Finite Group Databasebeta

← #15901316

Subgroups — #15901316

Order 3328.

1012 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88050C211yesmaximal, characteristic, complemented
2#13118495C211yesmaximal, characteristic
3#13117705C211yesmaximal, characteristic, complemented
4#13116226C212yesmaximal, complemented
5#13116076C212yesmaximal, complemented
6#13115767C212yesmaximal, complemented
7#13115476C212yesmaximal, complemented
8#13115265C211yesmaximal, characteristic
9#13115013C211yesmaximal, characteristic, complemented
10#13114772C211yesmaximal, characteristic, complemented
11C2×C13×#480C211yesmaximal, characteristic
12C23×Dic26C2211yescharacteristic
13C2×C13×#121C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C2×#88190C2212yes—
16C2×#88180—21no—
17C23×#1055C2211yescharacteristic
18C13×#480—24no—
19C2×C13×#107C2211yescharacteristic
20C2×C13×#107C2211yescharacteristic
21C2×C13×#102C2211yescharacteristic
22C2×C13×#102C2211yescharacteristic
23C2×C13×C22⋊C8C2211yescharacteristic
24#11718881—21no—
25#11718881—21no—
26C2×#88072C2212yes—
27#11718873C2211yescharacteristic
28#11718872—21no—
29C4×C22⋊Dic13C2211yescharacteristic
30C2×#88056C2212yes—
31C2×#88056C2212yes—
32C2×#88054C2212yes—
33#11718646C2212yes—
34#11718642—22no—
35C4⋊C4×Dic13C2212yes—
36#11718635C2212yes—
37C2×#88050—24no—
38C2×#88050C2212yes—
39C2×#88050C2212yes—
40#11718608—22no—
41#11718142C2212yes—
42#11718121C2212yes—
43C22×D4×C13—21no—
44C22×D4×C13C2311yescharacteristic
45C23×C4×C13—21no—
46C23×C4×C13C2311yescharacteristic
47C23×Dic13—21no—
48C22×Dic26—22no—
49C22×Dic26—24no—
50C22×Dic26—22no—
51C22×Dic26—21no—
52C22×Dic26—42no—
53C22×Dic26C2312yes—
54C22×C8×C13C2311yescharacteristic
55C22×C8×C13C2311yescharacteristic
56C13×#121—22no—
57C13×#121—22no—
58C2×C13×C4⋊C4—21no—
59C2×C13×C4⋊C4—22no—
60C2×C13×C4⋊C4D414yes—
61C2×C13×C4⋊C4C2311yescharacteristic
62C2×C13×C4⋊C4C2311yescharacteristic
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C22⋊C4—21no—
65C2×C22⋊Dic13—21no—
66C2×C22⋊Dic13—22no—
67C2×C22⋊Dic13—22no—
68C2×C22⋊Dic13D411yescharacteristic
69C2×C22⋊Dic13D411yescharacteristic
70#88190—24no—
71#88190—42no—
72#88180—44no—
73C2×C4⋊Dic13—22no—
74C2×C4⋊Dic13C2312yes—
75C2×C4×Dic13—22no—
76C2×C4×Dic13C2312yes—
77C22×#1055—22no—
78C22×#1055—42no—
79C22×#1055C2312yes—
80C13×#107—22no—
81C13×#107—22no—
82C13×#102—22no—
83C13×#102—22no—
84C13×C22⋊C8—22no—
85#88072—42no—
86#88071—21no—
87#88071—22no—
88#88071—22no—
89#88071—22no—
90#88071—22no—
91#88071D412yes—
92#88056—24no—
93#88056—42no—
94#88054—42no—
95#88050—24no—
96#88050—24no—
97#24481—131nomaximal, Sylow
98C24×C13—21no—
99C24×C13C2×D411yescharacteristic
100C2×D4×C13—21no—
101C2×D4×C13—22no—
102C2×D4×C13—21no—
103C2×D4×C13—22no—
104C2×D4×C13—42no—
105C2×D4×C13—42no—
106C2×D4×C13—42no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—24no—
110C22×C4×C13—21no—
111C22×C4×C13—21no—
112C22×C4×C13—21no—
113C22×C4×C13—42no—
114C22×C4×C13—42no—
115C22×C4×C13C2×D411yescharacteristic
116C22×C4×C13C2×D412yes—
117C22×C4×C13#4112yes—
118C22×C4×C13C2411yescharacteristic
119C22×Dic13—22no—
120C22×Dic13—22no—
121C22×Dic13—44no—
122C22×Dic13—42no—
123C22×Dic13C2×D412yes—
124C22×Dic13C2×D412yes—
125C2×Dic26—28no—
126C2×Dic26—22no—
127C2×Dic26—22no—
128C2×Dic26—22no—
129C2×Dic26—22no—
130C2×Dic26—22no—
131C2×Dic26—48no—
132C2×Dic26—44no—
133C2×Dic26—44no—
134C2×Dic26—42no—
135C2×Dic26#4114yes—
136C2×C8×C13—21no—
137C2×C8×C13—21no—
138C2×C8×C13—22no—
139C2×C8×C13—22no—
140C13×C4⋊C4—24no—
141C13×C4⋊C4—21no—
142C13×C4⋊C4—21no—
143C13×C4⋊C4—21no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4C2×D412yes—
146C13×C4⋊C4C2×D412yes—
147C13×C22⋊C4—42no—
148C22⋊Dic13—22no—
149C22⋊Dic13—22no—
150C22⋊Dic13—44no—
151C22⋊Dic13—44no—
152C22⋊Dic13—42no—
153C4⋊Dic13—22no—
154C4⋊Dic13—22no—
155C4⋊Dic13—44no—
156C4×Dic13—22no—
157C4×Dic13—22no—
158C4×Dic13—44no—
159C2×#1055—22no—
160C2×#1055—22no—
161C2×#1055—22no—
162C2×#1055—44no—
163C2×#480D1311yescharacteristic
164C22×#103—131no—
165#2436—131no—
166#2344—131no—
167#2068—132no—
168#2042—132no—
169#1983—132no—
170#1928—132no—
171#1893—131no—
172#1850—131no—
173#1787—131no—
174C23×C13—22no—
175C23×C13—22no—
176C23×C13—41no—
177C23×C13—42no—
178C23×C13—41no—
179C23×C13—41no—
180C23×C13—42no—
181C23×C13#12112yes—
182C23×C13#13512yes—
183C23×C13AGL(1,ℤ/8)12yes—
184C23×C13C22×D411yescharacteristic
185D4×C13—41no—
186D4×C13—41no—
187D4×C13—42no—
188D4×C13—84no—
189C2×C4×C13—22no—
190C2×C4×C13—22no—
191C2×C4×C13—22no—
192C2×C4×C13—24no—
193C2×C4×C13—22no—
194C2×C4×C13—22no—
195C2×C4×C13—41no—
196C2×C4×C13—41no—
197C2×C4×C13—41no—
198C2×C4×C13—41no—
199C2×C4×C13—48no—
200C2×C4×C13—41no—
201C2×C4×C13—41no—
202C2×C4×C13—41no—
203C2×C4×C13#12112yes—
204C2×C4×C13C22×D411yescharacteristic
205C2×C4×C13C2×#4112yes—
206C2×C4×C13C2×#4111yescharacteristic
207C2×C4×C13H(5,𝔽2)12yes—
208C2×Dic13—22no—
209C2×Dic13—22no—
210C2×Dic13—24no—
211C2×Dic13—22no—
212C2×Dic13—22no—
213C2×Dic13—22no—
214C2×Dic13—22no—
215C2×Dic13—48no—
216C2×Dic13—42no—
217C2×Dic13—42no—
218C2×Dic13—42no—
219C2×Dic13—42no—
220C2×Dic13—42no—
221Dic26—24no—
222Dic26—24no—
223Dic26—48no—
224Dic26—42no—
225Dic26—42no—
226C8×C13—41no—
227C8×C13—41no—
228#1055—42no—
229C23×Q8—131no—
230C23×C8—131no—
231C2×#124—132no—
232C2×#121C2×D1311yescharacteristic
233C2×#117—261no—
234C22×C4⋊C4C2×D1311yescharacteristic
235#480—24no—
236C2×#107C2×D1311yescharacteristic
237C2×#107C2×D1311yescharacteristic
238C2×C4⋊C8—132no—
239C2×#103—132no—
240C2×#103—132no—
241C2×#103—132no—
242C2×#103—132no—
243C2×#103—264no—
244C2×#102—132no—
245C2×#102C2×D1311yescharacteristic
246C2×#102C2×D1311yescharacteristic
247C2×C22⋊C8C2×D1311yescharacteristic
248#393—262no—
249#391—132no—
250#389—132no—
251#387—262no—
252#386—261no—
253#386—261no—
254#381—261no—
255#380—131no—
256C4×C4⋊C4—132no—
257C4×C22⋊C4—131no—
258#340—132no—
259#336—132no—
260C22×C13—22no—
261C22×C13—22no—
262C22×C13—24no—
263C22×C13—24no—
264C22×C13—81no—
265C22×C13—81no—
266C22×C13—81no—
267C22×C13—82no—
268C22×C13—82no—
269C22×C13#46314yes—
270C22×C13C2×#12111yescharacteristic
271C22×C13#54612yes—
272C22×C13C2×#13511yescharacteristic
273C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C13#57812yes—
275C4×C13—22no—
276C4×C13—42no—
277C4×C13—42no—
278C4×C13—81no—
279C4×C13#52511yescharacteristic
280C4×C13#53511yescharacteristic
281C4×C13#55112yes—
282Dic13—44no—
283Dic13—42no—
284Dic13—42no—
285Dic13—82no—
286C22×Q8—132no—
287C22×Q8—264no—
288C22×Q8—261no—
289C22×Q8—262no—
290C22×Q8—262no—
291C22×Q8—522no—
292C22×D4—21no—
293C22×D4C22×D1311yescharacteristic
294C23×C4—21no—
295C23×C4—261no—
296C23×C4C22×D1311yescharacteristic
297C22×C8—132no—
298C22×C8—262no—
299C22×C8—522no—
300C22×C8C22×D1311yescharacteristic
301C22×C8C22×D1311yescharacteristic
302#124—264no—
303#124—522no—
304#121—22no—
305#121—22no—
306#117—524no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—132no—
310C2×C4⋊C4—262no—
311C2×C4⋊C4C22⋊D1314yes—
312C2×C4⋊C4C22×D1311yescharacteristic
313C2×C4⋊C4C22×D1311yescharacteristic
314C2×C4⋊C4C22×D1311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—131no—
317C2×C22⋊C4—131no—
318C2×C22⋊C4—262no—
319C2×C22⋊C4—262no—
320C2×C22⋊C4—261no—
321C2×C42—132no—
322C2×C42—262no—
323#107—22no—
324#107—22no—
325C4⋊C8—522no—
326#103—264no—
327#103—264no—
328#103—264no—
329#103—522no—
330#102—22no—
331#102—22no—
332#102—522no—
333C22⋊C8—22no—
334#95—132no—
335#95—262no—
336#95—262no—
337#95—262no—
338#95—262no—
339#95—261no—
340C2×C13—24no—
341C2×C13—161no—
342C2×C13#241111yescharacteristic
343C2×C13C2×#46311yescharacteristic
344C2×C13#304411yescharacteristic
345C2×C13#330012yes—
346C2×C13#330812yes—
347C24—21no—
348C24D4×D1311yescharacteristic
349C2×Q8—134no—
350C2×Q8—268no—
351C2×Q8—262no—
352C2×Q8—262no—
353C2×Q8—262no—
354C2×Q8—262no—
355C2×Q8—262no—
356C2×Q8—524no—
357C2×Q8—528no—
358C2×Q8—524no—
359C2×Q8—522no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—132no—
376C22×C4—132no—
377C22×C4—262no—
378C22×C4—262no—
379C22×C4—524no—
380C22×C4—522no—
381C22×C4D4×D1311yescharacteristic
382C22×C4#612912yes—
383C22×C4C2×C22⋊D1312yes—
384C22×C4C23×D1311yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—262no—
390C2×C8—262no—
391C2×C8—262no—
392C2×C8—524no—
393C4⋊C4—24no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—262no—
399C4⋊C4—262no—
400C4⋊C4—524no—
401C4⋊C4C2×C22⋊D1312yes—
402C4⋊C4C2×C22⋊D1312yes—
403C22⋊C4—42no—
404C22⋊C4—262no—
405C22⋊C4—262no—
406C22⋊C4—524no—
407C22⋊C4—524no—
408C22⋊C4—522no—
409C42—262no—
410C42—262no—
411C42—524no—
412C13#2448111yescharacteristic, complemented, Sylow
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—41no—
419C23—42no—
420C23#8816512yes—
421C23#8816612yes—
422C23#8819312yes—
423C23C2×D4×D1311yescharacteristic
424Q8—264no—
425Q8—264no—
426Q8—528no—
427Q8—522no—
428Q8—522no—
429D4—41no—
430D4—41no—
431D4—42no—
432D4—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—48no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—262no—
448C2×C4—262no—
449C2×C4—262no—
450C2×C4—262no—
451C2×C4—262no—
452C2×C4—262no—
453C2×C4—264no—
454C2×C4—528no—
455C2×C4—522no—
456C2×C4—522no—
457C2×C4—522no—
458C2×C4—522no—
459C2×C4—522no—
460C2×C4#8819312yes—
461C2×C4C2×#612911yescharacteristic
462C2×C4#41×D1312yes—
463C2×C4#8825612yes—
464C2×C4C22×C22⋊D1311yescharacteristic
465C8—41no—
466C8—41no—
467C8—522no—
468C22—22no—
469C22—22no—
470C22—24no—
471C22—24no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—81no—
476C22—82no—
477C22#1171881414yes—
478C22#1171935912yes—
479C22C2×#8816511yescharacteristic
480C22C2×#8816611yescharacteristic
481C22#1171948412yes—
482C22C2×#8819311yescharacteristic
483C4—22no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—524no—
488C4—522no—
489C4—522no—
490C4—1042no—
491C4#1171936312yes—
492C4#1171953311yescharacteristic
493C4#1171953511yescharacteristic
494C2—24no—
495C2—161no—
496C2#1311822211yescharacteristic
497C2C2×#1171881411yescharacteristic
498C2#1312436811yescharacteristic
499C2#1312672112yes—
500C2#1312682512yes—
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#480C2
socleC23×C13C22×D4
prime radicalC2×C13×#121C22
hypercenterC2×#480D13
hypocenterC13#24481
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
1. C22×#88050C2semidirect
3. #13117705C2semidirect
4. #13116226C2semidirect
5. #13116076C2semidirect
6. #13115767C2semidirect
7. #13115476C2semidirect
9. #13115013C2semidirect
10. #13114772C2semidirect
412. C13#24481semidirect

Maximal subgroups

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

subgroupindex
C22×#880502
#131184952
#131177052
#131162262
#131160762
#131157672
#131154762
#131152652
#131150132
#131147722
C2×C13×#4802
#2448113