Finite Group Databasebeta

← #16795615

Subgroups — #16795615

Order 3328.

598 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#13131751C211yesmaximal, characteristic
2#13131740C211yesmaximal, characteristic, complemented
3#13130459C211yesmaximal, characteristic, complemented
4#13130453C211yesmaximal, characteristic, complemented
5#13130449C211yesmaximal, characteristic, complemented
6#13130439C211yesmaximal, characteristic, complemented
7#13130429C211yesmaximal, characteristic, complemented
8#13130427C211yesmaximal, characteristic, complemented
9#13130423C211yesmaximal, characteristic, complemented
10SD16×#1057C211yesmaximal, characteristic, complemented
11#13130409C211yesmaximal, characteristic, complemented
12#13130167C211yesmaximal, characteristic
13#13130144C211yesmaximal, characteristic, complemented
14#13130057C211yesmaximal, characteristic, complemented
15#13124900C211yesmaximal, characteristic
16#41×Dic13C2211yescharacteristic
17#11719327C2211yescharacteristic
18#11719158C2211yescharacteristic
19Q8×#1057C2211yescharacteristic
20Q8×#1057C2211yescharacteristic
21Q8×#1057C2211yescharacteristic
22D4×#1057—21no—
23D4×#1057—21no—
24D4×#1057C2211yescharacteristic
25D4×#1057C2211yescharacteristic
26#11719130C2211yescharacteristic
27#11719125C2211yescharacteristic
28#11719119—21no—
29#11719119C2211yescharacteristic
30C2×C4⋊#1057—21no—
31C2×C4⋊#1057C2211yescharacteristic
32#11719080—21no—
33#11719080C2211yescharacteristic
34#11719078—21no—
35#11719074C2211yescharacteristic
36#11719068—21no—
37#11719068—21no—
38C13×#452C411yescharacteristic
39#11718900C411yescharacteristic
40Q16×Dic13C2211yescharacteristic
41#11718843C411yescharacteristic
42#11718823C411yescharacteristic
43SD16×Dic13C2211yescharacteristic
44#11718706C411yescharacteristic
45#11718452C411yescharacteristic
46#11718451C2211yescharacteristic
47#11718420C411yescharacteristic
48#11718419C2211yescharacteristic
49#11718384C411yescharacteristic
50#11718379C2211yescharacteristic
51Q8⋊#1057C2211yescharacteristic
52Q8⋊#1057C2211yescharacteristic
53Q8⋊#1057C2211yescharacteristic
54Q8⋊#1057C2211yescharacteristic
55D4⋊#1057C2211yescharacteristic
56D4⋊#1057C2211yescharacteristic
57#11718256C2211yescharacteristic
58#11718256C2211yescharacteristic
59#11718253C2211yescharacteristic
60#11718253C2211yescharacteristic
61#1057⋊C8C2211yescharacteristic
62#1057⋊C8C2211yescharacteristic
63#11718251C2211yescharacteristic
64C8×#1057C2211yescharacteristic
65#11718244C2211yescharacteristic
66#11718244C2211yescharacteristic
67C2×C13×#41C2×C411yescharacteristic
68C22×#1057—21no—
69C22×#1057—21no—
70C22×#1057—21no—
71C2×C13×Q16C2×C411yescharacteristic
72C2×C13×SD16C2×C411yescharacteristic
73C13×C22⋊Q8C2×C411yescharacteristic
74Q8×Dic13C2×C411yescharacteristic
75Q8×Dic13C2311yescharacteristic
76Q8×Dic13C2311yescharacteristic
77#88198C2×C411yescharacteristic
78C2×#6107C2×C411yescharacteristic
79D4×Dic13—21no—
80D4×Dic13—21no—
81D4×Dic13C2×C411yescharacteristic
82D4×Dic13C2311yescharacteristic
83C2×#6105C2×C411yescharacteristic
84#88180—21no—
85#88180C2×C411yescharacteristic
86#88178C2×C411yescharacteristic
87C2×C4×Dic13—21no—
88C2×C4×Dic13C2311yescharacteristic
89#88143—21no—
90#88142C2×C411yescharacteristic
91#88141—21no—
92#88141C2311yescharacteristic
93#88132—21no—
94#88131—21no—
95C22⋊#1057—21no—
96C22⋊#1057—21no—
97C22⋊#1057—21no—
98C22⋊#1057—21no—
99C22⋊#1057D411yescharacteristic
100C22⋊#1057D411yescharacteristic
101#88112—21no—
102#88112—21no—
103#88112—21no—
104#88110—21no—
105#88110—21no—
106#88110—21no—
107C4⋊#1057—21no—
108C4⋊#1057—21no—
109C4⋊#1057—21no—
110C4⋊#1057—21no—
111C4⋊#1057—21no—
112C4⋊#1057D411yescharacteristic
113C4⋊#1057D411yescharacteristic
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×#102C2×C411yescharacteristic
128C13×C22⋊C8C2×C411yescharacteristic
129#88075C2×C411yescharacteristic
130#88075C2×C411yescharacteristic
131#88075C2×C411yescharacteristic
132#88072C2×C411yescharacteristic
133C22⋊#1055C2×C411yescharacteristic
134#88058C2×C411yescharacteristic
135#88057C2×C411yescharacteristic
136#88056C2×C411yescharacteristic
137#88056C2×C411yescharacteristic
138#88054C2×C411yescharacteristic
139#88054C2×C411yescharacteristic
140C8×Dic13C2311yescharacteristic
141C8×Dic13C2311yescharacteristic
142#88048C2×C411yescharacteristic
143#88047C2×C411yescharacteristic
144#17921—131nomaximal, Sylow
145C13×#41—22no—
146C13×#41—22no—
147C2×Q8×C13C22×C411yescharacteristic
148C2×Q8×C13C22×C411yescharacteristic
149C2×D4×C13—21no—
150C2×D4×C13C22×C411yescharacteristic
151C22×C4×C13—21no—
152C22×C4×C13C22×C411yescharacteristic
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13#4111yescharacteristic
157C2×Dic26—21no—
158C2×Dic26C22×C411yescharacteristic
159C2×#1057—21no—
160C2×#1057—21no—
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—81no—
166C2×#1057—81no—
167C2×#1057—81no—
168C2×#1057C2×D411yescharacteristic
169C2×#1057C2×D411yescharacteristic
170C2×#1057C2×D411yescharacteristic
171C2×#1057C2×D411yescharacteristic
172C13×Q16—22no—
173C13×SD16—22no—
174C2×C8×C13C22×C411yescharacteristic
175C2×C8×C13C22×C411yescharacteristic
176C13×C4⋊C4—21no—
177C13×C4⋊C4C22×C411yescharacteristic
178C13×C22⋊C4—21no—
179C22⋊Dic13—21no—
180C22⋊Dic13—21no—
181C22⋊Dic13—21no—
182C22⋊Dic13—21no—
183#6107—22no—
184#6105—22no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13C22×C411yescharacteristic
189C4⋊Dic13C22×C411yescharacteristic
190C4⋊Dic13C22×C411yescharacteristic
191#6101—21no—
192#6101—21no—
193#6101—21no—
194#6101—21no—
195#6101—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13C22×C411yescharacteristic
202C4×Dic13C2×D411yescharacteristic
203C4×Dic13#4111yescharacteristic
204C4×Dic13C2411yescharacteristic
205C2×#1055C22×C411yescharacteristic
206C2×#1055C22×C411yescharacteristic
207#3107—131no—
208#2988—131no—
209#2985—131no—
210#1587—131no—
211#1585—131no—
212#1584—131no—
213#1582—131no—
214#1579—131no—
215SD16⋊C8—131no—
216C8×Q16—131no—
217C8×SD16—131no—
218#1573—131no—
219#1488—131no—
220#1481—131no—
221#1454—131no—
222C23×C13—21no—
223C23×C13C4×D411yescharacteristic
224Q8×C13—21no—
225Q8×C13—21no—
226Q8×C13—21no—
227Q8×C13#13112yes—
228D4×C13—24no—
229D4×C13—21no—
230D4×C13#13112yes—
231C2×C4×C13—24no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13C4×D411yescharacteristic
238C2×C4×C13C23×C411yescharacteristic
239C2×Dic13—21no—
240C2×Dic13—21no—
241C2×Dic13—21no—
242C2×Dic13—21no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—41no—
246C2×Dic13—41no—
247C2×Dic13—41no—
248C2×Dic13—81no—
249C2×Dic13C4×D411yescharacteristic
250C2×Dic13C4×D411yescharacteristic
251C2×Dic13C22×D411yescharacteristic
252C2×Dic13C2×#4111yescharacteristic
253Dic26—21no—
254Dic26—21no—
255Dic26—81no—
256#1057—21no—
257#1057—21no—
258#1057—21no—
259#1057—21no—
260#1057—41no—
261#1057—41no—
262#1057—41no—
263#1057—41no—
264#1057—41no—
265#1057—41no—
266C8×C13—21no—
267C8×C13—21no—
268#1055—21no—
269#1055—21no—
270#533—131no—
271C4×#41—131no—
272#485—131no—
273#485—131no—
274#475—131no—
275#474—131no—
276#462—131no—
277#461—131no—
278#452—131no—
279#452#26111yescharacteristic
280#446—131no—
281#446—131no—
282#446—261no—
283#446—261no—
284C8×Q8—131no—
285C8×Q8—131no—
286C8×Q8—131no—
287C8×Q8—131no—
288C4×Q16—131no—
289C4×Q16—131no—
290C4×SD16—131no—
291C4×SD16—131no—
292C8×D4—131no—
293C8×D4—131no—
294C8×D4—261no—
295C8×D4—261no—
296#433—261no—
297#433—261no—
298#432—131no—
299#431—131no—
300#431—131no—
301#431—131no—
302#431—261no—
303C2×C4⋊C8—131no—
304C2×C4⋊C8—261no—
305#335—131no—
306#335—131no—
307#334—131no—
308#334—131no—
309Q8⋊C8—131no—
310Q8⋊C8—131no—
311Q8⋊C8—131no—
312Q8⋊C8—131no—
313Q8⋊C8—131no—
314Q8⋊C8—131no—
315D4⋊C8—131no—
316D4⋊C8—131no—
317#322—131no—
318#322—131no—
319#322—131no—
320C82—131no—
321C22×C13—21no—
322C22×C13—22no—
323C22×C13—21no—
324C22×C13—21no—
325C22×C13—41no—
326C22×C13—41no—
327C22×C13C2×C4×D411yescharacteristic
328C4×C13—21no—
329C4×C13—22no—
330C4×C13—41no—
331C4×C13—41no—
332C4×C13C2×#13111yescharacteristic
333C4×C13C2×#13111yescharacteristic
334Dic13—21no—
335Dic13—21no—
336Dic13—21no—
337Dic13—41no—
338Dic13—41no—
339Dic13—41no—
340Dic13—41no—
341Dic13#57711yescharacteristic
342Dic13#57811yescharacteristic
343C2×#41C2×#26111yescharacteristic
344C2×Q16—131no—
345C2×Q16C2×#26111yescharacteristic
346C2×SD16—131no—
347C2×SD16C2×#26111yescharacteristic
348C22×C8—261no—
349C22×C8—261no—
350C22×C8—261no—
351C4⋊Q8—131no—
352C4⋊Q8—131no—
353#125—261no—
354C22⋊Q8—131no—
355C22⋊Q8—261no—
356C22⋊Q8C2×#26111yescharacteristic
357C4×Q8—131no—
358C4×Q8—131no—
359C4×Q8—131no—
360C4×Q8—131no—
361C4×Q8—261no—
362C4×D4—131no—
363C4×D4—131no—
364C4×D4—261no—
365C4×D4—261no—
366#117—131no—
367#117—261no—
368#117—261no—
369C2×C42—131no—
370C2×C42—261no—
371#107—131no—
372#107—131no—
373#106—131no—
374#106—131no—
375C4⋊C8—131no—
376C4⋊C8—131no—
377C4⋊C8—131no—
378C4⋊C8—131no—
379C4⋊C8—131no—
380C4⋊C8—131no—
381C4⋊C8—131no—
382C4⋊C8—131no—
383C4⋊C8—261no—
384C4⋊C8—261no—
385C4⋊C8—261no—
386C4⋊C8—261no—
387C4⋊C8—261no—
388C4⋊C8—261no—
389C4⋊C8—261no—
390C4⋊C8—261no—
391#103—131no—
392#103—131no—
393#103—131no—
394#103—131no—
395#103—131no—
396#103C2×#26111yescharacteristic
397#102—131no—
398#102C2×#26111yescharacteristic
399C22⋊C8—131no—
400C22⋊C8—131no—
401C22⋊C8—131no—
402C22⋊C8—261no—
403C22⋊C8—261no—
404C22⋊C8—261no—
405C22⋊C8—261no—
406C22⋊C8C2×#26111yescharacteristic
407#97—261no—
408#97—261no—
409#97—261no—
410C4×C8—131no—
411C4×C8—131no—
412C4×C8—131no—
413C4×C8—131no—
414C4×C8—131no—
415C4×C8—131no—
416C4×C8—261no—
417C4×C8—261no—
418C4×C8—261no—
419C4×C8—261no—
420C2×C13—41no—
421C2×C13—42no—
422C2×C13#292911yescharacteristic
423C2×C13#294711yescharacteristic
424C2×C13#295711yescharacteristic
425#41—22no—
426#41—22no—
427C2×Q8—131no—
428C2×Q8—261no—
429C2×Q8C22×#26111yescharacteristic
430C2×Q8C22×#26111yescharacteristic
431C2×D4—21no—
432C2×D4C22×#26111yescharacteristic
433C22×C4—21no—
434C22×C4—131no—
435C22×C4—131no—
436C22×C4—261no—
437C22×C4—261no—
438C22×C4C22×#26111yescharacteristic
439Q16—22no—
440Q16—262no—
441SD16—22no—
442SD16—262no—
443C2×C8—131no—
444C2×C8—131no—
445C2×C8—131no—
446C2×C8—131no—
447C2×C8—131no—
448C2×C8—131no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—1041no—
456C2×C8—1041no—
457C2×C8—1041no—
458C2×C8C22×#26111yescharacteristic
459C2×C8C22×#26111yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—131no—
462C4⋊C4—131no—
463C4⋊C4—131no—
464C4⋊C4—261no—
465C4⋊C4—261no—
466C4⋊C4—261no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4C22×#26111yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—261no—
475C22⋊C4—261no—
476C22⋊C4—261no—
477C22⋊C4—261no—
478C42—131no—
479C42—131no—
480C42—131no—
481C42—131no—
482C42—261no—
483C42—261no—
484C42—261no—
485C42—261no—
486C42—261no—
487C13#1792111yescharacteristic, complemented, Sylow
488C23—21no—
489C23D4×#26111yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—261no—
494Q8—261no—
495Q8—1041no—
496Q8#8823812yes—
497D4—24no—
498D4—21no—
499D4#8823812yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—131no—
507C2×C4—131no—
508C2×C4—131no—
509C2×C4—131no—
510C2×C4—261no—
511C2×C4—261no—
512C2×C4—261no—
513C2×C4—261no—
514C2×C4—261no—
515C2×C4—261no—
516C2×C4—521no—
517C2×C4—521no—
518C2×C4—521no—
519C2×C4—1041no—
520C2×C4D4×#26111yescharacteristic
521C2×C4C23×#26111yescharacteristic
522C8—21no—
523C8—21no—
524C8—261no—
525C8—261no—
526C8—261no—
527C8—261no—
528C8—261no—
529C8—261no—
530C8—521no—
531C8—521no—
532C8—521no—
533C8—521no—
534C8—521no—
535C8—521no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×D4×#26111yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—131no—
548C4—131no—
549C4—261no—
550C4—261no—
551C4—261no—
552C4—521no—
553C4—521no—
554C4—521no—
555C4—521no—
556C4C2×#8823811yescharacteristic
557C4C2×#8823811yescharacteristic
558C2—41no—
559C2—42no—
560C2#1313155111yescharacteristic
561C2#1313159411yescharacteristic
562C2#1313161511yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC13×#452C4
socleC22×C13C2×C4×D4
prime radicalC2×C13×#41C2×C4
cosocleC4×Dic13C24
hypercenter#452#261
hypocenterC13#17921
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #13131740C2semidirect
3. #13130459C2semidirect
4. #13130453C2semidirect
5. #13130449C2semidirect
6. #13130439C2semidirect
7. #13130429C2semidirect
8. #13130427C2semidirect
9. #13130423C2semidirect
10. SD16×#1057C2semidirect
11. #13130409C2semidirect
13. #13130144C2semidirect
14. #13130057C2semidirect
487. C13#17921semidirect

Maximal subgroups

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

subgroupindex
#131317512
#131317402
#131304592
#131304532
#131304492
#131304392
#131304292
#131304272
#131304232
SD16×#10572
#131304092
#131301672
#131301442
#131300572
#131249002
#1792113