Finite Group Databasebeta

← #14757429

Subgroups — #14757429

Order 2816.

1622 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×#10623199C211yesmaximal, characteristic
2C2×#10622417C212yesmaximal, complemented
3C2×#10622385C212yesmaximal, complemented
4C2×D11×#97C211yesmaximal, characteristic, complemented
5#13071021C211yesmaximal, characteristic, complemented
6#13069438C212yesmaximal, complemented
7#13069434C212yesmaximal, complemented
8#13069433C211yesmaximal, characteristic, complemented
9#13069432C211yesmaximal, characteristic, complemented
10#13069430C211yesmaximal, characteristic, complemented
11C11×#1959C211yesmaximal, characteristic, complemented
12C22×#4343C2211yescharacteristic
13C22×#4341C2211yescharacteristic
14C2×C11×#124C2211yescharacteristic
15C2×#66997C2211yescharacteristic
16C2×#66993C2211yescharacteristic
17C22×C8×D11C2212yes—
18#10623199—24no—
19#10623199C418yescomplemented
20C2×#66945C2212yes—
21C2×#66937—22no—
22C2×(C22⋊D11)⋊C4—22no—
23C2×#66928C2211yescharacteristic
24C2×C42×D11C2211yescharacteristic
25C2×C11×#103C2212yes—
26C2×C11×#102C2212yes—
27C2×C11×#97C2211yescharacteristic
28C11×#384C2211yescharacteristic
29C2×#66899C2212yes—
30C2×#66896C2212yes—
31C2×#66885C2212yes—
32C2×#66880C2212yes—
33C2×#66879C2212yes—
34#10622600—22no—
35#10622590C2212yes—
36C4⋊C4×Dic11—22no—
37C2×#66874C2212yes—
38C2×#66873C2212yes—
39#10622548C2211yescharacteristic
40C2×#66867C2211yescharacteristic
41#10622417—28no—
42#10622385—28no—
43D11×#97—24no—
44C22×Q8×C11C2311yescharacteristic
45C22×D4×C11C2311yescharacteristic
46C22×C22⋊D11—21no—
47C22×C22⋊D11—22no—
48C23×Dic11—21no—
49C2×#4343—22no—
50C2×#4343—24no—
51C2×#4343—24no—
52C2×#4343C2×C414yes—
53C2×#4341—24no—
54C2×#4341—22no—
55C2×#4341—24no—
56C2×#4341C2×C414yes—
57C22×D44—21no—
58C22×D44C2311yescharacteristic
59C22×C4×D11—21no—
60C22×C4×D11D412yes—
61C22×C4×D11C2311yescharacteristic
62C22×Dic22C2311yescharacteristic
63C22×C8×C11C2312yes—
64C11×#124—22no—
65C11×#124C2×C414yes—
66C2×C11×C4⋊C4—22no—
67C2×C11×C4⋊C4C2312yes—
68C2×C11×C22⋊C4—22no—
69C2×C42×C11C2311yescharacteristic
70#66997—22no—
71#66997C2×C414yes—
72#66993—22no—
73#66993C2×C414yes—
74C2×(C2×D11)⋊C4—22no—
75C2×(C2×D11)⋊C4—22no—
76C2×C4⋊Dic11—22no—
77C2×C4⋊Dic11C2312yes—
78C2×Dic11⋊C4—22no—
79C2×C4×Dic11—22no—
80C2×C4×Dic11D412yes—
81C2×C4×Dic11C2311yescharacteristic
82C22×C11⋊C8C2312yes—
83C2×C8×D11—24no—
84C2×C8×D11—24no—
85C2×C8×D11D418yescomplemented
86#66945—24no—
87#66945—24no—
88#66937—216no—
89(C22⋊D11)⋊C4—216no—
90#66928—22no—
91#66928C2×C414yes—
92C42×D11C2×C414yes—
93C42×D11D414yes—
94C11×#103—24no—
95C11×#102—24no—
96C11×#97—22no—
97#66899—24no—
98#66896—24no—
99C11⋊#95—24no—
100#66885—24no—
101#66880—24no—
102#66879—24no—
103#66874—24no—
104#66873—24no—
105#66867—22no—
106#21199—111nomaximal, Sylow
107C24×C11—21no—
108C23×D11—21no—
109C23×D11C2×D411yescharacteristic
110C2×Q8×C11—21no—
111C2×Q8×C11—22no—
112C2×Q8×C11—22no—
113C2×Q8×C11C22×C412yes—
114C2×D4×C11—22no—
115C2×D4×C11—21no—
116C2×D4×C11—22no—
117C2×D4×C11C22×C412yes—
118C22×C4×C11—21no—
119C22×C4×C11—22no—
120C22×C4×C11C2×D412yes—
121C22×C4×C11C2411yescharacteristic
122C2×C22⋊D11—28no—
123C2×C22⋊D11—24no—
124C2×C22⋊D11—44no—
125C2×C22⋊D11—44no—
126C2×C22⋊D11—42no—
127C2×C22⋊D11—42no—
128C22×Dic11—24no—
129C22×Dic11—21no—
130C22×Dic11—22no—
131C22×Dic11—42no—
132C22×Dic11—42no—
133C22×Dic11C2×D412yes—
134C22×Dic11C2×D411yescharacteristic
135#4343—216no—
136#4343—48no—
137#4341—216no—
138#4341—48no—
139C2×D44—24no—
140C2×D44—21no—
141C2×D44—22no—
142C2×D44—22no—
143C2×D44—42no—
144C2×D44—42no—
145C2×D44C22×C412yes—
146C2×C4×D11—24no—
147C2×C4×D11—42no—
148C2×C4×D11—42no—
149C2×C4×D11—42no—
150C2×C4×D11—42no—
151C2×C4×D11C22⋊C418yes—
152C2×C4×D11C22×C412yes—
153C2×C4×D11C2×D412yes—
154C2×C4×D11C2×D414yes—
155C2×C4×D11#4114yes—
156C2×Dic22—22no—
157C2×Dic22—21no—
158C2×Dic22—22no—
159C2×Dic22C22×C412yes—
160C2×C8×C11—22no—
161C2×C8×C11—22no—
162C2×C8×C11C2×D414yes—
163C11×C4⋊C4—22no—
164C11×C4⋊C4—22no—
165C11×C4⋊C4—44no—
166C11×C22⋊C4—28no—
167C42×C11C22×C412yes—
168C42×C11C2×D412yes—
169(C2×D11)⋊C4—28no—
170(C2×D11)⋊C4—44no—
171C4⋊Dic11—22no—
172C4⋊Dic11—22no—
173C4⋊Dic11—44no—
174Dic11⋊C4—28no—
175C4×Dic11—44no—
176C4×Dic11C22⋊C418yes—
177C4×Dic11C22×C412yes—
178C4×Dic11C2×D412yes—
179C2×C11⋊C8—22no—
180C2×C11⋊C8—22no—
181C2×C11⋊C8C2×D414yes—
182C8×D11—216no—
183C2×#532—111no—
184C2×#416—112no—
185C2×#416—112no—
186C22×#97—111no—
187#2285—111no—
188#1961—112no—
189#1960—112no—
190#1959—111no—
191#1959—111no—
192#1959—111no—
193#1959D1111yescharacteristic, complemented
194C23×C11—22no—
195C23×C11—41no—
196C23×C11—41no—
197C23×C11—42no—
198C23×C11—42no—
199C23×C11C22×D411yescharacteristic
200C22×D11—22no—
201C22×D11—22no—
202C22×D11—22no—
203C22×D11—41no—
204C22×D11—41no—
205C22×D11—42no—
206C22×D11—42no—
207C22×D11C2×C22⋊C412yes—
208C22×D11#12412yes—
209C22×D11#12712yes—
210Q8×C11—24no—
211Q8×C11—42no—
212D4×C11—24no—
213D4×C11—42no—
214C2×C4×C11—22no—
215C2×C4×C11—22no—
216C2×C4×C11—22no—
217C2×C4×C11—41no—
218C2×C4×C11—41no—
219C2×C4×C11—42no—
220C2×C4×C11—44no—
221C2×C4×C11C2×C22⋊C414yes—
222C2×C4×C11C23×C411yescharacteristic
223C2×C4×C11C22×D412yes—
224C2×C4×C11C22×D411yescharacteristic
225C2×C4×C11C2×#4112yes—
226C22⋊D11—416no—
227C22⋊D11—48no—
228C2×Dic11—22no—
229C2×Dic11—22no—
230C2×Dic11—22no—
231C2×Dic11—41no—
232C2×Dic11—41no—
233C2×Dic11—48no—
234C2×Dic11—42no—
235C2×Dic11—44no—
236C2×Dic11C2×C22⋊C414yes—
237C2×Dic11C2×C22⋊C412yes—
238C2×Dic11#12412yes—
239C2×Dic11#12712yes—
240D44—24no—
241D44—48no—
242D44—42no—
243C4×D11—48no—
244C4×D11—48no—
245C4×D11C4×D418yes—
246C4×D11#12118yes—
247Dic22—24no—
248Dic22—42no—
249C8×C11—24no—
250C11⋊C8—24no—
251C22×#41—111no—
252C22×#41—111no—
253C23×C8—112no—
254#532—118no—
255#532—224no—
256C2×C4⋊Q8—111no—
257C2×#127—111no—
258C2×#124—111no—
259C2×#124C2×D1111yescharacteristic
260C2×#121—222no—
261C2×C4×D4—222no—
262C2×#117—112no—
263C22×C42—111no—
264#416—228no—
265#416—228no—
266C2×#103—112no—
267C2×#103—112no—
268C2×#103—112no—
269C2×#103C2×D1112yes—
270C2×#102—112no—
271C2×#102—112no—
272C2×#102—112no—
273C2×#102C2×D1112yes—
274C2×#97—112no—
275C2×#97—111no—
276C2×#97—224no—
277C2×#97C2×D1111yescharacteristic
278#384—111no—
279#384C2×D1111yescharacteristic
280#382—112no—
281#380—222no—
282C4×C4⋊C4—222no—
283C22×C11—44no—
284C22×C11—42no—
285C22×C11—82no—
286C22×C11—81no—
287C22×C11—81no—
288C22×C11C22×C22⋊C411yescharacteristic
289C22×C11C2×#12411yescharacteristic
290C22×C11C2×#12711yescharacteristic
291C22×C11#57512yes—
292C22×C11#57512yes—
293C2×D11—24no—
294C2×D11—22no—
295C2×D11—22no—
296C2×D11—24no—
297C2×D11—44no—
298C2×D11—42no—
299C2×D11—82no—
300C2×D11—81no—
301C2×D11—81no—
302C2×D11#39014yes—
303C4×C11—24no—
304C4×C11—42no—
305C4×C11—82no—
306C4×C11C2×C4×D412yes—
307C4×C11C2×#12112yes—
308Dic11—24no—
309Dic11—42no—
310Dic11—82no—
311Dic11#39014yes—
312C2×#41—114no—
313C2×#41—114no—
314C2×#41—222no—
315C2×#41—222no—
316C2×#41—224no—
317C2×#41—224no—
318C2×#41—224no—
319C2×#41—224no—
320C22×Q8—111no—
321C22×Q8C22×D1111yescharacteristic
322C22×D4—111no—
323C22×D4—221no—
324C22×D4—222no—
325C22×D4—221no—
326C22×D4C22×D1111yescharacteristic
327C23×C4—111no—
328C23×C4—112no—
329C23×C4—221no—
330C23×C4—221no—
331C22×C8—112no—
332C22×C8—118no—
333C22×C8—224no—
334C22×C8—224no—
335C22×C8C22×D1112yes—
336C4⋊Q8—114no—
337C4⋊Q8—222no—
338#127—114no—
339#127—222no—
340#124—22no—
341#124—114no—
342#124—222no—
343#124C4×D1114yes—
344#121—2216no—
345C4×D4—2216no—
346#117—224no—
347#117—224no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—112no—
350C2×C4⋊C4—222no—
351C2×C4⋊C4—222no—
352C2×C4⋊C4C22×D1112yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—222no—
355C2×C22⋊C4—222no—
356C2×C42—112no—
357C2×C42—111no—
358C2×C42—114no—
359C2×C42—114no—
360C2×C42—222no—
361C2×C42C22×D1111yescharacteristic
362#103—24no—
363#103—224no—
364#103—224no—
365#103—224no—
366#102—24no—
367#102—224no—
368#102—224no—
369#102—224no—
370#97—22no—
371#97—224no—
372#97—222no—
373#95—224no—
374C2×C11—82no—
375C2×C11C2×#39011yescharacteristic
376C2×C11#289612yes—
377C2×C11#313512yes—
378C2×C11#315011yescharacteristic
379C2×C11#315011yescharacteristic
380D11—24no—
381D11—82no—
382C24—21no—
383C24—111no—
384C24—221no—
385#41—2216no—
386#41—2216no—
387#41—448no—
388#41—448no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—112no—
393C2×Q8—221no—
394C2×Q8—222no—
395C2×Q8—222no—
396C2×Q8C2×C4×D1112yes—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—112no—
401C2×D4—224no—
402C2×D4—228no—
403C2×D4—224no—
404C2×D4—222no—
405C2×D4—221no—
406C2×D4—222no—
407C2×D4—442no—
408C2×D4—442no—
409C2×D4—444no—
410C2×D4—444no—
411C2×D4—442no—
412C2×D4—442no—
413C2×D4C2×C4×D1112yes—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—111no—
417C22×C4—112no—
418C22×C4—112no—
419C22×C4—114no—
420C22×C4—114no—
421C22×C4—112no—
422C22×C4—118no—
423C22×C4—224no—
424C22×C4—224no—
425C22×C4—221no—
426C22×C4—222no—
427C22×C4—442no—
428C22×C4—442no—
429C22×C4—442no—
430C22×C4—442no—
431C22×C4—442no—
432C22×C4—442no—
433C22×C4D4×D1112yes—
434C22×C4C23×D1111yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—114no—
438C2×C8—2216no—
439C2×C8—222no—
440C2×C8—222no—
441C2×C8D4×D1114yes—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—228no—
446C4⋊C4—222no—
447C4⋊C4—222no—
448C4⋊C4—444no—
449C22⋊C4—28no—
450C22⋊C4—228no—
451C22⋊C4—444no—
452C42—118no—
453C42—112no—
454C42—112no—
455C42—444no—
456C42C2×C4×D1112yes—
457C42D4×D1112yes—
458C11#2119911yescharacteristic, complemented, Sylow
459C23—22no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—112no—
465C23—112no—
466C23—112no—
467C23—222no—
468C23—222no—
469C23—222no—
470C23—441no—
471C23—442no—
472C23—442no—
473C23—441no—
474C23C2×D4×D1111yescharacteristic
475Q8—24no—
476Q8—42no—
477Q8—224no—
478Q8—442no—
479D4—24no—
480D4—42no—
481D4—224no—
482D4—448no—
483D4—4416no—
484D4—448no—
485D4—442no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—112no—
494C2×C4—112no—
495C2×C4—112no—
496C2×C4—118no—
497C2×C4—118no—
498C2×C4—114no—
499C2×C4—222no—
500C2×C4—222no—
501C2×C4—222no—
502C2×C4—448no—
503C2×C4—448no—
504C2×C4—448no—
505C2×C4—441no—
506C2×C4—441no—
507C2×C4—442no—
508C2×C4—444no—
509C2×C4C22⋊C4×D1114yes—
510C2×C4C22×C4×D1111yescharacteristic
511C2×C4C2×D4×D1112yes—
512C2×C4C2×D4×D1111yescharacteristic
513C2×C4#41×D1112yes—
514C8—24no—
515C8—224no—
516C22—44no—
517C22—42no—
518C22—81no—
519C22—81no—
520C22—82no—
521C22—114no—
522C22—222no—
523C22—222no—
524C22—224no—
525C22—224no—
526C22—444no—
527C22—442no—
528C22—881no—
529C22—881no—
530C22—882no—
531C22C2×C22⋊C4×D1111yescharacteristic
532C22D11×#12411yescharacteristic
533C22D11×#12711yescharacteristic
534C22#1062330412yes—
535C22#1062330812yes—
536C4—24no—
537C4—42no—
538C4—82no—
539C4—114no—
540C4—224no—
541C4—442no—
542C4—882no—
543C4C4×D4×D1112yes—
544C4D11×#12112yes—
545C2—82no—
546C2—224no—
547C2—882no—
548C2D11×#39011yescharacteristic
549C2#1307713212yes—
550C2#1307895812yes—
551C2#1307905811yescharacteristic
552C2#1307906011yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1959C2
socleC23×C11C22×D4
prime radicalC2×#10623199C2
cosocleC22×C4×C11C24
hypercenter#1959D11
hypocenterC11#21199
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#10622417C2semidirect
3. C2×#10622385C2semidirect
4. C2×D11×#97C2semidirect
5. #13071021C2semidirect
6. #13069438C2semidirect
7. #13069434C2semidirect
8. #13069433C2semidirect
9. #13069432C2semidirect
10. #13069430C2semidirect
11. C11×#1959C2semidirect
19. #10623199C4semidirect
85. C2×C8×D11D4semidirect
193. #1959D11semidirect
458. C11#21199semidirect

Maximal subgroups

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

subgroupindex
C2×#106231992
C2×#106224172
C2×#106223852
C2×D11×#972
#130710212
#130694382
#130694342
#130694332
#130694322
#130694302
C11×#19592
#2119911