Finite Group Databasebeta

← #16796195

Subgroups — #16796195

Order 3328.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131787C211yesmaximal, characteristic
2#13131739C211yesmaximal, characteristic, complemented
3#13130476C211yesmaximal, characteristic, complemented
4#13130470C211yesmaximal, characteristic, complemented
5#13130463C211yesmaximal, characteristic, complemented
6#13130461C211yesmaximal, characteristic, complemented
7#13130461C211yesmaximal, characteristic, complemented
8#13130454C211yesmaximal, characteristic, complemented
9Q16×#1057C211yesmaximal, characteristic, complemented
10#13130420C211yesmaximal, characteristic, complemented
11#13130411C211yesmaximal, characteristic, complemented
12#13130189C211yesmaximal, characteristic
13C2×Q8⋊#1057C211yesmaximal, characteristic, complemented
14#13130041C211yesmaximal, characteristic, complemented
15#13124917C211yesmaximal, characteristic
16C2×Q8×Dic13C2211yescharacteristic
17#11719329C2211yescharacteristic
18#11719158—21no—
19#11719158—21no—
20#11719158C2211yescharacteristic
21#11719158C2211yescharacteristic
22#11719158C2211yescharacteristic
23Q8×#1057C2211yescharacteristic
24Q8×#1057C2211yescharacteristic
25Q8×#1057C2211yescharacteristic
26#11719130—21no—
27#11719130C2211yescharacteristic
28#11719125—21no—
29#11719125C2211yescharacteristic
30C2×C4⋊#1057C2211yescharacteristic
31C2×C4×#1057C2211yescharacteristic
32#11719079—21no—
33#11719077—21no—
34#11719076C2211yescharacteristic
35#11719073C2211yescharacteristic
36#11719064—21no—
37#11719063—21no—
38C13×#451C411yescharacteristic
39#11718888C411yescharacteristic
40#11718846C411yescharacteristic
41#11718846C411yescharacteristic
42Q16×Dic13C2211yescharacteristic
43Q16×Dic13C2211yescharacteristic
44#11718704C411yescharacteristic
45#11718457C411yescharacteristic
46#11718457C411yescharacteristic
47#11718449C2211yescharacteristic
48#11718449C2211yescharacteristic
49#11718397C411yescharacteristic
50#11718379C2211yescharacteristic
51Q8⋊#1057C2211yescharacteristic
52Q8⋊#1057C2211yescharacteristic
53Q8⋊#1057C2211yescharacteristic
54Q8⋊#1057C2211yescharacteristic
55Q8⋊#1057C2211yescharacteristic
56Q8⋊#1057C2211yescharacteristic
57#11718255C2211yescharacteristic
58#11718255C2211yescharacteristic
59#11718254C2211yescharacteristic
60#11718254C2211yescharacteristic
61#1057⋊C8C2211yescharacteristic
62#1057⋊C8C2211yescharacteristic
63C8×#1057C2211yescharacteristic
64C8×#1057C2211yescharacteristic
65#11718243C2211yescharacteristic
66#11718243C2211yescharacteristic
67C22×Q8×C13C2×C411yescharacteristic
68C22×#1057—21no—
69C22×#1057—21no—
70C2×#6118—21no—
71C2×D13⋊C8—21no—
72C2×C13×Q16C2×C411yescharacteristic
73C2×C13×Q16C2×C411yescharacteristic
74C13×C22⋊Q8C2×C411yescharacteristic
75#88201C2×C411yescharacteristic
76Q8×Dic13—21no—
77Q8×Dic13—21no—
78Q8×Dic13C2×C411yescharacteristic
79Q8×Dic13C2×C411yescharacteristic
80Q8×Dic13C2311yescharacteristic
81Q8×Dic13C2311yescharacteristic
82Q8×Dic13C2311yescharacteristic
83C2×#6106C2×C411yescharacteristic
84C2×#6106C2×C411yescharacteristic
85#88184C2×C411yescharacteristic
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13C2×C411yescharacteristic
88C2×C4×Dic13—21no—
89C2×C4×Dic13C2311yescharacteristic
90#88152—21no—
91#88147C2311yescharacteristic
92#88146—21no—
93#88144C2×C411yescharacteristic
94#88140—21no—
95#88135—21no—
96C22⋊#1057—21no—
97C22⋊#1057D411yescharacteristic
98C22⋊#1057D411yescharacteristic
99#88112—21no—
100#88112—21no—
101#88112—21no—
102#88112D411yescharacteristic
103#88112D411yescharacteristic
104#88111—21no—
105#88111—21no—
106#88110—21no—
107#88110—21no—
108#88110—21no—
109#88110—21no—
110C4⋊#1057—21no—
111C4⋊#1057—21no—
112C4⋊#1057—21no—
113C4⋊#1057—21no—
114C4⋊#1057C2311yescharacteristic
115C4⋊#1057C2311yescharacteristic
116C4⋊#1057C2311yescharacteristic
117C4⋊#1057C2311yescharacteristic
118C4×#1057—21no—
119C4×#1057—21no—
120C4×#1057—21no—
121C4×#1057—21no—
122C4×#1057C2311yescharacteristic
123C4×#1057C2311yescharacteristic
124C4×#1057C2311yescharacteristic
125C4×#1057C2311yescharacteristic
126C13×#103C2×C411yescharacteristic
127C13×#103C2×C411yescharacteristic
128C13×C22⋊C8C2×C411yescharacteristic
129#88075C2×C411yescharacteristic
130#88075C2×C411yescharacteristic
131#88075C2×C411yescharacteristic
132#88075C2×C411yescharacteristic
133C22⋊#1055C2×C411yescharacteristic
134#88061C2×C411yescharacteristic
135#88061C2×C411yescharacteristic
136#88058C2×C411yescharacteristic
137#88058C2×C411yescharacteristic
138#88054C2×C411yescharacteristic
139#88054C2×C411yescharacteristic
140C8×Dic13C2311yescharacteristic
141C8×Dic13C2311yescharacteristic
142#88048C2×C411yescharacteristic
143#88048C2×C411yescharacteristic
144#18017—131nomaximal, Sylow
145C2×Q8×C13—21no—
146C2×Q8×C13—22no—
147C2×Q8×C13—41no—
148C2×Q8×C13C22×C411yescharacteristic
149C2×Q8×C13C22×C411yescharacteristic
150C2×Q8×C13C22×C411yescharacteristic
151C22×C4×C13—21no—
152C22×C4×C13C22×C411yescharacteristic
153C2×C22⋊D13—21no—
154C22×Dic13—21no—
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13#4111yescharacteristic
157C2×D52C22×C411yescharacteristic
158C2×C4×D13—21no—
159C2×#1057—24no—
160C2×#1057—21no—
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—81no—
167C2×#1057C2×D411yescharacteristic
168C2×#1057C2×D411yescharacteristic
169C2×#1057C2×D411yescharacteristic
170C2×#1057C2×D411yescharacteristic
171#6118—81no—
172D13⋊C8—81no—
173C13×Q16—22no—
174C13×Q16—41no—
175C2×C8×C13C22×C411yescharacteristic
176C2×C8×C13C22×C411yescharacteristic
177C13×C4⋊C4—21no—
178C13×C4⋊C4C22×C411yescharacteristic
179C13×C22⋊C4—21no—
180#6106—22no—
181#6106—41no—
182#6103—21no—
183#6103—21no—
184#6103—21no—
185#6103—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13C22×C411yescharacteristic
194C4⋊Dic13C22×C411yescharacteristic
195C4⋊Dic13C22×C411yescharacteristic
196#6101—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13C22×C411yescharacteristic
203C4×Dic13C2×D411yescharacteristic
204C4×Dic13#4111yescharacteristic
205C4×Dic13C2411yescharacteristic
206C2×#1055C22×C411yescharacteristic
207C2×#1055C22×C411yescharacteristic
208#3108—131no—
209#2994—131no—
210#2984—131no—
211#1591—131no—
212#1590—131no—
213#1588—131no—
214#1587—131no—
215#1587—131no—
216#1585—131no—
217C8×Q16—131no—
218C8×SD16—131no—
219#1573—131no—
220#1494—131no—
221C2×Q8⋊C8—131no—
222#1450—131no—
223C23×C13C4×D411yescharacteristic
224C22×D13—21no—
225Q8×C13—21no—
226Q8×C13—21no—
227Q8×C13—21no—
228Q8×C13—21no—
229Q8×C13—21no—
230Q8×C13—42no—
231Q8×C13#13112yes—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—42no—
239C2×C4×C13C4×D411yescharacteristic
240C2×C4×C13C23×C411yescharacteristic
241C22⋊D13—42no—
242C2×Dic13—22no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—42no—
249C2×Dic13C4×D411yescharacteristic
250C2×Dic13C4×D411yescharacteristic
251C2×Dic13C22×D411yescharacteristic
252C2×Dic13C2×#4111yescharacteristic
253D52—21no—
254D52—21no—
255C4×D13—41no—
256C4×D13—41no—
257#1057—22no—
258#1057—22no—
259#1057—21no—
260#1057—21no—
261#1057—21no—
262#1057—21no—
263#1057—41no—
264#1057—41no—
265#1057—41no—
266#1057—41no—
267C8×C13—21no—
268C8×C13—21no—
269#1055—21no—
270#1055—21no—
271#529—131no—
272C2×C4×Q8—131no—
273#483—131no—
274#480—131no—
275#479—131no—
276#479—131no—
277#464—131no—
278#464—131no—
279#451#26111yescharacteristic
280#448—131no—
281#446—131no—
282#446—131no—
283#446—131no—
284#446—261no—
285#446—261no—
286C8×Q8—131no—
287C8×Q8—131no—
288C8×Q8—131no—
289C4×Q16—131no—
290C4×Q16—131no—
291C4×SD16—131no—
292C4×SD16—131no—
293#436—131no—
294#435—261no—
295C8×D4—131no—
296C8×D4—261no—
297#433—261no—
298#432—131no—
299#432—261no—
300#431—131no—
301#431—131no—
302#431—261no—
303#424—261no—
304C2×C4⋊C8—131no—
305C2×C4×C8—131no—
306#335—131no—
307#335—131no—
308#334—131no—
309#334—131no—
310Q8⋊C8—131no—
311Q8⋊C8—131no—
312Q8⋊C8—131no—
313Q8⋊C8—131no—
314Q8⋊C8—131no—
315Q8⋊C8—131no—
316D4⋊C8—131no—
317D4⋊C8—131no—
318#322—131no—
319#322—131no—
320C82—131no—
321C82—131no—
322C22×C13—21no—
323C22×C13—21no—
324C22×C13#44312yes—
325C22×C13C2×C4×D411yescharacteristic
326C2×D13—41no—
327C2×D13—41no—
328C2×D13—41no—
329C4×C13—21no—
330C4×C13—22no—
331C4×C13—41no—
332C4×C13—41no—
333C4×C13—41no—
334C4×C13C2×#13111yescharacteristic
335C4×C13#56811yescharacteristic
336Dic13—21no—
337Dic13—21no—
338Dic13—21no—
339Dic13—41no—
340Dic13—41no—
341Dic13—41no—
342Dic13#57711yescharacteristic
343Dic13#57811yescharacteristic
344C22×Q8C2×#26111yescharacteristic
345C2×Q16C2×#26111yescharacteristic
346C2×Q16C2×#26111yescharacteristic
347C2×SD16—131no—
348C2×SD16—131no—
349C2×M16—261no—
350C22×C8—261no—
351C22×C8—261no—
352C22×C8—261no—
353#126—261no—
354#125—131no—
355#124—131no—
356#123—261no—
357C22⋊Q8C2×#26111yescharacteristic
358#121—131no—
359C4×Q8—131no—
360C4×Q8—131no—
361C4×Q8—131no—
362C4×Q8—131no—
363C4×Q8—131no—
364C4×Q8—261no—
365C4×Q8—261no—
366C4×D4—131no—
367C4×D4—261no—
368#117—261no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—261no—
371C2×C42—131no—
372C2×C42—261no—
373#107—131no—
374#107—131no—
375#106—131no—
376#106—131no—
377C4⋊C8—131no—
378C4⋊C8—131no—
379C4⋊C8—131no—
380C4⋊C8—131no—
381C4⋊C8—131no—
382C4⋊C8—131no—
383C4⋊C8—131no—
384C4⋊C8—131no—
385C4⋊C8—261no—
386C4⋊C8—261no—
387C4⋊C8—261no—
388C4⋊C8—261no—
389C4⋊C8—261no—
390C4⋊C8—261no—
391C4⋊C8—261no—
392#103—131no—
393#103—131no—
394#103—131no—
395#103—131no—
396#103C2×#26111yescharacteristic
397#103C2×#26111yescharacteristic
398#102—131no—
399#102—131no—
400C22⋊C8—131no—
401C22⋊C8—131no—
402C22⋊C8—131no—
403C22⋊C8—261no—
404C22⋊C8—261no—
405C22⋊C8—261no—
406C22⋊C8C2×#26111yescharacteristic
407#97—261no—
408#97—261no—
409#97—261no—
410#97—261no—
411C4×C8—131no—
412C4×C8—131no—
413C4×C8—131no—
414C4×C8—131no—
415C4×C8—131no—
416C4×C8—131no—
417C4×C8—261no—
418C4×C8—261no—
419C4×C8—261no—
420C4×C8—261no—
421C2×C13—22no—
422C2×C13#293311yescharacteristic
423C2×C13#294711yescharacteristic
424C2×C13C2×#44311yescharacteristic
425D13—81no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—41no—
429C2×Q8C22×#26111yescharacteristic
430C2×Q8C22×#26111yescharacteristic
431C2×Q8C22×#26111yescharacteristic
432C2×D4—131no—
433C2×D4—261no—
434C22×C4—21no—
435C22×C4—131no—
436C22×C4—131no—
437C22×C4—261no—
438C22×C4—261no—
439C22×C4C22×#26111yescharacteristic
440Q16—22no—
441Q16—41no—
442SD16—262no—
443SD16—521no—
444M16—1041no—
445C2×C8—131no—
446C2×C8—131no—
447C2×C8—131no—
448C2×C8—131no—
449C2×C8—131no—
450C2×C8—131no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—264no—
454C2×C8—261no—
455C2×C8—261no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—1041no—
459C2×C8—1041no—
460C2×C8C22×#26111yescharacteristic
461C2×C8C22×#26111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—131no—
464C4⋊C4—131no—
465C4⋊C4—131no—
466C4⋊C4—261no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4—261no—
474C4⋊C4C22×#26111yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—261no—
477C22⋊C4—261no—
478C22⋊C4—261no—
479C22⋊C4—261no—
480C42—131no—
481C42—131no—
482C42—131no—
483C42—131no—
484C42—261no—
485C42—261no—
486C42—261no—
487C42—261no—
488C42—261no—
489C13#1801711yescharacteristic, complemented, Sylow
490C23—261no—
491C23D4×#26111yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8#8823812yes—
499D4—261no—
500D4—261no—
501D4—522no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—42no—
509C2×C4—131no—
510C2×C4—131no—
511C2×C4—131no—
512C2×C4—131no—
513C2×C4—261no—
514C2×C4—261no—
515C2×C4—262no—
516C2×C4—261no—
517C2×C4—261no—
518C2×C4—261no—
519C2×C4—521no—
520C2×C4—521no—
521C2×C4—522no—
522C2×C4D4×#26111yescharacteristic
523C2×C4C23×#26111yescharacteristic
524C8—21no—
525C8—21no—
526C8—261no—
527C8—261no—
528C8—261no—
529C8—261no—
530C8—262no—
531C8—262no—
532C8—261no—
533C8—261no—
534C8—521no—
535C8—521no—
536C8—521no—
537C8—521no—
538C22—21no—
539C22—21no—
540C22—521no—
541C22—521no—
542C22—521no—
543C22#1171911112yes—
544C22C2×D4×#26111yescharacteristic
545C4—21no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—131no—
551C4—131no—
552C4—261no—
553C4—261no—
554C4—261no—
555C4—521no—
556C4—521no—
557C4—521no—
558C4C2×#8823811yescharacteristic
559C4#1171962111yescharacteristic
560C2—22no—
561C2—1041no—
562C2#1313156811yescharacteristic
563C2#1313159411yescharacteristic
564C2C2×#1171911111yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC13×#451C4
socleC22×C13C2×C4×D4
prime radical#88184C2×C4
cosocleC4×Dic13C24
hypercenter#451#261
hypocenterC13#18017
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #13131739C2semidirect
3. #13130476C2semidirect
4. #13130470C2semidirect
5. #13130463C2semidirect
6. #13130461C2semidirect
7. #13130461C2semidirect
8. #13130454C2semidirect
9. Q16×#1057C2semidirect
10. #13130420C2semidirect
11. #13130411C2semidirect
13. C2×Q8⋊#1057C2semidirect
14. #13130041C2semidirect
489. C13#18017semidirect

Maximal subgroups

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

subgroupindex
#131317872
#131317392
#131304762
#131304702
#131304632
#131304612
#131304542
Q16×#10572
#131304202
#131304112
#131301892
C2×Q8⋊#10572
#131300412
#131249172
#1801713