Finite Group Databasebeta

← #15829240

Subgroups — #15829240

Order 3328.

916 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719273C211yesmaximal, characteristic
2#13123925C211yesmaximal, characteristic, complemented
3#13113206C212yesmaximal, complemented
4#13113203C212yesmaximal, complemented
5#13113189C211yesmaximal, characteristic
6#13113187C211yesmaximal, characteristic
7#13113179C211yesmaximal, characteristic, complemented
8#13113168C212yesmaximal, complemented
9#13113159C211yesmaximal, characteristic, complemented
10#13113148C212yesmaximal, complemented
11C13×#2984C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C13C2211yescharacteristic
13C2×#88201C411yescharacteristic
14C2×#88198C411yescharacteristic
15#11719273—21no—
16#11719273—21no—
17#11719273—22no—
18#11719273—21no—
19#11719273—21no—
20#11719273—22no—
21C2×#88152C412yes—
22C2×#88150C412yes—
23C2×#88147C411yescharacteristic
24C2×#88141C411yescharacteristic
25C2×#88130C2212yes—
26C2×#88129C2211yescharacteristic
27C2×#88126C2212yes—
28C2×C4×Dic26C2211yescharacteristic
29C13×#446—21no—
30C13×#446—22no—
31C8×Q8×C13—21no—
32C13×#432C2212yes—
33C13×#432C2211yescharacteristic
34C13×#431C2212yes—
35C13×#431C2211yescharacteristic
36#11718685—21no—
37#11718685—22no—
38Q8×#1055—21no—
39#11718587C2212yes—
40#11718587C2211yescharacteristic
41#11718586C2212yes—
42#11718586C2211yescharacteristic
43#11718061C2211yescharacteristic
44#11718061C2212yes—
45#11718061C2211yescharacteristic
46#11718061C2211yescharacteristic
47#11718061C2211yescharacteristic
48#11718061C2212yes—
49#11718060C2212yes—
50#11718060C2212yes—
51#11718060C2212yes—
52#11718060C2212yes—
53C22×Q8×C13C2×C411yescharacteristic
54C22×D52C2×C411yescharacteristic
55C22×C4×D13C2×C412yes—
56C22×Dic26C2×C411yescharacteristic
57C22×Dic26C2×C411yescharacteristic
58C4×Q8×C13D411yescharacteristic
59C4×Q8×C13D411yescharacteristic
60C4×Q8×C13D412yes—
61C4×Q8×C13D411yescharacteristic
62C4×Q8×C13D411yescharacteristic
63C4×Q8×C13D412yes—
64C2×C13×C4⋊C4C2×C411yescharacteristic
65C2×C13×C4⋊C4C2×C412yes—
66C2×C42×C13C2311yescharacteristic
67C2×C42×C13C2312yes—
68#88201—21no—
69#88201—21no—
70#88201—22no—
71#88198—21no—
72#88198—21no—
73#88198—22no—
74C2×#6103C2×C412yes—
75C2×#6103C2×C412yes—
76C2×#6103C2×C412yes—
77C2×#6103C2×C412yes—
78C2×#6103C2312yes—
79C2×C4⋊Dic13C2×C411yescharacteristic
80C2×#6101C2×C412yes—
81C2×#6101C2×C412yes—
82C2×#6101C2×C412yes—
83C2×#6101C2311yescharacteristic
84C2×#6101C2311yescharacteristic
85C2×C4×Dic13C2×C411yescharacteristic
86C2×C4×Dic13C2×C411yescharacteristic
87C2×C4×Dic13C2×C412yes—
88#88152—22no—
89#88152—22no—
90#88152—22no—
91#88152—22no—
92#88150—22no—
93#88150—22no—
94#88150—22no—
95#88150—22no—
96#88147—22no—
97#88147—21no—
98#88147—21no—
99#88141—22no—
100#88141—21no—
101#88141—21no—
102#88130—22no—
103#88130—22no—
104#88130—22no—
105#88130—22no—
106#88129—21no—
107#88129—21no—
108#88129—21no—
109#88129—21no—
110#88126—22no—
111#88126—22no—
112#88126—22no—
113#88126—22no—
114C4×Dic26—21no—
115C4×Dic26—21no—
116C4×Dic26—21no—
117C4×Dic26—21no—
118C13×C4⋊C8—21no—
119C13×C4⋊C8—22no—
120C13×C4⋊C8—21no—
121C13×C4⋊C8—22no—
122C13×C22⋊C8C2311yescharacteristic
123C13×C22⋊C8C2312yes—
124C13×C22⋊C8C2311yescharacteristic
125C13×#97—21no—
126C13×#97—22no—
127C4×C8×C13—21no—
128C4×C8×C13—22no—
129C22⋊#1055C2311yescharacteristic
130C22⋊#1055C2312yes—
131C22⋊#1055C2311yescharacteristic
132C4⋊#1055—21no—
133C4⋊#1055—22no—
134C4⋊#1055—21no—
135C4⋊#1055—22no—
136#88043—21no—
137#88043—22no—
138C4×#1055—21no—
139C4×#1055—22no—
140#17428—131nomaximal, Sylow
141C23×D13C22×C412yes—
142C2×Q8×C13—42no—
143C2×Q8×C13C22⋊C411yescharacteristic
144C2×Q8×C13C22⋊C411yescharacteristic
145C2×Q8×C13C22⋊C412yes—
146C22×C4×C13C22×C411yescharacteristic
147C22×C4×C13C22×C412yes—
148C22×C4×C13C22×C411yescharacteristic
149C22×C4×C13C22×C412yes—
150C22×C4×C13C2411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C22×Dic13C22×C411yescharacteristic
153C22×Dic13C22×C411yescharacteristic
154C22×Dic13C22×C412yes—
155C22×Dic13C22×C411yescharacteristic
156C2×D52—21no—
157C2×D52—21no—
158C2×D52—42no—
159C2×C4×D13—22no—
160C2×C4×D13—22no—
161C2×C4×D13—24no—
162C2×C4×D13—42no—
163C2×Dic26—21no—
164C2×Dic26—21no—
165C2×Dic26—21no—
166C2×Dic26—21no—
167C2×Dic26—22no—
168C2×Dic26—41no—
169C2×Dic26—42no—
170C2×C8×C13—21no—
171C2×C8×C13—21no—
172C2×C8×C13—22no—
173C13×C4⋊C4C22⋊C412yes—
174C13×C4⋊C4C22⋊C411yescharacteristic
175C13×C4⋊C4C22⋊C411yescharacteristic
176C13×C4⋊C4C22⋊C412yes—
177C13×C4⋊C4C22⋊C412yes—
178C13×C4⋊C4C22⋊C412yes—
179C13×C4⋊C4C22⋊C412yes—
180C42×C13C2×D411yescharacteristic
181C42×C13C2×D411yescharacteristic
182C42×C13C2×D412yes—
183C42×C13C2×D412yes—
184C42×C13C2×D411yescharacteristic
185C42×C13C2×D412yes—
186C42×C13C2×D411yescharacteristic
187C42×C13C2×D412yes—
188#6103—22no—
189#6103—22no—
190#6103—22no—
191#6103—22no—
192#6103—22no—
193#6103—22no—
194#6103—22no—
195#6103—22no—
196#6103—22no—
197#6103—22no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200#6101—22no—
201#6101—22no—
202#6101—22no—
203#6101—22no—
204#6101—22no—
205#6101—22no—
206#6101—21no—
207#6101—21no—
208#6101—21no—
209#6101—21no—
210C4×Dic13—21no—
211C4×Dic13—21no—
212C4×Dic13—21no—
213C4×Dic13—21no—
214C4×Dic13—22no—
215C4×Dic13—22no—
216C2×#1055—21no—
217C2×#1055—22no—
218C2×#1055—21no—
219C2×#556—131no—
220#2984—131no—
221#2984D1311yescharacteristic, complemented
222#1473—132no—
223#1473—132no—
224#1469—131no—
225#1469—131no—
226#1467—131no—
227#1464—132no—
228#1461—131no—
229#1458—132no—
230C23×C13C23×C411yescharacteristic
231C22×D13—22no—
232C22×D13—22no—
233C22×D13—22no—
234C22×D13—42no—
235C22×D13—42no—
236Q8×C13—22no—
237Q8×C13—22no—
238Q8×C13—42no—
239C2×C4×C13—21no—
240C2×C4×C13—22no—
241C2×C4×C13—21no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—42no—
245C2×C4×C13—41no—
246C2×C4×C13C2×C22⋊C411yescharacteristic
247C2×C4×C13C2×C22⋊C412yes—
248C2×C4×C13C2×C22⋊C412yes—
249C2×C4×C13C2×C22⋊C412yes—
250C2×C4×C13C2×C22⋊C412yes—
251C2×C4×C13C2×C22⋊C411yescharacteristic
252C2×C4×C13C2×C22⋊C411yescharacteristic
253C2×C4×C13C2×C22⋊C411yescharacteristic
254C2×C4×C13C22×D411yescharacteristic
255C2×C4×C13C22×D411yescharacteristic
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—22no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—42no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13#13112yes—
268D52—42no—
269D52—42no—
270C4×D13—44no—
271C4×D13—44no—
272Dic26—42no—
273Dic26—42no—
274Dic26—42no—
275Dic26—42no—
276C8×C13—41no—
277C8×C13—41no—
278C8×C13—42no—
279#1055—41no—
280#1055—42no—
281#1055—41no—
282#556—261no—
283#556—262no—
284#556—261no—
285#556—261no—
286#556—262no—
287#556—261no—
288C2×C4⋊Q8—131no—
289C2×#126—132no—
290C2×#126—132no—
291C2×#124—131no—
292C2×#124—131no—
293C2×C22⋊Q8—132no—
294C2×C4×Q8—131no—
295C2×C4×Q8—131no—
296C2×C4×Q8C2×D1311yescharacteristic
297C2×C4×D4—131no—
298C2×#117—132no—
299#446—22no—
300#446—21no—
301#446—262no—
302#446—261no—
303C8×Q8—21no—
304C8×Q8—261no—
305#432—132no—
306#432—131no—
307#432C2×D1312yes—
308#432C2×D1311yescharacteristic
309#431—132no—
310#431—131no—
311#431C2×D1312yes—
312#431C2×D1311yescharacteristic
313#324—131no—
314#324—132no—
315#324—131no—
316#324—131no—
317#324—131no—
318#324—132no—
319#323—132no—
320#323—132no—
321#323—132no—
322#323—132no—
323C22×C13—21no—
324C22×C13—21no—
325C22×C13C22×C22⋊C411yescharacteristic
326C22×C13C2×#13111yescharacteristic
327C22×C13#56811yescharacteristic
328C2×D13—42no—
329C2×D13—42no—
330C2×D13—42no—
331C2×D13—82no—
332C2×D13—82no—
333C4×C13—21no—
334C4×C13—21no—
335C4×C13—22no—
336C4×C13—22no—
337C4×C13—21no—
338C4×C13—21no—
339C4×C13—21no—
340C4×C13—22no—
341C4×C13—22no—
342C4×C13#41312yes—
343Dic13—22no—
344Dic13—41no—
345Dic13—41no—
346Dic13—41no—
347Dic13—42no—
348C22×Q8—131no—
349C22×Q8—131no—
350C22×Q8C4×D1311yescharacteristic
351C22×D4—131no—
352C23×C4—132no—
353C4⋊Q8—261no—
354C4⋊Q8—261no—
355C4⋊Q8—262no—
356#126—262no—
357#126—262no—
358#126—262no—
359#126—262no—
360#126—262no—
361#126—262no—
362#126—262no—
363#126—262no—
364#124—261no—
365#124—261no—
366#124—262no—
367#124—261no—
368#124—261no—
369#124—261no—
370#124—261no—
371C22⋊Q8—262no—
372C22⋊Q8—262no—
373C22⋊Q8—262no—
374C22⋊Q8—262no—
375C4×Q8—262no—
376C4×Q8—261no—
377C4×Q8—261no—
378C4×Q8—261no—
379C4×Q8—261no—
380C4×Q8—261no—
381C4×Q8—261no—
382C4×Q8D5211yescharacteristic
383C4×Q8D5212yes—
384C4×Q8D5211yescharacteristic
385C4×Q8C22⋊D1311yescharacteristic
386C4×Q8C22⋊D1312yes—
387C4×Q8C22⋊D1311yescharacteristic
388C4×D4—262no—
389C4×D4—261no—
390C4×D4—261no—
391#117—262no—
392#117—262no—
393#117—262no—
394#117—262no—
395C2×C4⋊C4—132no—
396C2×C4⋊C4—132no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—132no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4C4×D1311yescharacteristic
402C2×C4⋊C4C4×D1312yes—
403C2×C22⋊C4—132no—
404C2×C22⋊C4—132no—
405C2×C22⋊C4—132no—
406C2×C22⋊C4—132no—
407C2×C22⋊C4—132no—
408C2×C42—132no—
409C2×C42—131no—
410C2×C42—131no—
411C2×C42C22×D1312yes—
412C2×C42C22×D1311yescharacteristic
413C4⋊C8—22no—
414C4⋊C8—21no—
415C4⋊C8—22no—
416C4⋊C8—21no—
417C4⋊C8—262no—
418C4⋊C8—261no—
419C4⋊C8—262no—
420C4⋊C8—261no—
421C22⋊C8—131no—
422C22⋊C8—132no—
423C22⋊C8—131no—
424C22⋊C8C22×D1311yescharacteristic
425C22⋊C8C22×D1312yes—
426C22⋊C8C22×D1311yescharacteristic
427#97—22no—
428#97—21no—
429#97—262no—
430#97—261no—
431C4×C8—22no—
432C4×C8—21no—
433C4×C8—262no—
434C4×C8—261no—
435C2×C13—21no—
436C2×C13—21no—
437C2×C13#288011yescharacteristic
438C2×C13#288411yescharacteristic
439C2×C13C2×#41311yescharacteristic
440D13—82no—
441C24—132no—
442C2×Q8—42no—
443C2×Q8—262no—
444C2×Q8—261no—
445C2×Q8—261no—
446C2×Q8—261no—
447C2×Q8—261no—
448C2×Q8—521no—
449C2×Q8—522no—
450C2×Q8#610311yescharacteristic
451C2×Q8#610311yescharacteristic
452C2×Q8#610312yes—
453C2×D4—261no—
454C2×D4—261no—
455C2×D4—522no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—131no—
459C22×C4—131no—
460C22×C4—132no—
461C22×C4—264no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—522no—
465C22×C4C2×C4×D1311yescharacteristic
466C22×C4C2×C4×D1312yes—
467C22×C4C2×C4×D1311yescharacteristic
468C22×C4C2×C4×D1312yes—
469C22×C4C23×D1311yescharacteristic
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—21no—
473C2×C8—261no—
474C2×C8—262no—
475C2×C8—261no—
476C4⋊C4—262no—
477C4⋊C4—262no—
478C4⋊C4—262no—
479C4⋊C4—262no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—262no—
483C4⋊C4—262no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4#610312yes—
489C4⋊C4#610311yescharacteristic
490C4⋊C4#610311yescharacteristic
491C4⋊C4#610312yes—
492C4⋊C4#610312yes—
493C4⋊C4#610312yes—
494C4⋊C4#610312yes—
495C22⋊C4—262no—
496C22⋊C4—262no—
497C22⋊C4—262no—
498C22⋊C4—262no—
499C22⋊C4—262no—
500C22⋊C4—262no—
501C22⋊C4—262no—
502C22⋊C4—262no—
503C22⋊C4—262no—
504C22⋊C4—262no—
505C42—262no—
506C42—262no—
507C42—261no—
508C42—261no—
509C42—261no—
510C42—261no—
511C42C2×D5212yes—
512C42C2×D5211yescharacteristic
513C42C2×D5212yes—
514C42C2×D5211yescharacteristic
515C42C2×C22⋊D1312yes—
516C42C2×C22⋊D1311yescharacteristic
517C42C2×C22⋊D1312yes—
518C42C2×C22⋊D1311yescharacteristic
519C13#1742811yescharacteristic, complemented, Sylow
520C23—262no—
521C23—262no—
522C23—262no—
523C23—522no—
524C23—522no—
525C23C22×C4×D1311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—522no—
530Q8—522no—
531Q8—522no—
532Q8—522no—
533D4—522no—
534D4—522no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—132no—
543C2×C4—261no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—262no—
551C2×C4—524no—
552C2×C4—524no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—522no—
556C2×C4C2×#610311yescharacteristic
557C2×C4C2×#610312yes—
558C2×C4C2×#610312yes—
559C2×C4C2×#610312yes—
560C2×C4C2×#610312yes—
561C2×C4C2×#610311yescharacteristic
562C2×C4C2×#610311yescharacteristic
563C2×C4C2×#610311yescharacteristic
564C2×C4C22×D5211yescharacteristic
565C2×C4C22×C22⋊D1311yescharacteristic
566C8—41no—
567C8—42no—
568C8—41no—
569C8—521no—
570C8—522no—
571C8—521no—
572C22—21no—
573C22—21no—
574C22—522no—
575C22—522no—
576C22—522no—
577C22—1042no—
578C22—1042no—
579C22D13×#13111yescharacteristic
580C22#1171945511yescharacteristic
581C22C22×#610311yescharacteristic
582C4—22no—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—262no—
592C4—521no—
593C4—521no—
594C4—521no—
595C4—522no—
596C4#1171879312yes—
597C2—21no—
598C2—21no—
599C2—1042no—
600C2#1312328311yescharacteristic
601C2#1312329911yescharacteristic
602C2C2×#1171879311yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC23×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2984C2
socleC22×C13C22×C22⋊C4
prime radicalC22×D52C2×C4
cosocleC22×C4×C13C24
hypercenter#2984D13
hypocenterC13#17428
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #13123925C2semidirect
3. #13113206C2semidirect
4. #13113203C2semidirect
7. #13113179C2semidirect
8. #13113168C2semidirect
9. #13113159C2semidirect
10. #13113148C2semidirect
11. C13×#2984C2semidirect
221. #2984D13semidirect
519. C13#17428semidirect

Maximal subgroups

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

subgroupindex
C2×#117192732
#131239252
#131132062
#131132032
#131131892
#131131872
#131131792
#131131682
#131131592
#131131482
C13×#29842
#1742813