Finite Group Databasebeta

← #16752978

Subgroups — #16752978

Order 3328.

1706 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128701C211yesmaximal, characteristic
2#13127764C214yesmaximal, complemented
3#13127742C214yesmaximal, complemented
4#13127650C214yesmaximal, complemented
5#13127576C214yesmaximal, complemented
6#13126066C214yesmaximal, complemented
7#13125987C211yesmaximal, characteristic, complemented
8#13125964C211yesmaximal, characteristic
9#13125007C214yesmaximal, complemented
10#13124875C211yesmaximal, characteristic, complemented
11#13124867C211yesmaximal, characteristic
12#13123921C211yesmaximal, characteristic, complemented
13C2×Q82×C13C211yesmaximal, characteristic, complemented
14Q82×C13—22no—
15Q82×C13—22no—
16Q82×C13C2214yes—
17Q82×C13C2214yes—
18C2×C13×C4⋊Q8—21no—
19C2×C13×C4⋊Q8—22no—
20C2×C13×C4⋊Q8C2211yescharacteristic
21C2×C13×C4⋊Q8C2212yes—
22C2×C4×Q8×C13—21no—
23C2×C4×Q8×C13C2211yescharacteristic
24C2×C4×Q8×C13C2212yes—
25C2×C4×Q8×C13C2211yescharacteristic
26#11719521—21no—
27#11719290C2212yes—
28#11719289C2212yes—
29#11719286—22no—
30#11719284C2212yes—
31#11719273—22no—
32#11719272C2212yes—
33#11719220—22no—
34#11719209—21no—
35#11719184C2214yes—
36#11719182C2211yescharacteristic
37#11719175C2211yescharacteristic
38#11718889C2212yes—
39#11718888C2212yes—
40#11718755C2218yes—
41#11718754C2212yes—
42#11718752C2214yes—
43#11718750C2218yes—
44#11718749C2212yes—
45#11718748C2218yes—
46#11718745C2214yes—
47#11718741C2218yes—
48#11718740C2212yes—
49#11718739C2212yes—
50#11718692C2212yes—
51#11718691C2218yes—
52C4×#6107C2214yes—
53#11718689C2218yes—
54#11718688C2212yes—
55#11718687C2218yes—
56C4×#6106C2214yes—
57#11718685C2214yes—
58#11718685C2214yes—
59#11718683C2218yes—
60#11718682C2212yes—
61#11718681C2212yes—
62#11718631C2212yes—
63#11718631C2214yes—
64#11718629C2212yes—
65#11718629C2214yes—
66#11718587C2214yes—
67#11718586C2211yescharacteristic
68#11718586C2211yescharacteristic
69C22×Q8×C13—21no—
70C22×Q8×C13—22no—
71C22×Q8×C13D411yescharacteristic
72C22×Q8×C13C2311yescharacteristic
73C2×#6127—21no—
74C13×C4⋊Q8—24no—
75C13×C4⋊Q8—28no—
76C13×C4⋊Q8—24no—
77C13×C4⋊Q8—28no—
78C13×C4⋊Q8—21no—
79C13×C4⋊Q8—21no—
80C13×C4⋊Q8—24no—
81C13×C4⋊Q8C2312yes—
82C13×C4⋊Q8C2312yes—
83C13×C4⋊Q8C2318yes—
84C4×Q8×C13—24no—
85C4×Q8×C13—24no—
86C4×Q8×C13—21no—
87C4×Q8×C13—21no—
88C4×Q8×C13—24no—
89C4×Q8×C13D412yes—
90C4×Q8×C13D412yes—
91C4×Q8×C13C2312yes—
92C4×Q8×C13C2312yes—
93C4×Q8×C13C2318yes—
94C4×Q8×C13C2312yes—
95C4×Q8×C13C2312yes—
96C2×C13×C4⋊C4—22no—
97C2×C13×C4⋊C4—21no—
98C2×C13×C4⋊C4—21no—
99C2×C13×C4⋊C4—22no—
100C2×C13×C4⋊C4D411yescharacteristic
101C2×C13×C4⋊C4D412yes—
102C2×C13×C4⋊C4C2311yescharacteristic
103C2×C13×C4⋊C4C2312yes—
104C2×C42×C13—22no—
105C2×C42×C13—21no—
106C2×C42×C13C2311yescharacteristic
107C2×C42×C13C2312yes—
108#88201—22no—
109#88200—22no—
110#88200—22no—
111#88198—22no—
112C2×#6107C2314yes—
113C2×#6106C2314yes—
114#88184—21no—
115#88184C2312yes—
116#88183—24no—
117C4×C22⋊D13—24no—
118#88180—21no—
119#88178—21no—
120#88178C2312yes—
121#88152—24no—
122#88150—24no—
123#88148—24no—
124#88147—22no—
125#88146—22no—
126C4⋊C4×D13—22no—
127#88143—24no—
128#88141—22no—
129#88130—24no—
130#88129—21no—
131#88129C2314yes—
132#88128C2311yescharacteristic
133C4×D52—21no—
134C4×D52C2312yes—
135C4×D52C2314yes—
136#88126—24no—
137#88124—21no—
138#88124C2314yes—
139#88123C2311yescharacteristic
140#88123C2311yescharacteristic
141#88123C2311yescharacteristic
142C4×Dic26—21no—
143C4×Dic26C2312yes—
144C4×Dic26C2314yes—
145#88075C2314yes—
146#88075C2314yes—
147C22⋊#1055C2312yes—
148C22⋊#1055C2312yes—
149#88050C2318yes—
150#88050C2314yes—
151#88049C2314yes—
152#88049C2318yes—
153#88048C2314yes—
154#88048C2318yes—
155#88047C2314yes—
156#88047C2318yes—
157C4⋊#1055C2312yes—
158C4⋊#1055C2312yes—
159C4⋊#1055C2318yes—
160#88043C2318yes—
161C4×#1055C2312yes—
162C4×#1055C2312yes—
163#61830—131nomaximal, Sylow
164C2×Q8×C13—24no—
165C2×Q8×C13—24no—
166C2×Q8×C13—21no—
167C2×Q8×C13—28no—
168C2×Q8×C13—22no—
169C2×Q8×C13—22no—
170C2×Q8×C13—82no—
171C2×Q8×C13—84no—
172C2×Q8×C13C2×D412yes—
173C2×Q8×C13C2×D412yes—
174C2×Q8×C13C2412yes—
175C22×C4×C13—21no—
176C22×C4×C13—22no—
177C22×C4×C13—21no—
178C22×C4×C13C2×D412yes—
179C22×C4×C13C2×D411yescharacteristic
180C22×C4×C13C2×D412yes—
181C22×C4×C13C2×D411yescharacteristic
182C22×C4×C13C2411yescharacteristic
183C2×C22⋊D13—22no—
184#6127—44no—
185C2×D52—21no—
186C2×D52—21no—
187C2×D52C2412yes—
188C2×C4×D13—22no—
189C2×C4×D13—24no—
190C2×Dic26—21no—
191C2×Dic26—21no—
192C2×Dic26C2412yes—
193C13×C4⋊C4—28no—
194C13×C4⋊C4—22no—
195C13×C4⋊C4—22no—
196C13×C4⋊C4—28no—
197C13×C4⋊C4—22no—
198C13×C4⋊C4—22no—
199C13×C4⋊C4—21no—
200C13×C4⋊C4—22no—
201C13×C4⋊C4C2×D412yes—
202C13×C4⋊C4C2×D414yes—
203C13×C4⋊C4C2×D412yes—
204C13×C4⋊C4C2×D414yes—
205C13×C4⋊C4C2412yes—
206C13×C4⋊C4C2414yes—
207C42×C13—28no—
208C42×C13—22no—
209C42×C13—22no—
210C42×C13C2×D411yescharacteristic
211C42×C13C2×D411yescharacteristic
212C42×C13C2×D414yes—
213C42×C13C2411yescharacteristic
214C42×C13C2414yes—
215C42×C13C2411yescharacteristic
216C22⋊Dic13—22no—
217#6107—28no—
218#6106—28no—
219#6103—22no—
220#6103—24no—
221#6103—22no—
222#6103—24no—
223#6103—24no—
224C4⋊Dic13—21no—
225C4⋊Dic13—21no—
226C4⋊Dic13—21no—
227C4⋊Dic13—21no—
228C4⋊Dic13C2412yes—
229C4⋊Dic13C2412yes—
230#6101—22no—
231#6101—22no—
232#6101—24no—
233#6101—24no—
234#6101—24no—
235C4×Dic13—22no—
236C4×Dic13—24no—
237C2×#1055C2414yes—
238C2×#1055C2414yes—
239#3507—131no—
240C2×Q82D1311yescharacteristic, complemented
241#3387—134no—
242#3385—134no—
243#3375—134no—
244#3369—134no—
245#3239—134no—
246#3230—131no—
247#3227—131no—
248#3117—134no—
249#3104—131no—
250#3103—131no—
251#2983—131no—
252C23×C13C22×D411yescharacteristic
253C22×D13—22no—
254Q8×C13—24no—
255Q8×C13—24no—
256Q8×C13—24no—
257Q8×C13—22no—
258Q8×C13—82no—
259Q8×C13—88no—
260Q8×C13—82no—
261Q8×C132−1+414yes—
262C2×C4×C13—24no—
263C2×C4×C13—24no—
264C2×C4×C13—22no—
265C2×C4×C13—22no—
266C2×C4×C13—24no—
267C2×C4×C13—22no—
268C2×C4×C13—41no—
269C2×C4×C13—42no—
270C2×C4×C13—44no—
271C2×C4×C13—42no—
272C2×C4×C13#13714yes—
273C2×C4×C13C22×D414yes—
274C2×C4×C13C22×D411yescharacteristic
275C2×C4×C13C22×D414yes—
276C2×C4×C13C22×D411yescharacteristic
277C2×C4×C13C22×D411yescharacteristic
278C2×C4×C13C22×D411yescharacteristic
279C2×C4×C13C22×D411yescharacteristic
280C2×C4×C13C2511yescharacteristic
281C22⋊D13—44no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—22no—
285D52—24no—
286D52—42no—
287D52—81no—
288C4×D13—44no—
289C4×D13—84no—
290Dic26—24no—
291Dic26—42no—
292Dic26—81no—
293#1055—24no—
294#1055—24no—
295#559—132no—
296Q82—22no—
297Q82—22no—
298Q82C2×D1314yes—
299Q82C2×D1314yes—
300#557—132no—
301#556—132no—
302#556—262no—
303#554—132no—
304#552—262no—
305#539—262no—
306#536—261no—
307#533—131no—
308#532—131no—
309C2×C4⋊Q8—21no—
310C2×C4⋊Q8—22no—
311C2×C4⋊Q8C2×D1311yescharacteristic
312C2×C4⋊Q8C2×D1312yes—
313#529—134no—
314#520—261no—
315C2×C4×Q8—21no—
316C2×C4×Q8C2×D1311yescharacteristic
317C2×C4×Q8C2×D1312yes—
318C2×C4×Q8C2×D1311yescharacteristic
319#501—138no—
320#500—132no—
321#498—132no—
322#489—138no—
323#487—132no—
324#486—132no—
325#484—132no—
326#484—134no—
327#481—132no—
328#481—134no—
329#479—138no—
330#477—134no—
331#477—132no—
332#476—134no—
333#475—138no—
334#475—132no—
335#474—138no—
336#464—138no—
337#462—132no—
338#460—132no—
339#451—132no—
340#448—132no—
341#446—134no—
342#446—134no—
343#441—138no—
344#440—138no—
345C4×Q16—134no—
346C4×SD16—134no—
347#432—134no—
348#431—131no—
349#431—131no—
350C22×C13—21no—
351C22×C13—21no—
352C22×C13C2×#13712yes—
353C22×C13C23×D411yescharacteristic
354C2×D13—42no—
355C2×D13—42no—
356C2×D13—42no—
357C4×C13—21no—
358C4×C13—24no—
359C4×C13—21no—
360C4×C13—24no—
361C4×C13—21no—
362C4×C13—24no—
363C4×C13—44no—
364C4×C13—42no—
365C4×C13C2×#13712yes—
366C4×C13C2×#13712yes—
367C4×C13C2×2−1+412yes—
368Dic13—42no—
369Dic13—42no—
370Dic13—42no—
371C2×#41—261no—
372C22×Q8—22no—
373C22×Q8—21no—
374C22×Q8C22⋊D1311yescharacteristic
375C22×Q8C22×D1311yescharacteristic
376C2×Q16—134no—
377C2×SD16—134no—
378C4⋊Q8—24no—
379C4⋊Q8—24no—
380C4⋊Q8—28no—
381C4⋊Q8—28no—
382C4⋊Q8—24no—
383C4⋊Q8—21no—
384C4⋊Q8—21no—
385C4⋊Q8—131no—
386C4⋊Q8—131no—
387C4⋊Q8—131no—
388C4⋊Q8—262no—
389C4⋊Q8C22×D1318yes—
390C4⋊Q8C22×D1312yes—
391C4⋊Q8C22×D1312yes—
392#127—131no—
393#126—264no—
394#126—264no—
395#125—134no—
396#125—264no—
397#125—261no—
398#124—134no—
399#124—262no—
400#124—261no—
401#123—264no—
402#123—264no—
403C22⋊Q8—132no—
404C22⋊Q8—262no—
405C22⋊Q8—262no—
406C22⋊Q8—261no—
407C22⋊Q8—264no—
408#121—132no—
409#121—261no—
410C4×Q8—24no—
411C4×Q8—24no—
412C4×Q8—24no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—134no—
416C4×Q8—132no—
417C4×Q8—262no—
418C4×Q8—261no—
419C4×Q8C22⋊D1312yes—
420C4×Q8C22⋊D1312yes—
421C4×Q8C22×D1312yes—
422C4×Q8C22×D1312yes—
423C4×Q8C22×D1318yes—
424C4×Q8C22×D1312yes—
425C4×Q8C22×D1312yes—
426C4×D4—134no—
427C4×D4—132no—
428C4×D4—264no—
429C4×D4—262no—
430C4×D4—261no—
431#117—261no—
432#117—262no—
433#117—264no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—262no—
439C2×C4⋊C4C22⋊D1311yescharacteristic
440C2×C4⋊C4C22⋊D1312yes—
441C2×C4⋊C4C22×D1311yescharacteristic
442C2×C4⋊C4C22×D1312yes—
443C2×C42—22no—
444C2×C42—21no—
445C2×C42C22×D1312yes—
446C2×C42C22×D1311yescharacteristic
447#107—134no—
448#107—138no—
449#106—134no—
450#106—138no—
451C4⋊C8—132no—
452C4⋊C8—132no—
453C4⋊C8—138no—
454#103—134no—
455#103—134no—
456#103—134no—
457#103—138no—
458#102—134no—
459#102—138no—
460C22⋊C8—132no—
461C22⋊C8—132no—
462#97—138no—
463C4×C8—132no—
464C4×C8—132no—
465C2×C13—22no—
466C2×C13#348611yescharacteristic
467C2×C13C22×#13711yescharacteristic
468C2×C13#358511yescharacteristic
469D13—82no—
470#41—524no—
471C2×Q8—24no—
472C2×Q8—24no—
473C2×Q8—21no—
474C2×Q8—28no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—84no—
478C2×Q8—82no—
479C2×Q8—132no—
480C2×Q8—261no—
481C2×Q8—261no—
482C2×Q8C2×C22⋊D1312yes—
483C2×Q8C2×C22⋊D1312yes—
484C2×Q8C23×D1312yes—
485C2×D4—132no—
486C2×D4—262no—
487C2×D4—261no—
488C2×D4—261no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—262no—
493C22×C4—264no—
494C22×C4C2×C22⋊D1312yes—
495C22×C4C2×C22⋊D1312yes—
496C22×C4C2×C22⋊D1311yescharacteristic
497C22×C4C2×C22⋊D1311yescharacteristic
498C22×C4C23×D1311yescharacteristic
499Q16—268no—
500SD16—268no—
501C2×C8—134no—
502C2×C8—134no—
503C4⋊C4—28no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—132no—
512C4⋊C4—132no—
513C4⋊C4—261no—
514C4⋊C4—264no—
515C4⋊C4—261no—
516C4⋊C4—262no—
517C4⋊C4—264no—
518C4⋊C4—261no—
519C4⋊C4—262no—
520C4⋊C4—264no—
521C4⋊C4—261no—
522C4⋊C4C2×C22⋊D1312yes—
523C4⋊C4C2×C22⋊D1312yes—
524C4⋊C4C2×C22⋊D1314yes—
525C4⋊C4C2×C22⋊D1314yes—
526C4⋊C4C23×D1312yes—
527C4⋊C4C23×D1314yes—
528C22⋊C4—262no—
529C22⋊C4—262no—
530C22⋊C4—264no—
531C22⋊C4—264no—
532C22⋊C4—262no—
533C22⋊C4—264no—
534C42—28no—
535C42—22no—
536C42—22no—
537C42—264no—
538C42—262no—
539C42C2×C22⋊D1314yes—
540C42C2×C22⋊D1311yescharacteristic
541C42C2×C22⋊D1311yescharacteristic
542C42C23×D1311yescharacteristic
543C42C23×D1311yescharacteristic
544C42C23×D1314yes—
545C13#6183011yescharacteristic, complemented, Sylow
546C23—262no—
547C23C22×C22⋊D1311yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—24no—
551Q8—22no—
552Q8—88no—
553Q8—82no—
554Q8—82no—
555Q8—264no—
556Q8—522no—
557Q8—1041no—
558Q8#8825414yes—
559D4—264no—
560D4—524no—
561D4—522no—
562D4—1041no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—262no—
574C2×C4—262no—
575C2×C4—262no—
576C2×C4—524no—
577C2×C4—1044no—
578C2×C4#8819614yes—
579C2×C4C22×C22⋊D1314yes—
580C2×C4C22×C22⋊D1314yes—
581C2×C4C22×C22⋊D1311yescharacteristic
582C2×C4C22×C22⋊D1311yescharacteristic
583C2×C4C22×C22⋊D1311yescharacteristic
584C2×C4C22×C22⋊D1311yescharacteristic
585C2×C4C22×C22⋊D1311yescharacteristic
586C2×C4C24×D1311yescharacteristic
587C8—264no—
588C8—264no—
589C22—21no—
590C22—21no—
591C22—522no—
592C22—522no—
593C22—522no—
594C22C2×#8819612yes—
595C22C23×C22⋊D1311yescharacteristic
596C4—24no—
597C4—21no—
598C4—24no—
599C4—24no—
600C4—21no—
601C4—21no—
602C4—42no—
603C4—44no—
604C4—522no—
605C4—522no—
606C4—522no—
607C4C2×#8819612yes—
608C4C2×#8819612yes—
609C4C2×#8825412yes—
610C2—22no—
611C2—1042no—
612C2#1312848711yescharacteristic
613C2C22×#8819611yescharacteristic
614C2#1312961611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D13
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
FittingC2×Q82×C13C2
socleC22×C13C23×D4
prime radical#11719182C22
hypercenterC2×Q82D13
hypocenterC13#61830
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
2. #13127764C2semidirect
3. #13127742C2semidirect
4. #13127650C2semidirect
5. #13127576C2semidirect
6. #13126066C2semidirect
7. #13125987C2semidirect
9. #13125007C2semidirect
10. #13124875C2semidirect
12. #13123921C2semidirect
13. C2×Q82×C13C2semidirect
240. C2×Q82D13semidirect
545. C13#61830semidirect

Maximal subgroups

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

subgroupindex
#131287012
#131277642
#131277422
#131276502
#131275762
#131260662
#131259872
#131259642
#131250072
#131248752
#131248672
#131239212
C2×Q82×C132
#6183013