Finite Group Databasebeta

← #14770340

Subgroups — #14770340

Order 2816.

1014 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623261C211yesmaximal, characteristic
2M16×Dic22C211yesmaximal, characteristic, complemented
3#13077567C211yesmaximal, characteristic, complemented
4#13069607C212yesmaximal, complemented
5#13069607C212yesmaximal, complemented
6#13069606C211yesmaximal, characteristic, complemented
7#13068607C212yesmaximal
8#13068605C211yesmaximal, characteristic
9C4×#66887C211yesmaximal, characteristic
10#13068407C212yesmaximal
11C11×#1997C211yesmaximal, characteristic, complemented
12C2×C11×C4⋊Q8C2211yescharacteristic
13C2×#66999C414yes—
14#10623261—21no—
15#10623261—21no—
16#10623261—22no—
17#10623261—22no—
18#10623261—21no—
19#10623261—21no—
20C2×#66950C412yes—
21C2×#66946C2212yes—
22C2×C4⋊C4×D11C412yes—
23C2×#66943C2212yes—
24C2×C4×D44C2211yescharacteristic
25C2×C4×Dic22C2211yescharacteristic
26C11×#423C2211yescharacteristic
27C4×C11×M16C2211yescharacteristic
28C2×#66887C2212yes—
29C2×#66887C2211yescharacteristic
30C2×#66887C2211yescharacteristic
31#10622731C2212yes—
32#10622731C2212yes—
33M16×Dic11C2212yes—
34M16×Dic11C2212yes—
35#10622537C2211yescharacteristic
36C4×#4318C2211yescharacteristic
37#10622444—21no—
38#10622443—21no—
39#10622441—22no—
40#10622314—22no—
41#10622292—21no—
42C8×Dic22—21no—
43C11×#343C2212yes—
44C11×#343C2212yes—
45#10622192C2212yes—
46#10622192C2211yescharacteristic
47#10622192C2211yescharacteristic
48#10622168C2212yes—
49#10622168C2212yes—
50C22×Q8×C11C2×C412yes—
51C22×D44C2311yescharacteristic
52C22×C4×D11C2×C412yes—
53C22×C4×D11C2×C414yes—
54C22×Dic22C2×C411yescharacteristic
55C2×C11×M16C2312yes—
56C2×C11×M16C2311yescharacteristic
57C2×C11×M16C2311yescharacteristic
58C11×C4⋊Q8—21no—
59C11×C4⋊Q8—21no—
60C11×C4⋊Q8—21no—
61C11×C4⋊Q8—21no—
62C2×C11×C4⋊C4C2×C412yes—
63C2×C11×C4⋊C4C2312yes—
64C2×C42×C11C2311yescharacteristic
65#66999—24no—
66#66999—24no—
67#66999—24no—
68#66999—24no—
69C2×(C2×D11)⋊C4C2×C412yes—
70C2×(C2×D11)⋊C4C2×C414yes—
71C2×C4⋊Dic11C2×C414yes—
72C2×C4⋊Dic11C2×C411yescharacteristic
73C2×C4⋊Dic11C2311yescharacteristic
74C2×Dic11⋊C4C2×C414yes—
75C2×Dic11⋊C4C2×C412yes—
76C2×C4×Dic11C2312yes—
77C2×#4318C2312yes—
78C2×#4318C2311yescharacteristic
79C2×#4318C2311yescharacteristic
80#66950—22no—
81#66950—22no—
82#66950—24no—
83#66946—22no—
84#66946—22no—
85#66946—22no—
86#66946—22no—
87C4⋊C4×D11—24no—
88C4⋊C4×D11—22no—
89C4⋊C4×D11—22no—
90#66943—22no—
91#66943—22no—
92#66943—22no—
93#66943—22no—
94C4×D44—21no—
95C4×D44—21no—
96C4×D44—21no—
97C4×D44—21no—
98C4×Dic22D411yescharacteristic
99C4×Dic22D411yescharacteristic
100C4×Dic22D412yes—
101C4×Dic22D412yes—
102C4×Dic22D411yescharacteristic
103C4×Dic22D411yescharacteristic
104C11×C4⋊C8—21no—
105C11×C4⋊C8—21no—
106C11×#97—21no—
107C4×C8×C11—21no—
108#66887—24no—
109#66887—22no—
110#66887Q814yescomplemented
111#66879—22no—
112#66879—22no—
113Dic11⋊C8—22no—
114Dic11⋊C8—22no—
115Dic11⋊C8—22no—
116Dic11⋊C8—22no—
117C8×Dic11—22no—
118C8×Dic11—22no—
119C4⋊(C11⋊C8)—21no—
120C4⋊(C11⋊C8)—21no—
121#66867—21no—
122C4×C11⋊C8—21no—
123#22515—111nomaximal, Sylow
124C23×D11C22×C412yes—
125C2×Q8×C11—22no—
126C2×Q8×C11—22no—
127C2×Q8×C11—82no—
128C22×C4×C11C22×C411yescharacteristic
129C22×C4×C11C22×C414yes—
130C22×C4×C11C22×C411yescharacteristic
131C22×C4×C11C2411yescharacteristic
132C22×Dic11C22×C412yes—
133C22×Dic11C22×C412yes—
134C22×Dic11C22×C412yes—
135C2×D44—21no—
136C2×D44—21no—
137C2×D44—21no—
138C2×D44—21no—
139C2×D44C2×Q812yes—
140C2×D44#4112yes—
141C2×C4×D11—24no—
142C2×C4×D11—24no—
143C2×C4×D11—22no—
144C2×C4×D11—22no—
145C2×C4×D11—44no—
146C2×C4×D11—84no—
147C2×Dic22—42no—
148C2×Dic22C22⋊C411yescharacteristic
149C2×Dic22C22⋊C411yescharacteristic
150C2×Dic22C22⋊C412yes—
151C11×M16—22no—
152C11×M16—21no—
153C11×M16—21no—
154C11×M16—21no—
155C11×M16—22no—
156C11×M16C2×Q812yes—
157C2×C8×C11—21no—
158C2×C8×C11—22no—
159C2×C8×C11—21no—
160C11×C4⋊C4—22no—
161C11×C4⋊C4—22no—
162C11×C4⋊C4—22no—
163C11×C4⋊C4—22no—
164C42×C11C2×D411yescharacteristic
165C42×C11C2×D411yescharacteristic
166C42×C11C2×D411yescharacteristic
167C42×C11C2×D411yescharacteristic
168(C2×D11)⋊C4—24no—
169(C2×D11)⋊C4—24no—
170(C2×D11)⋊C4—22no—
171(C2×D11)⋊C4—22no—
172C4⋊Dic11—24no—
173C4⋊Dic11—24no—
174C4⋊Dic11—21no—
175C4⋊Dic11—21no—
176C4⋊Dic11C22⋊C412yes—
177C4⋊Dic11C22⋊C411yescharacteristic
178C4⋊Dic11C22⋊C411yescharacteristic
179Dic11⋊C4—24no—
180Dic11⋊C4—24no—
181Dic11⋊C4C22⋊C412yes—
182Dic11⋊C4C22⋊C412yes—
183Dic11⋊C4C22⋊C412yes—
184Dic11⋊C4C22⋊C412yes—
185C4×Dic11C2×D412yes—
186C4×Dic11C2×D412yes—
187C4×Dic11C2×D412yes—
188C4×Dic11C2×D412yes—
189#4318—22no—
190#4318—21no—
191#4318—21no—
192#4318—22no—
193#4318—21no—
194#4318C2×Q812yes—
195C2×C11⋊C8—21no—
196C2×C11⋊C8—21no—
197C2×C11⋊C8—22no—
198C2×#554—111no—
199Q8×M16—111no—
200Q8×M16—111no—
201#1998—112no—
202#1998—112no—
203#1997—111no—
204#1997D1111yescharacteristic, complemented
205#1800—111no—
206#1800—112no—
207C4×#100—111no—
208C4×#100—112no—
209C23×C11C23×C411yescharacteristic
210C22×D11—22no—
211C22×D11—22no—
212C22×D11—42no—
213C22×D11—42no—
214C22×D11C4×Q814yes—
215Q8×C11—82no—
216Q8×C11—82no—
217C2×C4×C11—22no—
218C2×C4×C11—24no—
219C2×C4×C11—41no—
220C2×C4×C11—44no—
221C2×C4×C11C2×C22⋊C411yescharacteristic
222C2×C4×C11C2×C22⋊C411yescharacteristic
223C2×C4×C11C2×C22⋊C411yescharacteristic
224C2×C4×C11C2×C22⋊C411yescharacteristic
225C2×C4×C11C22×D411yescharacteristic
226C2×C4×C11C22×D411yescharacteristic
227C2×C4×C11C22×Q811yescharacteristic
228C2×C4×C11C2×#4111yescharacteristic
229C2×C4×C112−1+411yescharacteristic
230C2×C4×C112−1+411yescharacteristic
231C2×Dic11—22no—
232C2×Dic11—42no—
233C2×Dic11—42no—
234C2×Dic11—42no—
235C2×Dic11C2×C22⋊C412yes—
236C2×Dic11C2×C22⋊C412yes—
237C2×Dic11C2×C22⋊C412yes—
238C2×Dic11C2×C22⋊C412yes—
239D44—22no—
240D44—22no—
241D44—22no—
242D44—22no—
243C4×D11—44no—
244C4×D11—44no—
245C4×D11—84no—
246C4×D11—84no—
247Dic22—22no—
248Dic22—22no—
249Dic22—42no—
250C8×C11—22no—
251C8×C11—41no—
252C8×C11—42no—
253C11⋊C8—22no—
254C11⋊C8—42no—
255C11⋊C8—41no—
256#554—221no—
257#554—221no—
258#554—222no—
259#554—222no—
260#554—221no—
261#554—221no—
262C2×C4⋊Q8C2×D1111yescharacteristic
263C2×#125—112no—
264C2×C22⋊Q8—114no—
265C2×C22⋊Q8—112no—
266C2×C4×Q8—111no—
267C2×C4×D4—112no—
268C2×C4×D4—111no—
269C22×C4⋊C4—112no—
270#446—221no—
271#446—222no—
272#446—222no—
273#446—221no—
274C8×Q8—221no—
275C8×Q8—221no—
276#423—112no—
277#423—112no—
278#423—111no—
279#423C2×D1111yescharacteristic
280C2×#100—111no—
281C2×#100—112no—
282C2×#100—111no—
283C4×M16—112no—
284C4×M16—112no—
285C4×M16—111no—
286C4×M16C2×D1111yescharacteristic
287#343—112no—
288#343—111no—
289#343—111no—
290#343—112no—
291#343—112no—
292#343C2×D1112yes—
293#343C2×D1112yes—
294C22×C11—21no—
295C22×C11—21no—
296C22×C11C22×C22⋊C411yescharacteristic
297C22×C11C2×C4×Q811yescharacteristic
298C22×C11#51911yescharacteristic
299C2×D11—24no—
300C2×D11—44no—
301C2×D11—42no—
302C2×D11—42no—
303C2×D11—44no—
304C4×C11—21no—
305C4×C11—21no—
306C4×C11—21no—
307C4×C11—44no—
308C4×C11#41312yes—
309C4×C11#54811yescharacteristic
310C4×C11#54811yescharacteristic
311C4×C11D4×Q811yescharacteristic
312C4×C11D4×Q811yescharacteristic
313Dic11—22no—
314Dic11—22no—
315Dic11—22no—
316Dic11—22no—
317Dic11—42no—
318C22×Q8—111no—
319C22×Q8C4×D1112yes—
320C22×D4—111no—
321C23×C4—112no—
322C23×C4—114no—
323C2×M16—111no—
324C2×M16—112no—
325C2×M16—111no—
326C2×M16C22×D1112yes—
327C2×M16C22×D1111yescharacteristic
328C2×M16C22×D1111yescharacteristic
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333#125—222no—
334#125—222no—
335#125—222no—
336#125—222no—
337C22⋊Q8—222no—
338C22⋊Q8—222no—
339C22⋊Q8—224no—
340C22⋊Q8—224no—
341C22⋊Q8—224no—
342C22⋊Q8—224no—
343C22⋊Q8—224no—
344C4×Q8—111no—
345C4×Q8—111no—
346C4×Q8—112no—
347C4×Q8—112no—
348C4×Q8—111no—
349C4×Q8—111no—
350C4×D4—222no—
351C4×D4—222no—
352C4×D4—222no—
353C4×D4—222no—
354C4×D4—221no—
355C4×D4—221no—
356C4×D4—221no—
357C4×D4—221no—
358C2×C4⋊C4—111no—
359C2×C4⋊C4—112no—
360C2×C4⋊C4—114no—
361C2×C4⋊C4—114no—
362C2×C4⋊C4—111no—
363C2×C4⋊C4—224no—
364C2×C4⋊C4—222no—
365C2×C4⋊C4—222no—
366C2×C4⋊C4C4×D1112yes—
367C2×C4⋊C4C22×D1112yes—
368C2×C22⋊C4—114no—
369C2×C22⋊C4—112no—
370C2×C42—112no—
371C2×C42C22×D1111yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—222no—
375C4⋊C8—222no—
376C4⋊C8—222no—
377C4⋊C8—222no—
378C4⋊C8—221no—
379C4⋊C8—221no—
380#100—114no—
381#100—222no—
382#100—224no—
383#97—21no—
384#97—222no—
385#97—222no—
386#97—221no—
387C4×C8—21no—
388C4×C8—222no—
389C4×C8—222no—
390C4×C8—221no—
391C2×C11—21no—
392C2×C11—21no—
393C2×C11Q8×C22⋊C411yescharacteristic
394C2×C11C2×#41311yescharacteristic
395C2×C11#288811yescharacteristic
396D11—44no—
397C24—112no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—82no—
401C2×Q8—111no—
402C2×Q8—111no—
403C2×Q8—112no—
404C2×Q8—442no—
405C2×D4—112no—
406C2×D4—112no—
407C2×D4—221no—
408C2×D4—221no—
409C2×D4—221no—
410C2×D4—221no—
411C22×C4—112no—
412C22×C4—112no—
413C22×C4—112no—
414C22×C4—224no—
415C22×C4—224no—
416C22×C4—222no—
417C22×C4—222no—
418C22×C4—444no—
419C22×C4—884no—
420C22×C4C2×C4×D1111yescharacteristic
421C22×C4C2×C4×D1114yes—
422C22×C4C2×C4×D1111yescharacteristic
423C22×C4C23×D1111yescharacteristic
424M16—21no—
425M16—21no—
426M16—21no—
427M16—22no—
428M16—22no—
429M16—112no—
430M16—221no—
431M16—222no—
432M16—222no—
433M16—221no—
434M16—221no—
435M16Q8×D1112yes—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—221no—
440C2×C8—222no—
441C2×C8—221no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—112no—
447C4⋊C4—111no—
448C4⋊C4—111no—
449C4⋊C4—112no—
450C4⋊C4—112no—
451C4⋊C4—112no—
452C4⋊C4—112no—
453C4⋊C4—224no—
454C4⋊C4—224no—
455C4⋊C4—224no—
456C4⋊C4—224no—
457C4⋊C4—221no—
458C4⋊C4—221no—
459C22⋊C4—224no—
460C22⋊C4—224no—
461C22⋊C4—222no—
462C22⋊C4—222no—
463C42—112no—
464C42—112no—
465C42—112no—
466C42—112no—
467C42D4×D1111yescharacteristic
468C42D4×D1111yescharacteristic
469C42D4×D1111yescharacteristic
470C42D4×D1111yescharacteristic
471C11#2251511yescharacteristic, complemented, Sylow
472C23—114no—
473C23—222no—
474C23—222no—
475C23—442no—
476C23—442no—
477C23C22×C4×D1111yescharacteristic
478Q8—82no—
479Q8—82no—
480Q8—222no—
481Q8—222no—
482Q8—442no—
483D4—222no—
484D4—222no—
485D4—222no—
486D4—222no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—112no—
492C2×C4—112no—
493C2×C4—112no—
494C2×C4—112no—
495C2×C4—222no—
496C2×C4—444no—
497C2×C4—442no—
498C2×C4—442no—
499C2×C4—444no—
500C2×C4—442no—
501C2×C4—884no—
502C2×C4—884no—
503C2×C4C22⋊C4×D1111yescharacteristic
504C2×C4C22⋊C4×D1111yescharacteristic
505C2×C4C22⋊C4×D1111yescharacteristic
506C2×C4C22⋊C4×D1111yescharacteristic
507C2×C4C2×D4×D1111yescharacteristic
508C2×C4C2×D4×D1111yescharacteristic
509C2×C4C2×Q8×D1111yescharacteristic
510C2×C4#6704011yescharacteristic
511C2×C4#41×D1111yescharacteristic
512C2×C4#6704311yescharacteristic
513C8—22no—
514C8—41no—
515C8—42no—
516C8—222no—
517C8—442no—
518C8—441no—
519C22—21no—
520C22—21no—
521C22—224no—
522C22—444no—
523C22—444no—
524C22—442no—
525C22—442no—
526C22C2×C22⋊C4×D1111yescharacteristic
527C22C4×Q8×D1111yescharacteristic
528C22#1062310311yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—44no—
533C4—222no—
534C4—222no—
535C4—222no—
536C4—222no—
537C4—442no—
538C4#1062236412yes—
539C4#1062316311yescharacteristic
540C4#1062316411yescharacteristic
541C4#1062320811yescharacteristic
542C4#1062325811yescharacteristic
543C2—21no—
544C2—21no—
545C2—444no—
546C2#1307145011yescharacteristic
547C2C2×#1062236411yescharacteristic
548C2#1307709211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D11
commutatorC23×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1997C2
socleC22×C11C22×C22⋊C4
prime radicalC22×D44C23
cosocleC22×C4×C11C24
hypercenter#1997D11
hypocenterC11#22515
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. M16×Dic22C2semidirect
3. #13077567C2semidirect
4. #13069607C2semidirect
5. #13069607C2semidirect
6. #13069606C2semidirect
11. C11×#1997C2semidirect
110. #66887Q8semidirect
204. #1997D11semidirect
471. C11#22515semidirect

Maximal subgroups

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

subgroupindex
C2×#106232612
M16×Dic222
#130775672
#130696072
#130696062
#130686072
#130686052
C4×#668872
#130684072
C11×#19972
#2251511