Finite Group Databasebeta

← #15841610

Subgroups — #15841610

Order 3328.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125761C211yesmaximal, characteristic
2#13124867C211yesmaximal, characteristic
3#13124308C211yesmaximal, characteristic
4#13114057C214yesmaximal, complemented
5#13113914C214yesmaximal, complemented
6#13113517C211yesmaximal, characteristic, complemented
7#13113265C211yesmaximal, characteristic, complemented
8#13113231C211yesmaximal, characteristic, complemented
9C2×Q82×C13C211yesmaximal, characteristic, complemented
10Q82×C13—22no—
11Q82×C13—22no—
12Q82×C13—22no—
13Q82×C13C2214yes—
14C2×C13×C4⋊Q8—22no—
15C2×C13×C4⋊Q8—21no—
16C2×C13×C4⋊Q8—21no—
17C2×C13×C4⋊Q8C2211yescharacteristic
18C2×C4×Q8×C13—21no—
19C2×C4×Q8×C13—21no—
20C2×C4×Q8×C13C2211yescharacteristic
21C2×C4×Q8×C13C2211yescharacteristic
22C2×#88196—21no—
23#11719182C2211yescharacteristic
24#11718888—21no—
25#11718888—22no—
26#11718746C2212yes—
27#11718744—22no—
28#11718742C2212yes—
29#11718692C2212yes—
30#11718689—22no—
31#11718688C2212yes—
32#11718688C2212yes—
33C4×#6106—22no—
34#11718681C2212yes—
35#11718629—21no—
36#11718627—21no—
37#11718586C2211yescharacteristic
38#11718586C2211yescharacteristic
39#11718585C2211yescharacteristic
40#11718138C2214yes—
41Q8⋊#1055C2214yes—
42Q8⋊#1055C2214yes—
43C22×Q8×C13—22no—
44C22×Q8×C13D411yescharacteristic
45C22×Q8×C13D411yescharacteristic
46C22×Q8×C13D412yes—
47C2×#6127—21no—
48C13×C4⋊Q8—28no—
49C13×C4⋊Q8—24no—
50C13×C4⋊Q8—24no—
51C13×C4⋊Q8—28no—
52C13×C4⋊Q8—24no—
53C13×C4⋊Q8—24no—
54C13×C4⋊Q8—21no—
55C13×C4⋊Q8—21no—
56C13×C4⋊Q8D412yes—
57C13×C4⋊Q8C2312yes—
58C4×Q8×C13—24no—
59C4×Q8×C13—24no—
60C4×Q8×C13—24no—
61C4×Q8×C13—24no—
62C4×Q8×C13—21no—
63C4×Q8×C13—21no—
64C4×Q8×C13—21no—
65C4×Q8×C13—21no—
66C4×Q8×C13D412yes—
67C4×Q8×C13D412yes—
68C4×Q8×C13C2312yes—
69C4×Q8×C13C2312yes—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D411yescharacteristic
78C2×C42×C13—22no—
79C2×C42×C13—21no—
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×#6107—22no—
83#88196—44no—
84C2×#6106—22no—
85C2×#6106—22no—
86C2×#6106—24no—
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—
96#88075—24no—
97C22⋊#1055—21no—
98C22⋊#1055—21no—
99#88050—22no—
100#88049—22no—
101#88049—22no—
102#88048—22no—
103C4⋊#1055—21no—
104C4⋊#1055—21no—
105C4⋊#1055C2312yes—
106C4⋊#1055C2312yes—
107C4×#1055—21no—
108C4×#1055C2312yes—
109#18768—131nomaximal, Sylow
110C2×Q8×C13—24no—
111C2×Q8×C13—28no—
112C2×Q8×C13—21no—
113C2×Q8×C13—21no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—82no—
117C2×Q8×C13—84no—
118C2×Q8×C13SD1614yescomplemented
119C2×Q8×C13C2×D412yes—
120C2×Q8×C13C2×D412yes—
121C2×Q8×C13C2×D414yes—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—22no—
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13C2×D411yescharacteristic
128C22×C4×C13C2×D411yescharacteristic
129C22×C4×C13C2×D411yescharacteristic
130C2×C22⋊D13—41no—
131#6127—42no—
132#6127—42no—
133C2×D52—21no—
134C2×D52—21no—
135C2×D52—21no—
136C2×C4×D13—41no—
137C2×Dic26—21no—
138C13×C4⋊C4—28no—
139C13×C4⋊C4—28no—
140C13×C4⋊C4—22no—
141C13×C4⋊C4—22no—
142C13×C4⋊C4—22no—
143C13×C4⋊C4—22no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4—21no—
146C13×C4⋊C4—22no—
147C13×C4⋊C4—22no—
148C13×C4⋊C4—22no—
149C13×C4⋊C4—22no—
150C13×C4⋊C4C2×D412yes—
151C13×C4⋊C4C2×D412yes—
152C42×C13—28no—
153C42×C13—22no—
154C42×C13—22no—
155C42×C13—22no—
156C42×C13—22no—
157C42×C13C2×D411yescharacteristic
158C42×C13C2×D411yescharacteristic
159C42×C13C2×D411yescharacteristic
160C42×C13C2411yescharacteristic
161#6107—44no—
162#6106—44no—
163#6106—44no—
164#6106—84no—
165#6103—41no—
166#6103—41no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169#6099—42no—
170C2×#1055—22no—
171C2×#1055—22no—
172C2×#1055—22no—
173C2×Q82D1311yescharacteristic, complemented
174#3206—131no—
175#3103—131no—
176#3037—131no—
177#1645—134no—
178#1623—134no—
179#1548—131no—
180#1487—131no—
181#1479—131no—
182C23×C13#12011yescharacteristic
183C22×D13—41no—
184Q8×C13—24no—
185Q8×C13—22no—
186Q8×C13—22no—
187Q8×C13—24no—
188Q8×C13—44no—
189Q8×C13—82no—
190Q8×C13—88no—
191Q8×C13C2×SD1614yes—
192Q8×C13#13714yescomplemented
193C2×C4×C13—24no—
194C2×C4×C13—22no—
195C2×C4×C13—22no—
196C2×C4×C13—24no—
197C2×C4×C13—22no—
198C2×C4×C13—24no—
199C2×C4×C13—22no—
200C2×C4×C13—22no—
201C2×C4×C13—41no—
202C2×C4×C13—44no—
203C2×C4×C13—42no—
204C2×C4×C13—42no—
205C2×C4×C13#12011yescharacteristic
206C2×C4×C13#12011yescharacteristic
207C2×C4×C13#12011yescharacteristic
208C2×C4×C13C2×SD1612yes—
209C2×C4×C13#13712yes—
210C2×C4×C13C22×D411yescharacteristic
211C2×C4×C13C22×D411yescharacteristic
212C2×C4×C13C22×D411yescharacteristic
213C22⋊D13—82no—
214C2×Dic13—41no—
215D52—42no—
216D52—41no—
217D52—42no—
218D52—41no—
219C4×D13—82no—
220Dic26—41no—
221Dic26—41no—
222#1055—42no—
223#1055—42no—
224#1055—42no—
225C2×#137—261no—
226Q82—22no—
227Q82—22no—
228Q82—22no—
229Q82C2×D1314yes—
230#532—131no—
231C2×C4⋊Q8—22no—
232C2×C4⋊Q8—21no—
233C2×C4⋊Q8—21no—
234C2×C4⋊Q8C2×D1311yescharacteristic
235C2×C4×Q8—21no—
236C2×C4×Q8—21no—
237C2×C4×Q8C2×D1311yescharacteristic
238C2×C4×Q8C2×D1311yescharacteristic
239#492—132no—
240#487—132no—
241#481—261no—
242#475—132no—
243#464—262no—
244#462—132no—
245#461—262no—
246#460—132no—
247#460—132no—
248#448—261no—
249#448—262no—
250C4×SD16—262no—
251#431—131no—
252#431—131no—
253#423—131no—
254#417—261no—
255#333—134no—
256Q8⋊C8—134no—
257Q8⋊C8—134no—
258C22×C13—21no—
259C22×C13—21no—
260C22×C13#44812yes—
261C22×C13C2×#12011yescharacteristic
262C2×D13—81no—
263C2×D13—81no—
264C2×D13—81no—
265C4×C13—24no—
266C4×C13—24no—
267C4×C13—21no—
268C4×C13—21no—
269C4×C13—21no—
270C4×C13—44no—
271C4×C13—42no—
272C4×C13—42no—
273C4×C13#44812yes—
274C4×C13C22×SD1611yescharacteristic
275C4×C13C2×#13711yescharacteristic
276C4×C13C2×#13712yes—
277Dic13—81no—
278C2×#41—261no—
279C22×Q8—22no—
280C22×Q8C22⋊D1311yescharacteristic
281C22×Q8C22⋊D1311yescharacteristic
282C22×Q8C22⋊D1312yes—
283#137—524no—
284C2×Q16—262no—
285C2×SD16—262no—
286C2×SD16—262no—
287C2×SD16—264no—
288C2×M16—261no—
289C4⋊Q8—28no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—28no—
293C4⋊Q8—24no—
294C4⋊Q8—24no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—131no—
298C4⋊Q8C22⋊D1312yes—
299C4⋊Q8C22×D1312yes—
300#127—131no—
301#124—261no—
302#121—261no—
303#121—261no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8C22⋊D1312yes—
313C4×Q8C22⋊D1312yes—
314C4×Q8C22×D1312yes—
315C4×Q8C22×D1312yes—
316C4×D4—261no—
317C4×D4—261no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C22⋊D1311yescharacteristic
325C2×C4⋊C4C22⋊D1311yescharacteristic
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42C22×D1311yescharacteristic
330#106—262no—
331C4⋊C8—132no—
332C4⋊C8—132no—
333C4⋊C8—261no—
334C4⋊C8—261no—
335#103—262no—
336#103—264no—
337#103—262no—
338#102—262no—
339#102—262no—
340C22⋊C8—261no—
341C22⋊C8—261no—
342C4×C8—132no—
343C4×C8—261no—
344C2×C13—22no—
345C2×C13C2×#44811yescharacteristic
346C2×C13#300411yescharacteristic
347C2×C13#301911yescharacteristic
348D13—161no—
349#41—522no—
350#41—522no—
351C2×Q8—24no—
352C2×Q8—28no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—82no—
358C2×Q8—84no—
359C2×Q8—261no—
360C2×Q8#610614yescomplemented
361C2×Q8C2×C22⋊D1312yes—
362C2×Q8C2×C22⋊D1312yes—
363C2×Q8C2×C22⋊D1314yes—
364C2×D4—261no—
365C2×D4—261no—
366C2×D4—261no—
367C2×D4—521no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—521no—
374C22×C4C2×C22⋊D1311yescharacteristic
375C22×C4C2×C22⋊D1311yescharacteristic
376C22×C4C2×C22⋊D1311yescharacteristic
377Q16—524no—
378SD16—524no—
379SD16—524no—
380SD16—1044no—
381M16—522no—
382C2×C8—262no—
383C2×C8—262no—
384C2×C8—262no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—261no—
398C4⋊C4—261no—
399C4⋊C4C2×C22⋊D1312yes—
400C4⋊C4C2×C22⋊D1312yes—
401C22⋊C4—521no—
402C22⋊C4—521no—
403C42—28no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42C2×C22⋊D1311yescharacteristic
409C42C2×C22⋊D1311yescharacteristic
410C42C2×C22⋊D1311yescharacteristic
411C42C23×D1311yescharacteristic
412C13#1876811yescharacteristic, complemented, Sylow
413C23—521no—
414C23#8820711yescharacteristic
415Q8—24no—
416Q8—22no—
417Q8—22no—
418Q8—24no—
419Q8—44no—
420Q8—82no—
421Q8—88no—
422Q8—521no—
423Q8—521no—
424Q8C2×#610614yes—
425Q8#8819614yescomplemented
426D4—522no—
427D4—521no—
428D4—522no—
429D4—521no—
430D4—1042no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—521no—
444C2×C4—1042no—
445C2×C4C2×#610612yes—
446C2×C4#8819612yes—
447C2×C4#8820711yescharacteristic
448C2×C4#8820711yescharacteristic
449C2×C4#8820711yescharacteristic
450C2×C4C22×C22⋊D1311yescharacteristic
451C2×C4C22×C22⋊D1311yescharacteristic
452C2×C4C22×C22⋊D1311yescharacteristic
453C8—522no—
454C8—522no—
455C8—522no—
456C22—21no—
457C22—21no—
458C22—1041no—
459C22—1041no—
460C22—1041no—
461C22#1171888812yes—
462C22C2×#8820711yescharacteristic
463C4—21no—
464C4—21no—
465C4—24no—
466C4—24no—
467C4—21no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—1041no—
472C4#1171888812yes—
473C4C22×#610611yescharacteristic
474C4C2×#8819611yescharacteristic
475C4C2×#8819612yes—
476C2—22no—
477C2—2081no—
478C2C2×#1171888811yescharacteristic
479C2#1312407011yescharacteristic
480C2#1312419311yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×Q82×C13C2
socleC22×C13C2×#120
prime radical#11719182C22
hypercenterC2×Q82D13
hypocenterC13#18768
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #13114057C2semidirect
5. #13113914C2semidirect
6. #13113517C2semidirect
7. #13113265C2semidirect
8. #13113231C2semidirect
9. C2×Q82×C13C2semidirect
118. C2×Q8×C13SD16semidirect
173. C2×Q82D13semidirect
192. Q8×C13#137semidirect
360. C2×Q8#6106semidirect
412. C13#18768semidirect
425. Q8#88196semidirect

Maximal subgroups

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

subgroupindex
#131257612
#131248672
#131243082
#131140572
#131139142
#131135172
#131132652
#131132312
C2×Q82×C132
#1876813