180—21no—
71#88180—21no—
72#88180—21no—
73#88180—21no—
74C2×C4⋊Dic13—21no—
75C2×C4⋊Dic13—21no—
76C2×C4⋊Dic13—21no—
77C2×C4⋊Dic13—21no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13C2312yes—
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4×Dic13—21no—
83C2×#6099—21no—
84C2×#6099—22no—
85C2×#6099C2×C414yes—
86C2×#6099C2312yes—
87C22×#1055—21no—
88#88161—216no—
89M16×D13—216no—
90#88155—28no—
91#88155C2×C4116yescomplemented
92C2×#6094—24no—
93C2×C8×D13—24no—
94C13×#108—28no—
95#88061—24no—
96#88061—22no—
97#88061—22no—
98#88058—24no—
99#88057—24no—
100#88056—24no—
101#88056—22no—
102#88056—22no—
103#20679—131nomaximal, Sylow
104C24×C13#4111yescharacteristic
105C23×D13—21no—
106C22×C4×C13C22×C412yes—
107C22×C4×C13C2×D411yescharacteristic
108C22×C4×C13C2×D411yescharacteristic
109C22×C4×C13C2×D411yescharacteristic
110C22×C4×C13C2×D411yescharacteristic
111C22×C4×C13C2×D412yes—
112C22×C4×C13C2×Q812yes—
113C22×C4×C13#4112yes—
114C22×C4×C13#4111yescharacteristic
115C22×C4×C13C2411yescharacteristic
116C2×C22⋊D13—24no—
117C2×C22⋊D13—22no—
118C2×C22⋊D13—24no—
119C2×C22⋊D13—22no—
120C22×Dic13—21no—
121C22×Dic13—21no—
122C22×Dic13—21no—
123C22×Dic13—22no—
124C22×Dic13—41no—
125C22×Dic13—41no—
126C22×Dic13—82no—
127#6127—24no—
128#6127—24no—
129#6127—24no—
130#6127—24no—
131#6127—24no—
132#6127—22no—
133#6127—22no—
134#6127C4⋊C418yes—
135#6127C22×C414yes—
136#6127C2×Q814yes—
137C2×D52—22no—
138C2×D52—21no—
139C2×D52C4⋊C414yes—
140C2×D52C2×D412yes—
141C2×C4×D13—24no—
142C2×C4×D13—22no—
143C2×C4×D13—24no—
144C2×C4×D13—22no—
145C2×Dic26—22no—
146C2×Dic26—21no—
147C2×Dic26C4⋊C414yes—
148C2×Dic26C2×D412yes—
149C13×M16—22no—
150C13×M16—22no—
151C13×M16—22no—
152C13×M16—22no—
153C13×M16—42no—
154C13×M16—42no—
155C13×M16C22⋊C418yes—
156C13×M16C22⋊C418yes—
157C2×C8×C13—22no—
158C2×C8×C13—22no—
159C2×C8×C13—42no—
160C2×C8×C13—42no—
161C2×C8×C13C22×C414yes—
162C2×C8×C13C2×D414yes—
163C2×C8×C13C2×D414yes—
164C22⋊Dic13—41no—
165C22⋊Dic13—41no—
166C4⋊Dic13—22no—
167C4⋊Dic13—22no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—41no—
173C4⋊Dic13—41no—
174C4×Dic13—41no—
175C4×Dic13—41no—
176#6099—24no—
177#6099—22no—
178#6099C22×C414yes—
179C2×#1055—24no—
180C2×#1055—22no—
181#6094—216no—
182C8×D13—216no—
183C2×#568—131no—
184C2×#430D1311yescharacteristic, complemented
185C2×#428—131no—
186#2896—131no—
187C2×#417—131no—
188#1816—134no—
189#1815—134no—
190#1742—131no—
191C2×#343—131no—
192C23×C13—41no—
193C23×C13—41no—
194C23×C13#11712yes—
195C23×C13#12111yescharacteristic
196C23×C13#12111yescharacteristic
197C23×C13C22⋊Q812yes—
198C23×C13C2×#4111yescharacteristic
199C22×D13—22no—
200C22×D13—24no—
201C22×D13—81no—
202C22×D13—81no—
203C2×C4×C13C2×C22⋊C412yes—
204C2×C4×C13C2×C22⋊C412yes—
205C2×C4×C13C2×C4⋊C412yes—
206C2×C4×C13C2×C4⋊C412yes—
207C2×C4×C13#11712yes—
208C2×C4×C13#12112yes—
209C2×C4×C13#12112yes—
210C2×C4×C13#12111yescharacteristic
211C2×C4×C13#12111yescharacteristic
212C2×C4×C13C22⋊Q812yes—
213C2×C4×C13C22⋊Q812yes—
214C2×C4×C13C22⋊Q812yes—
215C2×C4×C13C23×C411yescharacteristic
216C2×C4×C13C22×D411yescharacteristic
217C2×C4×C13C22×D411yescharacteristic
218C2×C4×C13C22×D411yescharacteristic
219C2×C4×C13C22×Q811yescharacteristic
220C2×C4×C13C2×#4111yescharacteristic
221C22⋊D13—24no—
222C22⋊D13—24no—
223C22⋊D13—28no—
224C2×Dic13—22no—
225C2×Dic13—24no—
226C2×Dic13—41no—
227C2×Dic13—41no—
228C2×Dic13—42no—
229C2×Dic13—82no—
230C2×Dic13—82no—
231C2×Dic13—81no—
232C2×Dic13—81no—
233C2×Dic13—82no—
234D52—22no—
235D52—24no—
236D52C2×C4⋊C414yes—
237C4×D13—24no—
238C4×D13—24no—
239C4×D13—28no—
240Dic26—22no—
241Dic26—24no—
242Dic26C2×C4⋊C414yes—
243C8×C13—42no—
244C8×C13—42no—
245C8×C13C2×C22⋊C418yes—
246#1055—24no—
247C22×#41—131no—
248#568—138no—
249#568—264no—
250C2×#131—134no—
251C2×#131—262no—
252C22×M16—131no—
253C22×M16—262no—
254C22×M16C2×D1311yescharacteristic
255C22×M16C2×D1311yescharacteristic
256C22×M16C2×D1311yescharacteristic
257C2×#117—131no—
258C22×C4⋊C4—131no—
259#430—24no—
260C2×#108C2×D1314yescomplemented
261#428—262no—
262#428—262no—
263C2×#107—132no—
264C2×#106—132no—
265#419—261no—
266#418—261no—
267#417—132no—
268#417—134no—
269#417—132no—
270#417—262no—
271#416—262no—
272C2×#103—261no—
273C2×#102—261no—
274#343—132no—
275#343—134no—
276#343—132no—
277C22×C13—21no—
278C22×C13—21no—
279C22×C13—21no—
280C22×C13—21no—
281C22×C13—21no—
282C22×C13—21no—
283C22×C13—42no—
284C22×C13—42no—
285C22×C13#38012yes—
286C22×C13#38012yes—
287C22×C13C2×#11711yescharacteristic
288C22×C13C2×#12111yescharacteristic
289C22×C13C2×C22⋊Q811yescharacteristic
290C2×D13—24no—
291C2×D13—42no—
292C2×D13—42no—
293C2×D13—84no—
294C4×C13#38012yes—
295C4×C13#38012yes—
296C4×C13C22×C22⋊C411yescharacteristic
297C4×C13C22×C4⋊C411yescharacteristic
298C4×C13C2×#12111yescharacteristic
299C4×C13C2×C22⋊Q811yescharacteristic
300Dic13—24no—
301Dic13—81no—
302Dic13—82no—
303Dic13—81no—
304C2×#41—134no—
305C2×#41—132no—
306C2×#41—134no—
307C2×#41—132no—
308C2×#41—261no—
309C2×#41—262no—
310C2×#41—261no—
311C2×#41—262no—
312C22×Q8—131no—
313C22×D4—131no—
314C22×D4—261no—
315C23×C4—261no—
316C23×C4—261no—
317C23×C4C22×D1311yescharacteristic
318#131—1316no—
319#131—2616no—
320#131—268no—
321C2×M16—21no—
322C2×M16—21no—
323C2×M16—22no—
324C2×M16—22no—
325C2×M16—22no—
326C2×M16—22no—
327C2×M16—134no—
328C2×M16—132no—
329C2×M16—261no—
330C2×M16—262no—
331C2×M16—2616no—
332C2×M16—264no—
333C2×M16C4×D1314yes—
334C2×M16D5212yes—
335C2×M16D5212yes—
336C2×M16C22⋊D1314yes—
337C2×M16C22×D1312yes—
338C2×M16C22×D1312yes—
339C22×C8—21no—
340C22×C8—21no—
341C22×C8—261no—
342C22×C8—264no—
343C22×C8C22×D1312yes—
344#117—261no—
345#117—261no—
346#117—261no—
347#117—261no—
348C2×C4⋊C4—132no—
349C2×C4⋊C4—131no—
350C2×C4⋊C4—131no—
351C2×C4⋊C4—261no—
352C2×C4⋊C4—261no—
353C2×C4⋊C4—261no—
354C2×C4⋊C4—261no—
355C2×C4⋊C4—261no—
356C2×C22⋊C4—261no—
357C2×C42—261no—
358#108—28no—
359#107—264no—
360#106—264no—
361#103—264no—
362#103—262no—
363#103—262no—
364#102—264no—
365#102—262no—
366#102—262no—
367C2×C13—22no—
368C2×C13—21no—
369C2×C13—21no—
370C2×C13—22no—
371C2×C13C2×#38011yescharacteristic
372C2×C13#306811yescharacteristic
373C2×C13#306911yescharacteristic
374D13—44no—
375C24—261no—
376C24#613111yescharacteristic
377#41—138no—
378#41—134no—
379#41—134no—
380#41—264no—
381#41—264no—
382#41—264no—
383#41—264no—
384#41—264no—
385#41—262no—
386#41—262no—
387C2×Q8—134no—
388C2×Q8—132no—
389C2×Q8—261no—
390C2×Q8—262no—
391C2×D4—134no—
392C2×D4—132no—
393C2×D4—264no—
394C2×D4—262no—
395C2×D4—264no—
396C2×D4—262no—
397C2×D4—261no—
398C2×D4—262no—
399C22×C4—261no—
400C22×C4—262no—
401C22×C4—264no—
402C22×C4—264no—
403C22×C4—262no—
404C22×C4—261no—
405C22×C4—261no—
406C22×C4—262no—
407C22×C4—521no—
408C22×C4—521no—
409C22×C4—1042no—
410C22×C4C2×C4×D1312yes—
411C22×C4C2×D5211yescharacteristic
412C22×C4C2×D5211yescharacteristic
413C22×C4D4×D1311yescharacteristic
414C22×C4D4×D1311yescharacteristic
415C22×C4#612912yes—
416C22×C4Q8×D1312yes—
417C22×C4#613111yescharacteristic
418C22×C4C2×C22⋊D1312yes—
419C22×C4C23×D1311yescharacteristic
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—42no—
425M16—42no—
426M16—134no—
427M16—264no—
428M16—262no—
429M16—2616no—
430M16#610318yes—
431M16#610318yes—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—264no—
437C2×C8—262no—
438C2×C8—2616no—
439C2×C8C2×C4×D1314yes—
440C2×C8C2×D5214yes—
441C2×C8C2×C22⋊D1314yes—
442C4⋊C4—261no—
443C4⋊C4—262no—
444C4⋊C4—261no—
445C4⋊C4—262no—
446C4⋊C4—261no—
447C4⋊C4—261no—
448C4⋊C4—521no—
449C4⋊C4—521no—
450C22⋊C4—521no—
451C22⋊C4—521no—
452C42—521no—
453C42—521no—
454C13#2067911yescharacteristic, complemented, Sylow
455C23—41no—
456C23—41no—
457C23—262no—
458C23—264no—
459C23—1041no—
460C23—1041no—
461C23#8814612yes—
462C23#8814911yescharacteristic
463C23#8814911yescharacteristic
464C23#8820012yes—
465C23C2×#613111yescharacteristic
466Q8—134no—
467Q8—262no—
468Q8—264no—
469D4—134no—
470D4—264no—
471D4—264no—
472D4—268no—
473D4—262no—
474D4—264no—
475C2×C4—262no—
476C2×C4—264no—
477C2×C4—264no—
478C2×C4—264no—
479C2×C4—268no—
480C2×C4—521no—
481C2×C4—521no—
482C2×C4—522no—
483C2×C4—1042no—
484C2×C4—1042no—
485C2×C4—1041no—
486C2×C4—1041no—
487C2×C4—1042no—
488C2×C4C4⋊C4×D1312yes—
489C2×C4C4⋊C4×D1312yes—
490C2×C4#8814612yes—
491C2×C4#8814911yescharacteristic
492C2×C4#8814911yescharacteristic
493C2×C4#8815112yes—
494C2×C4#8815112yes—
495C2×C4C2×#610312yes—
496C2×C4C2×#610312yes—
497C2×C4#8819212yes—
498C2×C4#8819212yes—
499C2×C4#8820012yes—
500C2×C4C22×C4×D1311yescharacteristic
501C2×C4C22×D5211yescharacteristic
502C2×C4C2×D4×D1311yescharacteristic
503C2×C4C2×#612911yescharacteristic
504C2×C4C2×Q8×D1311yescharacteristic
505C2×C4C22×C22⋊D1311yescharacteristic
506C8—42no—
507C8—42no—
508C8—264no—
509C8C2×#610318yes—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—264no—
519C22—522no—
520C22—522no—
521C22—1044no—
522C22#1171864812yes—
523C22#1171864812yes—
524C22C2×#8814611yescharacteristic
525C22C2×#8814911yescharacteristic
526C22C2×#8820011yescharacteristic
527C4—264no—
528C4—1041no—
529C4—1042no—
530C4—1041no—
531C4#1171864812yes—
532C4#1171864812yes—
533C4C2×C4⋊C4×D1311yescharacteristic
534C4C2×#8815111yescharacteristic
535C4C22×#610311yescharacteristic
536C4C2×#8819211yescharacteristic
537C2—22no—
538C2—21no—
539C2—21no—
540C2—22no—
541C2—524no—
542C2C2×#1171864811yescharacteristic
543C2#1312456511yescharacteristic
544C2#1312457411yescharacteristic
subgroupG/H
centerC22C2×#88149
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#430C2
socleC22×C13C2×#121
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC2×#430D13
hypocenterC13#20679
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
2. C2×#11718782C2semidirect
5. #13114897C2semidirect
6. #13114892C2semidirect
7. #13114574C2semidirect
8. C2×#11718146C2semidirect
9. C2×C13×#430C2semidirect
18. #11719450C4semidirect
23. C2×C13×#108C22semidirect
34. C2×#88058C22semidirect
35. C2×#88057C22semidirect
37. #11718146C22semidirect
38. #11718146C22semidirect
39. #11718146C22semidirect
91. #88155C2×C4semidirect
184. C2×#430D13semidirect
260. C2×#108C2×D13semidirect
454. C13#20679semidirect

Maximal subgroups

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

subgroupindex
C2×#117194502
C2×#117187822
#131233562
C2×#117187942
#131148972
#131148922
#131145742
C2×#117181462
C2×C13×#4302
#2067913