Finite Group Databasebeta

← C2×#13078222 #15620905

Subgroups — C2×#13078222 #15620905

Order 2816.

2918 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67003C211yesmaximal, characteristic, complemented
2C23×#4318C211yesmaximal, characteristic, complemented
3C2×#10623322C211yesmaximal, characteristic, complemented
4#13078222C2116yesmaximal, direct product factor
5C2×#10622665C212yesmaximal, complemented
6C2×#10622652C212yesmaximal, complemented
7C2×#10622662C212yesmaximal, complemented
8C2×#10622655C212yesmaximal, complemented
9C2×#10622611C211yesmaximal, characteristic, complemented
10C2×#10622846C211yesmaximal, characteristic, complemented
11C2×#10622615C211yesmaximal, characteristic, complemented
12C2×C11×#525C211yesmaximal, characteristic, complemented
13C22×C11×#41C2211yescharacteristic
14C24×C4×C11C2211yescharacteristic
15C22×#4339C2211yescharacteristic
16C11×#525—24no—
17C11×#525C2218yescomplemented
18C2×C11×#123—21no—
19C2×C11×C22⋊Q8C2212yes—
20C2×C11×#121C2212yes—
21C2×C11×#120—21no—
22C2×C4×D4×C11—22no—
23C2×C11×#117C2211yescharacteristic
24C2×#67006—21no—
25C2×#67003—28no—
26C2×#67003C2218yescomplemented
27C22×#4326C2211yescharacteristic, complemented
28C22×#4325C2211yescharacteristic, complemented
29C22×#4324C2211yescharacteristic, complemented
30C22×#4323C2211yescharacteristic, complemented
31#10623322—24no—
32#10623322C2218yescomplemented
33C2×#66983C2212yes—
34C2×#66982—21no—
35C2×C4×C22⋊D11—22no—
36C2×#66979C2211yescharacteristic
37C2×#66977C2212yes—
38C22×#4318—28no—
39C22×#4318C2212yes—
40C22×#4318C2212yes—
41C22×#4318C2218yescomplemented
42C23×C11⋊C8C2211yescharacteristic
43C23×C11⋊C8C2211yescharacteristic
44#10622846C2218yescomplemented
45C2×#66899C2212yescomplemented
46C2×#66896C2212yescomplemented
47#10622665C22116yescomplemented
48#10622662C22116yescomplemented
49#10622655C22116yescomplemented
50#10622652C22116yescomplemented
51#10622615C2218yescomplemented
52#10622611C2218yescomplemented
53C2×#66874C2212yescomplemented
54C2×#66873C2212yescomplemented
55C2×#66872C2212yescomplemented
56C2×#66871C2212yescomplemented
57C25×C11D411yescharacteristic
58C2×C11×#41—22no—
59C2×C11×#41—24no—
60C2×C11×#41—24no—
61C2×C11×#41C2314yes—
62C22×Q8×C11C2311yescharacteristic
63C22×D4×C11—22no—
64C22×D4×C11—21no—
65C22×D4×C11C2311yescharacteristic
66C23×C4×C11—24no—
67C23×C4×C11—24no—
68C23×C4×C11—21no—
69C23×C4×C11D412yes—
70C23×C4×C11D411yescharacteristic
71C23×C4×C11D414yes—
72C23×C4×C11C2312yes—
73C23×C4×C11C2314yes—
74C23×C4×C11C2311yescharacteristic
75C22×C22⋊D11—22no—
76C22×C22⋊D11—21no—
77C2×#4339—22no—
78C2×#4339—24no—
79C2×#4339—24no—
80C2×#4339C2314yes—
81C22×D44C2311yescharacteristic
82C22×C4×D11—21no—
83C22×Dic22C2311yescharacteristic
84C11×#123—28no—
85C11×C22⋊Q8—24no—
86C11×C22⋊Q8C2318yes—
87C11×#121—24no—
88C11×#121C2318yes—
89C11×#120—28no—
90C4×D4×C11—216no—
91C11×#117—22no—
92C11×#117C2314yes—
93C2×C11×C4⋊C4—22no—
94C2×C11×C4⋊C4C2312yes—
95C2×C11×C22⋊C4—21no—
96C2×C11×C22⋊C4—22no—
97C2×C11×C22⋊C4—22no—
98C2×C42×C11—21no—
99#67006—28no—
100C2×C22⋊Dic11—22no—
101C2×C22⋊Dic11—21no—
102#67003—232no—
103C2×#4326—24no—
104C2×#4326C2314yescomplemented
105C2×#4325—24no—
106C2×#4325C2314yescomplemented
107C2×#4324—24no—
108C2×#4324C2314yescomplemented
109C2×#4323—24no—
110C2×#4323C2314yescomplemented
111#66983—24no—
112#66983C2318yes—
113#66982—28no—
114C4×C22⋊D11—216no—
115C2×(C2×D11)⋊C4—22no—
116#66979—22no—
117#66979C2314yes—
118C2×C4⋊Dic11C2312yes—
119#66977—24no—
120#66977C2318yes—
121C2×Dic11⋊C4—22no—
122C2×C4×Dic11—21no—
123C2×#4318—28no—
124C2×#4318—28no—
125C2×#4318—232no—
126C2×#4318C2318yes—
127C2×#4318C2318yes—
128C22×C11⋊C8—24no—
129C22×C11⋊C8D418yescomplemented
130C22×C11⋊C8C2312yes—
131C22×C11⋊C8C2314yes—
132C22×C11⋊C8C2311yescharacteristic
133C22×C11⋊C8C2311yescharacteristic
134C22×C11⋊C8C2314yes—
135#66899C2318yescomplemented
136#66896C2318yescomplemented
137#66874C2318yescomplemented
138#66873C2318yescomplemented
139#66872C2318yescomplemented
140#66871C2318yescomplemented
141C2×#3054—111nomaximal, Sylow
142C24×C11—24no—
143C24×C11—21no—
144C24×C11—44no—
145C24×C11C2×D412yes—
146C24×C11C2×D411yescharacteristic
147C24×C11C2×D414yes—
148C23×D11—21no—
149C11×#41—28no—
150C11×#41—28no—
151C11×#41—48no—
152C2×Q8×C11—21no—
153C2×Q8×C11—22no—
154C2×Q8×C11—22no—
155C2×Q8×C11C2412yes—
156C2×D4×C11—28no—
157C2×D4×C11—24no—
158C2×D4×C11—22no—
159C2×D4×C11—21no—
160C2×D4×C11—22no—
161C2×D4×C11—48no—
162C2×D4×C11—44no—
163C2×D4×C11C2412yes—
164C22×C4×C11—24no—
165C22×C4×C11—28no—
166C22×C4×C11—216no—
167C22×C4×C11—24no—
168C22×C4×C11—28no—
169C22×C4×C11—24no—
170C22×C4×C11—28no—
171C22×C4×C11—22no—
172C22×C4×C11—21no—
173C22×C4×C11—24no—
174C22×C4×C11—44no—
175C22×C4×C11C2×D412yes—
176C22×C4×C11C2×D414yes—
177C22×C4×C11C2×D418yes—
178C22×C4×C11C2×D414yes—
179C22×C4×C11C2×D412yes—
180C22×C4×C11C2×D411yescharacteristic
181C22×C4×C11C2×D414yes—
182C22×C4×C11#4114yes—
183C22×C4×C11C2414yes—
184C22×C4×C11C2411yescharacteristic
185C22×C4×C11C2412yes—
186C2×C22⋊D11—28no—
187C2×C22⋊D11—24no—
188C2×C22⋊D11—48no—
189C2×C22⋊D11—44no—
190C22×Dic11—22no—
191C22×Dic11—21no—
192#4339—28no—
193#4339—28no—
194#4339—48no—
195C2×D44—21no—
196C2×D44—22no—
197C2×D44—22no—
198C2×D44C2412yes—
199C2×C4×D11—24no—
200C2×C4×D11—44no—
201C2×Dic22—22no—
202C2×Dic22—21no—
203C2×Dic22—22no—
204C2×Dic22C2412yes—
205C11×C4⋊C4—28no—
206C11×C4⋊C4—22no—
207C11×C4⋊C4C2414yes—
208C11×C22⋊C4—24no—
209C11×C22⋊C4—28no—
210C11×C22⋊C4—28no—
211C42×C11—24no—
212C22⋊Dic11—28no—
213C22⋊Dic11—24no—
214#4326—28no—
215#4325—28no—
216#4324—28no—
217#4323—28no—
218(C2×D11)⋊C4—28no—
219C4⋊Dic11—22no—
220C4⋊Dic11C2414yes—
221Dic11⋊C4—28no—
222C4×Dic11—24no—
223#4318—216no—
224#4318—216no—
225C2×C11⋊C8—24no—
226C2×C11⋊C8—28no—
227C2×C11⋊C8C2×D414yes—
228C2×C11⋊C8C2×D414yes—
229C2×C11⋊C8C2×D4116yescomplemented
230C2×C11⋊C8C2414yes—
231C2×C11⋊C8C2412yes—
232C2×C11⋊C8C2412yes—
233C22×#135—111no—
234C23×M16—111no—
235C2×#525—111no—
236C2×#525D1111yescharacteristic, complemented
237#3054—1116no—
238C2×#470—112no—
239C2×#469—112no—
240C2×#468—112no—
241C2×#468—112no—
242C2×#427—111no—
243C2×#416—111no—
244C2×#416—111no—
245C23×C11—24no—
246C23×C11—28no—
247C23×C11—24no—
248C23×C11—24no—
249C23×C11—24no—
250C23×C11—22no—
251C23×C11—48no—
252C23×C11—416no—
253C23×C11—42no—
254C23×C11—44no—
255C23×C11#12114yes—
256C23×C11C22×D414yes—
257C23×C11C22×D412yes—
258C23×C11C22×D411yescharacteristic
259C22×D11—22no—
260C22×D11—42no—
261C22×D11—44no—
262Q8×C11—22no—
263Q8×C11—22no—
264Q8×C11—42no—
265D4×C11—22no—
266D4×C11—22no—
267D4×C11—416no—
268D4×C11—48no—
269D4×C11—42no—
270C2×C4×C11—28no—
271C2×C4×C11—24no—
272C2×C4×C11—216no—
273C2×C4×C11—28no—
274C2×C4×C11—24no—
275C2×C4×C11—24no—
276C2×C4×C11—24no—
277C2×C4×C11—22no—
278C2×C4×C11—42no—
279C2×C4×C11—44no—
280C2×C4×C11—48no—
281C2×C4×C11#12114yes—
282C2×C4×C11#12118yes—
283C2×C4×C11C22×D412yes—
284C2×C4×C11C22×D414yes—
285C2×C4×C11C22×D414yes—
286C2×C4×C11C22×D412yes—
287C2×C4×C11C22×D411yescharacteristic
288C2×C4×C11C2×#4112yes—
289C2×C4×C11C2×#4114yes—
290C2×C4×C11C2511yescharacteristic
291C22⋊D11—416no—
292C22⋊D11—48no—
293C2×Dic11—24no—
294C2×Dic11—22no—
295C2×Dic11—42no—
296C2×Dic11—44no—
297D44—22no—
298D44—22no—
299D44—42no—
300C4×D11—48no—
301Dic22—22no—
302Dic22—22no—
303Dic22—42no—
304C11⋊C8—24no—
305C11⋊C8C22×D414yes—
306C11⋊C8C22×D414yes—
307C22×#41—111no—
308C22×#41C2×D1111yescharacteristic
309C24×C4C2×D1111yescharacteristic
310C2×#135—118no—
311C2×#135—228no—
312C22×Q16—111no—
313C22×SD16—111no—
314C22×SD16—111no—
315C22×D8—111no—
316C22×M16—112no—
317C22×M16—112no—
318C22×M16—118no—
319C22×M16—228no—
320C23×C8—111no—
321C23×C8—111no—
322#525—24no—
323#525—118no—
324#525—224no—
325#525C2×D1118yescomplemented
326C2×#123—21no—
327C2×#123—221no—
328C2×C22⋊Q8—112no—
329C2×C22⋊Q8C2×D1112yes—
330C2×#121—112no—
331C2×#121C2×D1112yes—
332C2×#120—21no—
333C2×#120—221no—
334C2×C4×D4—22no—
335C2×C4×D4—222no—
336C2×#117—111no—
337C2×#117C2×D1111yescharacteristic
338#470—1116no—
339#469—1116no—
340#468—1116no—
341#468—1116no—
342#427—118no—
343C2×#107—112no—
344C2×#106—112no—
345#416—118no—
346#416—118no—
347C2×#103—112no—
348C2×#103—112no—
349C2×#102—112no—
350C2×#102—112no—
351C22×C11—28no—
352C22×C11—24no—
353C22×C11—24no—
354C22×C11—24no—
355C22×C11—24no—
356C22×C11—416no—
357C22×C11—48no—
358C22×C11—42no—
359C22×C11—44no—
360C22×C11—84no—
361C22×C11C2×#12114yes—
362C22×C11C2×#12112yes—
363C22×C11#57514yes—
364C22×C11C23×D411yescharacteristic
365C2×D11—42no—
366C2×D11—44no—
367C2×D11—84no—
368C4×C11—24no—
369C4×C11—42no—
370C4×C11—44no—
371C4×C11C2×#12114yes—
372C4×C11C2×#12112yes—
373C4×C11C23×D411yescharacteristic
374C4×C11C22×#4111yescharacteristic
375Dic11—42no—
376Dic11—44no—
377C25C22⋊D1111yescharacteristic
378C2×#41—22no—
379C2×#41—24no—
380C2×#41—24no—
381C2×#41—114no—
382C2×#41—222no—
383C2×#41—224no—
384C2×#41—224no—
385C2×#41C22×D1114yes—
386C22×Q8—111no—
387C22×Q8C22×D1111yescharacteristic
388C22×D4—22no—
389C22×D4—21no—
390C22×D4—111no—
391C22×D4—222no—
392C22×D4—221no—
393C22×D4C22×D1111yescharacteristic
394C23×C4—24no—
395C23×C4—24no—
396C23×C4—21no—
397C23×C4—221no—
398C23×C4C22⋊D1111yescharacteristic
399C23×C4C22⋊D1114yes—
400C23×C4C22⋊D1112yes—
401C23×C4C22×D1112yes—
402C23×C4C22×D1114yes—
403C23×C4C22×D1111yescharacteristic
404#135—2232no—
405C2×Q16—114no—
406C2×Q16—224no—
407C2×SD16—114no—
408C2×SD16—114no—
409C2×SD16—224no—
410C2×SD16—224no—
411C2×D8—114no—
412C2×D8—224no—
413C2×M16—118no—
414C2×M16—118no—
415C2×M16—228no—
416C2×M16—228no—
417C2×M16—2232no—
418C22×C8—118no—
419C22×C8—111no—
420C22×C8—111no—
421C22×C8—114no—
422C22×C8—112no—
423C22×C8—114no—
424C22×C8—224no—
425#123—28no—
426#123—228no—
427C22⋊Q8—24no—
428C22⋊Q8—118no—
429C22⋊Q8—224no—
430C22⋊Q8C22×D1118yes—
431#121—24no—
432#121—118no—
433#121—224no—
434#121C22×D1118yes—
435#120—28no—
436#120—228no—
437C4×D4—216no—
438C4×D4—2216no—
439#117—22no—
440#117—114no—
441#117—222no—
442#117C22×D1114yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—112no—
445C2×C4⋊C4—222no—
446C2×C4⋊C4C22×D1112yes—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—221no—
451C2×C22⋊C4—222no—
452C2×C22⋊C4—222no—
453C2×C42—21no—
454C2×C42—221no—
455#107—118no—
456#106—118no—
457#103—118no—
458#103—118no—
459#102—118no—
460#102—118no—
461C2×C11—24no—
462C2×C11—44no—
463C2×C11—82no—
464C2×C11#305414yescomplemented
465C2×C11C22×#12111yescharacteristic
466C2×C11C2×#57512yes—
467D11—82no—
468C24—24no—
469C24—21no—
470C24—44no—
471C24—221no—
472C24C2×C22⋊D1112yes—
473C24C2×C22⋊D1111yescharacteristic
474C24C2×C22⋊D1114yes—
475#41—28no—
476#41—28no—
477#41—48no—
478#41—228no—
479#41—228no—
480#41—448no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—112no—
485C2×Q8—221no—
486C2×Q8—222no—
487C2×Q8—222no—
488C2×Q8C23×D1112yes—
489C2×D4—28no—
490C2×D4—24no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—48no—
495C2×D4—44no—
496C2×D4—112no—
497C2×D4—228no—
498C2×D4—224no—
499C2×D4—222no—
500C2×D4—222no—
501C2×D4—221no—
502C2×D4—448no—
503C2×D4—444no—
504C2×D4C23×D1112yes—
505C22×C4—24no—
506C22×C4—28no—
507C22×C4—216no—
508C22×C4—24no—
509C22×C4—28no—
510C22×C4—24no—
511C22×C4—28no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—24no—
515C22×C4—44no—
516C22×C4—222no—
517C22×C4—221no—
518C22×C4—224no—
519C22×C4—444no—
520C22×C4D4×D1114yes—
521C22×C4#434114yes—
522C22×C4C2×C22⋊D1112yes—
523C22×C4C2×C22⋊D1118yes—
524C22×C4C2×C22⋊D1114yes—
525C22×C4C2×C22⋊D1114yes—
526C22×C4C2×C22⋊D1112yes—
527C22×C4C2×C22⋊D1111yescharacteristic
528C22×C4C23×D1114yes—
529C22×C4C23×D1111yescharacteristic
530C22×C4C23×D1112yes—
531Q16—228no—
532SD16—228no—
533SD16—228no—
534D8—228no—
535M16—2216no—
536M16—2216no—
537C2×C8—114no—
538C2×C8—114no—
539C2×C8—1116no—
540C2×C8—112no—
541C2×C8—112no—
542C2×C8—114no—
543C2×C8—224no—
544C2×C8—228no—
545C4⋊C4—28no—
546C4⋊C4—22no—
547C4⋊C4—114no—
548C4⋊C4—222no—
549C4⋊C4—228no—
550C4⋊C4C23×D1114yes—
551C22⋊C4—24no—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—224no—
555C22⋊C4—228no—
556C22⋊C4—228no—
557C42—24no—
558C42—224no—
559C11C2×#305411yescharacteristic, complemented, Sylow
560C23—24no—
561C23—28no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—22no—
566C23—48no—
567C23—416no—
568C23—42no—
569C23—44no—
570C23—222no—
571C23—442no—
572C23—444no—
573C23#6699214yes—
574C23C22×C22⋊D1114yes—
575C23C22×C22⋊D1112yes—
576C23C22×C22⋊D1111yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—42no—
580Q8—222no—
581Q8—222no—
582Q8—442no—
583D4—22no—
584D4—22no—
585D4—416no—
586D4—48no—
587D4—42no—
588D4—222no—
589D4—222no—
590D4—4416no—
591D4—448no—
592D4—442no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—216no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—224no—
605C2×C4—222no—
606C2×C4—442no—
607C2×C4—444no—
608C2×C4—448no—
609C2×C4#6699214yes—
610C2×C4#6699218yes—
611C2×C4C2×D4×D1112yes—
612C2×C4C2×D4×D1114yes—
613C2×C4C2×#434112yes—
614C2×C4C2×#434114yes—
615C2×C4C22×C22⋊D1114yes—
616C2×C4C22×C22⋊D1112yes—
617C2×C4C22×C22⋊D1111yescharacteristic
618C2×C4C24×D1111yescharacteristic
619C8—114no—
620C8—114no—
621C8—224no—
622C22—28no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—416no—
628C22—48no—
629C22—44no—
630C22—42no—
631C22—84no—
632C22—444no—
633C22—442no—
634C22—884no—
635C22C2×#6699214yes—
636C22C2×#6699212yes—
637C22#1062335314yes—
638C22C23×C22⋊D1111yescharacteristic
639C4—24no—
640C4—44no—
641C4—42no—
642C4—444no—
643C4—442no—
644C4C2×#6699214yes—
645C4C2×#6699212yes—
646C4C22×D4×D1111yescharacteristic
647C4C22×#434111yescharacteristic
648C2—24no—
649C2—44no—
650C2—82no—
651C2—882no—
652C2#1307822214yesdirect product factor
653C2C22×#6699211yescharacteristic
654C2C2×#1062335312yes—
subgroupG/H
centerC22×C4C2×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingC2×C11×#525C2
socleC23×C11C22×D4
hypercenterC2×#525D11
hypocenterC11C2×#3054
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C22×#67003C2semidirect
2. C23×#4318C2semidirect
3. C2×#10623322C2semidirect
4. #13078222C2direct (N × H)
5. C2×#10622665C2semidirect
6. C2×#10622652C2semidirect
7. C2×#10622662C2semidirect
8. C2×#10622655C2semidirect
9. C2×#10622611C2semidirect
10. C2×#10622846C2semidirect
11. C2×#10622615C2semidirect
12. C2×C11×#525C2semidirect
17. C11×#525C22semidirect
26. C2×#67003C22semidirect
27. C22×#4326C22semidirect
28. C22×#4325C22semidirect
29. C22×#4324C22semidirect
30. C22×#4323C22semidirect
32. #10623322C22semidirect
41. C22×#4318C22semidirect
44. #10622846C22semidirect
45. C2×#66899C22semidirect
46. C2×#66896C22semidirect
47. #10622665C22semidirect
48. #10622662C22semidirect
49. #10622655C22semidirect
50. #10622652C22semidirect
51. #10622615C22semidirect
52. #10622611C22semidirect
53. C2×#66874C22semidirect
54. C2×#66873C22semidirect
55. C2×#66872C22semidirect
56. C2×#66871C22semidirect
104. C2×#4326C23semidirect
106. C2×#4325C23semidirect
108. C2×#4324C23semidirect
110. C2×#4323C23semidirect
129. C22×C11⋊C8D4semidirect
135. #66899C23semidirect
136. #66896C23semidirect
137. #66874C23semidirect
138. #66873C23semidirect
139. #66872C23semidirect
140. #66871C23semidirect
229. C2×C11⋊C8C2×D4semidirect
236. C2×#525D11semidirect
325. #525C2×D11semidirect
464. C2×C11#3054semidirect
559. C11C2×#3054semidirect
652. C2#13078222direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#670032
C23×#43182
C2×#106233222
#130782222
C2×#106226652
C2×#106226522
C2×#106226622
C2×#106226552
C2×#106226112
C2×#106228462
C2×#106226152
C2×C11×#5252
C2×#305411