Finite Group Databasebeta

← #14915161

Subgroups — #14915161

Order 2816.

2070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623262C211yesmaximal, characteristic
2#13071741C214yesmaximal
3#13071739C214yesmaximal
4#13071719C214yesmaximal, complemented
5#13071715C214yesmaximal, complemented
6#13071355C211yesmaximal, characteristic, complemented
7#13071336C211yesmaximal, characteristic, complemented
8#13071292C212yesmaximal
9#13071224C211yesmaximal, characteristic
10C4×#66923C211yesmaximal, characteristic, complemented
11#13071154C212yesmaximal
12#13071091C212yesmaximal, complemented
13#13070981C212yesmaximal, complemented
14C4×#66925C211yesmaximal, characteristic
15C4×C11×C4⋊Q8C211yesmaximal, characteristic, complemented
16C2×C11×C4⋊Q8C2211yescharacteristic
17C2×C4×Q8×C11C2212yes—
18C2×#66999C2212yes—
19C2×Q8×Dic11C2212yes—
20C2×Q8×Dic11C2212yes—
21#10623262C4116yescomplemented
22C2×#66951C2214yes—
23C2×#66942C2214yes—
24C2×#66925C2211yescharacteristic
25C2×#66925C2211yescharacteristic
26C2×C42×D11C2211yescharacteristic
27C2×#66923C2211yescharacteristic
28C11×#391C2212yes—
29C11×#389C2214yes—
30C11×#384C2211yescharacteristic
31C4×C11×C4⋊C4C2214yes—
32C43×C11C2211yescharacteristic
33#10622843C2212yes—
34#10622603C2214yes—
35#10622603C2214yes—
36#10622600C2212yes—
37#10622594C2214yes—
38#10622594C2214yes—
39#10622591C2214yes—
40#10622591C2214yes—
41C4⋊C4×Dic11C2212yes—
42C4⋊C4×Dic11C2214yes—
43C4⋊C4×Dic11C2214yes—
44C4⋊C4×Dic11C2212yes—
45#10622586C2212yes—
46#10622585C2214yes—
47#10622585C2214yes—
48#10622553C2214yes—
49C4×(C2×D11)⋊C4C2212yes—
50C4×(C2×D11)⋊C4C2212yes—
51#10622549C2211yescharacteristic
52C4×C4⋊Dic11C2212yes—
53C4×C4⋊Dic11C2214yes—
54#10622546C2214yes—
55#10622546C2214yes—
56#10622545C2214yes—
57#10622545C2211yescharacteristic
58#10622545C2211yescharacteristic
59C4×Dic11⋊C4C2214yes—
60C4×Dic11⋊C4C2212yes—
61C4×Dic11⋊C4C2212yes—
62C42×Dic11C2211yescharacteristic
63#10622541C2212yes—
64#10622281C2214yes—
65#10622279C2214yes—
66#10622268C2214yes—
67#10622267C2214yes—
68#10622266C2214yes—
69#10622262C2214yes—
70#10622262C2214yes—
71#10622261C2214yes—
72C22×Q8×C11C2312yes—
73C22×C4×D11C2312yes—
74C22×C4×D11C2311yescharacteristic
75C11×C4⋊Q8C2×C418yescomplemented
76C4×Q8×C11—28no—
77C2×C11×C4⋊C4C2314yes—
78C2×C11×C4⋊C4C2312yes—
79C2×C11×C4⋊C4C2314yes—
80C2×C11×C4⋊C4C2314yes—
81C2×C42×C11C2311yescharacteristic
82C2×C42×C11C2312yes—
83C2×C42×C11C2314yes—
84C2×C42×C11C2312yes—
85C2×C42×C11C2311yescharacteristic
86C2×C42×C11C2311yescharacteristic
87#66999C2×C4116yes—
88Q8×Dic11—28no—
89Q8×Dic11C2×C4116yescomplemented
90C2×(C2×D11)⋊C4C2311yescharacteristic
91C2×(C2×D11)⋊C4C2311yescharacteristic
92C2×(C2×D11)⋊C4C2314yes—
93C2×(C2×D11)⋊C4C2312yes—
94C2×(C2×D11)⋊C4C2314yes—
95C2×C4⋊Dic11C2314yes—
96C2×C4⋊Dic11C2312yes—
97C2×C4⋊Dic11C2312yes—
98C2×C4⋊Dic11C2314yes—
99C2×Dic11⋊C4C2311yescharacteristic
100C2×Dic11⋊C4C2311yescharacteristic
101C2×Dic11⋊C4C2314yes—
102C2×Dic11⋊C4C2314yes—
103C2×Dic11⋊C4C2314yes—
104C2×Dic11⋊C4C2312yes—
105C2×Dic11⋊C4C2314yes—
106C2×C4×Dic11C2312yes—
107C2×C4×Dic11C2311yescharacteristic
108C2×C4×Dic11C2312yes—
109C2×C4×Dic11C2314yes—
110C2×C4×Dic11C2314yes—
111C2×C4×Dic11C2312yes—
112C2×C4×Dic11C2314yes—
113C2×C4×Dic11C2314yes—
114C2×C4×Dic11C2311yescharacteristic
115C2×C4×Dic11C2314yes—
116C2×C4×Dic11C2311yescharacteristic
117#66951C2×C4132yes—
118#66942C2×C4132yescomplemented
119#66925—24no—
120#66925C2×C418yes—
121C42×D11—24no—
122#66923C2×C418yescomplemented
123C11×#95C2314yes—
124C11×#95C2314yes—
125C11⋊#95C2314yes—
126C11⋊#95C2314yes—
127C11⋊#95C2314yes—
128C11⋊#95C2314yes—
129C11⋊#95C2314yes—
130C11⋊#95C2314yes—
131C11⋊#95C2314yes—
132C11⋊#95C2312yes—
133C11⋊#95C2312yes—
134C11⋊#95C2312yes—
135C11⋊#95C2312yes—
136C11⋊#95C2314yes—
137C11⋊#95C2314yes—
138#34281—111nomaximal, Sylow
139C23×D11C2411yescharacteristic
140C2×Q8×C11—24no—
141C2×Q8×C11—24no—
142C2×Q8×C11C22×C418yes—
143C22×C4×C11C2414yes—
144C22×C4×C11C2412yes—
145C22×C4×C11C2414yes—
146C22×C4×C11C2411yescharacteristic
147C22×C4×C11C2411yescharacteristic
148C22×C4×C11C2412yes—
149C22×C4×C11C2411yescharacteristic
150C22×Dic11C2411yescharacteristic
151C22×Dic11C2414yes—
152C22×Dic11C2411yescharacteristic
153C22×Dic11C2412yes—
154C22×Dic11C2414yes—
155C22×Dic11C2411yescharacteristic
156C22×Dic11C2412yes—
157C2×C4×D11—24no—
158C2×C4×D11—22no—
159C2×C4×D11—22no—
160C2×C4×D11—24no—
161C2×C4×D11—42no—
162C2×C4×D11C22×C414yes—
163C11×C4⋊C4—28no—
164C11×C4⋊C4—24no—
165C11×C4⋊C4—44no—
166C11×C4⋊C4C22×C4116yes—
167C42×C11—22no—
168C42×C11—28no—
169C42×C11—24no—
170C42×C11—42no—
171C42×C11C22×C414yes—
172C42×C11#4114yes—
173(C2×D11)⋊C4—22no—
174(C2×D11)⋊C4—22no—
175(C2×D11)⋊C4—44no—
176(C2×D11)⋊C4C22×C418yes—
177(C2×D11)⋊C4C22×C4116yes—
178C4⋊Dic11—28no—
179C4⋊Dic11—24no—
180C4⋊Dic11C22×C418yes—
181C4⋊Dic11C22×C4116yes—
182Dic11⋊C4—22no—
183Dic11⋊C4—22no—
184Dic11⋊C4—28no—
185Dic11⋊C4—44no—
186Dic11⋊C4C22×C4116yes—
187Dic11⋊C4C22×C418yes—
188Dic11⋊C4C22×C4116yes—
189C4×Dic11—24no—
190C4×Dic11—28no—
191C4×Dic11—22no—
192C4×Dic11—24no—
193C4×Dic11—44no—
194C4×Dic11—42no—
195C4×Dic11—44no—
196C4×Dic11—44no—
197C4×Dic11C22×C414yes—
198C4×Dic11C22×C4116yes—
199C4×Dic11#4114yes—
200C2×#541—111no—
201#2373—114no—
202#2373—114no—
203#2371—114no—
204#2370—114no—
205#2330—111no—
206#2327—111no—
207#2323—112no—
208#2314—111no—
209C4×C4⋊Q8D1111yescharacteristic, complemented
210C4×#125—111no—
211C4×C22⋊Q8—112no—
212#2295—112no—
213#2276—112no—
214C4×#117—111no—
215C23×C11C2511yescharacteristic
216C22×D11—21no—
217C22×D11—21no—
218C22×D11—22no—
219C22×D11—22no—
220C22×D11C23×C412yes—
221Q8×C11—216no—
222C2×C4×C11—24no—
223C2×C4×C11—22no—
224C2×C4×C11—24no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—22no—
228C2×C4×C11—22no—
229C2×C4×C11—22no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—24no—
233C2×C4×C11C23×C418yes—
234C2×C4×C11C23×C414yes—
235C2×C4×C11C23×C412yes—
236C2×C4×C11C2×#4112yes—
237C2×C4×C11C2×#4112yes—
238C2×C4×C11C2×#4112yes—
239C2×C4×C11C2×#4112yes—
240C2×C4×C112−1+414yes—
241C2×Dic11—21no—
242C2×Dic11—21no—
243C2×Dic11—22no—
244C2×Dic11—24no—
245C2×Dic11—22no—
246C2×Dic11—22no—
247C2×Dic11—24no—
248C2×Dic11—22no—
249C2×Dic11—21no—
250C2×Dic11—21no—
251C2×Dic11—24no—
252C2×Dic11—22no—
253C2×Dic11—24no—
254C2×Dic11—24no—
255C2×Dic11—22no—
256C2×Dic11C23×C414yes—
257C2×Dic11C23×C418yes—
258C2×Dic11C23×C412yes—
259C2×Dic11C2×#4114yes—
260C4×D11—28no—
261C4×D11—48no—
262#541—1116no—
263C2×C4⋊Q8C2×D1111yescharacteristic
264C2×#126—114no—
265C2×#125—114no—
266C2×#125—111no—
267C2×C22⋊Q8—112no—
268C2×C4×Q8—112no—
269C2×C4×Q8—112no—
270C2×C4×Q8C2×D1112yes—
271C2×#117—111no—
272C2×#117—111no—
273C22×C42—111no—
274#391—112no—
275#391C2×D1112yes—
276#389—112no—
277#389—114no—
278#389—114no—
279#389—112no—
280#389C2×D1114yes—
281#388—114no—
282#388—114no—
283#388—114no—
284#387—114no—
285#387—114no—
286#387—114no—
287#387—114no—
288#387—114no—
289#387—114no—
290#387—114no—
291#387—114no—
292#387—114no—
293#385—114no—
294#384C2×D1111yescharacteristic
295#382—111no—
296#380—112no—
297C4×C4⋊C4—112no—
298C4×C4⋊C4—112no—
299C4×C4⋊C4—114no—
300C4×C4⋊C4—114no—
301C4×C4⋊C4—112no—
302C4×C4⋊C4—114no—
303C4×C4⋊C4—112no—
304C4×C4⋊C4—112no—
305C4×C4⋊C4—114no—
306C4×C4⋊C4—114no—
307C4×C4⋊C4—114no—
308C4×C4⋊C4C2×D1114yes—
309C4×C22⋊C4—112no—
310C4×C22⋊C4—112no—
311C4×C22⋊C4—114no—
312#376—111no—
313#376—114no—
314#376—111no—
315#376—114no—
316C43—111no—
317C43C2×D1111yescharacteristic
318C22×C11C24×C411yescharacteristic
319C22×C11C22×#4111yescharacteristic
320C22×C11C22×#4111yescharacteristic
321C22×C11C2×2−1+412yes—
322C22×C11#58512yes—
323C2×D11—22no—
324C2×D11—24no—
325C2×D11—42no—
326C2×D11—41no—
327C2×D11—41no—
328C4×C11—28no—
329C4×C11—24no—
330C4×C11—24no—
331C4×C11—22no—
332C4×C11—22no—
333C4×C11—44no—
334C4×C11C4×#4114yes—
335Dic11—22no—
336Dic11—24no—
337Dic11—28no—
338Dic11—24no—
339Dic11—44no—
340Dic11—41no—
341Dic11—41no—
342C22×Q8C22×D1112yes—
343C23×C4—111no—
344C23×C4—112no—
345C4⋊Q8C4×D1118yescomplemented
346#126—1132no—
347#125—1132no—
348#125—118no—
349C22⋊Q8—1116no—
350C4×Q8—28no—
351C4×Q8—1116no—
352C4×Q8—228no—
353#117—118no—
354#117—224no—
355C2×C4⋊C4—112no—
356C2×C4⋊C4—112no—
357C2×C4⋊C4—114no—
358C2×C4⋊C4—114no—
359C2×C4⋊C4—114no—
360C2×C4⋊C4—111no—
361C2×C4⋊C4—111no—
362C2×C4⋊C4—112no—
363C2×C4⋊C4—114no—
364C2×C4⋊C4—114no—
365C2×C4⋊C4—114no—
366C2×C4⋊C4C22×D1112yes—
367C2×C4⋊C4C22×D1114yes—
368C2×C4⋊C4C22×D1114yes—
369C2×C4⋊C4C22×D1114yes—
370C2×C22⋊C4—112no—
371C2×C22⋊C4—114no—
372C2×C22⋊C4—111no—
373C2×C22⋊C4—111no—
374C2×C22⋊C4—114no—
375C2×C42—111no—
376C2×C42—111no—
377C2×C42—114no—
378C2×C42—111no—
379C2×C42—114no—
380C2×C42—112no—
381C2×C42—112no—
382C2×C42—112no—
383C2×C42—114no—
384C2×C42—114no—
385C2×C42—114no—
386C2×C42—224no—
387C2×C42C22×D1114yes—
388C2×C42C22×D1111yescharacteristic
389C2×C42C22×D1112yes—
390C2×C42C22×D1112yes—
391C2×C42C22×D1111yescharacteristic
392C2×C42C22×D1111yescharacteristic
393#95—112no—
394#95—112no—
395#95—112no—
396#95—114no—
397#95—112no—
398#95—114no—
399#95—114no—
400#95—114no—
401#95—114no—
402#95—114no—
403#95—114no—
404#95—114no—
405#95—114no—
406#95C22×D1114yes—
407#95C22×D1114yes—
408C2×C11C2×C4×#4111yescharacteristic
409C2×C11C4×2−1+412yes—
410C2×C11#349512yes—
411C2×C11#350911yescharacteristic
412C2×C11#351211yescharacteristic
413D11—42no—
414C24—111no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8C2×C4×D1118yes—
418C22×C4—112no—
419C22×C4—112no—
420C22×C4—114no—
421C22×C4—111no—
422C22×C4—114no—
423C22×C4—111no—
424C22×C4—114no—
425C22×C4—111no—
426C22×C4—224no—
427C22×C4—222no—
428C22×C4—222no—
429C22×C4—224no—
430C22×C4—442no—
431C22×C4C23×D1114yes—
432C22×C4C23×D1112yes—
433C22×C4C23×D1114yes—
434C22×C4C23×D1111yescharacteristic
435C22×C4C23×D1111yescharacteristic
436C22×C4C23×D1112yes—
437C22×C4C23×D1111yescharacteristic
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—44no—
441C4⋊C4—118no—
442C4⋊C4—118no—
443C4⋊C4—1116no—
444C4⋊C4—1116no—
445C4⋊C4—1116no—
446C4⋊C4—228no—
447C4⋊C4—222no—
448C4⋊C4—222no—
449C4⋊C4—228no—
450C4⋊C4—224no—
451C4⋊C4—444no—
452C4⋊C4C2×C4×D11116yes—
453C22⋊C4—118no—
454C22⋊C4—1116no—
455C22⋊C4—222no—
456C22⋊C4—222no—
457C22⋊C4—444no—
458C42—28no—
459C42—24no—
460C42—22no—
461C42—42no—
462C42—114no—
463C42—1116no—
464C42—114no—
465C42—224no—
466C42—228no—
467C42—222no—
468C42—224no—
469C42—444no—
470C42—442no—
471C42—444no—
472C42—444no—
473C42C2×C4×D1114yes—
474C42#434114yes—
475C11#3428111yescharacteristic, complemented, Sylow
476C23—112no—
477C23—221no—
478C23—222no—
479C23—221no—
480C23—222no—
481C23C24×D1111yescharacteristic
482Q8—216no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—114no—
495C2×C4—118no—
496C2×C4—112no—
497C2×C4—114no—
498C2×C4—221no—
499C2×C4—222no—
500C2×C4—224no—
501C2×C4—222no—
502C2×C4—221no—
503C2×C4—222no—
504C2×C4—224no—
505C2×C4—222no—
506C2×C4—228no—
507C2×C4—221no—
508C2×C4—221no—
509C2×C4—222no—
510C2×C4—224no—
511C2×C4—224no—
512C2×C4—224no—
513C2×C4—222no—
514C2×C4—448no—
515C2×C4C22×C4×D1118yes—
516C2×C4C22×C4×D1114yes—
517C2×C4C22×C4×D1112yes—
518C2×C4C2×#434112yes—
519C2×C4C2×#434112yes—
520C2×C4C2×#434112yes—
521C2×C4#6704014yes—
522C2×C4#41×D1112yes—
523C22—222no—
524C22—224no—
525C22—441no—
526C22—441no—
527C22—442no—
528C22C23×C4×D1111yescharacteristic
529C22C22×#434111yescharacteristic
530C22C2×#6704012yes—
531C22C2×#41×D1111yescharacteristic
532C22#1062345312yes—
533C4—22no—
534C4—22no—
535C4—28no—
536C4—24no—
537C4—24no—
538C4—44no—
539C4—222no—
540C4—224no—
541C4—228no—
542C4—224no—
543C4—441no—
544C4—441no—
545C4—444no—
546C4C4×#434114yes—
547C2—442no—
548C2C2×C4×#434111yescharacteristic
549C2C4×#6704012yes—
550C2#1308236312yes—
551C2#1308250011yescharacteristic
552C2#1308253611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC4×C11×C4⋊Q8C2
socle, cosocleC23×C11C25
prime radicalC23×D11C24
hypercenterC4×C4⋊Q8D11
hypocenterC11#34281
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #13071719C2semidirect
5. #13071715C2semidirect
6. #13071355C2semidirect
7. #13071336C2semidirect
10. C4×#66923C2semidirect
12. #13071091C2semidirect
13. #13070981C2semidirect
15. C4×C11×C4⋊Q8C2semidirect
21. #10623262C4semidirect
75. C11×C4⋊Q8C2×C4semidirect
89. Q8×Dic11C2×C4semidirect
118. #66942C2×C4semidirect
122. #66923C2×C4semidirect
209. C4×C4⋊Q8D11semidirect
345. C4⋊Q8C4×D11semidirect
475. C11#34281semidirect

Maximal subgroups

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

subgroupindex
C2×#106232622
#130717412
#130717392
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C4×#669232
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C4×#669252
C4×C11×C4⋊Q82
#3428111