Finite Group Databasebeta

← C2×#13078298 #15621010

Subgroups — C2×#13078298 #15621010

Order 2816.

2934 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67002C211yesmaximal, characteristic, complemented
2C22×#67001C211yesmaximal, characteristic, complemented
3C2×#10623361C211yesmaximal, characteristic, complemented
4#13078298C2116yesmaximal, direct product factor
5C2×#10622652C212yesmaximal, complemented
6C2×#10622662C212yesmaximal, complemented
7C2×#10622651C212yesmaximal, complemented
8C2×#10622661C212yesmaximal, complemented
9C2×#10622577C211yesmaximal, characteristic, complemented
10C2×#10622846C211yesmaximal, characteristic, complemented
11C22×#66873C211yesmaximal, characteristic, complemented
12C2×C11×#537C211yesmaximal, characteristic, complemented
13C22×C11×#41C2211yescharacteristic
14C22×C11×#41C2211yescharacteristic
15C23×D44C2211yescharacteristic
16C11×#537—24no—
17C11×#537C2218yescomplemented
18C2×C11×C22⋊Q8—21no—
19C2×C11×C22⋊Q8C2212yescomplemented
20C2×C11×#121—21no—
21C2×C11×#121—22no—
22C2×C11×#121C2212yescomplemented
23C22×C11×C4⋊C4C2211yescharacteristic
24#10623361—24no—
25#10623361C2218yescomplemented
26C2×#67002—28no—
27C2×#67002C2218yescomplemented
28C2×#67001—28no—
29C2×#67001C2218yescomplemented
30C2×#67000—21no—
31C22×#4325C2212yes—
32C2×#66993—21no—
33C2×#66990—22no—
34C22×#4323C2212yes—
35C2×#66983C2212yes—
36C2×#66983C2212yes—
37C2×#66979C2211yescharacteristic
38C22×#4318C2211yescharacteristic
39C22×#4318C2212yescomplemented
40C23×C11⋊C8C2211yescharacteristic
41C23×C11⋊C8C2212yescomplemented
42#10622846C2218yescomplemented
43C2×#66899C2211yescharacteristic, complemented
44C2×#66899C2211yescharacteristic, complemented
45C2×#66896C2211yescharacteristic, complemented
46C2×#66896C2211yescharacteristic, complemented
47#10622662C22116yescomplemented
48#10622661C22116yescomplemented
49#10622652C22116yescomplemented
50#10622651C22116yescomplemented
51C2×#66873C2212yes—
52C2×#66873C2212yes—
53C2×#66873C2218yescomplemented
54#10622577C2218yescomplemented
55C2×#66872C2212yescomplemented
56C2×#66871C2212yescomplemented
57C24×D11—21no—
58C2×C11×#41—24no—
59C2×C11×#41—22no—
60C2×C11×#41—24no—
61C2×C11×#41—24no—
62C2×C11×#41—24no—
63C2×C11×#41D414yes—
64C2×C11×#41C2314yes—
65C2×C11×#41C2314yes—
66C22×Q8×C11D411yescharacteristic
67C22×Q8×C11C2311yescharacteristic
68C22×D4×C11—21no—
69C22×D4×C11—22no—
70C22×D4×C11D412yes—
71C22×D4×C11D411yescharacteristic
72C22×D4×C11C2311yescharacteristic
73C23×C4×C11—21no—
74C23×C4×C11—21no—
75C23×C4×C11C2312yes—
76C23×C4×C11C2311yescharacteristic
77C22×C22⋊D11—22no—
78C22×C22⋊D11—22no—
79C22×D44—24no—
80C22×D44—24no—
81C22×D44—21no—
82C22×D44—22no—
83C22×D44C2312yes—
84C22×D44C2314yes—
85C11×C22⋊Q8—28no—
86C11×C22⋊Q8—24no—
87C11×C22⋊Q8C2318yescomplemented
88C11×#121—28no—
89C11×#121—216no—
90C11×#121—24no—
91C11×#121C2318yescomplemented
92C2×C11×C4⋊C4—21no—
93C2×C11×C4⋊C4—22no—
94C2×C11×C4⋊C4—22no—
95C2×C11×C4⋊C4C2312yes—
96C2×C11×C4⋊C4C2314yes—
97C2×C11×C22⋊C4—22no—
98C2×C11×C22⋊C4—22no—
99C2×C22⋊Dic11—21no—
100#67002—232no—
101#67001—232no—
102#67000—28no—
103C2×#4325—28no—
104C2×#4325C2318yes—
105#66993—28no—
106#66990—216no—
107C2×#4323—28no—
108C2×#4323C2318yes—
109#66983—24no—
110#66983—24no—
111#66983C2318yes—
112#66983C2318yes—
113C2×(C2×D11)⋊C4—22no—
114C2×(C2×D11)⋊C4—22no—
115#66979—22no—
116#66979C2314yes—
117C2×C4⋊Dic11C2311yescharacteristic
118C2×C4⋊Dic11C2311yescharacteristic
119C2×C4×Dic11—21no—
120C2×#4318—28no—
121C2×#4318D418yescomplemented
122C2×#4318C2314yes—
123C2×#4318C2318yescomplemented
124C22×C11⋊C8—24no—
125C22×C11⋊C8—28no—
126C22×C11⋊C8C2311yescharacteristic
127C22×C11⋊C8C2311yescharacteristic
128C22×C11⋊C8C2314yes—
129C22×C11⋊C8C2312yes—
130C22×C11⋊C8C2318yescomplemented
131#66899C2314yescomplemented
132#66899C2314yescomplemented
133#66896C2314yescomplemented
134#66896C2314yescomplemented
135#66873C2318yes—
136#66873C2318yes—
137#66872C2318yescomplemented
138#66871C2318yescomplemented
139C2×#3062—111nomaximal, Sylow
140C24×C11—21no—
141C24×C11C2×D412yes—
142C24×C11C2×D411yescharacteristic
143C23×D11—22no—
144C23×D11—24no—
145C23×D11—44no—
146C23×D11—82no—
147C23×D11—82no—
148C11×#41—216no—
149C11×#41—28no—
150C11×#41—28no—
151C11×#41—28no—
152C11×#41—28no—
153C11×#41—48no—
154C2×Q8×C11—24no—
155C2×Q8×C11—21no—
156C2×Q8×C11—22no—
157C2×Q8×C11—22no—
158C2×Q8×C11C2×D414yes—
159C2×Q8×C11C2412yes—
160C2×D4×C11—28no—
161C2×D4×C11—24no—
162C2×D4×C11—28no—
163C2×D4×C11—24no—
164C2×D4×C11—22no—
165C2×D4×C11—21no—
166C2×D4×C11—22no—
167C2×D4×C11—44no—
168C2×D4×C11—84no—
169C2×D4×C11C2×D418yes—
170C2×D4×C11C2×D414yes—
171C2×D4×C11C2412yes—
172C22×C4×C11—24no—
173C22×C4×C11—24no—
174C22×C4×C11—22no—
175C22×C4×C11—21no—
176C22×C4×C11—24no—
177C22×C4×C11—24no—
178C22×C4×C11—44no—
179C22×C4×C11—82no—
180C22×C4×C11C2×D412yes—
181C22×C4×C11C2×D414yes—
182C22×C4×C11C2×D411yescharacteristic
183C22×C4×C11C2×D412yes—
184C22×C4×C11C2×D414yes—
185C22×C4×C11#4114yes—
186C22×C4×C11C2414yes—
187C22×C4×C11C2411yescharacteristic
188C22×C4×C11C2412yes—
189C2×C22⋊D11—28no—
190C2×C22⋊D11—28no—
191C2×C22⋊D11—84no—
192C2×C22⋊D11—84no—
193C22×Dic11—21no—
194C22×Dic11—21no—
195C2×D44—22no—
196C2×D44—24no—
197C2×D44—24no—
198C2×D44—28no—
199C2×D44—24no—
200C2×D44—28no—
201C2×D44—44no—
202C2×D44—48no—
203C2×D44C2414yes—
204C11×C4⋊C4—24no—
205C11×C4⋊C4—22no—
206C11×C4⋊C4—28no—
207C11×C4⋊C4C2414yes—
208C11×C22⋊C4—28no—
209C11×C22⋊C4—28no—
210C22⋊Dic11—24no—
211#4325—216no—
212#4323—216no—
213(C2×D11)⋊C4—28no—
214(C2×D11)⋊C4—28no—
215C4⋊Dic11—21no—
216C4⋊Dic11—21no—
217C4⋊Dic11C2412yes—
218C4⋊Dic11C2412yes—
219C4×Dic11—24no—
220#4318—216no—
221#4318C2×D4116yescomplemented
222C2×C11⋊C8—216no—
223C2×C11⋊C8—22no—
224C2×C11⋊C8—22no—
225C2×C11⋊C8—28no—
226C2×C11⋊C8C2×D418yes—
227C2×C11⋊C8C2412yes—
228C2×C11⋊C8C2412yes—
229C2×C11⋊C8C2414yes—
230C22×AGL(1,ℤ/8)—111no—
231C22×#131—111no—
232C2×#537D1111yescharacteristic, complemented
233C2×#535—111no—
234#3062—1116no—
235C2×#469—112no—
236C2×#468—112no—
237C2×#466—112no—
238C2×#465—112no—
239C2×#428—111no—
240C2×#416—111no—
241C22×#102—111no—
242C23×C11—22no—
243C23×C11—22no—
244C23×C11—24no—
245C23×C11—24no—
246C23×C11—24no—
247C23×C11—22no—
248C23×C11—42no—
249C23×C11—42no—
250C23×C11—42no—
251C23×C11#12114yes—
252C23×C11C22×D414yes—
253C23×C11C22×D412yes—
254C23×C11C22×D411yescharacteristic
255C22×D11—24no—
256C22×D11—48no—
257C22×D11—44no—
258C22×D11—44no—
259C22×D11—44no—
260C22×D11—84no—
261C22×D11—84no—
262C22×D11—816no—
263Q8×C11—28no—
264Q8×C11—22no—
265Q8×C11—22no—
266Q8×C11—42no—
267D4×C11—216no—
268D4×C11—28no—
269D4×C11—22no—
270D4×C11—22no—
271D4×C11—48no—
272D4×C11—42no—
273D4×C11—88no—
274C2×C4×C11—24no—
275C2×C4×C11—28no—
276C2×C4×C11—24no—
277C2×C4×C11—24no—
278C2×C4×C11—24no—
279C2×C4×C11—22no—
280C2×C4×C11—42no—
281C2×C4×C11—44no—
282C2×C4×C11—48no—
283C2×C4×C11—84no—
284C2×C4×C11#12114yes—
285C2×C4×C11#12118yes—
286C2×C4×C11C22×D412yes—
287C2×C4×C11C22×D414yes—
288C2×C4×C11C22×D414yes—
289C2×C4×C11C22×D412yes—
290C2×C4×C11C22×D411yescharacteristic
291C2×C4×C11C2×#4112yes—
292C2×C4×C11C2×#4114yes—
293C2×C4×C11C2511yescharacteristic
294C22⋊D11—88no—
295C22⋊D11—88no—
296C2×Dic11—22no—
297C2×Dic11—22no—
298C2×Dic11—42no—
299C2×Dic11—42no—
300D44—24no—
301D44—24no—
302D44—44no—
303D44—48no—
304C11⋊C8—22no—
305C11⋊C8—22no—
306C11⋊C8C22×D418yes—
307C22×#41C2×D1111yescharacteristic
308C22×#41C2×D1111yescharacteristic
309C23×D4—111no—
310C2×AGL(1,ℤ/8)—118no—
311C2×AGL(1,ℤ/8)—228no—
312C22×SD16—112no—
313C22×D8—112no—
314C2×#131—118no—
315C2×#131—228no—
316C22×M16—111no—
317C22×M16—112no—
318C23×C8—111no—
319C23×C8—112no—
320#537—24no—
321#537C2×D1118yescomplemented
322#535—118no—
323#535—224no—
324C2×#127—221no—
325C2×#124—221no—
326C2×C22⋊Q8—21no—
327C2×C22⋊Q8C2×D1112yescomplemented
328C2×#121—21no—
329C2×#121—22no—
330C2×#121—112no—
331C2×#121—112no—
332C2×#121C2×D1112yescomplemented
333C2×#120—222no—
334C2×#117—111no—
335C22×C4⋊C4C2×D1111yescharacteristic
336#469—1116no—
337#468—1116no—
338#466—1116no—
339#465—1116no—
340#428—118no—
341C2×#107—112no—
342C2×#106—112no—
343#416—118no—
344C2×#103—111no—
345C2×#103—111no—
346C2×#102—111no—
347C2×#102—111no—
348C2×#102—112no—
349C2×#102—112no—
350C2×#102—118no—
351C22×C11—28no—
352C22×C11—22no—
353C22×C11—22no—
354C22×C11—24no—
355C22×C11—24no—
356C22×C11—24no—
357C22×C11—48no—
358C22×C11—42no—
359C22×C11—42no—
360C22×C11—42no—
361C22×C11—84no—
362C22×C11C2×#12114yes—
363C22×C11C2×#12112yes—
364C22×C11#57612yes—
365C22×C11#57712yes—
366C22×C11C23×D411yescharacteristic
367C2×D11—44no—
368C2×D11—44no—
369C2×D11—44no—
370C2×D11—816no—
371C2×D11—88no—
372C4×C11—24no—
373C4×C11—42no—
374C4×C11—44no—
375C4×C11C2×#12112yes—
376C4×C11C2×#12114yes—
377C4×C11C23×D411yescharacteristic
378C4×C11C22×#4111yescharacteristic
379Dic11—42no—
380Dic11—42no—
381C25—221no—
382C2×#41—24no—
383C2×#41—24no—
384C2×#41—22no—
385C2×#41—24no—
386C2×#41—24no—
387C2×#41C22⋊D1114yes—
388C2×#41C22×D1114yes—
389C2×#41C22×D1114yes—
390C22×Q8C22⋊D1111yescharacteristic
391C22×Q8C22×D1111yescharacteristic
392C22×D4—21no—
393C22×D4—22no—
394C22×D4—112no—
395C22×D4—114no—
396C22×D4—222no—
397C22×D4—222no—
398C22×D4—224no—
399C22×D4—224no—
400C22×D4—221no—
401C22×D4—222no—
402C22×D4C22⋊D1112yes—
403C22×D4C22⋊D1111yescharacteristic
404C22×D4C22×D1111yescharacteristic
405C23×C4—21no—
406C23×C4—21no—
407C23×C4C22×D1112yes—
408C23×C4C22×D1111yescharacteristic
409AGL(1,ℤ/8)—2232no—
410C2×SD16—118no—
411C2×SD16—228no—
412C2×D8—118no—
413C2×D8—228no—
414#131—2232no—
415C2×M16—118no—
416C2×M16—114no—
417C2×M16—118no—
418C2×M16—228no—
419C22×C8—112no—
420C22×C8—111no—
421C22×C8—111no—
422C22×C8—114no—
423C22×C8—118no—
424C22×C8—228no—
425C22×C8—224no—
426#127—228no—
427#124—228no—
428C22⋊Q8—28no—
429C22⋊Q8—24no—
430C22⋊Q8C22×D1118yescomplemented
431#121—28no—
432#121—216no—
433#121—24no—
434#121—118no—
435#121—118no—
436#121—224no—
437#121—224no—
438#121C22×D1118yescomplemented
439#120—2216no—
440#117—114no—
441#117—222no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—22no—
445C2×C4⋊C4—111no—
446C2×C4⋊C4—111no—
447C2×C4⋊C4C22×D1112yes—
448C2×C4⋊C4C22×D1114yes—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—22no—
451C2×C22⋊C4—221no—
452C2×C22⋊C4—222no—
453C2×C22⋊C4—222no—
454C2×C42—221no—
455#107—118no—
456#106—118no—
457#103—114no—
458#103—114no—
459#102—114no—
460#102—114no—
461#102—118no—
462#102—118no—
463C2×C11—24no—
464C2×C11—44no—
465C2×C11—82no—
466C2×C11#306214yescomplemented
467C2×C11C22×#12111yescharacteristic
468C2×C11C2×#57611yescharacteristic
469C2×C11C2×#57711yescharacteristic
470D11—84no—
471C24—21no—
472C24—222no—
473C24—224no—
474C24—444no—
475C24—882no—
476C24—882no—
477C24C2×C22⋊D1112yes—
478C24C2×C22⋊D1111yescharacteristic
479#41—216no—
480#41—28no—
481#41—28no—
482#41—28no—
483#41—28no—
484#41—48no—
485C2×Q8—24no—
486C2×Q8—21no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8C2×C22⋊D1114yes—
490C2×Q8C23×D1112yes—
491C2×D4—28no—
492C2×D4—24no—
493C2×D4—28no—
494C2×D4—24no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—44no—
499C2×D4—84no—
500C2×D4—114no—
501C2×D4—228no—
502C2×D4—228no—
503C2×D4—224no—
504C2×D4—228no—
505C2×D4—224no—
506C2×D4—228no—
507C2×D4—222no—
508C2×D4—224no—
509C2×D4—444no—
510C2×D4—448no—
511C2×D4—884no—
512C2×D4—884no—
513C2×D4C2×C22⋊D1118yes—
514C2×D4C2×C22⋊D1114yes—
515C2×D4C23×D1112yes—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—24no—
521C22×C4—24no—
522C22×C4—44no—
523C22×C4—82no—
524C22×C4—221no—
525C22×C4—221no—
526C22×C4D4×D1114yes—
527C22×C4#434114yes—
528C22×C4C2×C22⋊D1114yes—
529C22×C4C2×C22⋊D1112yes—
530C22×C4C2×C22⋊D1112yes—
531C22×C4C2×C22⋊D1111yescharacteristic
532C22×C4C23×D1114yes—
533C22×C4C23×D1112yes—
534C22×C4C23×D1111yescharacteristic
535SD16—2216no—
536D8—2216no—
537M16—1116no—
538M16—2216no—
539C2×C8—118no—
540C2×C8—114no—
541C2×C8—112no—
542C2×C8—112no—
543C2×C8—2216no—
544C2×C8—222no—
545C2×C8—222no—
546C2×C8—228no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—28no—
550C4⋊C4—112no—
551C4⋊C4—112no—
552C4⋊C4—221no—
553C4⋊C4—221no—
554C4⋊C4C23×D1114yes—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—224no—
558C22⋊C4—228no—
559C22⋊C4—228no—
560C42—224no—
561C11C2×#306211yescharacteristic, complemented, Sylow
562C23—22no—
563C23—22no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—224no—
572C23—448no—
573C23—444no—
574C23—444no—
575C23—444no—
576C23—884no—
577C23—884no—
578C23—8816no—
579C23#6699214yes—
580C23C22×C22⋊D1114yes—
581C23C22×C22⋊D1112yes—
582C23C22×C22⋊D1111yescharacteristic
583Q8—28no—
584Q8—22no—
585Q8—22no—
586Q8—42no—
587D4—216no—
588D4—28no—
589D4—22no—
590D4—22no—
591D4—48no—
592D4—42no—
593D4—88no—
594D4—224no—
595D4—224no—
596D4—444no—
597D4—448no—
598D4—888no—
599D4—888no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—84no—
610C2×C4—222no—
611C2×C4—222no—
612C2×C4—442no—
613C2×C4—442no—
614C2×C4#6699214yes—
615C2×C4#6699218yes—
616C2×C4C2×D4×D1112yes—
617C2×C4C2×D4×D1114yes—
618C2×C4C2×#434112yes—
619C2×C4C2×#434114yes—
620C2×C4C22×C22⋊D1114yes—
621C2×C4C22×C22⋊D1112yes—
622C2×C4C22×C22⋊D1111yescharacteristic
623C2×C4C24×D1111yescharacteristic
624C8—118no—
625C8—222no—
626C8—222no—
627C22—28no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—48no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—84no—
638C22—444no—
639C22—444no—
640C22—444no—
641C22—8816no—
642C22—888no—
643C22C2×#6699214yes—
644C22C2×#6699212yes—
645C22#1062336612yes—
646C22#1062336812yes—
647C22C23×C22⋊D1111yescharacteristic
648C4—24no—
649C4—44no—
650C4—42no—
651C4—442no—
652C4—442no—
653C4C2×#6699212yes—
654C4C2×#6699214yes—
655C4C22×D4×D1111yescharacteristic
656C4C22×#434111yescharacteristic
657C2—24no—
658C2—44no—
659C2—82no—
660C2—884no—
661C2#1307829814yesdirect product factor
662C2C22×#6699211yescharacteristic
663C2C2×#1062336611yescharacteristic
664C2C2×#1062336811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingC2×C11×#537C2
socleC23×C11C22×D4
hypercenterC2×#537D11
hypocenterC11C2×#3062
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C22×#67002C2semidirect
2. C22×#67001C2semidirect
3. C2×#10623361C2semidirect
4. #13078298C2direct (N × H)
5. C2×#10622652C2semidirect
6. C2×#10622662C2semidirect
7. C2×#10622651C2semidirect
8. C2×#10622661C2semidirect
9. C2×#10622577C2semidirect
10. C2×#10622846C2semidirect
11. C22×#66873C2semidirect
12. C2×C11×#537C2semidirect
17. C11×#537C22semidirect
19. C2×C11×C22⋊Q8C22semidirect
22. C2×C11×#121C22semidirect
25. #10623361C22semidirect
27. C2×#67002C22semidirect
29. C2×#67001C22semidirect
39. C22×#4318C22semidirect
41. C23×C11⋊C8C22semidirect
42. #10622846C22semidirect
43. C2×#66899C22semidirect
44. C2×#66899C22semidirect
45. C2×#66896C22semidirect
46. C2×#66896C22semidirect
47. #10622662C22semidirect
48. #10622661C22semidirect
49. #10622652C22semidirect
50. #10622651C22semidirect
53. C2×#66873C22semidirect
54. #10622577C22semidirect
55. C2×#66872C22semidirect
56. C2×#66871C22semidirect
87. C11×C22⋊Q8C23semidirect
91. C11×#121C23semidirect
121. C2×#4318D4semidirect
123. C2×#4318C23semidirect
130. C22×C11⋊C8C23semidirect
131. #66899C23semidirect
132. #66899C23semidirect
133. #66896C23semidirect
134. #66896C23semidirect
137. #66872C23semidirect
138. #66871C23semidirect
221. #4318C2×D4semidirect
232. C2×#537D11semidirect
321. #537C2×D11semidirect
327. C2×C22⋊Q8C2×D11semidirect
332. C2×#121C2×D11semidirect
430. C22⋊Q8C22×D11semidirect
438. #121C22×D11semidirect
466. C2×C11#3062semidirect
561. C11C2×#3062semidirect
661. C2#13078298direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#670022
C22×#670012
C2×#106233612
#130782982
C2×#106226522
C2×#106226622
C2×#106226512
C2×#106226612
C2×#106225772
C2×#106228462
C22×#668732
C2×C11×#5372
C2×#306211