Finite Group Databasebeta

← #15841508

Subgroups — #15841508

Order 3328.

1192 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125964C211yesmaximal, characteristic, complemented
2#13124867C211yesmaximal, characteristic, complemented
3#13124300C211yesmaximal, characteristic
4#13114099C214yesmaximal, complemented
5#13113939C214yesmaximal, complemented
6#13113521C211yesmaximal, characteristic, complemented
7#13113265C211yesmaximal, characteristic, complemented
8#13113227C211yesmaximal, characteristic, complemented
9C2×D4×Q8×C13C211yesmaximal, characteristic, complemented
10C23×Q8×C13—21no—
11D4×Q8×C13—22no—
12D4×Q8×C13—22no—
13D4×Q8×C13—22no—
14D4×Q8×C13C2214yes—
15C2×C13×C4⋊Q8—21no—
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—21no—
18C2×C13×C22⋊Q8—22no—
19C2×C4×Q8×C13C2211yescharacteristic
20C2×C4×D4×C13—21no—
21C2×C4×D4×C13C2211yescharacteristic
22C2×#88186—21no—
23#11719182C2211yescharacteristic
24#11718888—21no—
25#11718749C2212yescomplemented
26#11718745—22no—
27#11718740C2212yescomplemented
28#11718688C2212yescomplemented
29C4×#6106—22no—
30#11718681C2212yescomplemented
31#11718680C2212yes—
32#11718677—22no—
33#11718676C2212yes—
34#11718629—21no—
35#11718629—22no—
36#11718627—21no—
37#11718586C2211yescharacteristic
38#11718586C2211yescharacteristic
39#11718585C2211yescharacteristic
40#11718139C2214yescomplemented
41Q8⋊#1055C2214yescomplemented
42D4⋊#1055C2214yes—
43C22×Q8×C13—22no—
44C22×Q8×C13—24no—
45C22×Q8×C13—24no—
46C22×Q8×C13—21no—
47C22×Q8×C13—48no—
48C22×Q8×C13D411yescharacteristic
49C22×D4×C13D411yescharacteristic
50C23×C4×C13—22no—
51C23×C4×C13—21no—
52C2×#6127—21no—
53C13×C4⋊Q8—24no—
54C13×C4⋊Q8—24no—
55C13×C4⋊Q8—21no—
56C13×C4⋊Q8—21no—
57C13×C4⋊Q8D412yes—
58C13×C4⋊Q8C2312yes—
59C13×C22⋊Q8—24no—
60C13×C22⋊Q8—28no—
61C13×C22⋊Q8—24no—
62C13×C22⋊Q8—28no—
63C4×Q8×C13—21no—
64C4×Q8×C13—21no—
65C4×Q8×C13D412yes—
66C4×Q8×C13C2312yes—
67C4×D4×C13—24no—
68C4×D4×C13—24no—
69C4×D4×C13—21no—
70C4×D4×C13—21no—
71C4×D4×C13D412yes—
72C4×D4×C13C2312yes—
73C2×C13×C4⋊C4—22no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D411yescharacteristic
78C2×C13×C4⋊C4D412yes—
79C2×C13×C22⋊C4—22no—
80C2×C13×C22⋊C4—21no—
81C2×C42×C13—21no—
82C2×C42×C13C2311yescharacteristic
83C2×#6106—22no—
84C2×#6105—22no—
85#88186—44no—
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—22no—
96C22⋊#1055—21no—
97C22⋊#1055—21no—
98#88050—22no—
99#88049—22no—
100#88049—22no—
101#88049—24no—
102#88048—22no—
103#88048—24no—
104C4⋊#1055—21no—
105C4⋊#1055—21no—
106C4⋊#1055C2312yes—
107C4⋊#1055C2312yes—
108C4×#1055—21no—
109C4×#1055C2312yes—
110#18758—131nomaximal, Sylow
111C24×C13—21no—
112C2×Q8×C13—24no—
113C2×Q8×C13—28no—
114C2×Q8×C13—21no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—48no—
118C2×Q8×C13—44no—
119C2×Q8×C13—416no—
120C2×Q8×C13—48no—
121C2×Q8×C13—48no—
122C2×Q8×C13C2×D412yes—
123C2×D4×C13—21no—
124C2×D4×C13—22no—
125C2×D4×C13SD1614yescomplemented
126C2×D4×C13C2×D412yes—
127C22×C4×C13—21no—
128C22×C4×C13—28no—
129C22×C4×C13—21no—
130C22×C4×C13—22no—
131C22×C4×C13—21no—
132C22×C4×C13—22no—
133C22×C4×C13—22no—
134C22×C4×C13—48no—
135C22×C4×C13—44no—
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C2×C22⋊D13—41no—
140#6127—42no—
141#6127—42no—
142C2×D52—21no—
143C2×D52—21no—
144C2×D52—21no—
145C2×C4×D13—41no—
146C2×Dic26—21no—
147C13×C4⋊C4—28no—
148C13×C4⋊C4—22no—
149C13×C4⋊C4—22no—
150C13×C4⋊C4—21no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—22no—
153C13×C4⋊C4—22no—
154C13×C4⋊C4—22no—
155C13×C4⋊C4C2×D412yes—
156C13×C4⋊C4C2×D412yes—
157C13×C4⋊C4C2×D414yes—
158C13×C22⋊C4—28no—
159C13×C22⋊C4—22no—
160C13×C22⋊C4—22no—
161C42×C13—22no—
162C42×C13—22no—
163C42×C13C2×D411yescharacteristic
164C42×C13C2×D411yescharacteristic
165C42×C13C2×D411yescharacteristic
166C42×C13C2411yescharacteristic
167#6106—44no—
168#6105—44no—
169#6104—44no—
170#6103—41no—
171#6103—41no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174#6099—42no—
175C2×#1055—22no—
176C2×#1055—22no—
177C2×#1055—22no—
178C2×D4×Q8D1311yescharacteristic, complemented
179#3227—131no—
180#3103—131no—
181#3036—131no—
182#1651—134no—
183#1627—134no—
184#1549—131no—
185#1487—131no—
186#1478—131no—
187C23×C13—22no—
188C23×C13—24no—
189C23×C13—42no—
190C23×C13—42no—
191C23×C13#12011yescharacteristic
192C22×D13—41no—
193Q8×C13—24no—
194Q8×C13—22no—
195Q8×C13—416no—
196Q8×C13—44no—
197Q8×C13—44no—
198Q8×C13AGL(1,ℤ/8)14yescomplemented
199D4×C13—22no—
200D4×C13—24no—
201D4×C13C2×SD1614yes—
202C2×C4×C13—24no—
203C2×C4×C13—22no—
204C2×C4×C13—22no—
205C2×C4×C13—22no—
206C2×C4×C13—24no—
207C2×C4×C13—22no—
208C2×C4×C13—41no—
209C2×C4×C13—416no—
210C2×C4×C13—44no—
211C2×C4×C13—42no—
212C2×C4×C13—44no—
213C2×C4×C13—42no—
214C2×C4×C13—44no—
215C2×C4×C13#12011yescharacteristic
216C2×C4×C13#12011yescharacteristic
217C2×C4×C13#12011yescharacteristic
218C2×C4×C13C2×SD1612yes—
219C2×C4×C13#13712yes—
220C2×C4×C13C22×D411yescharacteristic
221C2×C4×C13C22×D411yescharacteristic
222C2×C4×C13C22×D411yescharacteristic
223C22⋊D13—82no—
224C2×Dic13—41no—
225D52—42no—
226D52—41no—
227D52—42no—
228D52—41no—
229C4×D13—82no—
230Dic26—41no—
231Dic26—41no—
232#1055—42no—
233#1055—42no—
234#1055—42no—
235C23×Q8—21no—
236C2×AGL(1,ℤ/8)—261no—
237D4×Q8—22no—
238D4×Q8—22no—
239D4×Q8—22no—
240D4×Q8C2×D1314yes—
241#532—131no—
242C2×C4⋊Q8—21no—
243C2×C4⋊Q8C2×D1311yescharacteristic
244C2×C22⋊Q8—21no—
245C2×C22⋊Q8—22no—
246C2×C4×Q8C2×D1311yescharacteristic
247C2×C4×D4—21no—
248C2×C4×D4C2×D1311yescharacteristic
249#498—132no—
250#486—132no—
251#481—261no—
252#481—262no—
253#476—262no—
254#475—132no—
255#463—262no—
256#461—132no—
257#460—132no—
258#459—132no—
259#448—261no—
260C4×SD16—262no—
261#431—131no—
262#431—131no—
263#423—131no—
264#417—261no—
265#332—134no—
266Q8⋊C8—134no—
267D4⋊C8—134no—
268C22×C13—21no—
269C22×C13—21no—
270C22×C13—24no—
271C22×C13—48no—
272C22×C13—42no—
273C22×C13—42no—
274C22×C13#44812yes—
275C22×C13C2×#12011yescharacteristic
276C2×D13—81no—
277C2×D13—81no—
278C2×D13—81no—
279C4×C13—24no—
280C4×C13—21no—
281C4×C13—21no—
282C4×C13—21no—
283C4×C13—44no—
284C4×C13—42no—
285C4×C13—42no—
286C4×C13#44812yes—
287C4×C13C22×SD1611yescharacteristic
288C4×C13C2×AGL(1,ℤ/8)12yes—
289C4×C13C2×#13711yescharacteristic
290Dic13—81no—
291C2×#41—261no—
292C22×Q8—22no—
293C22×Q8—24no—
294C22×Q8—24no—
295C22×Q8—21no—
296C22×Q8—48no—
297C22×Q8C22⋊D1311yescharacteristic
298C22×D4C22⋊D1311yescharacteristic
299C23×C4—22no—
300C23×C4—21no—
301AGL(1,ℤ/8)—524no—
302C2×SD16—262no—
303C2×SD16—262no—
304C2×D8—262no—
305C2×M16—261no—
306C4⋊Q8—24no—
307C4⋊Q8—24no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—131no—
311C4⋊Q8C22⋊D1312yes—
312C4⋊Q8C22×D1312yes—
313#127—131no—
314#124—261no—
315C22⋊Q8—24no—
316C22⋊Q8—28no—
317C22⋊Q8—24no—
318C22⋊Q8—28no—
319#121—261no—
320#121—261no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C22⋊D1312yes—
324C4×Q8C22×D1312yes—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—261no—
330C4×D4—261no—
331C4×D4C22⋊D1312yes—
332C4×D4C22×D1312yes—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C22⋊D1311yescharacteristic
337C2×C4⋊C4C22⋊D1311yescharacteristic
338C2×C4⋊C4C22⋊D1312yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C42—21no—
342C2×C42C22×D1311yescharacteristic
343#106—262no—
344#106—264no—
345C4⋊C8—132no—
346C4⋊C8—132no—
347C4⋊C8—261no—
348C4⋊C8—261no—
349#103—262no—
350#103—262no—
351#102—262no—
352#102—262no—
353#102—264no—
354C22⋊C8—261no—
355C22⋊C8—261no—
356C4×C8—132no—
357C4×C8—261no—
358C2×C13—22no—
359C2×C13—44no—
360C2×C13C2×#44811yescharacteristic
361C2×C13#300311yescharacteristic
362C2×C13#302011yescharacteristic
363D13—161no—
364C24—21no—
365#41—522no—
366#41—522no—
367C2×Q8—24no—
368C2×Q8—28no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—48no—
373C2×Q8—44no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—261no—
378C2×Q8C2×C22⋊D1312yes—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—261no—
382C2×D4—261no—
383C2×D4—261no—
384C2×D4—521no—
385C2×D4#610614yescomplemented
386C2×D4C2×C22⋊D1312yes—
387C22×C4—21no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—521no—
397C22×C4C2×C22⋊D1311yescharacteristic
398C22×C4C2×C22⋊D1311yescharacteristic
399C22×C4C2×C22⋊D1311yescharacteristic
400SD16—524no—
401SD16—524no—
402D8—524no—
403M16—522no—
404C2×C8—262no—
405C2×C8—262no—
406C2×C8—262no—
407C4⋊C4—28no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—261no—
416C4⋊C4—261no—
417C4⋊C4C2×C22⋊D1312yes—
418C4⋊C4C2×C22⋊D1312yes—
419C4⋊C4C2×C22⋊D1314yes—
420C22⋊C4—28no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—521no—
424C22⋊C4—521no—
425C42—22no—
426C42—22no—
427C42C2×C22⋊D1311yescharacteristic
428C42C2×C22⋊D1311yescharacteristic
429C42C2×C22⋊D1311yescharacteristic
430C42C23×D1311yescharacteristic
431C13#1875811yescharacteristic, complemented, Sylow
432C23—22no—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—521no—
437C23#8820711yescharacteristic
438Q8—24no—
439Q8—22no—
440Q8—416no—
441Q8—44no—
442Q8—44no—
443Q8—521no—
444Q8—521no—
445Q8#8818614yescomplemented
446D4—22no—
447D4—24no—
448D4—522no—
449D4—522no—
450D4—521no—
451D4—521no—
452D4—1042no—
453D4C2×#610614yes—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—416no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—521no—
468C2×C4—1042no—
469C2×C4C2×#610612yes—
470C2×C4#8819612yes—
471C2×C4#8820711yescharacteristic
472C2×C4#8820711yescharacteristic
473C2×C4#8820711yescharacteristic
474C2×C4C22×C22⋊D1311yescharacteristic
475C2×C4C22×C22⋊D1311yescharacteristic
476C2×C4C22×C22⋊D1311yescharacteristic
477C8—522no—
478C8—522no—
479C8—522no—
480C22—21no—
481C22—21no—
482C22—24no—
483C22—48no—
484C22—42no—
485C22—42no—
486C22—1041no—
487C22—1041no—
488C22—1041no—
489C22#1171888812yes—
490C22C2×#8820711yescharacteristic
491C4—21no—
492C4—21no—
493C4—24no—
494C4—21no—
495C4—42no—
496C4—44no—
497C4—42no—
498C4—1041no—
499C4#1171888812yes—
500C4C2×#8818612yes—
501C4C22×#610611yescharacteristic
502C4C2×#8819611yescharacteristic
503C2—22no—
504C2—44no—
505C2—2081no—
506C2C2×#1171888811yescharacteristic
507C2#1312406311yescharacteristic
508C2#1312420011yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×D4×Q8×C13C2
socleC22×C13C2×#120
prime radical#13124300C2
hypercenterC2×D4×Q8D13
hypocenterC13#18758
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #13125964C2semidirect
2. #13124867C2semidirect
4. #13114099C2semidirect
5. #13113939C2semidirect
6. #13113521C2semidirect
7. #13113265C2semidirect
8. #13113227C2semidirect
9. C2×D4×Q8×C13C2semidirect
25. #11718749C22semidirect
27. #11718740C22semidirect
28. #11718688C22semidirect
30. #11718681C22semidirect
40. #11718139C22semidirect
41. Q8⋊#1055C22semidirect
125. C2×D4×C13SD16semidirect
178. C2×D4×Q8D13semidirect
198. Q8×C13AGL(1,ℤ/8)semidirect
385. C2×D4#6106semidirect
431. C13#18758semidirect
445. Q8#88186semidirect

Maximal subgroups

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

subgroupindex
#131259642
#131248672
#131243002
#131140992
#131139392
#131135212
#131132652
#131132272
C2×D4×Q8×C132
#1875813