Finite Group Databasebeta

← C2×#13076621 #14959925

Subgroups — C2×#13076621 #14959925

Order 2816.

3322 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66947C211yesmaximal, characteristic, complemented
2C22×#66948C211yesmaximal, characteristic, complemented
3#13076621C2116yesmaximal, direct product factor
4C2×#10622605C211yesmaximal, characteristic, complemented
5C2×#10622289C212yesmaximal, complemented
6C2×#10622285C212yesmaximal, complemented
7C2×#10622285C212yesmaximal, complemented
8C2×#10622604C211yesmaximal, characteristic, complemented
9C2×#10622284C211yesmaximal, characteristic, complemented
10C2×#10622284C212yesmaximal, complemented
11C2×#10622280C211yesmaximal, characteristic, complemented
12C2×C11×#400C211yesmaximal, characteristic, complemented
13C23×D44C2211yescharacteristic
14C23×D44C2211yescharacteristic
15C23×D44C2212yes—
16C23×D44C2212yes—
17C23×C4×D11C2211yescharacteristic
18C22×C11×C4⋊C4C2211yescharacteristic
19C22×C11×C4⋊C4C2211yescharacteristic
20C22×(C2×D11)⋊C4C2211yescharacteristic
21C22×(C2×D11)⋊C4C2211yescharacteristic
22C22×(C2×D11)⋊C4C2211yescharacteristic
23C22×(C2×D11)⋊C4C2212yes—
24C22×(C2×D11)⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2212yes—
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2212yes—
28C22×(C2×D11)⋊C4C2212yes—
29C22×Dic11⋊C4C2211yescharacteristic
30C22×Dic11⋊C4C2212yes—
31C22×Dic11⋊C4C2212yes—
32C2×#66948—28no—
33C2×#66948C2218yescomplemented
34C2×#66947—28no—
35C2×#66947C2218yescomplemented
36C11×#400C2218yescomplemented
37C2×C11×#95C2211yescharacteristic
38C2×C11×#95C2212yes—
39C2×C11×#95C2212yes—
40C2×C11⋊#95C2211yescharacteristic
41#10622605C2218yescomplemented
42#10622604C2218yescomplemented
43#10622289C22116yescomplemented
44#10622285C22116yescomplemented
45#10622285C22116yescomplemented
46#10622284C2218yescomplemented
47#10622284C22116yescomplemented
48#10622280C2218yescomplemented
49C24×D11C2311yescharacteristic
50C24×D11C2312yes—
51C24×D11C2311yescharacteristic
52C23×C4×C11C2312yes—
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2312yes—
55C23×C4×C11C2311yescharacteristic
56C23×C4×C11C2311yescharacteristic
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2312yes—
59C23×Dic11C2311yescharacteristic
60C22×D44—24no—
61C22×D44—28no—
62C22×D44—24no—
63C22×D44—44no—
64C22×D44—44no—
65C22×D44—44no—
66C22×D44—44no—
67C22×D44—44no—
68C22×D44—44no—
69C22×D44—44no—
70C22×D44C2314yes—
71C22×D44C2318yes—
72C22×D44C2314yes—
73C22×D44C2318yes—
74C22×C4×D11—44no—
75C22×C4×D11D418yescomplemented
76C22×C4×D11C2314yes—
77C2×C11×C4⋊C4—24no—
78C2×C11×C4⋊C4—24no—
79C2×C11×C4⋊C4C2314yes—
80C2×C11×C4⋊C4C2314yes—
81C2×(C2×D11)⋊C4—24no—
82C2×(C2×D11)⋊C4—24no—
83C2×(C2×D11)⋊C4—24no—
84C2×(C2×D11)⋊C4—28no—
85C2×(C2×D11)⋊C4—44no—
86C2×(C2×D11)⋊C4—44no—
87C2×(C2×D11)⋊C4—44no—
88C2×(C2×D11)⋊C4—44no—
89C2×(C2×D11)⋊C4—44no—
90C2×(C2×D11)⋊C4C2314yes—
91C2×(C2×D11)⋊C4C2314yes—
92C2×(C2×D11)⋊C4C2314yes—
93C2×(C2×D11)⋊C4C2318yes—
94C2×(C2×D11)⋊C4C2318yes—
95C2×(C2×D11)⋊C4C2318yes—
96C2×(C2×D11)⋊C4C2318yes—
97C2×(C2×D11)⋊C4C2318yes—
98C2×(C2×D11)⋊C4C2318yes—
99C2×Dic11⋊C4—24no—
100C2×Dic11⋊C4—44no—
101C2×Dic11⋊C4—44no—
102C2×Dic11⋊C4C2314yes—
103C2×Dic11⋊C4C2318yes—
104C2×Dic11⋊C4C2318yes—
105#66948—232no—
106#66947—232no—
107C11×#95C2314yes—
108C11×#95C2318yes—
109C11×#95C2318yes—
110C11⋊#95C2314yes—
111C2×#2813—111nomaximal, Sylow
112C24×C11C2411yescharacteristic
113C23×D11—22no—
114C23×D11—24no—
115C23×D11—24no—
116C23×D11—22no—
117C23×D11—22no—
118C23×D11—24no—
119C23×D11—24no—
120C23×D11—24no—
121C23×D11—24no—
122C23×D11—24no—
123C23×D11—24no—
124C23×D11—24no—
125C23×D11—24no—
126C23×D11—22no—
127C23×D11—22no—
128C23×D11—24no—
129C23×D11C2×D414yes—
130C23×D11C2412yes—
131C23×D11C2414yes—
132C23×D11C2412yes—
133C22×C4×C11—24no—
134C22×C4×C11—22no—
135C22×C4×C11—22no—
136C22×C4×C11—24no—
137C22×C4×C11—24no—
138C22×C4×C11—24no—
139C22×C4×C11—22no—
140C22×C4×C11—24no—
141C22×C4×C11—24no—
142C22×C4×C11—24no—
143C22×C4×C11—24no—
144C22×C4×C11—22no—
145C22×C4×C11C2×D414yes—
146C22×C4×C11C2414yes—
147C22×C4×C11C2412yes—
148C22×C4×C11C2414yes—
149C22×C4×C11C2412yes—
150C22×C4×C11C2412yes—
151C22×Dic11—22no—
152C22×Dic11—24no—
153C22×Dic11—24no—
154C22×Dic11—24no—
155C22×Dic11—24no—
156C22×Dic11—24no—
157C22×Dic11C2×D414yes—
158C22×Dic11C2414yes—
159C22×Dic11C2412yes—
160C22×Dic11C2412yes—
161C2×D44—28no—
162C2×D44—216no—
163C2×D44—28no—
164C2×D44—48no—
165C2×D44—48no—
166C2×D44—48no—
167C2×D44—48no—
168C2×D44—48no—
169C2×D44—48no—
170C2×D44—48no—
171C2×D44—416no—
172C2×D44—416no—
173C2×D44—432no—
174C2×D44—432no—
175C2×C4×D11—416no—
176C2×C4×D11—48no—
177C2×C4×D11C2×D4116yescomplemented
178C11×C4⋊C4—28no—
179C11×C4⋊C4—28no—
180(C2×D11)⋊C4—28no—
181(C2×D11)⋊C4—216no—
182(C2×D11)⋊C4—28no—
183(C2×D11)⋊C4—28no—
184(C2×D11)⋊C4—48no—
185(C2×D11)⋊C4—48no—
186(C2×D11)⋊C4—48no—
187(C2×D11)⋊C4—48no—
188(C2×D11)⋊C4—48no—
189Dic11⋊C4—28no—
190Dic11⋊C4—48no—
191Dic11⋊C4—48no—
192C22×#123—111no—
193C22×#121—111no—
194#2813—1116no—
195C2×#400D1111yescharacteristic, complemented
196C2×#399—111no—
197C2×#396—112no—
198C2×#394—112no—
199C2×#394—112no—
200C2×#394—111no—
201C2×#392—111no—
202C2×#392—112no—
203C2×#380—111no—
204C23×C11C22×D412yes—
205C23×C11H(5,𝔽2)14yes—
206C23×C11H(5,𝔽2)14yes—
207C23×C11H(5,𝔽2)12yes—
208C23×C112−1+412yes—
209C23×C11C2511yescharacteristic
210C22×D11—28no—
211C22×D11—28no—
212C22×D11—28no—
213C22×D11—22no—
214C22×D11—24no—
215C22×D11—24no—
216C22×D11—22no—
217C22×D11—22no—
218C22×D11—24no—
219C22×D11—24no—
220C22×D11—24no—
221C22×D11—24no—
222C22×D11—24no—
223C22×D11—24no—
224C22×D11—24no—
225C22×D11—24no—
226C22×D11—22no—
227C22×D11—22no—
228C22×D11—24no—
229C22×D11—48no—
230C22×D11—48no—
231C22×D11—48no—
232C22×D11—44no—
233C22×D11—48no—
234C22×D11—44no—
235C22×D11—44no—
236C22×D11—48no—
237C22×D11—48no—
238C22×D11—48no—
239C22×D11—48no—
240C22×D11—48no—
241C22×D11—48no—
242C22×D11—48no—
243C22×D11C22×D414yes—
244C2×C4×C11—28no—
245C2×C4×C11—24no—
246C2×C4×C11—22no—
247C2×C4×C11—22no—
248C2×C4×C11—24no—
249C2×C4×C11—24no—
250C2×C4×C11—24no—
251C2×C4×C11—22no—
252C2×C4×C11—24no—
253C2×C4×C11—24no—
254C2×C4×C11—24no—
255C2×C4×C11—24no—
256C2×C4×C11—22no—
257C2×C4×C11—44no—
258C2×C4×C11—48no—
259C2×C4×C11—44no—
260C2×C4×C11—48no—
261C2×C4×C11C22×D414yes—
262C2×Dic11—28no—
263C2×Dic11—24no—
264C2×Dic11—22no—
265C2×Dic11—24no—
266C2×Dic11—24no—
267C2×Dic11—24no—
268C2×Dic11—24no—
269C2×Dic11—44no—
270C2×Dic11—48no—
271C2×Dic11C22×D414yes—
272D44—416no—
273D44—416no—
274D44—432no—
275D44—432no—
276C4×D11—416no—
277C23×D4—111no—
278C23×D4—112no—
279C23×D4—111no—
280C23×D4—112no—
281C24×C4—111no—
282C2×#123—118no—
283C2×#123—228no—
284C2×#121—118no—
285C2×#121—228no—
286C22×C4⋊C4—111no—
287C22×C4⋊C4—112no—
288C22×C4⋊C4—112no—
289C22×C4⋊C4C2×D1111yescharacteristic
290C22×C4⋊C4C2×D1111yescharacteristic
291C22×C22⋊C4—111no—
292C22×C22⋊C4—111no—
293C22×C22⋊C4—111no—
294C22×C22⋊C4—112no—
295C22×C22⋊C4—112no—
296C22×C22⋊C4—112no—
297C22×C22⋊C4—112no—
298C22×C22⋊C4—112no—
299C22×C22⋊C4—112no—
300#400C2×D1118yescomplemented
301#399—118no—
302#396—1116no—
303#394—118no—
304#394—1116no—
305#394—1116no—
306#392—118no—
307#392—1116no—
308#380—118no—
309C2×#95—111no—
310C2×#95C2×D1112yes—
311C2×#95C2×D1111yescharacteristic
312C2×#95C2×D1112yes—
313C22×C11#53414yes—
314C22×C11#53514yes—
315C22×C11#53714yes—
316C22×C11#56018yes—
317C22×C11#56218yes—
318C22×C11C23×D411yescharacteristic
319C22×C11C2×H(5,𝔽2)12yes—
320C22×C11C2×H(5,𝔽2)12yes—
321C22×C11C2×H(5,𝔽2)11yescharacteristic
322C22×C11C2×2−1+411yescharacteristic
323C2×D11—24no—
324C2×D11—24no—
325C2×D11—24no—
326C2×D11—44no—
327C2×D11—44no—
328C2×D11—416no—
329C2×D11—42no—
330C2×D11—44no—
331C2×D11—42no—
332C2×D11—44no—
333C2×D11—42no—
334C2×D11—44no—
335C2×D11—44no—
336C2×D11—44no—
337C2×D11—44no—
338C2×D11—44no—
339C2×D11—44no—
340C2×D11—44no—
341C2×D11—816no—
342C2×D11—88no—
343C4×C11—24no—
344C4×C11—42no—
345C4×C11—44no—
346C4×C11—42no—
347C4×C11—44no—
348Dic11—24no—
349Dic11—42no—
350Dic11—44no—
351C25—111no—
352C25—111no—
353C25—112no—
354C22×D4—118no—
355C22×D4—114no—
356C22×D4—114no—
357C22×D4—118no—
358C22×D4—228no—
359C22×D4—224no—
360C22×D4—224no—
361C22×D4—444no—
362C22×D4—444no—
363C22×D4—444no—
364C22×D4—444no—
365C22×D4—444no—
366C22×D4—444no—
367C22×D4—444no—
368C23×C4—111no—
369C23×C4—118no—
370C23×C4—114no—
371C23×C4—111no—
372C23×C4—112no—
373C23×C4—444no—
374C23×C4C22×D1111yescharacteristic
375C23×C4C22×D1112yes—
376C23×C4C22×D1111yescharacteristic
377C23×C4C22×D1112yes—
378C23×C4C22×D1111yescharacteristic
379#123—2232no—
380#121—2232no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—114no—
384C2×C4⋊C4—118no—
385C2×C4⋊C4—118no—
386C2×C4⋊C4—224no—
387C2×C4⋊C4—444no—
388C2×C4⋊C4—444no—
389C2×C4⋊C4C22×D1114yes—
390C2×C4⋊C4C22×D1114yes—
391C2×C22⋊C4—118no—
392C2×C22⋊C4—114no—
393C2×C22⋊C4—114no—
394C2×C22⋊C4—114no—
395C2×C22⋊C4—118no—
396C2×C22⋊C4—118no—
397C2×C22⋊C4—118no—
398C2×C22⋊C4—118no—
399C2×C22⋊C4—118no—
400C2×C22⋊C4—228no—
401C2×C22⋊C4—224no—
402C2×C22⋊C4—224no—
403C2×C22⋊C4—224no—
404C2×C22⋊C4—444no—
405C2×C22⋊C4—444no—
406C2×C22⋊C4—444no—
407C2×C22⋊C4—444no—
408C2×C22⋊C4—444no—
409#95—114no—
410#95C22×D1114yes—
411#95C22×D1118yes—
412#95C22×D1118yes—
413C2×C11#281318yescomplemented
414C2×C11C2×#53411yescharacteristic
415C2×C11C2×#53511yescharacteristic
416C2×C11C2×#53711yescharacteristic
417C2×C11C2×#56012yes—
418C2×C11C2×#56212yes—
419D11—44no—
420D11—82no—
421D11—84no—
422C24—114no—
423C24—112no—
424C24—112no—
425C24—114no—
426C24—222no—
427C24—222no—
428C24—224no—
429C24—224no—
430C24—222no—
431C24—224no—
432C24—222no—
433C24—224no—
434C24—224no—
435C24—224no—
436C24—224no—
437C24—224no—
438C24—222no—
439C24—224no—
440C24—224no—
441C24—224no—
442C24C23×D1111yescharacteristic
443C2×D4—2216no—
444C2×D4—228no—
445C2×D4—228no—
446C2×D4—448no—
447C2×D4—448no—
448C2×D4—448no—
449C2×D4—448no—
450C2×D4—448no—
451C2×D4—448no—
452C2×D4—448no—
453C2×D4—4416no—
454C2×D4—4416no—
455C2×D4—4432no—
456C2×D4—4432no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—114no—
470C22×C4—112no—
471C22×C4—1116no—
472C22×C4—114no—
473C22×C4—112no—
474C22×C4—224no—
475C22×C4—222no—
476C22×C4—224no—
477C22×C4—224no—
478C22×C4—224no—
479C22×C4—224no—
480C22×C4—4416no—
481C22×C4—448no—
482C22×C4D4×D1114yes—
483C22×C4C23×D1114yes—
484C22×C4C23×D1112yes—
485C22×C4C23×D1114yes—
486C22×C4C23×D1112yes—
487C22×C4C23×D1112yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—228no—
491C4⋊C4—448no—
492C4⋊C4—448no—
493C22⋊C4—2216no—
494C22⋊C4—228no—
495C22⋊C4—228no—
496C22⋊C4—228no—
497C22⋊C4—448no—
498C22⋊C4—448no—
499C22⋊C4—448no—
500C22⋊C4—448no—
501C22⋊C4—448no—
502C11C2×#281311yescharacteristic, complemented, Sylow
503C23—114no—
504C23—228no—
505C23—228no—
506C23—222no—
507C23—222no—
508C23—224no—
509C23—224no—
510C23—228no—
511C23—222no—
512C23—224no—
513C23—222no—
514C23—224no—
515C23—224no—
516C23—224no—
517C23—224no—
518C23—224no—
519C23—224no—
520C23—224no—
521C23—224no—
522C23—222no—
523C23—448no—
524C23—448no—
525C23—448no—
526C23—448no—
527C23—448no—
528C23—444no—
529C23—448no—
530C23—448no—
531C23—444no—
532C23—448no—
533C23—444no—
534C23—448no—
535C23—448no—
536C23—448no—
537C23C2×D4×D1112yes—
538C23#6704012yes—
539C23#6704212yes—
540C23#6704214yes—
541C23#6704214yes—
542C23C24×D1111yescharacteristic
543D4—4416no—
544D4—4416no—
545D4—4432no—
546D4—4432no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—114no—
565C2×C4—228no—
566C2×C4—224no—
567C2×C4—222no—
568C2×C4—224no—
569C2×C4—224no—
570C2×C4—224no—
571C2×C4—224no—
572C2×C4—4416no—
573C2×C4—448no—
574C2×C4—444no—
575C2×C4C2×D4×D1114yes—
576C22—224no—
577C22—224no—
578C22—224no—
579C22—444no—
580C22—444no—
581C22—4416no—
582C22—444no—
583C22—442no—
584C22—444no—
585C22—442no—
586C22—442no—
587C22—444no—
588C22—444no—
589C22—444no—
590C22—444no—
591C22—444no—
592C22—444no—
593C22—444no—
594C22—888no—
595C22—8816no—
596C22#1062316014yes—
597C22#1062318414yes—
598C22#1062320514yes—
599C22#1062323118yes—
600C22#1062324218yes—
601C22C22×D4×D1111yescharacteristic
602C22C2×#6704011yescharacteristic
603C22C2×#6704212yes—
604C22C2×#6704212yes—
605C22C2×#6704211yescharacteristic
606C4—24no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—44no—
611C4—224no—
612C4—442no—
613C4—444no—
614C2—444no—
615C2—884no—
616C2—882no—
617C2#1307662118yesdirect product factor
618C2C2×#1062318411yescharacteristic
619C2C2×#1062320511yescharacteristic
620C2C2×#1062316011yescharacteristic
621C2C2×#1062324212yes—
622C2C2×#1062323112yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#400C2
socleC24×C11C24
hypercenterC2×#400D11
hypocenterC11C2×#2813
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
1. C22×#66947C2semidirect
2. C22×#66948C2semidirect
3. #13076621C2direct (N × H)
4. C2×#10622605C2semidirect
5. C2×#10622289C2semidirect
6. C2×#10622285C2semidirect
7. C2×#10622285C2semidirect
8. C2×#10622604C2semidirect
9. C2×#10622284C2semidirect
10. C2×#10622284C2semidirect
11. C2×#10622280C2semidirect
12. C2×C11×#400C2semidirect
33. C2×#66948C22semidirect
35. C2×#66947C22semidirect
36. C11×#400C22semidirect
41. #10622605C22semidirect
42. #10622604C22semidirect
43. #10622289C22semidirect
44. #10622285C22semidirect
45. #10622285C22semidirect
46. #10622284C22semidirect
47. #10622284C22semidirect
48. #10622280C22semidirect
75. C22×C4×D11D4semidirect
177. C2×C4×D11C2×D4semidirect
195. C2×#400D11semidirect
300. #400C2×D11semidirect
413. C2×C11#2813semidirect
502. C11C2×#2813semidirect
617. C2#13076621direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669472
C22×#669482
#130766212
C2×#106226052
C2×#106222892
C2×#106222852
C2×#106226042
C2×#106222842
C2×#106222802
C2×C11×#4002
C2×#281311