Finite Group Databasebeta

← #15841442

Subgroups — #15841442

Order 3328.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125761C211yesmaximal, characteristic, complemented
2#13124836C211yesmaximal, characteristic
3#13124308C211yesmaximal, characteristic, complemented
4#13114050C214yesmaximal, complemented
5#13113908C214yesmaximal, complemented
6#13113517C211yesmaximal, characteristic, complemented
7#13113261C211yesmaximal, characteristic, complemented
8#13113231C211yesmaximal, characteristic, complemented
9C2×C13×#554C211yesmaximal, characteristic, complemented
10C13×#554—22no—
11C13×#554—22no—
12C13×#554—22no—
13C13×#554C2214yes—
14C2×C13×C4⋊Q8C2211yescharacteristic
15C2×C13×#125—21no—
16C2×C13×C22⋊Q8—21no—
17C2×C13×C22⋊Q8—22no—
18C2×C4×Q8×C13C2211yescharacteristic
19C2×C4×D4×C13—21no—
20C2×C4×D4×C13C2211yescharacteristic
21C22×C13×C4⋊C4—21no—
22C2×#88196—21no—
23#11719182C2211yescharacteristic
24#11718888—22no—
25#11718878—21no—
26#11718746C2212yescomplemented
27#11718744—22no—
28#11718742C2212yescomplemented
29#11718692C2212yescomplemented
30#11718689—22no—
31#11718688C2212yescomplemented
32#11718676C2212yes—
33C4×#6104—22no—
34#11718668C2212yes—
35#11718628—21no—
36#11718627—21no—
37#11718586C2211yescharacteristic
38#11718586C2211yescharacteristic
39#11718585C2211yescharacteristic
40#11718138C2214yescomplemented
41Q8⋊#1055C2214yescomplemented
42D4⋊#1055C2214yes—
43C22×Q8×C13D411yescharacteristic
44C22×Q8×C13D412yes—
45C22×D4×C13D411yescharacteristic
46C23×C4×C13—22no—
47C23×C4×C13—21no—
48C2×#6127—21no—
49C13×C4⋊Q8—21no—
50C13×C4⋊Q8—21no—
51C13×C4⋊Q8D412yes—
52C13×C4⋊Q8C2312yes—
53C13×#125—24no—
54C13×#125—24no—
55C13×C22⋊Q8—24no—
56C13×C22⋊Q8—28no—
57C13×C22⋊Q8—24no—
58C13×C22⋊Q8—28no—
59C4×Q8×C13—21no—
60C4×Q8×C13—21no—
61C4×Q8×C13D412yes—
62C4×Q8×C13C2312yes—
63C4×D4×C13—24no—
64C4×D4×C13—24no—
65C4×D4×C13—21no—
66C4×D4×C13—21no—
67C4×D4×C13D412yes—
68C4×D4×C13C2312yes—
69C2×C13×C4⋊C4—22no—
70C2×C13×C4⋊C4—24no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—24no—
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D411yescharacteristic
78C2×C13×C22⋊C4—22no—
79C2×C13×C22⋊C4—21no—
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×#6107—22no—
83#88196—44no—
84C2×#6106—22no—
85C2×#6106—24no—
86C2×#6104—22no—
87#88184—21no—
88#88184—21no—
89C2×#6099—21no—
90#88129—21no—
91#88128C2311yescharacteristic
92C4×D52—21no—
93C4×D52—21no—
94#88123C2311yescharacteristic
95#88075—24no—
96#88072—22no—
97C22⋊#1055—21no—
98C22⋊#1055—21no—
99#88050—22no—
100#88049—22no—
101#88049—22no—
102#88047—22no—
103C4⋊#1055—21no—
104C4⋊#1055—21no—
105C4⋊#1055C2312yes—
106C4⋊#1055C2312yes—
107C4×#1055—21no—
108C4×#1055C2312yes—
109#18752—131nomaximal, Sylow
110C24×C13—21no—
111C2×Q8×C13—21no—
112C2×Q8×C13—22no—
113C2×Q8×C13—22no—
114C2×Q8×C13—22no—
115C2×Q8×C13—82no—
116C2×Q8×C13C2×D412yes—
117C2×Q8×C13C2×D414yes—
118C2×D4×C13—24no—
119C2×D4×C13—21no—
120C2×D4×C13C2×D412yes—
121C22×C4×C13—21no—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—22no—
125C22×C4×C13—28no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—42no—
129C22×C4×C13—42no—
130C22×C4×C13—84no—
131C22×C4×C13C2×D411yescharacteristic
132C22×C4×C13C2×D411yescharacteristic
133C22×C4×C13C2×D411yescharacteristic
134C2×C22⋊D13—41no—
135#6127—42no—
136#6127—42no—
137C2×D52—21no—
138C2×D52—21no—
139C2×D52—21no—
140C2×C4×D13—41no—
141C2×Dic26—21no—
142C13×C4⋊C4—28no—
143C13×C4⋊C4—28no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4—22no—
146C13×C4⋊C4—22no—
147C13×C4⋊C4—22no—
148C13×C4⋊C4—21no—
149C13×C4⋊C4—22no—
150C13×C4⋊C4—22no—
151C13×C4⋊C4—22no—
152C13×C4⋊C4C2×D412yes—
153C13×C4⋊C4C2×D412yes—
154C13×C22⋊C4—28no—
155C13×C22⋊C4—22no—
156C13×C22⋊C4—22no—
157C42×C13—22no—
158C42×C13—22no—
159C42×C13C2×D411yescharacteristic
160C42×C13C2×D411yescharacteristic
161C42×C13C2×D411yescharacteristic
162C42×C13C2411yescharacteristic
163#6107—44no—
164#6106—44no—
165#6106—84no—
166#6104—44no—
167#6103—41no—
168#6103—41no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171#6099—42no—
172C2×#1055—22no—
173C2×#1055—22no—
174C2×#1055—22no—
175C2×#554D1311yescharacteristic, complemented
176#3206—131no—
177#3100—131no—
178#3037—131no—
179#1644—134no—
180#1622—134no—
181#1548—131no—
182#1486—131no—
183#1479—131no—
184C23×C13—24no—
185C23×C13—22no—
186C23×C13—42no—
187C23×C13—42no—
188C23×C13#12011yescharacteristic
189C22×D13—41no—
190Q8×C13—22no—
191Q8×C13—22no—
192Q8×C13—44no—
193Q8×C13—42no—
194Q8×C13—82no—
195D4×C13—24no—
196D4×C13—22no—
197D4×C13#13714yescomplemented
198C2×C4×C13—22no—
199C2×C4×C13—22no—
200C2×C4×C13—22no—
201C2×C4×C13—22no—
202C2×C4×C13—24no—
203C2×C4×C13—41no—
204C2×C4×C13—42no—
205C2×C4×C13—44no—
206C2×C4×C13—42no—
207C2×C4×C13—42no—
208C2×C4×C13—42no—
209C2×C4×C13—42no—
210C2×C4×C13—44no—
211C2×C4×C13—88no—
212C2×C4×C13#12011yescharacteristic
213C2×C4×C13#12011yescharacteristic
214C2×C4×C13#12011yescharacteristic
215C2×C4×C13#13512yes—
216C2×C4×C13AGL(1,ℤ/8)12yes—
217C2×C4×C13C22×D411yescharacteristic
218C2×C4×C13C22×D411yescharacteristic
219C2×C4×C13C22×D411yescharacteristic
220C22⋊D13—82no—
221C2×Dic13—41no—
222D52—42no—
223D52—41no—
224D52—41no—
225D52—42no—
226C4×D13—82no—
227Dic26—41no—
228Dic26—41no—
229#1055—42no—
230#1055—42no—
231#1055—42no—
232C2×#137—261no—
233#554—22no—
234#554—22no—
235#554—22no—
236#554C2×D1314yes—
237#532—131no—
238C2×C4⋊Q8C2×D1311yescharacteristic
239C2×#125—21no—
240C2×C22⋊Q8—21no—
241C2×C22⋊Q8—22no—
242C2×C4×Q8C2×D1311yescharacteristic
243C2×C4×D4—21no—
244C2×C4×D4C2×D1311yescharacteristic
245C22×C4⋊C4—21no—
246#492—132no—
247#487—132no—
248#480—261no—
249#474—132no—
250#464—262no—
251#462—132no—
252#461—262no—
253#460—132no—
254#459—132no—
255#448—262no—
256#447—261no—
257C4×D8—262no—
258#431—131no—
259#431—131no—
260#423—131no—
261#417—261no—
262#333—134no—
263Q8⋊C8—134no—
264D4⋊C8—134no—
265C22×C13—21no—
266C22×C13—21no—
267C22×C13—24no—
268C22×C13—48no—
269C22×C13—42no—
270C22×C13—42no—
271C22×C13#44912yes—
272C22×C13C2×#12011yescharacteristic
273C2×D13—81no—
274C2×D13—81no—
275C2×D13—81no—
276C4×C13—21no—
277C4×C13—21no—
278C4×C13—21no—
279C4×C13—42no—
280C4×C13—42no—
281C4×C13—42no—
282C4×C13—44no—
283C4×C13#44912yes—
284C4×C13C2×#13511yescharacteristic
285C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
286C4×C13C2×#13712yes—
287Dic13—81no—
288C2×#41—261no—
289C22×Q8C22⋊D1311yescharacteristic
290C22×Q8C22⋊D1312yes—
291C22×D4C22⋊D1311yescharacteristic
292C23×C4—22no—
293C23×C4—21no—
294#137—524no—
295C2×Q16—262no—
296C2×SD16—262no—
297C2×SD16—264no—
298C2×D8—262no—
299C2×M16—261no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—131no—
303C4⋊Q8C22⋊D1312yes—
304C4⋊Q8C22×D1312yes—
305#127—131no—
306#125—24no—
307#125—24no—
308#124—261no—
309C22⋊Q8—24no—
310C22⋊Q8—28no—
311C22⋊Q8—24no—
312C22⋊Q8—28no—
313#121—261no—
314#121—261no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8C22⋊D1312yes—
318C4×Q8C22×D1312yes—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—261no—
324C4×D4—261no—
325C4×D4C22⋊D1312yes—
326C4×D4C22×D1312yes—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4C22⋊D1311yescharacteristic
335C2×C4⋊C4C22⋊D1311yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C42—21no—
339C2×C42C22×D1311yescharacteristic
340#107—262no—
341C4⋊C8—132no—
342C4⋊C8—132no—
343C4⋊C8—261no—
344C4⋊C8—261no—
345#103—264no—
346#103—262no—
347#102—262no—
348#102—262no—
349#102—262no—
350C22⋊C8—261no—
351C22⋊C8—261no—
352C4×C8—132no—
353C4×C8—261no—
354C2×C13—22no—
355C2×C13—44no—
356C2×C13C2×#44911yescharacteristic
357C2×C13#300411yescharacteristic
358C2×C13#301911yescharacteristic
359D13—161no—
360C24—21no—
361#41—522no—
362#41—522no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—82no—
368C2×Q8—261no—
369C2×Q8C2×C22⋊D1312yes—
370C2×Q8C2×C22⋊D1314yes—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—261no—
374C2×D4—261no—
375C2×D4—261no—
376C2×D4—521no—
377C2×D4C2×C22⋊D1312yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—84no—
388C22×C4—521no—
389C22×C4C2×C22⋊D1311yescharacteristic
390C22×C4C2×C22⋊D1311yescharacteristic
391C22×C4C2×C22⋊D1311yescharacteristic
392Q16—524no—
393SD16—524no—
394SD16—1044no—
395D8—524no—
396M16—522no—
397C2×C8—262no—
398C2×C8—262no—
399C2×C8—262no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—261no—
411C4⋊C4—261no—
412C4⋊C4C2×C22⋊D1312yes—
413C4⋊C4C2×C22⋊D1312yes—
414C22⋊C4—28no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—521no—
418C22⋊C4—521no—
419C42—22no—
420C42—22no—
421C42C2×C22⋊D1311yescharacteristic
422C42C2×C22⋊D1311yescharacteristic
423C42C2×C22⋊D1311yescharacteristic
424C42C23×D1311yescharacteristic
425C13#1875211yescharacteristic, complemented, Sylow
426C23—24no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—521no—
431C23#8820711yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—44no—
435Q8—42no—
436Q8—82no—
437Q8—521no—
438Q8—521no—
439D4—24no—
440D4—22no—
441D4—522no—
442D4—522no—
443D4—521no—
444D4—521no—
445D4—1042no—
446D4#8819614yescomplemented
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—88no—
461C2×C4—521no—
462C2×C4—1042no—
463C2×C4#8820312yes—
464C2×C4#8820412yes—
465C2×C4#8820711yescharacteristic
466C2×C4#8820711yescharacteristic
467C2×C4#8820711yescharacteristic
468C2×C4C22×C22⋊D1311yescharacteristic
469C2×C4C22×C22⋊D1311yescharacteristic
470C2×C4C22×C22⋊D1311yescharacteristic
471C8—522no—
472C8—522no—
473C8—522no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—1041no—
481C22—1041no—
482C22—1041no—
483C22#1171889912yes—
484C22C2×#8820711yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—42no—
492C4—1041no—
493C4#1171889912yes—
494C4C2×#8819612yes—
495C4C2×#8820311yescharacteristic
496C4C2×#8820411yescharacteristic
497C2—22no—
498C2—44no—
499C2—2081no—
500C2C2×#1171889911yescharacteristic
501C2#1312407011yescharacteristic
502C2#1312419311yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×C13×#554C2
socleC22×C13C2×#120
prime radical#13124836C2
hypercenterC2×#554D13
hypocenterC13#18752
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #13125761C2semidirect
3. #13124308C2semidirect
4. #13114050C2semidirect
5. #13113908C2semidirect
6. #13113517C2semidirect
7. #13113261C2semidirect
8. #13113231C2semidirect
9. C2×C13×#554C2semidirect
26. #11718746C22semidirect
28. #11718742C22semidirect
29. #11718692C22semidirect
31. #11718688C22semidirect
40. #11718138C22semidirect
41. Q8⋊#1055C22semidirect
175. C2×#554D13semidirect
197. D4×C13#137semidirect
425. C13#18752semidirect
446. D4#88196semidirect

Maximal subgroups

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

subgroupindex
#131257612
#131248362
#131243082
#131140502
#131139082
#131135172
#131132612
#131132312
C2×C13×#5542
#1875213