Finite Group Databasebeta

← #15845819

Subgroups — #15845819

Order 3328.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125211C211yesmaximal, characteristic
2#13124875C211yesmaximal, characteristic, complemented
3C2×#11718688C211yesmaximal, characteristic
4#13114404C212yesmaximal, complemented
5#13114376C212yesmaximal, complemented
6#13114280C212yesmaximal, complemented
7#13114252C212yesmaximal, complemented
8#13113588C211yesmaximal, characteristic, complemented
9#13113527C211yesmaximal, characteristic, complemented
10#13113505C211yesmaximal, characteristic
11Q8×C13×M16C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C13C2211yescharacteristic
13C22×#6106—21no—
14C2×#88128C2211yescharacteristic
15#11719175C2211yescharacteristic
16C13×#446—22no—
17C13×#446C2212yescomplemented
18C8×Q8×C13C2212yescomplemented
19C13×#423—21no—
20C13×#423C2211yescharacteristic
21C4×C13×M16—21no—
22C4×C13×M16C2211yescharacteristic
23#11718889—21no—
24#11718792—21no—
25#11718792—21no—
26#11718692C2211yescharacteristic, complemented
27#11718692C2211yescharacteristic, complemented
28C4×#6107—21no—
29C4×#6107—21no—
30#11718688—21no—
31#11718688—21no—
32#11718688C2212yes—
33#11718688C2212yes—
34#11718682C2211yescharacteristic, complemented
35#11718682C2211yescharacteristic, complemented
36#11718631—21no—
37C2×#88049—21no—
38#11718586C2211yescharacteristic
39C2×C4⋊#1055C2211yescharacteristic
40#11718374—22no—
41#11718355C2212yes—
42#11718355C2212yes—
43#11718078C2212yescomplemented
44#11718077C2212yes—
45#11718077C2212yes—
46#11718073C2212yescomplemented
47#11718050C2212yescomplemented
48#11718049C2212yescomplemented
49C22×Q8×C13D411yescharacteristic
50C22×D52—21no—
51C22×D52—21no—
52C22×D52—21no—
53C2×C13×M16—21no—
54C2×C13×M16D411yescharacteristic
55C2×C13×M16D411yescharacteristic
56C4×Q8×C13—21no—
57C4×Q8×C13—21no—
58C4×Q8×C13D411yescharacteristic
59C4×Q8×C13D411yescharacteristic
60C4×Q8×C13C2311yescharacteristic
61C4×Q8×C13C2311yescharacteristic
62C2×C13×C4⋊C4—21no—
63C2×C13×C4⋊C4D411yescharacteristic
64C2×C42×C13—21no—
65C2×C42×C13C2311yescharacteristic
66C2×#6107—21no—
67C2×#6107—21no—
68C2×#6106—22no—
69C2×#6106—22no—
70C2×#6106—44no—
71#88180—21no—
72#88178—21no—
73#88178—21no—
74C22×#1055—21no—
75#88128—21no—
76#88128—21no—
77#88128—21no—
78#88128C2312yes—
79#88124—21no—
80#88123C2311yescharacteristic
81#88123C2311yescharacteristic
82C4×Dic26—21no—
83C4×Dic26—21no—
84C13×C4⋊C8—22no—
85C13×C4⋊C8—22no—
86C13×C4⋊C8—21no—
87C13×C4⋊C8C2312yes—
88C13×#97—22no—
89C13×#97D412yes—
90C4×C8×C13—22no—
91C4×C8×C13C2312yes—
92#88075—21no—
93#88075—21no—
94C22⋊#1055—21no—
95#88061—22no—
96#88061—22no—
97#88061—22no—
98#88061—22no—
99#88050—21no—
100#88050—21no—
101#88049—22no—
102#88049—22no—
103#88047—21no—
104#88047—21no—
105C4⋊#1055—21no—
106C4⋊#1055C2311yescharacteristic
107C4⋊#1055C2311yescharacteristic
108C4⋊#1055C2312yes—
109C4×#1055—21no—
110#19174—131nomaximal, Sylow
111C23×D13—41no—
112C2×Q8×C13—21no—
113C2×Q8×C13—22no—
114C2×Q8×C13—41no—
115C2×Q8×C13C2×D411yescharacteristic
116C2×Q8×C13C2×D411yescharacteristic
117C22×C4×C13—21no—
118C22×C4×C13—21no—
119C22×C4×C13C2×D411yescharacteristic
120C22×C4×C13C2×D411yescharacteristic
121C22×C4×C13#4111yescharacteristic
122C2×D52—22no—
123C2×D52—22no—
124C2×D52—22no—
125C2×D52—42no—
126C2×D52—42no—
127C2×D52—42no—
128C2×D52—42no—
129C2×D52—41no—
130C2×D52—41no—
131C2×D52—41no—
132C2×D52—82no—
133C2×Dic26—21no—
134C2×Dic26—21no—
135C13×M16—22no—
136C13×M16—22no—
137C13×M16—42no—
138C2×C8×C13—22no—
139C2×C8×C13C2×D412yes—
140C2×C8×C13C2×D412yes—
141C13×C4⋊C4—21no—
142C13×C4⋊C4—21no—
143C13×C4⋊C4—21no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4—21no—
146C13×C4⋊C4C2×D411yescharacteristic
147C13×C4⋊C4C2×D411yescharacteristic
148C42×C13—21no—
149C42×C13—21no—
150C42×C13—21no—
151C42×C13—21no—
152C42×C13C2×D411yescharacteristic
153C42×C13C2×D411yescharacteristic
154C42×C13#4111yescharacteristic
155C42×C13C2411yescharacteristic
156C22⋊Dic13—41no—
157#6107—42no—
158#6107—42no—
159#6106—44no—
160#6106—44no—
161C4⋊Dic13—21no—
162C4⋊Dic13—21no—
163C4⋊Dic13—21no—
164C4⋊Dic13—21no—
165#6101—41no—
166#6101—41no—
167C4×Dic13—41no—
168C2×#1055—21no—
169C2×#1055—21no—
170C2×#1055—22no—
171C2×#1055—42no—
172#3138—131no—
173#3104—131no—
174C2×#460—131no—
175Q8×M16D1311yescharacteristic, complemented
176#1702—132no—
177#1698—132no—
178#1682—132no—
179#1678—132no—
180#1565—131no—
181#1550—131no—
182#1545—131no—
183C23×C13#12111yescharacteristic
184C22×D13—42no—
185C22×D13—82no—
186C22×D13—82no—
187C22×D13—82no—
188Q8×C13—21no—
189Q8×C13—21no—
190Q8×C13—21no—
191Q8×C13—42no—
192Q8×C13AGL(1,ℤ/8)12yes—
193C2×C4×C13—21no—
194C2×C4×C13—21no—
195C2×C4×C13—21no—
196C2×C4×C13—21no—
197C2×C4×C13—21no—
198C2×C4×C13—21no—
199C2×C4×C13—41no—
200C2×C4×C13—42no—
201C2×C4×C13—81no—
202C2×C4×C13#12111yescharacteristic
203C2×C4×C13#12111yescharacteristic
204C2×C4×C13#12111yescharacteristic
205C2×C4×C13#13512yes—
206C2×C4×C13#13712yes—
207C2×C4×C13C22×D411yescharacteristic
208C2×C4×C13C22×D411yescharacteristic
209C2×C4×C13C2×#4111yescharacteristic
210C2×Dic13—41no—
211C2×Dic13—41no—
212D52—42no—
213D52—42no—
214D52—42no—
215D52—42no—
216D52—82no—
217D52—82no—
218D52—82no—
219D52—82no—
220Dic26—41no—
221Dic26—41no—
222Dic26—81no—
223C8×C13—22no—
224C8×C13—22no—
225C8×C13—42no—
226#1055—42no—
227#1055—41no—
228#1055—41no—
229C22×SD16—261no—
230#533—131no—
231C2×#127—131no—
232C2×C4×Q8C2×D1311yescharacteristic
233#490—262no—
234#486—132no—
235#486—132no—
236#484—261no—
237#477—131no—
238#477—131no—
239#462—131no—
240#462—131no—
241#460—132no—
242#460—132no—
243#460—261no—
244#460—261no—
245#451—261no—
246#446—22no—
247#446C2×D1312yescomplemented
248C8×Q8C2×D1312yescomplemented
249C4×Q16—261no—
250C4×Q16—261no—
251#431—131no—
252#423—21no—
253#423C2×D1311yescharacteristic
254C2×C4⋊C8—131no—
255#418—261no—
256#418—261no—
257C2×#102—261no—
258C4×M16—21no—
259C4×M16C2×D1311yescharacteristic
260#335—132no—
261#334—132no—
262#333—132no—
263#332—132no—
264#331—132no—
265#331—132no—
266C22×C13—21no—
267C22×C13—21no—
268C22×C13#46412yes—
269C22×C13C2×#12111yescharacteristic
270C2×D13—82no—
271C2×D13—82no—
272C2×D13—82no—
273C2×D13—164no—
274C4×C13—21no—
275C4×C13—22no—
276C4×C13—21no—
277C4×C13—21no—
278C4×C13—41no—
279C4×C13—41no—
280C4×C13—41no—
281C4×C13#46412yes—
282C4×C13C2×#13511yescharacteristic
283C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C13C2×#13711yescharacteristic
285C4×C13#57511yescharacteristic
286Dic13—81no—
287Dic13—81no—
288C22×Q8D5211yescharacteristic
289C22×D4—261no—
290C22×D4—261no—
291C22×D4—261no—
292C2×Q16—261no—
293C2×Q16—261no—
294C2×SD16—262no—
295C2×SD16—262no—
296C2×SD16—524no—
297C2×M16—21no—
298C2×M16C22⋊D1311yescharacteristic
299C2×M16C22⋊D1311yescharacteristic
300C22×C8—261no—
301C4⋊Q8—131no—
302C4⋊Q8—131no—
303#127—132no—
304#127—261no—
305#127—261no—
306#127—261no—
307#125—261no—
308C22⋊Q8—261no—
309C22⋊Q8—261no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—261no—
313C4×Q8—261no—
314C4×Q8D5211yescharacteristic
315C4×Q8D5211yescharacteristic
316C4×Q8C22×D1311yescharacteristic
317C4×Q8C22×D1311yescharacteristic
318#117—261no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4D5211yescharacteristic
321C2×C42—21no—
322C2×C42C22×D1311yescharacteristic
323#107—261no—
324#107—261no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—21no—
328C4⋊C8—131no—
329C4⋊C8—131no—
330C4⋊C8—132no—
331C4⋊C8—261no—
332C4⋊C8C22×D1312yes—
333#103—261no—
334#103—261no—
335#103—261no—
336#103—261no—
337#102—262no—
338#102—262no—
339#102—262no—
340#102—262no—
341#102—262no—
342#102—262no—
343C22⋊C8—261no—
344#97—22no—
345#97C22⋊D1312yes—
346C4×C8—22no—
347C4×C8—261no—
348C4×C8C22×D1312yes—
349C2×C13—22no—
350C2×C13C2×#46411yescharacteristic
351C2×C13#305511yescharacteristic
352C2×C13#306811yescharacteristic
353D13—162no—
354C24—521no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—41no—
358C2×Q8—261no—
359C2×Q8—261no—
360C2×Q8C2×D5211yescharacteristic
361C2×Q8C2×D5211yescharacteristic
362C2×D4—262no—
363C2×D4—262no—
364C2×D4—262no—
365C2×D4—522no—
366C2×D4—522no—
367C2×D4—521no—
368C2×D4—521no—
369C2×D4—521no—
370C2×D4—522no—
371C2×D4—522no—
372C2×D4—1042no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C2×D5211yescharacteristic
376C22×C4#612711yescharacteristic
377C22×C4C2×C22⋊D1311yescharacteristic
378Q16—522no—
379Q16—522no—
380SD16—524no—
381SD16—524no—
382M16—22no—
383M16—22no—
384M16—42no—
385C2×C8—22no—
386C2×C8—261no—
387C2×C8—261no—
388C2×C8—262no—
389C2×C8—522no—
390C2×C8C2×C22⋊D1312yes—
391C2×C8C2×C22⋊D1312yes—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—261no—
398C4⋊C4—261no—
399C4⋊C4—261no—
400C4⋊C4—261no—
401C4⋊C4—521no—
402C4⋊C4—521no—
403C4⋊C4C2×D5211yescharacteristic
404C4⋊C4C2×D5211yescharacteristic
405C22⋊C4—521no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—521no—
411C42C2×D5211yescharacteristic
412C42#612711yescharacteristic
413C42C2×C22⋊D1311yescharacteristic
414C42C23×D1311yescharacteristic
415C13#1917411yescharacteristic, complemented, Sylow
416C23—522no—
417C23—1042no—
418C23—1042no—
419C23—1042no—
420C23#8818411yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—521no—
426Q8—521no—
427Q8—1041no—
428Q8#8816212yes—
429D4—522no—
430D4—522no—
431D4—522no—
432D4—522no—
433D4—1042no—
434D4—1042no—
435D4—1042no—
436D4—1042no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—521no—
447C2×C4—521no—
448C2×C4#8818411yescharacteristic
449C2×C4#8818411yescharacteristic
450C2×C4#8818411yescharacteristic
451C2×C4#8819612yes—
452C2×C4#8820412yes—
453C2×C4C22×D5211yescharacteristic
454C2×C4C2×#612711yescharacteristic
455C2×C4C22×C22⋊D1311yescharacteristic
456C8—22no—
457C8—22no—
458C8—42no—
459C8—522no—
460C8—521no—
461C8—521no—
462C22—21no—
463C22—21no—
464C22—1042no—
465C22—1042no—
466C22—1042no—
467C22—2084no—
468C22#1171868912yes—
469C22C2×#8818411yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—1041no—
478C4—1041no—
479C4#1171868912yes—
480C4C2×#8816211yescharacteristic
481C4#1171945311yescharacteristic
482C4C2×#8819611yescharacteristic
483C4C2×#8820411yescharacteristic
484C2—22no—
485C2—2082no—
486C2C2×#1171868911yescharacteristic
487C2#1312445511yescharacteristic
488C2#1312456411yescharacteristic
subgroupG/H
centerC22C2×#88184
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingQ8×C13×M16C2
socleC22×C13C2×#121
prime radicalC2×#88128C22
hypercenterQ8×M16D13
hypocenterC13#19174
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #13124875C2semidirect
4. #13114404C2semidirect
5. #13114376C2semidirect
6. #13114280C2semidirect
7. #13114252C2semidirect
8. #13113588C2semidirect
9. #13113527C2semidirect
11. Q8×C13×M16C2semidirect
17. C13×#446C22semidirect
18. C8×Q8×C13C22semidirect
26. #11718692C22semidirect
27. #11718692C22semidirect
34. #11718682C22semidirect
35. #11718682C22semidirect
43. #11718078C22semidirect
46. #11718073C22semidirect
47. #11718050C22semidirect
48. #11718049C22semidirect
175. Q8×M16D13semidirect
247. #446C2×D13semidirect
248. C8×Q8C2×D13semidirect
415. C13#19174semidirect

Maximal subgroups

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

subgroupindex
#131252112
#131248752
C2×#117186882
#131144042
#131143762
#131142802
#131142522
#131135882
#131135272
#131135052
Q8×C13×M162
#1917413