Finite Group Databasebeta

← #15620743

Subgroups — #15620743

Order 2816.

2674 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66962C211yesmaximal, characteristic, complemented
2C22×#67004C211yesmaximal, characteristic, complemented
3#13081945C211yesmaximal, characteristic
4C2×#10623047C211yesmaximal, characteristic, complemented
5#13078149C214yesmaximal, complemented
6#13078145C214yesmaximal, complemented
7#13078140C214yesmaximal, complemented
8#13078136C214yesmaximal, complemented
9#13078087C218yesmaximal, complemented
10#13077179C211yesmaximal, characteristic, complemented
11#13077133C211yesmaximal, characteristic, complemented
12C11×#2896C211yesmaximal, characteristic, complemented
13C22×C11×#41C2211yescharacteristic
14C22×#4339C2211yescharacteristic
15C23×Dic22C2211yescharacteristic
16C22×C11×M16C2211yescharacteristic
17C2×C11×#117C2211yescharacteristic
18#10623364—22no—
19#10623363C2214yes—
20#10623361C2212yes—
21#10623359—22no—
22#10623356C2212yes—
23C2×#67004—28no—
24C2×#67004C2214yes—
25C2×#67004C2214yescomplemented
26#10623347C2212yes—
27C22×#4326C2212yescomplemented
28#10623336C2212yes—
29#10623335—22no—
30C22×#4324C2212yescomplemented
31#10623322—22no—
32C2×#66979C2211yescharacteristic
33C22×#4318C2211yescharacteristic
34C2×#66962—28no—
35C2×#66962C2214yes—
36C2×#66962C2214yescomplemented
37C22×Dic44C2212yescomplemented
38C22×#4314C2212yescomplemented
39#10623047—22no—
40#10623047—22no—
41#10623047C2214yescomplemented
42#10623047C2214yescomplemented
43C2×#66940C2214yescomplemented
44C2×#66930—22no—
45C2×C4×Dic22—22no—
46C11×#419C2212yescomplemented
47C11×#418C2212yescomplemented
48C11×#417C2214yescomplemented
49C11×#416C2214yes—
50#10622732C2214yescomplemented
51#10622731C2214yescomplemented
52#10622715C2214yescomplemented
53#10622617C2212yescomplemented
54#10622615C2214yes—
55#10622614C2212yescomplemented
56#10622579C2214yescomplemented
57#10622411C22116yescomplemented
58#10622406C22116yescomplemented
59#10622380C22116yescomplemented
60#10622375C22116yescomplemented
61C2×C11×#41—21no—
62C2×C11×#41—21no—
63C2×C11×#41—24no—
64C2×C11×#41—24no—
65C2×C11×#41C2312yes—
66C2×C11×#41C2312yes—
67C22×Q8×C11C2311yescharacteristic
68C22×D4×C11—21no—
69C22×D4×C11C2311yescharacteristic
70C23×C4×C11—21no—
71C23×C4×C11C2311yescharacteristic
72C22×C22⋊D11—21no—
73C23×Dic11—21no—
74C2×#4339—21no—
75C2×#4339—21no—
76C2×#4339—24no—
77C2×#4339—24no—
78C2×#4339C2312yes—
79C2×#4339C2312yes—
80C22×D44C2311yescharacteristic
81C22×C4×D11—21no—
82C22×Dic22—21no—
83C22×Dic22—21no—
84C22×Dic22—21no—
85C22×Dic22—24no—
86C22×Dic22D418yescomplemented
87C22×Dic22C2311yescharacteristic
88C22×Dic22C2312yes—
89C22×Dic22C2312yes—
90C22×Dic22C2312yes—
91C2×C11×M16—24no—
92C2×C11×M16C2312yes—
93C2×C11×M16C2312yes—
94C22×C8×C11C2312yes—
95C11×#117—21no—
96C11×#117—21no—
97C11×#117C2312yes—
98C11×#117C2312yes—
99C2×C11×C4⋊C4C2312yes—
100C2×C11×C22⋊C4—21no—
101C2×C42×C11—21no—
102#67006—24no—
103C2×C22⋊Dic11D412yes—
104#67004—216no—
105#67004—216no—
106#67000C2314yes—
107#66999—24no—
108#66997C2314yes—
109C2×#4326—28no—
110C2×#4326C2318yes—
111#66993C2314yes—
112#66992—28no—
113#66991—24no—
114#66990—24no—
115#66989C2314yes—
116#66988—24no—
117C2×#4324—28no—
118C2×#4324C2318yes—
119#66983—24no—
120#66982—24no—
121C4×C22⋊D11—28no—
122#66979D412yes—
123#66979D412yes—
124#66979C2312yes—
125#66979C2312yes—
126C2×C4⋊Dic11D412yes—
127#66977—24no—
128C2×Dic11⋊C4—24no—
129C2×C4×Dic11—24no—
130C2×C4×Dic11C2312yes—
131C2×#4318—24no—
132C2×#4318C2312yes—
133C2×#4318C2312yes—
134C22×C11⋊C8C2312yes—
135#66962—216no—
136#66962—216no—
137C2×Dic44—28no—
138C2×Dic44C2318yes—
139C2×#4314—28no—
140C2×#4314C2318yes—
141#66940—28no—
142#66940C23116yes—
143#66930—28no—
144#66930—28no—
145C4×Dic22—28no—
146C4×Dic22—28no—
147C11×#103C2318yes—
148C11×#102C2318yes—
149Dic11⋊C8C23116yes—
150#66874C2318yes—
151#66873C2318yes—
152#60424—111nomaximal, Sylow
153C24×C11—21no—
154C24×C11C2×D411yescharacteristic
155C23×D11—21no—
156C11×#41—24no—
157C11×#41—24no—
158C11×#41—24no—
159C11×#41—24no—
160C11×#41—44no—
161C11×#41—44no—
162C2×Q8×C11—21no—
163C2×Q8×C11—22no—
164C2×Q8×C11—22no—
165C2×Q8×C11C2412yes—
166C2×D4×C11—22no—
167C2×D4×C11—22no—
168C2×D4×C11—22no—
169C2×D4×C11—21no—
170C2×D4×C11—22no—
171C2×D4×C11—44no—
172C2×D4×C11C2412yes—
173C22×C4×C11—22no—
174C22×C4×C11—21no—
175C22×C4×C11—22no—
176C22×C4×C11—22no—
177C22×C4×C11—44no—
178C22×C4×C11C2×D411yescharacteristic
179C22×C4×C11C2×D411yescharacteristic
180C22×C4×C11C2×D411yescharacteristic
181C22×C4×C11C2×D414yes—
182C22×C4×C11#4114yes—
183C22×C4×C11C2411yescharacteristic
184C22×C4×C11C2411yescharacteristic
185C22×C4×C11C2411yescharacteristic
186C2×C22⋊D11—28no—
187C2×C22⋊D11—22no—
188C2×C22⋊D11—22no—
189C2×C22⋊D11—44no—
190C22×Dic11—28no—
191C22×Dic11—22no—
192C22×Dic11—22no—
193C22×Dic11—22no—
194C22×Dic11—21no—
195C22×Dic11C2×D414yes—
196#4339—24no—
197#4339—24no—
198#4339—24no—
199#4339—24no—
200#4339—44no—
201#4339—44no—
202C2×D44—21no—
203C2×D44—22no—
204C2×D44—22no—
205C2×D44C2412yes—
206C2×C4×D11—22no—
207C2×C4×D11—22no—
208C2×C4×D11—44no—
209C2×Dic22—28no—
210C2×Dic22—28no—
211C2×Dic22—28no—
212C2×Dic22—22no—
213C2×Dic22—22no—
214C2×Dic22—22no—
215C2×Dic22—22no—
216C2×Dic22—21no—
217C2×Dic22—24no—
218C2×Dic22—24no—
219C2×Dic22—24no—
220C2×Dic22—22no—
221C2×Dic22C2×D418yes—
222C2×Dic22C2×D418yes—
223C2×Dic22C2×D418yes—
224C2×Dic22C2412yes—
225C2×Dic22C2414yes—
226C11×M16—24no—
227C11×M16—24no—
228C2×C8×C11—24no—
229C2×C8×C11C2414yes—
230C11×C4⋊C4—22no—
231C11×C4⋊C4C2414yes—
232C11×C22⋊C4—22no—
233C11×C22⋊C4—22no—
234C42×C11—22no—
235C42×C11—22no—
236C22⋊Dic11—28no—
237C22⋊Dic11C2×D414yes—
238C22⋊Dic11C2×D414yes—
239#4326—216no—
240#4324—216no—
241(C2×D11)⋊C4—28no—
242C4⋊Dic11C2×D414yes—
243C4⋊Dic11C2×D414yes—
244Dic11⋊C4—216no—
245Dic11⋊C4—28no—
246C4×Dic11—216no—
247C4×Dic11C2×D414yes—
248C4×Dic11C2414yes—
249#4318—24no—
250#4318—24no—
251C2×C11⋊C8—24no—
252C2×C11⋊C8C2414yes—
253Dic44—216no—
254#4314—216no—
255C22×#137—111no—
256C22×#137—111no—
257#3449—111no—
258C2×#519—111no—
259#3044—114no—
260#3044—114no—
261#3043—114no—
262#3043—114no—
263#3037—118no—
264#2905—111no—
265#2896—111no—
266#2896D1111yescharacteristic, complemented
267C23×C11—22no—
268C23×C11—42no—
269C23×C11—42no—
270C23×C11—82no—
271C23×C11#12114yes—
272C23×C11C22×D411yescharacteristic
273C23×C11C22×D411yescharacteristic
274C23×C11C22×D411yescharacteristic
275C22×D11—22no—
276C22×D11—42no—
277C22×D11—82no—
278Q8×C11—22no—
279Q8×C11—22no—
280Q8×C11—42no—
281D4×C11—22no—
282D4×C11—22no—
283D4×C11—44no—
284D4×C11—44no—
285D4×C11—42no—
286C2×C4×C11—24no—
287C2×C4×C11—22no—
288C2×C4×C11—44no—
289C2×C4×C11—42no—
290C2×C4×C11—44no—
291C2×C4×C11—44no—
292C2×C4×C11#12114yes—
293C2×C4×C11#12114yes—
294C2×C4×C11#12114yes—
295C2×C4×C11C22×D411yescharacteristic
296C2×C4×C11C22×D411yescharacteristic
297C2×C4×C11C22×D411yescharacteristic
298C2×C4×C11C22×D412yes—
299C2×C4×C11C22×D412yes—
300C2×C4×C11C22×D412yes—
301C2×C4×C11C2×#4112yes—
302C2×C4×C11C2×#4112yes—
303C2×C4×C11C2×#4112yes—
304C2×C4×C11C2511yescharacteristic
305C22⋊D11—44no—
306C22⋊D11—44no—
307C22⋊D11—88no—
308C2×Dic11—28no—
309C2×Dic11—28no—
310C2×Dic11—28no—
311C2×Dic11—24no—
312C2×Dic11—22no—
313C2×Dic11—44no—
314C2×Dic11—42no—
315C2×Dic11C22×D418yes—
316D44—22no—
317D44—22no—
318D44—42no—
319C4×D11—44no—
320C4×D11—44no—
321Dic22—28no—
322Dic22—28no—
323Dic22—28no—
324Dic22—24no—
325Dic22—22no—
326Dic22—22no—
327Dic22—42no—
328Dic22C22×D418yes—
329C8×C11—24no—
330C11⋊C8—24no—
331C22×#41—111no—
332C22×#41C2×D1111yescharacteristic
333C23×Q8—111no—
334C2×#137—114no—
335C2×#137—114no—
336C2×#137—114no—
337C2×#137—114no—
338C2×#137—228no—
339C2×#137—228no—
340C22×Q16—112no—
341C22×Q16—112no—
342C22×SD16—112no—
343C22×SD16—112no—
344C22×M16—111no—
345C22×M16C2×D1111yescharacteristic
346#537—222no—
347#536—112no—
348#536—112no—
349#535—112no—
350#535—112no—
351#534—222no—
352#532—114no—
353#525—222no—
354#525—222no—
355#519—114no—
356#519—114no—
357#519—222no—
358#519—222no—
359C2×C4×Q8—114no—
360C2×C4×Q8—222no—
361C2×#117—111no—
362C2×#117—222no—
363C2×#117C2×D1111yescharacteristic
364#464—1116no—
365#464—1116no—
366#462—1116no—
367#460—1116no—
368#424—114no—
369#424—114no—
370#423—114no—
371#419—112no—
372#419C2×D1112yescomplemented
373#418—112no—
374#418C2×D1112yescomplemented
375#417—114no—
376#417C2×D1114yescomplemented
377#416—114no—
378#416C2×D1114yes—
379C22×C11—22no—
380C22×C11—22no—
381C22×C11—22no—
382C22×C11—44no—
383C22×C11—44no—
384C22×C11—42no—
385C22×C11—84no—
386C22×C11C2×#12112yes—
387C22×C11C2×#12112yes—
388C22×C11C2×#12112yes—
389C22×C11C23×D411yescharacteristic
390C2×D11—44no—
391C2×D11—42no—
392C2×D11—84no—
393C4×C11—44no—
394C4×C11—42no—
395C4×C11C2×#12112yes—
396C4×C11C2×#12112yes—
397C4×C11C2×#12112yes—
398C4×C11C23×D411yescharacteristic
399C4×C11C22×#4111yescharacteristic
400Dic11—28no—
401Dic11—28no—
402Dic11—42no—
403C2×#41—21no—
404C2×#41—21no—
405C2×#41—24no—
406C2×#41—24no—
407C2×#41—112no—
408C2×#41—112no—
409C2×#41—221no—
410C2×#41—221no—
411C2×#41—224no—
412C2×#41—224no—
413C2×#41C22×D1112yes—
414C2×#41C22×D1112yes—
415C22×Q8—118no—
416C22×Q8—112no—
417C22×Q8—112no—
418C22×Q8—112no—
419C22×Q8—111no—
420C22×Q8—224no—
421C22×Q8—221no—
422C22×Q8—221no—
423C22×Q8—221no—
424C22×Q8C22×D1111yescharacteristic
425C22×D4—21no—
426C22×D4—111no—
427C22×D4—221no—
428C22×D4C22×D1111yescharacteristic
429C23×C4—21no—
430C23×C4—221no—
431C23×C4—221no—
432C23×C4C22×D1111yescharacteristic
433#137—2216no—
434#137—2216no—
435#137—2216no—
436#137—2216no—
437C2×Q16—118no—
438C2×Q16—118no—
439C2×Q16—228no—
440C2×Q16—228no—
441C2×SD16—118no—
442C2×SD16—118no—
443C2×SD16—228no—
444C2×SD16—228no—
445C2×M16—24no—
446C2×M16—112no—
447C2×M16—112no—
448C2×M16—224no—
449C2×M16C22×D1112yes—
450C2×M16C22×D1112yes—
451C22×C8—112no—
452C22×C8C22×D1112yes—
453C4⋊Q8—114no—
454#127—114no—
455#124—114no—
456#124—114no—
457#123—224no—
458#123—224no—
459C22⋊Q8—224no—
460C22⋊Q8—224no—
461#121—224no—
462#121—224no—
463#121—228no—
464#120—224no—
465#120—224no—
466C4×Q8—1116no—
467C4×Q8—228no—
468C4×Q8—228no—
469C4×Q8—228no—
470C4×D4—228no—
471#117—21no—
472#117—21no—
473#117—112no—
474#117—112no—
475#117—112no—
476#117—112no—
477#117—228no—
478#117—228no—
479#117C22×D1112yes—
480#117C22×D1112yes—
481C2×C4⋊C4—112no—
482C2×C4⋊C4—224no—
483C2×C4⋊C4C22×D1112yes—
484C2×C22⋊C4—21no—
485C2×C22⋊C4—112no—
486C2×C42—21no—
487C2×C42—112no—
488C2×C42—224no—
489C4⋊C8—1116no—
490#103—118no—
491#103C22×D1118yes—
492#102—118no—
493#102C22×D1118yes—
494C2×C11—22no—
495C2×C11—22no—
496C2×C11—22no—
497C2×C11—82no—
498C2×C11C22×#12111yescharacteristic
499C2×C11#358612yes—
500D11—82no—
501C24—21no—
502C24—221no—
503C24D4×D1111yescharacteristic
504#41—24no—
505#41—24no—
506#41—24no—
507#41—24no—
508#41—44no—
509#41—44no—
510#41—224no—
511#41—224no—
512#41—224no—
513#41—224no—
514#41—444no—
515#41—444no—
516C2×Q8—21no—
517C2×Q8—22no—
518C2×Q8—22no—
519C2×Q8—118no—
520C2×Q8—118no—
521C2×Q8—118no—
522C2×Q8—114no—
523C2×Q8—112no—
524C2×Q8—228no—
525C2×Q8—228no—
526C2×Q8—228no—
527C2×Q8—224no—
528C2×Q8—224no—
529C2×Q8—224no—
530C2×Q8—222no—
531C2×Q8—222no—
532C2×Q8—222no—
533C2×Q8—221no—
534C2×Q8—222no—
535C2×Q8—222no—
536C2×Q8C23×D1112yes—
537C2×D4—22no—
538C2×D4—22no—
539C2×D4—22no—
540C2×D4—22no—
541C2×D4—21no—
542C2×D4—44no—
543C2×D4—112no—
544C2×D4—228no—
545C2×D4—222no—
546C2×D4—222no—
547C2×D4—222no—
548C2×D4—222no—
549C2×D4—221no—
550C2×D4—444no—
551C2×D4C23×D1112yes—
552C22×C4—22no—
553C22×C4—21no—
554C22×C4—22no—
555C22×C4—22no—
556C22×C4—44no—
557C22×C4—114no—
558C22×C4—228no—
559C22×C4—222no—
560C22×C4—222no—
561C22×C4—222no—
562C22×C4—221no—
563C22×C4—222no—
564C22×C4—222no—
565C22×C4—444no—
566C22×C4#433914yes—
567C22×C4D4×D1111yescharacteristic
568C22×C4D4×D1111yescharacteristic
569C22×C4D4×D1111yescharacteristic
570C22×C4D4×D1114yes—
571C22×C4C23×D1111yescharacteristic
572C22×C4C23×D1111yescharacteristic
573C22×C4C23×D1111yescharacteristic
574Q16—2216no—
575Q16—2216no—
576SD16—2216no—
577SD16—2216no—
578M16—24no—
579M16—24no—
580M16—224no—
581M16—224no—
582C2×C8—24no—
583C2×C8—114no—
584C2×C8—224no—
585C2×C8C23×D1114yes—
586C4⋊C4—22no—
587C4⋊C4—114no—
588C4⋊C4—114no—
589C4⋊C4—2216no—
590C4⋊C4—228no—
591C4⋊C4C23×D1114yes—
592C22⋊C4—22no—
593C22⋊C4—22no—
594C22⋊C4—114no—
595C22⋊C4—114no—
596C22⋊C4—228no—
597C22⋊C4—228no—
598C42—22no—
599C42—22no—
600C42—114no—
601C42—114no—
602C42—2216no—
603C11#6042411yescharacteristic, complemented, Sylow
604C23—22no—
605C23—42no—
606C23—42no—
607C23—82no—
608C23—222no—
609C23—442no—
610C23—882no—
611C23#6693714yes—
612C23C2×D4×D1111yescharacteristic
613C23C2×D4×D1111yescharacteristic
614C23C2×D4×D1111yescharacteristic
615Q8—22no—
616Q8—22no—
617Q8—42no—
618Q8—118no—
619Q8—228no—
620Q8—228no—
621Q8—228no—
622Q8—224no—
623Q8—222no—
624Q8—222no—
625Q8—442no—
626D4—22no—
627D4—22no—
628D4—44no—
629D4—44no—
630D4—42no—
631D4—222no—
632D4—222no—
633D4—444no—
634D4—444no—
635D4—442no—
636D4—888no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—118no—
644C2×C4—228no—
645C2×C4—228no—
646C2×C4—228no—
647C2×C4—224no—
648C2×C4—222no—
649C2×C4—444no—
650C2×C4—442no—
651C2×C4—444no—
652C2×C4—444no—
653C2×C4#6693714yes—
654C2×C4#6693714yes—
655C2×C4#6693714yes—
656C2×C4C2×#433912yes—
657C2×C4C2×#433912yes—
658C2×C4C2×#433912yes—
659C2×C4C2×D4×D1111yescharacteristic
660C2×C4C2×D4×D1111yescharacteristic
661C2×C4C2×D4×D1111yescharacteristic
662C2×C4C2×D4×D1112yes—
663C2×C4C2×D4×D1112yes—
664C2×C4C2×D4×D1112yes—
665C2×C4C24×D1111yescharacteristic
666C8—24no—
667C8—224no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—84no—
675C22—444no—
676C22—442no—
677C22—884no—
678C22C2×#6693712yes—
679C22C2×#6693712yes—
680C22C2×#6693712yes—
681C22C22×D4×D1111yescharacteristic
682C4—44no—
683C4—42no—
684C4—228no—
685C4—228no—
686C4—442no—
687C4C2×#6693712yes—
688C4C2×#6693712yes—
689C4C2×#6693712yes—
690C4C22×#433911yescharacteristic
691C4C22×D4×D1111yescharacteristic
692C2—22no—
693C2—22no—
694C2—22no—
695C2—82no—
696C2—882no—
697C2C22×#6693711yescharacteristic
698C2#1308340012yes—
subgroupG/H
centerC22C22×D4×D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingC11×#2896C2
socleC22×C11C23×D4
prime radical#13081945C2
hypercenter#2896D11
hypocenterC11#60424
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. C22×#66962C2semidirect
2. C22×#67004C2semidirect
4. C2×#10623047C2semidirect
5. #13078149C2semidirect
6. #13078145C2semidirect
7. #13078140C2semidirect
8. #13078136C2semidirect
9. #13078087C2semidirect
10. #13077179C2semidirect
11. #13077133C2semidirect
12. C11×#2896C2semidirect
25. C2×#67004C22semidirect
27. C22×#4326C22semidirect
30. C22×#4324C22semidirect
36. C2×#66962C22semidirect
37. C22×Dic44C22semidirect
38. C22×#4314C22semidirect
41. #10623047C22semidirect
42. #10623047C22semidirect
43. C2×#66940C22semidirect
46. C11×#419C22semidirect
47. C11×#418C22semidirect
48. C11×#417C22semidirect
50. #10622732C22semidirect
51. #10622731C22semidirect
52. #10622715C22semidirect
53. #10622617C22semidirect
55. #10622614C22semidirect
56. #10622579C22semidirect
57. #10622411C22semidirect
58. #10622406C22semidirect
59. #10622380C22semidirect
60. #10622375C22semidirect
86. C22×Dic22D4semidirect
266. #2896D11semidirect
372. #419C2×D11semidirect
374. #418C2×D11semidirect
376. #417C2×D11semidirect
603. C11#60424semidirect

Maximal subgroups

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

subgroupindex
C22×#669622
C22×#670042
#130819452
C2×#106230472
#130781492
#130781452
#130781402
#130781362
#130780872
#130771792
#130771332
C11×#28962
#6042411