Finite Group Databasebeta

← C2×#13078749 #15621880

Subgroups — C2×#13078749 #15621880

Order 2816.

2582 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623131C211yesmaximal, characteristic, complemented
2C22×D4×Dic11C211yesmaximal, characteristic, complemented
3#13078749C2116yesmaximal, direct product factor
4C2×#10622334C211yesmaximal, characteristic, complemented
5C2×#10622646C211yesmaximal, characteristic, complemented
6C2×#10622376C212yesmaximal, complemented
7C2×#10622766C212yesmaximal, complemented
8C2×#10622831C211yesmaximal, characteristic, complemented
9C2×#10622765C212yesmaximal, complemented
10C2×#10622370C212yesmaximal, complemented
11C2×#10622332C211yesmaximal, characteristic, complemented
12C2×C11×#447C211yesmaximal, characteristic
13C23×D4×C11C2211yescharacteristic
14C24×Dic11—21no—
15C22×C11×D8C2212yes—
16C2×C11×#121C2211yescharacteristic
17C22×C22⋊Dic11—21no—
18C2×#66989C2211yescharacteristic, complemented
19C2×#66988—21no—
20C2×D4×Dic11—21no—
21C2×D4×Dic11—21no—
22C2×D4×Dic11—21no—
23C2×D4×Dic11—24no—
24C2×D4×Dic11C2212yes—
25C2×D4×Dic11C2212yes—
26C2×D4×Dic11C2218yescomplemented
27C2×D4×Dic11C2211yescharacteristic, complemented
28C22×#4324C2212yes—
29C22×C4⋊Dic11C2211yescharacteristic
30C2×#66977C2211yescharacteristic
31C22×C4×Dic11C2211yescharacteristic
32#10623131—24no—
33#10623131C2218yescomplemented
34C2×#66943C2211yescharacteristic, complemented
35C2×#66940C2211yescharacteristic, complemented
36C2×#66932—21no—
37C2×#66930—21no—
38C11×#447C2218yes—
39C2×C11×#102C2212yes—
40C2×C11×C22⋊C8C2211yescharacteristic
41#10622831C2218yescomplemented
42C2×#66896C2212yes—
43C2×#66896C2212yescomplemented
44C2×C22⋊(C11⋊C8)C2211yescharacteristic
45#10622766C22116yescomplemented
46#10622765C22116yescomplemented
47C2×#66881C2212yescomplemented
48C2×#66880C2212yescomplemented
49C2×#66879C2212yescomplemented
50C2×Dic11⋊C8C2212yescomplemented
51#10622646C2218yescomplemented
52C2×#66871C2212yescomplemented
53#10622376C22116yescomplemented
54#10622370C22116yescomplemented
55#10622334C2218yescomplemented
56#10622332C2218yescomplemented
57C25×C11—21no—
58C22×D4×C11—28no—
59C22×D4×C11—21no—
60C22×D4×C11—21no—
61C22×D4×C11—22no—
62C22×D4×C11C2314yes—
63C22×D4×C11C2311yescharacteristic
64C22×D4×C11C2312yes—
65C23×C4×C11C2311yescharacteristic
66C23×Dic11—21no—
67C23×Dic11—21no—
68C23×Dic11—22no—
69C23×Dic11—22no—
70C23×Dic11—28no—
71C23×Dic11—48no—
72C23×Dic11D411yescharacteristic
73C23×Dic11D411yescharacteristic
74C22×Dic22C2311yescharacteristic
75C2×C11×D8—28no—
76C2×C11×D8C2318yes—
77C22×C8×C11C2312yes—
78C11×#121—22no—
79C11×#121C2314yes—
80C2×C11×C4⋊C4C2311yescharacteristic
81C2×C11×C22⋊C4—21no—
82C2×C22⋊Dic11—21no—
83C2×C22⋊Dic11—21no—
84C2×C22⋊Dic11—21no—
85C2×C22⋊Dic11—21no—
86C2×C22⋊Dic11—22no—
87C2×C22⋊Dic11—22no—
88C2×C22⋊Dic11—28no—
89#66989—22no—
90#66989C2314yescomplemented
91#66988—28no—
92D4×Dic11—28no—
93D4×Dic11—24no—
94D4×Dic11—24no—
95D4×Dic11—28no—
96D4×Dic11—28no—
97D4×Dic11—22no—
98D4×Dic11C2318yes—
99D4×Dic11C2318yes—
100D4×Dic11C2314yescomplemented
101C2×#4324—28no—
102C2×#4324C2318yes—
103C2×C4⋊Dic11—21no—
104C2×C4⋊Dic11—22no—
105C2×C4⋊Dic11—21no—
106C2×C4⋊Dic11C2311yescharacteristic
107C2×C4⋊Dic11C2312yes—
108C2×C4⋊Dic11C2314yes—
109#66977—22no—
110#66977C2314yes—
111C2×Dic11⋊C4—21no—
112C2×Dic11⋊C4—21no—
113C2×Dic11⋊C4—21no—
114C2×C4×Dic11—21no—
115C2×C4×Dic11D411yescharacteristic
116C2×C4×Dic11D411yescharacteristic
117C2×C4×Dic11D414yes—
118C2×C4×Dic11C2311yescharacteristic
119C2×C4×Dic11C2311yescharacteristic
120C2×C4×Dic11C2314yes—
121C22×C11⋊C8C2312yes—
122#66943—22no—
123#66943C2314yescomplemented
124#66940—22no—
125#66940C2314yescomplemented
126#66932—28no—
127#66930—28no—
128C11×#102C2318yes—
129C11×C22⋊C8C2314yes—
130#66896C2318yes—
131#66896C2318yescomplemented
132C22⋊(C11⋊C8)C2314yes—
133#66881C2318yescomplemented
134#66880C2318yescomplemented
135#66879C2318yescomplemented
136Dic11⋊C8C2318yescomplemented
137#66871C2318yescomplemented
138C2×#3114—111nomaximal, Sylow
139C24×C11—28no—
140C24×C11—24no—
141C24×C11—22no—
142C24×C11—21no—
143C24×C11—44no—
144C24×C11—44no—
145C24×C11C2×D411yescharacteristic
146C2×D4×C11—28no—
147C2×D4×C11—216no—
148C2×D4×C11—216no—
149C2×D4×C11—24no—
150C2×D4×C11—24no—
151C2×D4×C11—22no—
152C2×D4×C11—21no—
153C2×D4×C11—24no—
154C2×D4×C11—24no—
155C2×D4×C11—44no—
156C2×D4×C11#4118yes—
157C2×D4×C11C2412yes—
158C2×D4×C11C2414yes—
159C22×C4×C11—24no—
160C22×C4×C11—21no—
161C22×C4×C11C2×D411yescharacteristic
162C22×C4×C11C2×D412yes—
163C22×C4×C11C2412yes—
164C22×C4×C11C2411yescharacteristic
165C22×Dic11—24no—
166C22×Dic11—28no—
167C22×Dic11—28no—
168C22×Dic11—24no—
169C22×Dic11—21no—
170C22×Dic11—21no—
171C22×Dic11—24no—
172C22×Dic11—22no—
173C22×Dic11—42no—
174C22×Dic11—44no—
175C22×Dic11—48no—
176C22×Dic11—48no—
177C22×Dic11—432no—
178C22×Dic11—44no—
179C22×Dic11C2×D411yescharacteristic
180C22×Dic11C2×D411yescharacteristic
181C22×Dic11C2×D411yescharacteristic
182C22×Dic11C2×D411yescharacteristic
183C22×Dic11C2×D414yes—
184C22×Dic11C2×D414yes—
185C2×Dic22—21no—
186C2×Dic22—24no—
187C2×Dic22C2412yes—
188C11×D8—216no—
189C2×C8×C11—24no—
190C2×C8×C11C2414yes—
191C11×C4⋊C4—21no—
192C11×C4⋊C4C2412yes—
193C11×C22⋊C4—24no—
194C22⋊Dic11—24no—
195C22⋊Dic11—24no—
196C22⋊Dic11—24no—
197C22⋊Dic11—28no—
198C22⋊Dic11—28no—
199C22⋊Dic11—24no—
200#4324—216no—
201C4⋊Dic11—24no—
202C4⋊Dic11—21no—
203C4⋊Dic11—24no—
204C4⋊Dic11—22no—
205C4⋊Dic11C2414yes—
206C4⋊Dic11C2412yes—
207Dic11⋊C4—24no—
208Dic11⋊C4—24no—
209Dic11⋊C4—24no—
210C4×Dic11—24no—
211C4×Dic11C2×D414yes—
212C4×Dic11C2×D414yes—
213C4×Dic11C2×D412yes—
214C4×Dic11C2×D412yes—
215C4×Dic11C2412yes—
216C4×Dic11C2412yes—
217C2×C11⋊C8—24no—
218C2×C11⋊C8C2414yes—
219C2×#529—111no—
220C22×C4×D4—111no—
221#3114—1116no—
222C2×#483—111no—
223C2×#480—111no—
224C2×#478—112no—
225C2×#463—112no—
226C2×#450—111no—
227C2×#447D1111yescharacteristic
228C2×#442—112no—
229C2×#440—112no—
230C2×#432—111no—
231C23×C11—22no—
232C23×C11—22no—
233C23×C11—216no—
234C23×C11—28no—
235C23×C11—24no—
236C23×C11—22no—
237C23×C11—432no—
238C23×C11—48no—
239C23×C11—48no—
240C23×C11—44no—
241C23×C11—44no—
242C23×C11—42no—
243C23×C11—42no—
244C23×C11—42no—
245C23×C11AGL(1,ℤ/8)14yes—
246C23×C11C22×D412yes—
247C23×C11C22×D411yescharacteristic
248D4×C11—216no—
249D4×C11—28no—
250D4×C11—24no—
251D4×C11—24no—
252D4×C11—48no—
253D4×C11—42no—
254D4×C11C2×#4118yes—
255C2×C4×C11—22no—
256C2×C4×C11—24no—
257C2×C4×C11—24no—
258C2×C4×C11—22no—
259C2×C4×C11—42no—
260C2×C4×C11C22×D412yes—
261C2×C4×C11C22×D411yescharacteristic
262C2×C4×C11C2×#4114yes—
263C2×C4×C11C2511yescharacteristic
264C2×Dic11—28no—
265C2×Dic11—22no—
266C2×Dic11—22no—
267C2×Dic11—22no—
268C2×Dic11—22no—
269C2×Dic11—22no—
270C2×Dic11—22no—
271C2×Dic11—24no—
272C2×Dic11—44no—
273C2×Dic11—48no—
274C2×Dic11—416no—
275C2×Dic11—416no—
276C2×Dic11—42no—
277C2×Dic11—48no—
278C2×Dic11—44no—
279C2×Dic11—42no—
280C2×Dic11C22×D412yes—
281C2×Dic11C22×D412yes—
282C2×Dic11C22×D412yes—
283C2×Dic11C22×D412yes—
284Dic22—24no—
285Dic22—42no—
286C8×C11—24no—
287C11⋊C8—24no—
288C23×D4C2×D1111yescharacteristic
289C24×C4—221no—
290C22×SD16—112no—
291C22×D8C2×D1112yes—
292#529—118no—
293#529—224no—
294C2×#126—221no—
295C2×#125—111no—
296C2×#124—111no—
297C2×#123—221no—
298C2×C22⋊Q8—111no—
299C2×#121C2×D1111yescharacteristic
300C2×C4×Q8—111no—
301C2×C4×D4—112no—
302C2×C4×D4—118no—
303C2×C4×D4—112no—
304C2×C4×D4—111no—
305C2×C4×D4—221no—
306C2×C4×D4—221no—
307C2×C4×D4—224no—
308C2×C4×D4—221no—
309C2×#117—221no—
310C22×C4⋊C4—111no—
311C22×C22⋊C4—221no—
312C22×C42—111no—
313#483—118no—
314#480—118no—
315#478—1116no—
316#463—1116no—
317#450—118no—
318#447C2×D1118yes—
319#442—1116no—
320#440—1116no—
321#432—118no—
322C2×#107—112no—
323C2×#106—112no—
324C2×C4⋊C8—112no—
325C2×#103—112no—
326C2×#102—112no—
327C2×#102—112no—
328C2×#102C2×D1112yes—
329C2×C22⋊C8—111no—
330C2×C22⋊C8C2×D1111yescharacteristic
331C2×#97—112no—
332C22×C11—28no—
333C22×C11—22no—
334C22×C11—22no—
335C22×C11—28no—
336C22×C11—432no—
337C22×C11—416no—
338C22×C11—44no—
339C22×C11—44no—
340C22×C11—42no—
341C22×C11—42no—
342C22×C11—42no—
343C22×C11—84no—
344C22×C11#52512yes—
345C22×C11C2×AGL(1,ℤ/8)14yes—
346C22×C11C2×AGL(1,ℤ/8)12yes—
347C22×C11#57512yes—
348C22×C11C23×D411yescharacteristic
349C4×C11—22no—
350C4×C11—42no—
351C4×C11#52512yes—
352C4×C11C22×#4112yes—
353Dic11—22no—
354Dic11—22no—
355Dic11—22no—
356Dic11—22no—
357Dic11—42no—
358Dic11—44no—
359Dic11—42no—
360C25—21no—
361C22×Q8—111no—
362C22×D4—28no—
363C22×D4—21no—
364C22×D4—22no—
365C22×D4—21no—
366C22×D4C22×D1114yes—
367C22×D4C22×D1111yescharacteristic
368C22×D4C22×D1112yes—
369C23×C4—111no—
370C23×C4—111no—
371C23×C4—221no—
372C23×C4—228no—
373C23×C4—222no—
374C23×C4—222no—
375C23×C4—221no—
376C23×C4—448no—
377C23×C4C22×D1111yescharacteristic
378C2×SD16—118no—
379C2×SD16—228no—
380C2×D8—28no—
381C2×D8C22×D1118yes—
382C22×C8—112no—
383C22×C8C22×D1112yes—
384#126—228no—
385#125—114no—
386#125—222no—
387#124—114no—
388#124—222no—
389#123—228no—
390C22⋊Q8—114no—
391C22⋊Q8—222no—
392#121—22no—
393#121C22×D1114yes—
394C4×Q8—114no—
395C4×Q8—222no—
396C4×D4—118no—
397C4×D4—118no—
398C4×D4—114no—
399C4×D4—228no—
400C4×D4—224no—
401C4×D4—224no—
402C4×D4—228no—
403C4×D4—222no—
404C4×D4—228no—
405#117—228no—
406C2×C4⋊C4—112no—
407C2×C4⋊C4—114no—
408C2×C4⋊C4—111no—
409C2×C4⋊C4—221no—
410C2×C4⋊C4—221no—
411C2×C4⋊C4—221no—
412C2×C4⋊C4—222no—
413C2×C4⋊C4—221no—
414C2×C4⋊C4—221no—
415C2×C4⋊C4C22×D1111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—222no—
418C2×C22⋊C4—222no—
419C2×C22⋊C4—228no—
420C2×C22⋊C4—221no—
421C2×C22⋊C4—221no—
422C2×C22⋊C4—221no—
423C2×C22⋊C4—221no—
424C2×C42—111no—
425C2×C42—111no—
426C2×C42—114no—
427C2×C42—111no—
428C2×C42—111no—
429C2×C42—114no—
430C2×C42—221no—
431#107—118no—
432#106—118no—
433C4⋊C8—118no—
434#103—118no—
435#102—118no—
436#102—118no—
437#102C22×D1118yes—
438C22⋊C8—114no—
439C22⋊C8C22×D1114yes—
440#97—118no—
441C2×C11—24no—
442C2×C11—48no—
443C2×C11—82no—
444C2×C11#311414yescomplemented
445C2×C11C2×#52511yescharacteristic
446C2×C11C22×AGL(1,ℤ/8)11yescharacteristic
447C2×C11C2×#57511yescharacteristic
448C24—28no—
449C24—24no—
450C24—22no—
451C24—21no—
452C24—44no—
453C24—44no—
454C24D4×D1111yescharacteristic
455C2×Q8—112no—
456C2×Q8—221no—
457C2×Q8—224no—
458C2×D4—28no—
459C2×D4—216no—
460C2×D4—216no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—44no—
468C2×D4#434118yes—
469C2×D4C23×D1112yes—
470C2×D4C23×D1114yes—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—114no—
474C22×C4—111no—
475C22×C4—111no—
476C22×C4—111no—
477C22×C4—111no—
478C22×C4—114no—
479C22×C4—224no—
480C22×C4—228no—
481C22×C4—228no—
482C22×C4—224no—
483C22×C4—224no—
484C22×C4—221no—
485C22×C4—221no—
486C22×C4—222no—
487C22×C4—442no—
488C22×C4—4432no—
489C22×C4—448no—
490C22×C4—448no—
491C22×C4—444no—
492C22×C4—444no—
493C22×C4D4×D1112yes—
494C22×C4D4×D1111yescharacteristic
495C22×C4C23×D1112yes—
496C22×C4C23×D1111yescharacteristic
497SD16—2216no—
498D8—216no—
499C2×C8—24no—
500C2×C8—114no—
501C2×C8—224no—
502C2×C8C23×D1114yes—
503C4⋊C4—21no—
504C4⋊C4—112no—
505C4⋊C4—114no—
506C4⋊C4—224no—
507C4⋊C4—224no—
508C4⋊C4—224no—
509C4⋊C4—224no—
510C4⋊C4—222no—
511C4⋊C4—221no—
512C4⋊C4—224no—
513C4⋊C4C23×D1112yes—
514C22⋊C4—24no—
515C22⋊C4—228no—
516C22⋊C4—228no—
517C22⋊C4—224no—
518C22⋊C4—224no—
519C22⋊C4—224no—
520C22⋊C4—224no—
521C42—114no—
522C42—112no—
523C42—112no—
524C42—114no—
525C42—112no—
526C42—112no—
527C42—224no—
528C11C2×#311411yescharacteristic, complemented, Sylow
529C23—22no—
530C23—22no—
531C23—216no—
532C23—28no—
533C23—24no—
534C23—22no—
535C23—432no—
536C23—48no—
537C23—48no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#6696414yes—
544C23C2×D4×D1112yes—
545C23C2×D4×D1111yescharacteristic
546Q8—224no—
547Q8—442no—
548D4—216no—
549D4—28no—
550D4—24no—
551D4—24no—
552D4—48no—
553D4—42no—
554D4C2×#434118yes—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—112no—
561C2×C4—112no—
562C2×C4—112no—
563C2×C4—112no—
564C2×C4—228no—
565C2×C4—222no—
566C2×C4—222no—
567C2×C4—222no—
568C2×C4—222no—
569C2×C4—222no—
570C2×C4—222no—
571C2×C4—224no—
572C2×C4—444no—
573C2×C4—4416no—
574C2×C4—4416no—
575C2×C4—448no—
576C2×C4—448no—
577C2×C4—444no—
578C2×C4—442no—
579C2×C4—442no—
580C2×C4C2×D4×D1112yes—
581C2×C4C2×D4×D1111yescharacteristic
582C2×C4C2×#434114yes—
583C2×C4C24×D1111yescharacteristic
584C8—24no—
585C8—224no—
586C22—28no—
587C22—22no—
588C22—22no—
589C22—28no—
590C22—432no—
591C22—44no—
592C22—416no—
593C22—44no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—84no—
598C22#1062311712yes—
599C22C2×#6696414yes—
600C22C2×#6696412yes—
601C22#1062330412yes—
602C22C22×D4×D1111yescharacteristic
603C4—22no—
604C4—42no—
605C4—222no—
606C4—222no—
607C4—222no—
608C4—222no—
609C4—444no—
610C4—442no—
611C4—442no—
612C4#1062311712yes—
613C4C22×#434112yes—
614C2—24no—
615C2—48no—
616C2—82no—
617C2#1307874914yesdirect product factor
618C2C2×#1062311711yescharacteristic
619C2C22×#6696411yescharacteristic
620C2C2×#1062330411yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
Fitting, prime radicalC2×C11×#447C2
socleC23×C11C22×D4
hypercenterC2×#447D11
hypocenterC11C2×#3114
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#10623131C2semidirect
2. C22×D4×Dic11C2semidirect
3. #13078749C2direct (N × H)
4. C2×#10622334C2semidirect
5. C2×#10622646C2semidirect
6. C2×#10622376C2semidirect
7. C2×#10622766C2semidirect
8. C2×#10622831C2semidirect
9. C2×#10622765C2semidirect
10. C2×#10622370C2semidirect
11. C2×#10622332C2semidirect
18. C2×#66989C22semidirect
26. C2×D4×Dic11C22semidirect
27. C2×D4×Dic11C22semidirect
33. #10623131C22semidirect
34. C2×#66943C22semidirect
35. C2×#66940C22semidirect
41. #10622831C22semidirect
43. C2×#66896C22semidirect
45. #10622766C22semidirect
46. #10622765C22semidirect
47. C2×#66881C22semidirect
48. C2×#66880C22semidirect
49. C2×#66879C22semidirect
50. C2×Dic11⋊C8C22semidirect
51. #10622646C22semidirect
52. C2×#66871C22semidirect
53. #10622376C22semidirect
54. #10622370C22semidirect
55. #10622334C22semidirect
56. #10622332C22semidirect
90. #66989C23semidirect
100. D4×Dic11C23semidirect
123. #66943C23semidirect
125. #66940C23semidirect
131. #66896C23semidirect
133. #66881C23semidirect
134. #66880C23semidirect
135. #66879C23semidirect
136. Dic11⋊C8C23semidirect
137. #66871C23semidirect
444. C2×C11#3114semidirect
528. C11C2×#3114semidirect
617. C2#13078749direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106231312
C22×D4×Dic112
#130787492
C2×#106223342
C2×#106226462
C2×#106223762
C2×#106227662
C2×#106228312
C2×#106227652
C2×#106223702
C2×#106223322
C2×C11×#4472
C2×#311411