Finite Group Databasebeta

← #15755003

Subgroups — #15755003

Order 3328.

1048 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13115798C211yesmaximal, characteristic, complemented
2#13114836C211yesmaximal, characteristic, complemented
3#13114836C211yesmaximal, characteristic, complemented
4C2×#11718139C211yesmaximal, characteristic, complemented
5C2×D4⋊#1055C211yesmaximal, characteristic, complemented
6C2×D4⋊#1055C211yesmaximal, characteristic, complemented
7C2×C13×#547C211yesmaximal, characteristic
8C22×C13×#41—21no—
9C13×#547—22no—
10C13×#547—22no—
11C13×#547—22no—
12C13×#547C414yescomplemented
13C2×C13×C4⋊Q8C2211yescharacteristic
14C2×C13×#123—22no—
15C2×C13×C22⋊Q8—21no—
16C2×C13×#121—21no—
17C2×C4×D4×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C22×C13×C4⋊C4—21no—
20C2×#88072—21no—
21C2×#88072—21no—
22#11718596C2211yescharacteristic
23C2×C4⋊#1055C2211yescharacteristic
24C2×C4⋊#1055C2211yescharacteristic
25C2×C4×#1055C2211yescharacteristic
26#11718139—22no—
27D4⋊#1055—22no—
28D4⋊#1055—22no—
29#11718121—22no—
30#11718121—21no—
31#11718121—21no—
32C2×C13×#41—24no—
33C2×C13×#41—24no—
34C2×C13×#41—42no—
35C2×C13×#41—42no—
36C2×C13×#41—42no—
37C2×C13×#41—42no—
38C22×Q8×C13—21no—
39C22×D4×C13—22no—
40C22×D4×C13D411yescharacteristic
41C22×D4×C13D411yescharacteristic
42C23×C4×C13—22no—
43C23×C4×C13—21no—
44C23×C4×C13—21no—
45C13×C4⋊Q8—21no—
46C13×C4⋊Q8—21no—
47C13×C4⋊Q8C2×C412yes—
48C13×C4⋊Q8D412yes—
49C13×#123—28no—
50C13×#123—28no—
51C13×C22⋊Q8—24no—
52C13×C22⋊Q8—24no—
53C13×#121—24no—
54C13×#121—24no—
55C4×D4×C13—21no—
56C4×D4×C13—21no—
57C4×D4×C13—21no—
58C4×D4×C13—21no—
59C4×D4×C13C2×C412yes—
60C4×D4×C13C2×C412yes—
61C4×D4×C13D412yes—
62C4×D4×C13D412yes—
63C2×C13×C4⋊C4—22no—
64C2×C13×C4⋊C4—24no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4—24no—
67C2×C13×C4⋊C4D411yescharacteristic
68C2×C13×C4⋊C4D411yescharacteristic
69C2×C13×C4⋊C4D412yes—
70C2×C13×C22⋊C4—22no—
71C2×C13×C22⋊C4—21no—
72C2×C13×C22⋊C4—21no—
73C2×C42×C13C2311yescharacteristic
74C2×C4⋊Dic13—21no—
75C2×C4⋊Dic13—21no—
76C2×C4⋊Dic13—21no—
77C22×#1055—21no—
78C22×#1055—21no—
79C22×#1055—21no—
80#88072—42no—
81#88072—42no—
82C4⋊#1055—21no—
83C4⋊#1055—21no—
84C4⋊#1055—21no—
85C4⋊#1055—21no—
86C4×#1055—21no—
87C4×#1055—21no—
88#8446—131nomaximal, Sylow
89C24×C13—21no—
90C24×C13—21no—
91C13×#41—48no—
92C13×#41—48no—
93C13×#41—48no—
94C13×#41—48no—
95C2×Q8×C13—22no—
96C2×Q8×C13—22no—
97C2×Q8×C13—41no—
98C2×Q8×C13—41no—
99C2×Q8×C13—41no—
100C2×Q8×C13—41no—
101C2×D4×C13—28no—
102C2×D4×C13—21no—
103C2×D4×C13—21no—
104C2×D4×C13—21no—
105C2×D4×C13—21no—
106C2×D4×C13—21no—
107C2×D4×C13—21no—
108C2×D4×C13—21no—
109C2×D4×C13—21no—
110C2×D4×C13—44no—
111C2×D4×C13—44no—
112C2×D4×C13C22⋊C412yes—
113C2×D4×C13C22⋊C412yes—
114C2×D4×C13D812yes—
115C2×D4×C13SD1612yes—
116C22×C4×C13—28no—
117C22×C4×C13—21no—
118C22×C4×C13—21no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—44no—
125C22×C4×C13—44no—
126C22×C4×C13—41no—
127C22×C4×C13—41no—
128C22×C4×C13—41no—
129C22×C4×C13—41no—
130C22×C4×C13—41no—
131C22×C4×C13—41no—
132C22×C4×C13—41no—
133C22×C4×C13—41no—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×Dic13—41no—
138C13×C4⋊C4—28no—
139C13×C4⋊C4—22no—
140C13×C4⋊C4—21no—
141C13×C4⋊C4—22no—
142C13×C4⋊C4—21no—
143C13×C4⋊C4—22no—
144C13×C4⋊C4C22⋊C412yes—
145C13×C4⋊C4C22⋊C412yes—
146C13×C4⋊C4C22⋊C414yes—
147C13×C22⋊C4—28no—
148C13×C22⋊C4—22no—
149C13×C22⋊C4—22no—
150C13×C22⋊C4—22no—
151C13×C22⋊C4—22no—
152C42×C13C22×C411yescharacteristic
153C42×C13C2×D411yescharacteristic
154C42×C13C2×D411yescharacteristic
155C42×C13C2×D411yescharacteristic
156C4⋊Dic13—42no—
157C4⋊Dic13—41no—
158C4⋊Dic13—41no—
159C4⋊Dic13—41no—
160C4⋊Dic13—41no—
161C2×#1055—22no—
162C2×#1055—41no—
163C2×#1055—41no—
164C2×#1055—41no—
165C2×#1055—41no—
166C2×#1055—41no—
167C2×#1055—41no—
168C2×#1055—41no—
169C2×#1055—41no—
170C2×#547D1311yescharacteristic
171#1990—131no—
172#1802—131no—
173#1802—131no—
174C2×#332—131no—
175C2×D4⋊C8—131no—
176C2×D4⋊C8—131no—
177C23×C13—22no—
178C23×C13—22no—
179C23×C13—22no—
180C23×C13—22no—
181C23×C13—41no—
182C23×C13—41no—
183C23×C13—41no—
184C23×C13—41no—
185C23×C13—41no—
186C23×C13—41no—
187C23×C13—41no—
188C23×C13—41no—
189C23×C13—41no—
190C23×C13—41no—
191C23×C13#12011yescharacteristic
192Q8×C13—42no—
193Q8×C13—42no—
194Q8×C13—42no—
195Q8×C13—42no—
196D4×C13—22no—
197D4×C13—22no—
198D4×C13—22no—
199D4×C13—22no—
200D4×C13—22no—
201D4×C13—416no—
202D4×C13—41no—
203D4×C13—41no—
204D4×C13#10214yescomplemented
205C2×C4×C13—21no—
206C2×C4×C13—21no—
207C2×C4×C13—22no—
208C2×C4×C13—22no—
209C2×C4×C13—24no—
210C2×C4×C13—416no—
211C2×C4×C13—41no—
212C2×C4×C13—42no—
213C2×C4×C13—41no—
214C2×C4×C13—41no—
215C2×C4×C13—42no—
216C2×C4×C13—41no—
217C2×C4×C13—42no—
218C2×C4×C13—42no—
219C2×C4×C13—42no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13C2×C22⋊C411yescharacteristic
226C2×C4×C13C2×C22⋊C411yescharacteristic
227C2×C4×C13C2×C22⋊C411yescharacteristic
228C2×C4×C13#12011yescharacteristic
229C2×C4×C13#12011yescharacteristic
230C2×C4×C13#12011yescharacteristic
231C2×C4×C13C2×D811yescharacteristic
232C2×C4×C13C2×SD1611yescharacteristic
233C2×C4×C13#13511yescharacteristic
234C2×C4×C13#13511yescharacteristic
235C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
236C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
237C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
238C2×C4×C13#13711yescharacteristic
239C2×Dic13—81no—
240C2×Dic13—81no—
241C2×Dic13—81no—
242#1055—42no—
243#1055—81no—
244#1055—81no—
245C22×#41—21no—
246#547—22no—
247#547—22no—
248#547—22no—
249#547Dic1314yescomplemented
250C2×C4⋊Q8C2×D1311yescharacteristic
251C2×#123—22no—
252C2×C22⋊Q8—21no—
253C2×#121—21no—
254C2×C4×D4C2×D1311yescharacteristic
255C2×C4×D4C2×D1311yescharacteristic
256C22×C4⋊C4—21no—
257C2×C4⋊C8—131no—
258C2×C4⋊C8—131no—
259C2×#102—261no—
260C2×#102—261no—
261C2×C4×C8—131no—
262#384—131no—
263#340—261no—
264#340—261no—
265#340—262no—
266#332—262no—
267D4⋊C8—262no—
268D4⋊C8—262no—
269C22×C13—24no—
270C22×C13—42no—
271C22×C13—42no—
272C22×C13—42no—
273C22×C13—42no—
274C22×C13—42no—
275C22×C13—42no—
276C22×C13—42no—
277C22×C13—42no—
278C22×C13—42no—
279C22×C13—81no—
280C22×C13—81no—
281C22×C13—81no—
282C22×C13—81no—
283C22×C13#37911yescharacteristic
284C22×C13#44711yescharacteristic
285C22×C13#44911yescharacteristic
286C22×C13#45011yescharacteristic
287C22×C13#45211yescharacteristic
288C22×C13#45411yescharacteristic
289C22×C13#45611yescharacteristic
290C4×C13—21no—
291C4×C13—21no—
292C4×C13—42no—
293C4×C13—44no—
294C4×C13—42no—
295C4×C13C2×#10211yescharacteristic
296C4×C13#41611yescharacteristic
297C4×C13#41711yescharacteristic
298C4×C13#41811yescharacteristic
299C4×C13#44711yescharacteristic
300C4×C13#44911yescharacteristic
301C4×C13#45011yescharacteristic
302C4×C13#45211yescharacteristic
303Dic13—161no—
304C2×#41—24no—
305C2×#41—24no—
306C2×#41—42no—
307C2×#41—42no—
308C2×#41—42no—
309C2×#41—42no—
310C22×Q8—21no—
311C22×D4—22no—
312C22×D4C22⋊D1311yescharacteristic
313C22×D4C22⋊D1311yescharacteristic
314C23×C4—22no—
315C23×C4—21no—
316C23×C4—21no—
317C22×C8—261no—
318C22×C8—261no—
319C22×C8—261no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8C2×Dic1312yes—
323C4⋊Q8C22⋊D1312yes—
324#123—28no—
325#123—28no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328#121—24no—
329#121—24no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4C2×Dic1312yes—
335C4×D4C2×Dic1312yes—
336C4×D4C22⋊D1312yes—
337C4×D4C22⋊D1312yes—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—261no—
343C2×C4⋊C4—261no—
344C2×C4⋊C4—261no—
345C2×C4⋊C4C22⋊D1311yescharacteristic
346C2×C4⋊C4C22⋊D1311yescharacteristic
347C2×C4⋊C4C22⋊D1312yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C42C22×D1311yescharacteristic
352C4⋊C8—261no—
353C4⋊C8—261no—
354C4⋊C8—261no—
355C4⋊C8—261no—
356#102—522no—
357#102—522no—
358C4×C8—261no—
359C4×C8—261no—
360C2×C13—44no—
361C2×C13—82no—
362C2×C13#162511yescharacteristic
363C2×C13#163511yescharacteristic
364C2×C13#164711yescharacteristic
365C2×C13#165411yescharacteristic
366C2×C13#178611yescharacteristic
367C2×C13#178811yescharacteristic
368C2×C13#179311yescharacteristic
369C24—21no—
370C24—21no—
371#41—48no—
372#41—48no—
373#41—48no—
374#41—48no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×D4—28no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4#610412yes—
393C2×D4#610512yes—
394C2×D4C22⋊Dic1312yes—
395C2×D4C22⋊Dic1312yes—
396C22×C4—28no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—521no—
415C22×C4C2×C22⋊D1311yescharacteristic
416C22×C4C2×C22⋊D1311yescharacteristic
417C22×C4C2×C22⋊D1311yescharacteristic
418C2×C8—262no—
419C2×C8—521no—
420C2×C8—521no—
421C2×C8—521no—
422C2×C8—521no—
423C2×C8—521no—
424C2×C8—521no—
425C2×C8—521no—
426C2×C8—521no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—522no—
434C4⋊C4—521no—
435C4⋊C4—521no—
436C4⋊C4—521no—
437C4⋊C4—521no—
438C4⋊C4C22⋊Dic1312yes—
439C4⋊C4C22⋊Dic1312yes—
440C4⋊C4C22⋊Dic1314yes—
441C22⋊C4—28no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42C22×Dic1311yescharacteristic
447C42C2×C22⋊D1311yescharacteristic
448C42C2×C22⋊D1311yescharacteristic
449C42C2×C22⋊D1311yescharacteristic
450C13#844611yescharacteristic, complemented, Sylow
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23#8820711yescharacteristic
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478D4#8807214yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—416no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1041no—
500C2×C4—1041no—
501C2×C4—1041no—
502C2×C4C2×#610411yescharacteristic
503C2×C4#8818611yescharacteristic
504C2×C4#8818611yescharacteristic
505C2×C4C2×#610511yescharacteristic
506C2×C4#8820311yescharacteristic
507C2×C4#8820411yescharacteristic
508C2×C4#8820411yescharacteristic
509C2×C4#8820511yescharacteristic
510C2×C4C2×C22⋊Dic1311yescharacteristic
511C2×C4C2×C22⋊Dic1311yescharacteristic
512C2×C4C2×C22⋊Dic1311yescharacteristic
513C2×C4#8820711yescharacteristic
514C2×C4#8820711yescharacteristic
515C2×C4#8820711yescharacteristic
516C8—522no—
517C8—1041no—
518C8—1041no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#1171887811yescharacteristic
534C22#1171887911yescharacteristic
535C22#1171889911yescharacteristic
536C22#1171890011yescharacteristic
537C22#1171890611yescharacteristic
538C22#1171890711yescharacteristic
539C22#1171890811yescharacteristic
540C4—21no—
541C4—21no—
542C4—42no—
543C4—44no—
544C4—42no—
545C4—2081no—
546C4C2×#8807211yescharacteristic
547C4#1171887611yescharacteristic
548C4#1171887811yescharacteristic
549C4#1171887911yescharacteristic
550C4#1171889311yescharacteristic
551C4#1171889411yescharacteristic
552C4#1171889911yescharacteristic
553C4#1171890011yescharacteristic
554C2—44no—
555C2—82no—
556C2#1311392511yescharacteristic
557C2#1311398711yescharacteristic
558C2#1311407111yescharacteristic
559C2#1311412011yescharacteristic
560C2#1311476711yescharacteristic
561C2#1311477611yescharacteristic
562C2#1311479511yescharacteristic
subgroupG/H
centerC23#88207
commutatorC42×C13C22×C4
FrattiniC2×C42C22×D13
Fitting, prime radicalC2×C13×#547C2
socleC23×C13#120
cosocleC2×C42×C13C23
hypercenterC2×#547D13
hypocenterC13#8446
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #13115798C2semidirect
2. #13114836C2semidirect
3. #13114836C2semidirect
4. C2×#11718139C2semidirect
5. C2×D4⋊#1055C2semidirect
6. C2×D4⋊#1055C2semidirect
12. C13×#547C4semidirect
204. D4×C13#102semidirect
249. #547Dic13semidirect
450. C13#8446semidirect
478. D4#88072semidirect

Maximal subgroups

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

subgroupindex
#131157982
#131148362
C2×#117181392
C2×D4⋊#10552
C2×C13×#5472
#844613