Finite Group Databasebeta

← #15865468

Subgroups — #15865468

Order 3328.

1014 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123426C211yesmaximal, characteristic, complemented
2#13123356C211yesmaximal, characteristic
3C2×#11718774C211yesmaximal, characteristic
4C2×#11718793C211yesmaximal, characteristic
5C2×#11718762C211yesmaximal, characteristic, complemented
6#13115499C214yesmaximal, complemented
7#13115492C214yesmaximal, complemented
8#13114574C211yesmaximal, characteristic, complemented
9C2×C13×#341C211yesmaximal, characteristic, complemented
10C22×#6127C2211yescharacteristic
11C22×C13×M16C2211yescharacteristic
12C22×C13×M16C2211yescharacteristic
13C23×C8×C13C2211yescharacteristic
14C2×#88180C2211yescharacteristic
15C2×#88180C2211yescharacteristic
16C22×#6099C2211yescharacteristic
17C2×#88064C2212yes—
18#11718794—22no—
19#11718793C418yescomplemented
20#11718792C2212yes—
21C2×#88063C2212yes—
22#11718782—22no—
23#11718782C2214yes—
24#11718781C2212yes—
25#11718774—21no—
26#11718774—21no—
27#11718774—22no—
28#11718774—22no—
29C2×#88061C2212yes—
30#11718767—22no—
31C2×#88056C2212yes—
32#11718762—22no—
33#11718762C2214yes—
34C2×#88055—21no—
35C2×C8×Dic13—21no—
36C13×#341C2214yes—
37#11718146C2214yes—
38C23×C4×C13C2311yescharacteristic
39C22×C22⋊D13—21no—
40C2×#6127—22no—
41C2×#6127—21no—
42C2×#6127—21no—
43C2×#6127—21no—
44C2×#6127—21no—
45C2×#6127—22no—
46C2×#6127—22no—
47C2×#6127C2×C414yes—
48C22×D52C2311yescharacteristic
49C22×C4×D13—21no—
50C22×Dic26C2311yescharacteristic
51C2×C13×M16—21no—
52C2×C13×M16—22no—
53C2×C13×M16—21no—
54C2×C13×M16—22no—
55C2×C13×M16—22no—
56C2×C13×M16C2×C414yes—
57C2×C13×M16C2312yes—
58C2×C13×M16C2312yes—
59C22×C8×C13—21no—
60C22×C8×C13—22no—
61C22×C8×C13—22no—
62C22×C8×C13—21no—
63C22×C8×C13D411yescharacteristic
64C22×C8×C13D411yescharacteristic
65C22×C8×C13D412yes—
66C22×C8×C13C2312yes—
67C2×C22⋊Dic13—21no—
68C2×C22⋊Dic13—21no—
69#88180—21no—
70#88180—21no—
71#88180—21no—
72#88180—21no—
73#88180—21no—
74#88180—21no—
75#88180D412yes—
76#88180C2312yes—
77C2×C4⋊Dic13D412yes—
78C2×C4⋊Dic13C2312yes—
79C2×C4×Dic13—21no—
80C2×C4×Dic13—21no—
81C2×#6099—21no—
82C2×#6099—22no—
83C2×#6099C2×C414yes—
84C2×#6099C2312yes—
85C22×#1055—21no—
86#88064C2×C418yescomplemented
87#88063C2×C418yescomplemented
88#88061—22no—
89#88061—22no—
90#88061—24no—
91#88058—24no—
92#88057—24no—
93#88056—22no—
94#88056—22no—
95#88056—24no—
96#88055—24no—
97C8×Dic13—24no—
98#21075—131nomaximal, Sylow
99C24×C13#4111yescharacteristic
100C23×D13—21no—
101C22×C4×C13C22×C412yes—
102C22×C4×C13C2×D411yescharacteristic
103C22×C4×C13C2×D411yescharacteristic
104C22×C4×C13C2×D411yescharacteristic
105C22×C4×C13C2×D411yescharacteristic
106C22×C4×C13#4112yes—
107C22×C4×C13#4112yes—
108C22×C4×C13#4112yes—
109C22×C4×C13#4111yescharacteristic
110C22×C4×C13C2411yescharacteristic
111C2×C22⋊D13—24no—
112C2×C22⋊D13—41no—
113C2×C22⋊D13—41no—
114C2×C22⋊D13—42no—
115C22×Dic13—22no—
116C22×Dic13—21no—
117C22×Dic13—22no—
118#6127—24no—
119#6127—24no—
120#6127—24no—
121#6127—24no—
122#6127—42no—
123#6127—42no—
124#6127—42no—
125#6127—42no—
126C2×D52—21no—
127C2×D52—21no—
128C2×D52—21no—
129C2×D52—21no—
130C2×D52—21no—
131C2×D52C22×C412yes—
132C2×C4×D13—24no—
133C2×C4×D13—41no—
134C2×C4×D13—41no—
135C2×C4×D13—42no—
136C2×Dic26—21no—
137C2×Dic26—21no—
138C2×Dic26—21no—
139C2×Dic26—21no—
140C2×Dic26—21no—
141C2×Dic26C22×C412yes—
142C13×M16—24no—
143C13×M16—22no—
144C13×M16—22no—
145C13×M16—22no—
146C13×M16—42no—
147C13×M16C22×C414yes—
148C2×C8×C13—24no—
149C2×C8×C13—22no—
150C2×C8×C13—22no—
151C2×C8×C13—24no—
152C2×C8×C13—22no—
153C2×C8×C13—22no—
154C2×C8×C13—22no—
155C2×C8×C13—22no—
156C2×C8×C13—42no—
157C2×C8×C13C2×D412yes—
158C2×C8×C13C2×D412yes—
159C22⋊Dic13—22no—
160C22⋊Dic13—22no—
161C22⋊Dic13—41no—
162C22⋊Dic13—41no—
163C4⋊Dic13—22no—
164C4⋊Dic13—22no—
165C4⋊Dic13—22no—
166C4⋊Dic13C2×D414yes—
167C4×Dic13—22no—
168C4×Dic13—22no—
169C4×Dic13—41no—
170C4×Dic13—41no—
171#6099—24no—
172#6099—22no—
173#6099C22×C414yes—
174C2×#1055—24no—
175C2×#1055—22no—
176#2911—131no—
177#2896—131no—
178C2×#416—131no—
179C2×#413—131no—
180C2×#405—131no—
181#1931—134no—
182#1930—134no—
183#1742—131no—
184C2×#341D1311yescharacteristic, complemented
185C23×C13—42no—
186C23×C13#11712yes—
187C23×C13#12111yescharacteristic
188C23×C13#12111yescharacteristic
189C23×C13#12612yes—
190C23×C13C2×#4111yescharacteristic
191C22×D13—22no—
192C22×D13—41no—
193C22×D13—41no—
194C22×D13—82no—
195C2×C4×C13#11712yes—
196C2×C4×C13C4×D412yes—
197C2×C4×C13C4×D412yes—
198C2×C4×C13C4×D412yes—
199C2×C4×C13C4×D412yes—
200C2×C4×C13#12111yescharacteristic
201C2×C4×C13#12111yescharacteristic
202C2×C4×C13#12312yes—
203C2×C4×C13#12312yes—
204C2×C4×C13#12412yes—
205C2×C4×C13#12412yes—
206C2×C4×C13#12612yes—
207C2×C4×C13C23×C411yescharacteristic
208C2×C4×C13C22×D411yescharacteristic
209C2×C4×C13C22×D411yescharacteristic
210C2×C4×C13C2×#4111yescharacteristic
211C2×C4×C13C2×#4111yescharacteristic
212C2×C4×C13C2×#4111yescharacteristic
213C22⋊D13—44no—
214C22⋊D13—44no—
215C2×Dic13—22no—
216C2×Dic13—24no—
217C2×Dic13—44no—
218C2×Dic13—41no—
219C2×Dic13—41no—
220C2×Dic13—42no—
221C2×Dic13—82no—
222D52—22no—
223D52—22no—
224D52—41no—
225D52—41no—
226C4×D13—44no—
227C4×D13—44no—
228Dic26—22no—
229Dic26—22no—
230Dic26—41no—
231Dic26—41no—
232C8×C13—24no—
233C8×C13—22no—
234C8×C13—22no—
235C8×C13—42no—
236#1055—24no—
237C22×#41—131no—
238C22×M16—131no—
239C22×M16C2×D1311yescharacteristic
240C22×M16C2×D1311yescharacteristic
241C23×C8C2×D1311yescharacteristic
242C2×#117—131no—
243C2×#117—131no—
244#428—134no—
245#428—262no—
246#427—262no—
247#419—132no—
248#418—132no—
249#417—262no—
250#416—262no—
251#416—261no—
252#416—261no—
253#416—262no—
254C2×#103—132no—
255C2×#102—132no—
256#413—138no—
257C2×#101—132no—
258C2×#100—132no—
259#405—134no—
260#405—262no—
261C2×#97—261no—
262C2×C4×C8—261no—
263#343—134no—
264#341C2×D1314yes—
265C22×C13—22no—
266C22×C13—22no—
267C22×C13—22no—
268C22×C13—42no—
269C22×C13—42no—
270C22×C13#38812yes—
271C22×C13#38812yes—
272C22×C13C2×#11711yescharacteristic
273C22×C13C2×#12111yescharacteristic
274C22×C13C2×#12611yescharacteristic
275C2×D13—42no—
276C2×D13—82no—
277C2×D13—82no—
278C2×D13—82no—
279C4×C13#38812yes—
280C4×C13#38812yes—
281C4×C13C2×C4×D411yescharacteristic
282C4×C13C2×C4×D411yescharacteristic
283C4×C13C2×#12311yescharacteristic
284C4×C13C2×#12411yescharacteristic
285Dic13—42no—
286Dic13—44no—
287Dic13—82no—
288C2×#41—134no—
289C2×#41—262no—
290C2×#41—261no—
291C2×#41—261no—
292C2×#41—262no—
293C2×#41—261no—
294C2×#41—261no—
295C2×#41—262no—
296C22×Q8—131no—
297C22×D4—131no—
298C22×D4—261no—
299C23×C4—261no—
300C23×C4C22×D1311yescharacteristic
301C2×M16—21no—
302C2×M16—22no—
303C2×M16—21no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—134no—
307C2×M16—132no—
308C2×M16—261no—
309C2×M16—262no—
310C2×M16C4×D1314yes—
311C2×M16C22×D1312yes—
312C2×M16C22×D1312yes—
313C22×C8—21no—
314C22×C8—22no—
315C22×C8—22no—
316C22×C8—21no—
317C22×C8—261no—
318C22×C8D5211yescharacteristic
319C22×C8D5211yescharacteristic
320C22×C8D5212yes—
321C22×C8C22×D1312yes—
322#117—132no—
323#117—132no—
324#117—261no—
325#117—261no—
326#117—261no—
327#117—261no—
328#117—261no—
329#117—261no—
330C2×C4⋊C4—132no—
331C2×C4⋊C4—132no—
332C2×C22⋊C4—261no—
333C2×C22⋊C4—261no—
334C2×C42—261no—
335C2×C42—261no—
336#107—264no—
337#106—264no—
338#103—264no—
339#103—262no—
340#103—262no—
341#102—264no—
342#102—262no—
343#102—262no—
344#101—138no—
345#100—138no—
346#97—264no—
347C4×C8—264no—
348C2×C13—21no—
349C2×C13—21no—
350C2×C13—22no—
351C2×C13—22no—
352C2×C13C2×#38811yescharacteristic
353C2×C13#306712yes—
354D13—82no—
355C24—261no—
356C24#613111yescharacteristic
357#41—264no—
358#41—264no—
359#41—264no—
360#41—264no—
361#41—522no—
362#41—522no—
363#41—522no—
364#41—522no—
365C2×Q8—132no—
366C2×Q8—261no—
367C2×Q8—261no—
368C2×Q8—261no—
369C2×Q8—261no—
370C2×Q8—261no—
371C2×D4—132no—
372C2×D4—264no—
373C2×D4—261no—
374C2×D4—261no—
375C2×D4—261no—
376C2×D4—261no—
377C2×D4—261no—
378C2×D4—521no—
379C2×D4—521no—
380C2×D4—522no—
381C22×C4—262no—
382C22×C4—261no—
383C22×C4—264no—
384C22×C4—262no—
385C22×C4—521no—
386C22×C4—521no—
387C22×C4—522no—
388C22×C4C2×C4×D1312yes—
389C22×C4C2×D5211yescharacteristic
390C22×C4C2×D5211yescharacteristic
391C22×C4#612712yes—
392C22×C4D4×D1311yescharacteristic
393C22×C4D4×D1311yescharacteristic
394C22×C4#612912yes—
395C22×C4#612912yes—
396C22×C4#613111yescharacteristic
397C22×C4C23×D1311yescharacteristic
398M16—24no—
399M16—22no—
400M16—22no—
401M16—22no—
402M16—42no—
403M16—134no—
404M16—264no—
405M16—262no—
406M16C2×C4×D1314yes—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—42no—
416C2×C8—264no—
417C2×C8—262no—
418C2×C8C2×D5212yes—
419C2×C8C2×D5212yes—
420C4⋊C4—134no—
421C4⋊C4—262no—
422C4⋊C4—262no—
423C4⋊C4—262no—
424C22⋊C4—262no—
425C22⋊C4—262no—
426C22⋊C4—521no—
427C22⋊C4—521no—
428C42—262no—
429C42—262no—
430C42—521no—
431C42—521no—
432C13#2107511yescharacteristic, complemented, Sylow
433C23—42no—
434C23—262no—
435C23—521no—
436C23—521no—
437C23—1042no—
438C23#8814612yes—
439C23#8814911yescharacteristic
440C23#8814911yescharacteristic
441C23#8815212yes—
442C23C2×#613111yescharacteristic
443Q8—262no—
444Q8—262no—
445Q8—521no—
446Q8—521no—
447D4—262no—
448D4—262no—
449D4—524no—
450D4—524no—
451D4—521no—
452D4—521no—
453C2×C4—262no—
454C2×C4—264no—
455C2×C4—524no—
456C2×C4—521no—
457C2×C4—521no—
458C2×C4—524no—
459C2×C4—524no—
460C2×C4—522no—
461C2×C4—1042no—
462C2×C4C4×D5212yes—
463C2×C4C4×D5212yes—
464C2×C4#8813512yes—
465C2×C4#8813512yes—
466C2×C4#8813912yes—
467C2×C4#8813912yes—
468C2×C4#8814012yes—
469C2×C4#8814012yes—
470C2×C4#8814612yes—
471C2×C4#8814911yescharacteristic
472C2×C4#8814911yescharacteristic
473C2×C4#8815212yes—
474C2×C4C22×C4×D1311yescharacteristic
475C2×C4C22×D5211yescharacteristic
476C2×C4C2×#612711yescharacteristic
477C2×C4C2×D4×D1311yescharacteristic
478C2×C4C2×#612911yescharacteristic
479C2×C4C2×#612911yescharacteristic
480C8—24no—
481C8—22no—
482C8—22no—
483C8—42no—
484C8—264no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—522no—
491C22—1042no—
492C22—1042no—
493C22—1042no—
494C22#1171833112yes—
495C22#1171833112yes—
496C22C2×#8814611yescharacteristic
497C22C2×#8814911yescharacteristic
498C22C2×#8815211yescharacteristic
499C4—522no—
500C4—524no—
501C4—1042no—
502C4#1171833112yes—
503C4#1171833112yes—
504C4C2×C4×D5211yescharacteristic
505C4C2×#8813511yescharacteristic
506C4C2×#8813911yescharacteristic
507C4C2×#8814011yescharacteristic
508C2—21no—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—1042no—
513C2C2×#1171833111yescharacteristic
514C2#1312455712yes—
subgroupG/H
centerC2×C4#88149
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#341C2
socleC22×C13C2×#121
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC2×#341D13
hypocenterC13#21075
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
1. #13123426C2semidirect
5. C2×#11718762C2semidirect
6. #13115499C2semidirect
7. #13115492C2semidirect
8. #13114574C2semidirect
9. C2×C13×#341C2semidirect
19. #11718793C4semidirect
86. #88064C2×C4semidirect
87. #88063C2×C4semidirect
184. C2×#341D13semidirect
432. C13#21075semidirect

Maximal subgroups

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

subgroupindex
#131234262
#131233562
C2×#117187742
C2×#117187932
C2×#117187622
#131154992
#131154922
#131145742
C2×C13×#3412
#2107513