Finite Group Databasebeta

← #15905548

Subgroups — #15905548

Order 3328.

942 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718745C211yesmaximal, characteristic, complemented
2C2×C4×#6104C211yesmaximal, characteristic
3C4⋊C4×D52C211yesmaximal, characteristic, complemented
4#13116318C212yesmaximal, complemented
5#13116210C212yesmaximal, complemented
6#13115798C211yesmaximal, characteristic, complemented
7#13115555C211yesmaximal, characteristic, complemented
8#13115187C212yesmaximal
9#13114843C212yesmaximal, complemented
10#13114836C211yesmaximal, characteristic, complemented
11C13×#2563C211yesmaximal, characteristic, complemented
12C2×C13×C4⋊Q8C2211yescharacteristic
13C2×C4×D4×C13C2211yescharacteristic
14C22×#6104C2211yescharacteristic
15C2×C4⋊C4×D13—21no—
16C2×C4×D52C2212yes—
17C2×C4×D52C2211yescharacteristic
18C13×#393—22no—
19C13×#390C2212yes—
20C13×#382C2212yes—
21C13×#380—21no—
22C13×#380—21no—
23C4×C13×C4⋊C4C2211yescharacteristic
24C2×#88072C2211yescharacteristic
25C2×#88072C2212yes—
26#11718745—24no—
27C4×#6104—24no—
28#11718648—21no—
29C2×#88049C2211yescharacteristic
30C2×#88049C2212yes—
31C2×#88048C2212yes—
32C2×#88047C2211yescharacteristic
33#11718596C2211yescharacteristic
34#11718589C2212yes—
35C2×C4⋊#1055C2211yescharacteristic
36C2×C4⋊#1055C2212yes—
37C2×C4×#1055C2211yescharacteristic
38#11718329—22no—
39#11718121C2212yes—
40#11718121C2212yes—
41#11718121C2212yes—
42#11718116C2212yes—
43C22×Q8×C13—21no—
44C22×D4×C13C2311yescharacteristic
45C23×C4×C13—21no—
46C22×D52C2311yescharacteristic
47C22×C4×D13—21no—
48C22×C4×D13—22no—
49C13×C4⋊Q8—22no—
50C13×C4⋊Q8—22no—
51C4×D4×C13—22no—
52C4×D4×C13—22no—
53C2×C13×C4⋊C4—21no—
54C2×C13×C4⋊C4D411yescharacteristic
55C2×C13×C4⋊C4D411yescharacteristic
56C2×C13×C4⋊C4C2311yescharacteristic
57C2×C13×C4⋊C4C2312yes—
58C2×C13×C22⋊C4—21no—
59C2×C13×C22⋊C4—22no—
60C2×C13×C22⋊C4—22no—
61C2×C42×C13C2312yes—
62C2×C42×C13C2311yescharacteristic
63C2×#6104—22no—
64C2×#6104—22no—
65C2×#6104Q814yescomplemented
66C2×#6103—21no—
67C2×#6103—22no—
68C2×C4⋊Dic13—21no—
69C2×C4⋊Dic13—21no—
70C2×C4⋊Dic13C2312yes—
71C2×C4⋊Dic13C2311yescharacteristic
72C2×#6101—22no—
73C22×#1055C2311yescharacteristic
74C22×#1055C2312yes—
75C22×#1055C2311yescharacteristic
76C4⋊C4×D13—44no—
77C4×D52—22no—
78C4×D52—24no—
79C4×D52—42no—
80C4×D52D414yes—
81C13×#95—22no—
82C13×#95—22no—
83C13×#95D412yes—
84#88072—22no—
85#88072—42no—
86#88071—22no—
87#88049—22no—
88#88049—24no—
89#88048—24no—
90#88047—22no—
91C4⋊#1055—22no—
92C4⋊#1055—42no—
93C4×#1055—22no—
94#24812—131nomaximal, Sylow
95C24×C13—21no—
96C23×D13—21no—
97C2×Q8×C13—42no—
98C2×Q8×C13—81no—
99C2×Q8×C13—81no—
100C2×D4×C13—21no—
101C2×D4×C13—21no—
102C2×D4×C13—21no—
103C2×D4×C13—21no—
104C2×D4×C13—21no—
105C2×D4×C13C2×Q812yes—
106C22×C4×C13—21no—
107C22×C4×C13—22no—
108C22×C4×C13—42no—
109C22×C4×C13—42no—
110C22×C4×C13—42no—
111C22×C4×C13C2×D412yes—
112C22×C4×C13C2×D412yes—
113C22×C4×C13C2×D411yescharacteristic
114C22×C4×C13C2×D411yescharacteristic
115C22×C4×C13C2411yescharacteristic
116C22×Dic13—21no—
117C22×Dic13—22no—
118C2×D52—21no—
119C2×D52—21no—
120C2×D52—21no—
121C2×D52C2×D412yes—
122C2×D52C2×Q812yes—
123C2×D52#4112yes—
124C2×C4×D13—24no—
125C2×C4×D13—42no—
126C2×C4×D13—42no—
127C2×C4×D13—44no—
128C2×C4×D13—44no—
129C2×C4×D13—82no—
130C13×C4⋊C4—22no—
131C13×C4⋊C4—22no—
132C13×C4⋊C4—21no—
133C13×C4⋊C4—22no—
134C13×C4⋊C4—21no—
135C13×C4⋊C4—22no—
136C13×C4⋊C4—42no—
137C13×C22⋊C4—42no—
138C13×C22⋊C4—82no—
139C13×C22⋊C4—82no—
140C42×C13—22no—
141C42×C13—22no—
142C42×C13C2×D412yes—
143C42×C13#4112yes—
144#6104—28no—
145#6103—24no—
146#6103—44no—
147C4⋊Dic13—21no—
148C4⋊Dic13—22no—
149C4⋊Dic13—22no—
150C4⋊Dic13—42no—
151C4⋊Dic13—81no—
152C4⋊Dic13C2×D412yes—
153#6101—44no—
154C2×#1055—21no—
155C2×#1055—21no—
156C2×#1055—21no—
157C2×#1055—22no—
158C2×#1055—22no—
159C2×#1055—21no—
160C2×#1055—21no—
161C2×#1055—22no—
162C2×#1055C2×Q812yes—
163C2×#476—131no—
164C2×C4×D8—131no—
165#2563D1311yescharacteristic, complemented
166D4×C4⋊C4—131no—
167#2085—132no—
168#2065—132no—
169#1990—131no—
170#1939—131no—
171#1879—132no—
172#1804—132no—
173#1802—131no—
174C23×C13—22no—
175C23×C13—41no—
176C23×C13—41no—
177C23×C13—41no—
178C23×C13—41no—
179C23×C13—41no—
180C23×C13—41no—
181C23×C13C22×D411yescharacteristic
182C22×D13—22no—
183C22×D13—22no—
184C22×D13—22no—
185C22×D13—41no—
186C22×D13—41no—
187C22×D13—41no—
188C22×D13—41no—
189Q8×C13—84no—
190D4×C13—22no—
191D4×C13—22no—
192D4×C13—41no—
193D4×C13—41no—
194C2×C4×C13—21no—
195C2×C4×C13—21no—
196C2×C4×C13—22no—
197C2×C4×C13—22no—
198C2×C4×C13—22no—
199C2×C4×C13—22no—
200C2×C4×C13—22no—
201C2×C4×C13—22no—
202C2×C4×C13—41no—
203C2×C4×C13—42no—
204C2×C4×C13—41no—
205C2×C4×C13—41no—
206C2×C4×C13—42no—
207C2×C4×C13—42no—
208C2×C4×C13—84no—
209C2×C4×C13#12112yes—
210C2×C4×C13#12112yes—
211C2×C4×C13#13512yes—
212C2×C4×C13AGL(1,ℤ/8)12yes—
213C2×C4×C13C22×D411yescharacteristic
214C2×C4×C13C22×Q811yescharacteristic
215C2×C4×C13C2×#4111yescharacteristic
216C2×C4×C13C2×#4111yescharacteristic
217C2×C4×C13H(5,𝔽2)11yescharacteristic
218C2×C4×C132−1+411yescharacteristic
219C2×Dic13—22no—
220C2×Dic13—41no—
221C2×Dic13—41no—
222C2×Dic13—42no—
223C2×Dic13—42no—
224C2×Dic13—42no—
225D52—22no—
226D52—22no—
227D52—22no—
228D52C22⋊Q814yes—
229C4×D13—48no—
230C4×D13—88no—
231#1055—22no—
232#1055—41no—
233#1055—42no—
234C22×D8—131no—
235C2×C4⋊Q8C2×D1311yescharacteristic
236C2×C4×D4—131no—
237C2×C4×D4—132no—
238C2×C4×D4C2×D1311yescharacteristic
239C22×C4⋊C4—261no—
240#476—264no—
241C4×D8—264no—
242C2×#107—131no—
243C2×#106—132no—
244C2×C4⋊C8—131no—
245C2×C4⋊C8—132no—
246C2×#102—131no—
247C2×#102—131no—
248C2×#102—132no—
249C2×#102—132no—
250C2×C4×C8—131no—
251#393—22no—
252#390C2×D1312yes—
253#389—132no—
254#385—262no—
255#384—131no—
256#382C2×D1312yes—
257#380—21no—
258#380—21no—
259#380—261no—
260C4×C4⋊C4C2×D1311yescharacteristic
261#340—132no—
262#340—132no—
263#340—132no—
264#336—132no—
265C22×C13—42no—
266C22×C13—42no—
267C22×C13—42no—
268C22×C13—81no—
269C22×C13—81no—
270C22×C13—81no—
271C22×C13—81no—
272C22×C13C2×#12111yescharacteristic
273C22×C13#54611yescharacteristic
274C22×C13D4×Q811yescharacteristic
275C22×C13C2×#13511yescharacteristic
276C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C13#57611yescharacteristic
278C22×C13#57711yescharacteristic
279C2×D13—24no—
280C2×D13—42no—
281C2×D13—42no—
282C2×D13—42no—
283C2×D13—42no—
284C2×D13—42no—
285C2×D13—42no—
286C4×C13—22no—
287C4×C13—42no—
288C4×C13—42no—
289C4×C13—81no—
290C4×C13—82no—
291C4×C13#46314yes—
292C4×C13C2×C22⋊Q811yescharacteristic
293C4×C13#53711yescharacteristic
294C4×C13#55211yescharacteristic
295C4×C13#55411yescharacteristic
296Dic13—42no—
297Dic13—82no—
298C22×Q8—21no—
299C22×D4—131no—
300C22×D4C22×D1311yescharacteristic
301C23×C4—21no—
302C23×C4—261no—
303C23×C4—262no—
304C2×D8—134no—
305C2×D8—262no—
306C2×D8—262no—
307C22×C8—131no—
308C22×C8—132no—
309C22×C8—131no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—134no—
315C4×D4—264no—
316C4×D4—262no—
317C4×D4—522no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—131no—
320C2×C4⋊C4—132no—
321C2×C4⋊C4—262no—
322C2×C4⋊C4—261no—
323C2×C4⋊C4—261no—
324C2×C4⋊C4—524no—
325C2×C4⋊C4C22⋊D1311yescharacteristic
326C2×C4⋊C4C22⋊D1311yescharacteristic
327C2×C4⋊C4C22×D1311yescharacteristic
328C2×C4⋊C4C22×D1312yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—261no—
333C2×C22⋊C4—262no—
334C2×C42C22×D1312yes—
335C2×C42C22×D1311yescharacteristic
336#107—262no—
337#106—264no—
338C4⋊C8—262no—
339C4⋊C8—522no—
340#102—262no—
341#102—264no—
342#102—262no—
343#102—522no—
344C4×C8—262no—
345#95—22no—
346#95—22no—
347#95—262no—
348#95C22⋊D1312yes—
349C2×C13—82no—
350C2×C13#241711yescharacteristic
351C2×C13C2×#46311yescharacteristic
352C2×C13#304511yescharacteristic
353C2×C13#330311yescharacteristic
354C2×C13#332511yescharacteristic
355C2×C13#338011yescharacteristic
356C2×C13#338411yescharacteristic
357D13—44no—
358C24—21no—
359C24—261no—
360C2×Q8—42no—
361C2×Q8—81no—
362C2×Q8—81no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—132no—
369C2×D4—132no—
370C2×D4—132no—
371C2×D4—261no—
372C2×D4—261no—
373C2×D4—261no—
374C2×D4Q8×D1312yes—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—261no—
381C22×C4—264no—
382C22×C4—262no—
383C22×C4—524no—
384C22×C4—524no—
385C22×C4—522no—
386C22×C4—522no—
387C22×C4—1042no—
388C22×C4C2×C22⋊D1312yes—
389C22×C4C2×C22⋊D1312yes—
390C22×C4C2×C22⋊D1311yescharacteristic
391C22×C4C2×C22⋊D1311yescharacteristic
392C22×C4C23×D1311yescharacteristic
393D8—268no—
394C2×C8—132no—
395C2×C8—261no—
396C2×C8—261no—
397C2×C8—261no—
398C2×C8—262no—
399C2×C8—262no—
400C2×C8—261no—
401C2×C8—262no—
402C2×C8—261no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—132no—
411C4⋊C4—261no—
412C4⋊C4—262no—
413C4⋊C4—262no—
414C4⋊C4—524no—
415C4⋊C4—522no—
416C4⋊C4—1041no—
417C22⋊C4—42no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—264no—
421C22⋊C4—524no—
422C42—22no—
423C42—22no—
424C42D4×D1312yes—
425C42#612912yes—
426C13#2481211yescharacteristic, complemented, Sylow
427C23—22no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—262no—
435C23—262no—
436C23—262no—
437C23—521no—
438C23—521no—
439C23—521no—
440C23—521no—
441C23C22×C22⋊D1311yescharacteristic
442Q8—84no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—134no—
448D4—262no—
449D4—262no—
450D4—262no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—262no—
467C2×C4—521no—
468C2×C4—521no—
469C2×C4—528no—
470C2×C4—522no—
471C2×C4—522no—
472C2×C4—522no—
473C2×C4—1048no—
474C2×C4#8819212yes—
475C2×C4#8819212yes—
476C2×C4#8820312yes—
477C2×C4#8820412yes—
478C2×C4C2×D4×D1311yescharacteristic
479C2×C4C2×#612911yescharacteristic
480C2×C4C2×Q8×D1311yescharacteristic
481C2×C4#8825411yescharacteristic
482C2×C4#41×D1311yescharacteristic
483C2×C4#8825611yescharacteristic
484C8—262no—
485C8—521no—
486C8—522no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—264no—
495C22—522no—
496C22—522no—
497C22—522no—
498C22—522no—
499C22—522no—
500C22—522no—
501C22C2×#8819211yescharacteristic
502C22Q8×C22⋊D1311yescharacteristic
503C22C2×#8820311yescharacteristic
504C22C2×#8820411yescharacteristic
505C22#1171953411yescharacteristic
506C22#1171954011yescharacteristic
507C22#1171954111yescharacteristic
508C4—22no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—81no—
513C4—522no—
514C4—1042no—
515C4#1171871814yes—
516C4D13×C22⋊Q811yescharacteristic
517C4#1171933411yescharacteristic
518C4#1171934011yescharacteristic
519C4#1171934411yescharacteristic
520C2—82no—
521C2—524no—
522C2#1311828311yescharacteristic
523C2C2×#1171871811yescharacteristic
524C2#1312437111yescharacteristic
525C2#1312675811yescharacteristic
526C2#1312707211yescharacteristic
527C2#1312769911yescharacteristic
528C2#1312773511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2563C2
socleC23×C13C22×D4
prime radicalC22×#6104C22
hypercenter#2563D13
hypocenterC13#24812
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#11718745C2semidirect
3. C4⋊C4×D52C2semidirect
4. #13116318C2semidirect
5. #13116210C2semidirect
6. #13115798C2semidirect
7. #13115555C2semidirect
9. #13114843C2semidirect
10. #13114836C2semidirect
11. C13×#2563C2semidirect
65. C2×#6104Q8semidirect
165. #2563D13semidirect
426. C13#24812semidirect

Maximal subgroups

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

subgroupindex
C2×#117187452
C2×C4×#61042
C4⋊C4×D522
#131163182
#131162102
#131157982
#131155552
#131151872
#131148432
#131148362
C13×#25632
#2481213