Finite Group Databasebeta

← C2×#13073551 #14924946

Subgroups — C2×#13073551 #14924946

Order 2816.

4566 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66938C211yesmaximal, characteristic, complemented
2C22×#66949C212yesmaximal, complemented
3C22×#66937C212yesmaximal, complemented
4C22×#66946C211yesmaximal, characteristic, complemented
5#13073551C2116yesmaximal, direct product factor
6C2×#10622605C211yesmaximal, characteristic, complemented
7C2×#10622284C211yesmaximal, characteristic, complemented
8C2×#10622573C211yesmaximal, characteristic, complemented
9C2×#10622601C212yesmaximal, complemented
10C2×#10622570C212yesmaximal, complemented
11C2×#10622262C211yesmaximal, characteristic, complemented
12C2×C11×#399C211yesmaximal, characteristic, complemented
13C23×C22⋊D11C2211yescharacteristic
14C23×C22⋊D11C2212yes—
15C23×D44C2211yescharacteristic
16C23×D44C2212yes—
17C23×C4×D11C2212yes—
18C23×Dic22C2211yescharacteristic
19C22×C11×C4⋊C4C2211yescharacteristic
20C22×C11×C4⋊C4C2212yes—
21C22×C11×C22⋊C4C2211yescharacteristic
22C22×C11×C22⋊C4C2212yes—
23C22×C22⋊Dic11C2211yescharacteristic
24C22×(C2×D11)⋊C4C2211yescharacteristic
25C22×(C2×D11)⋊C4C2211yescharacteristic
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2212yes—
28C22×(C2×D11)⋊C4C2212yes—
29C22×(C2×D11)⋊C4C2212yes—
30C22×Dic11⋊C4C2212yes—
31C22×Dic11⋊C4C2212yes—
32C22×C4×Dic11C2212yes—
33C22×C4×Dic11C2211yescharacteristic
34C2×#66949—216no—
35C2×#66949C22116yescomplemented
36C2×#66946—28no—
37C2×#66946C2218yescomplemented
38C2×#66938—28no—
39C2×#66938C2218yescomplemented
40C2×#66937—216no—
41C2×#66937C22116yescomplemented
42C11×#399C2218yescomplemented
43C2×C11×#95C2211yescharacteristic
44C2×C11⋊#95C2211yescharacteristic
45#10622605C2218yescomplemented
46#10622601C22116yescomplemented
47#10622573C2218yescomplemented
48#10622570C22116yescomplemented
49#10622284C2218yescomplemented
50#10622262C2218yescomplemented
51C25×C11C2311yescharacteristic
52C24×D11C2311yescharacteristic
53C24×D11C2312yes—
54C23×C4×C11C2312yes—
55C23×C4×C11C2311yescharacteristic
56C23×C4×C11C2312yes—
57C23×C4×C11C2311yescharacteristic
58C22×C22⋊D11—28no—
59C22×C22⋊D11—28no—
60C22×C22⋊D11—28no—
61C22×C22⋊D11—24no—
62C22×C22⋊D11—44no—
63C22×C22⋊D11C2314yes—
64C22×C22⋊D11C2318yes—
65C23×Dic11C2311yescharacteristic
66C23×Dic11C2311yescharacteristic
67C23×Dic11C2312yes—
68C23×Dic11C2311yescharacteristic
69C22×D44—28no—
70C22×D44—24no—
71C22×D44—28no—
72C22×D44—28no—
73C22×D44—44no—
74C22×D44C2314yes—
75C22×D44C2318yes—
76C22×C4×D11—28no—
77C22×C4×D11—28no—
78C22×C4×D11—28no—
79C22×C4×D11C2318yes—
80C22×Dic22—24no—
81C22×Dic22—28no—
82C22×Dic22C2314yes—
83C2×C11×C4⋊C4—24no—
84C2×C11×C4⋊C4—28no—
85C2×C11×C4⋊C4C2314yes—
86C2×C11×C4⋊C4C2318yes—
87C2×C11×C22⋊C4—28no—
88C2×C11×C22⋊C4—24no—
89C2×C11×C22⋊C4C2318yes—
90C2×C11×C22⋊C4C2314yes—
91C2×C22⋊Dic11—24no—
92C2×C22⋊Dic11C2314yes—
93C2×(C2×D11)⋊C4—24no—
94C2×(C2×D11)⋊C4—28no—
95C2×(C2×D11)⋊C4—28no—
96C2×(C2×D11)⋊C4—24no—
97C2×(C2×D11)⋊C4—28no—
98C2×(C2×D11)⋊C4—28no—
99C2×(C2×D11)⋊C4C2318yes—
100C2×(C2×D11)⋊C4C2314yes—
101C2×(C2×D11)⋊C4C2314yes—
102C2×(C2×D11)⋊C4C2318yes—
103C2×(C2×D11)⋊C4C2318yes—
104C2×(C2×D11)⋊C4C2318yes—
105C2×Dic11⋊C4—28no—
106C2×Dic11⋊C4D4116yescomplemented
107C2×Dic11⋊C4C2318yes—
108C2×Dic11⋊C4C2318yes—
109C2×C4×Dic11—44no—
110C2×C4×Dic11D418yescomplemented
111C2×C4×Dic11C2318yes—
112C2×C4×Dic11C2314yes—
113#66949—264no—
114#66946—232no—
115#66938—232no—
116#66937—264no—
117C11×#95C2314yes—
118C11⋊#95C2314yes—
119C2×#2540—111nomaximal, Sylow
120C24×C11—22no—
121C24×C11—24no—
122C24×C11—24no—
123C24×C11—22no—
124C24×C11—22no—
125C24×C11C2412yes—
126C24×C11C2411yescharacteristic
127C23×D11—24no—
128C23×D11—24no—
129C23×D11—24no—
130C23×D11—24no—
131C23×D11—24no—
132C23×D11—24no—
133C23×D11—22no—
134C23×D11—24no—
135C23×D11—24no—
136C23×D11—22no—
137C23×D11—22no—
138C23×D11#4118yes—
139C23×D11C2412yes—
140C23×D11C2414yes—
141C22×C4×C11—24no—
142C22×C4×C11—24no—
143C22×C4×C11—24no—
144C22×C4×C11—24no—
145C22×C4×C11—22no—
146C22×C4×C11—24no—
147C22×C4×C11—24no—
148C22×C4×C11—24no—
149C22×C4×C11C2×D414yes—
150C22×C4×C11C2×D418yes—
151C22×C4×C11C2414yes—
152C22×C4×C11C2412yes—
153C22×C4×C11C2412yes—
154C22×C4×C11C2414yes—
155C2×C22⋊D11—216no—
156C2×C22⋊D11—216no—
157C2×C22⋊D11—28no—
158C2×C22⋊D11—216no—
159C2×C22⋊D11—48no—
160C2×C22⋊D11—432no—
161C2×C22⋊D11—416no—
162C22×Dic11—24no—
163C22×Dic11—22no—
164C22×Dic11—24no—
165C22×Dic11—22no—
166C22×Dic11—24no—
167C22×Dic11—22no—
168C22×Dic11C2×D414yes—
169C22×Dic11C2×D418yes—
170C22×Dic11C2×D414yes—
171C22×Dic11C2×D418yes—
172C22×Dic11C2412yes—
173C22×Dic11C2414yes—
174C22×Dic11C2412yes—
175C22×Dic11C2412yes—
176C2×D44—232no—
177C2×D44—216no—
178C2×D44—216no—
179C2×D44—28no—
180C2×D44—216no—
181C2×D44—48no—
182C2×D44—432no—
183C2×C4×D11—216no—
184C2×C4×D11—216no—
185C2×C4×D11—216no—
186C2×C4×D11—432no—
187C2×Dic22—28no—
188C2×Dic22—216no—
189C2×Dic22—416no—
190C11×C4⋊C4—28no—
191C11×C4⋊C4—216no—
192C11×C22⋊C4—216no—
193C11×C22⋊C4—28no—
194C22⋊Dic11—28no—
195(C2×D11)⋊C4—216no—
196(C2×D11)⋊C4—28no—
197(C2×D11)⋊C4—216no—
198(C2×D11)⋊C4—216no—
199(C2×D11)⋊C4—28no—
200(C2×D11)⋊C4—216no—
201Dic11⋊C4—216no—
202Dic11⋊C4C2×D4132yescomplemented
203C4×Dic11—48no—
204C4×Dic11C2×D4116yescomplemented
205C22×#124—111no—
206C22×C22⋊Q8—112no—
207C22×#121—112no—
208C22×C4×D4—111no—
209#2540—1116no—
210C2×#399—111no—
211C2×#399D1111yescharacteristic, complemented
212C2×#392—111no—
213C2×#392—111no—
214C2×#390—112no—
215C2×#390—112no—
216C2×C4×C4⋊C4—111no—
217C23×C11—22no—
218C23×C11—24no—
219C23×C11—24no—
220C23×C11—22no—
221C23×C11—22no—
222C23×C11—44no—
223C23×C11—44no—
224C23×C11—44no—
225C23×C11—48no—
226C23×C11—48no—
227C23×C11C22×D412yes—
228C23×C11C22×D414yes—
229C23×C11C2×#4114yes—
230C23×C11H(5,𝔽2)12yes—
231C23×C112−1+412yes—
232C23×C11C2511yescharacteristic
233C22×D11—216no—
234C22×D11—216no—
235C22×D11—216no—
236C22×D11—24no—
237C22×D11—24no—
238C22×D11—24no—
239C22×D11—24no—
240C22×D11—24no—
241C22×D11—24no—
242C22×D11—22no—
243C22×D11—24no—
244C22×D11—24no—
245C22×D11—22no—
246C22×D11—22no—
247C22×D11—48no—
248C22×D11—48no—
249C22×D11—48no—
250C22×D11—48no—
251C22×D11—44no—
252C22×D11—44no—
253C22×D11—44no—
254C22×D11—48no—
255C22×D11—48no—
256C22×D11C2×#4118yes—
257C2×C4×C11—28no—
258C2×C4×C11—216no—
259C2×C4×C11—24no—
260C2×C4×C11—24no—
261C2×C4×C11—24no—
262C2×C4×C11—24no—
263C2×C4×C11—22no—
264C2×C4×C11—24no—
265C2×C4×C11—24no—
266C2×C4×C11—24no—
267C2×C4×C11—44no—
268C2×C4×C11—48no—
269C2×C4×C11C22×D414yes—
270C2×C4×C11C22×D418yes—
271C22⋊D11—432no—
272C22⋊D11—416no—
273C2×Dic11—28no—
274C2×Dic11—216no—
275C2×Dic11—24no—
276C2×Dic11—22no—
277C2×Dic11—24no—
278C2×Dic11—22no—
279C2×Dic11—24no—
280C2×Dic11—22no—
281C2×Dic11—44no—
282C2×Dic11#127116yes—
283C2×Dic11C22×D414yes—
284C2×Dic11C22×D418yes—
285C2×Dic11C22×D414yes—
286C2×Dic11C22×D418yes—
287D44—232no—
288D44—432no—
289C4×D11—432no—
290Dic22—416no—
291C23×Q8—111no—
292C23×D4—111no—
293C23×D4—112no—
294C23×D4—111no—
295C23×D4—112no—
296C24×C4—112no—
297C2×#124—118no—
298C2×#124—228no—
299C2×C22⋊Q8—1116no—
300C2×C22⋊Q8—2216no—
301C2×#121—1116no—
302C2×#121—2216no—
303C2×C4×D4—118no—
304C2×C4×D4—228no—
305C22×C4⋊C4—112no—
306C22×C4⋊C4—112no—
307C22×C4⋊C4C2×D1111yescharacteristic
308C22×C4⋊C4C2×D1112yes—
309C22×C22⋊C4—111no—
310C22×C22⋊C4—111no—
311C22×C22⋊C4—112no—
312C22×C22⋊C4—112no—
313C22×C22⋊C4—112no—
314C22×C22⋊C4—112no—
315C22×C22⋊C4—111no—
316C22×C22⋊C4C2×D1111yescharacteristic
317C22×C22⋊C4C2×D1112yes—
318C22×C42—111no—
319C22×C42—112no—
320#399—118no—
321#399C2×D1118yescomplemented
322#392—118no—
323#392—118no—
324#390—1116no—
325#390—1116no—
326C4×C4⋊C4—118no—
327C2×#95—111no—
328C2×#95C2×D1111yescharacteristic
329C22×C11—42no—
330C22×C11—42no—
331C22×C11—42no—
332C22×C11—44no—
333C22×C11—44no—
334C22×C11—88no—
335C22×C11#52514yes—
336C22×C11C2×#12714yes—
337C22×C11#53714yes—
338C22×C11#54618yes—
339C22×C11#54818yes—
340C22×C11C23×D411yescharacteristic
341C22×C11C23×D412yes—
342C22×C11C22×#4112yes—
343C22×C11C2×H(5,𝔽2)11yescharacteristic
344C22×C11C2×2−1+411yescharacteristic
345C2×D11—28no—
346C2×D11—28no—
347C2×D11—28no—
348C2×D11—44no—
349C2×D11—44no—
350C2×D11—44no—
351C2×D11—44no—
352C2×D11—432no—
353C2×D11—42no—
354C2×D11—42no—
355C2×D11—42no—
356C2×D11—44no—
357C2×D11—44no—
358C2×D11—88no—
359C4×C11—24no—
360C4×C11—28no—
361C4×C11—42no—
362C4×C11—44no—
363Dic11—24no—
364Dic11—28no—
365Dic11—42no—
366Dic11C2×#12718yes—
367C25—112no—
368C25—111no—
369C25C22×D1111yescharacteristic
370C22×Q8—114no—
371C22×Q8—228no—
372C22×Q8—224no—
373C22×D4—114no—
374C22×D4—118no—
375C22×D4—114no—
376C22×D4—118no—
377C22×D4—228no—
378C22×D4—228no—
379C22×D4—228no—
380C22×D4—228no—
381C22×D4—224no—
382C22×D4—228no—
383C22×D4—224no—
384C22×D4—228no—
385C22×D4—444no—
386C22×D4—444no—
387C23×C4—111no—
388C23×C4—111no—
389C23×C4—112no—
390C23×C4—111no—
391C23×C4—118no—
392C23×C4—228no—
393C23×C4—228no—
394C23×C4—228no—
395C23×C4C22×D1111yescharacteristic
396C23×C4C22×D1112yes—
397C23×C4C22×D1112yes—
398C23×C4C22×D1111yescharacteristic
399#124—2232no—
400C22⋊Q8—2264no—
401#121—2264no—
402C4×D4—2232no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—28no—
405C2×C4⋊C4—1116no—
406C2×C4⋊C4—118no—
407C2×C4⋊C4—118no—
408C2×C4⋊C4—228no—
409C2×C4⋊C4C22×D1114yes—
410C2×C4⋊C4C22×D1118yes—
411C2×C22⋊C4—28no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—114no—
414C2×C22⋊C4—118no—
415C2×C22⋊C4—118no—
416C2×C22⋊C4—118no—
417C2×C22⋊C4—118no—
418C2×C22⋊C4—114no—
419C2×C22⋊C4—114no—
420C2×C22⋊C4—228no—
421C2×C22⋊C4—228no—
422C2×C22⋊C4—228no—
423C2×C22⋊C4—224no—
424C2×C22⋊C4—224no—
425C2×C22⋊C4—228no—
426C2×C22⋊C4—224no—
427C2×C22⋊C4C22×D1118yes—
428C2×C22⋊C4C22×D1114yes—
429C2×C42—118no—
430C2×C42—114no—
431C2×C42—118no—
432C2×C42—444no—
433#95—114no—
434#95C22×D1114yes—
435C2×C11—82no—
436C2×C11#254018yescomplemented
437C2×C11C2×#52511yescharacteristic
438C2×C11C22×#12711yescharacteristic
439C2×C11C2×#53711yescharacteristic
440C2×C11C2×#54612yes—
441C2×C11C2×#54812yes—
442D11—48no—
443D11—82no—
444C24—22no—
445C24—24no—
446C24—22no—
447C24—24no—
448C24—22no—
449C24—118no—
450C24—114no—
451C24—112no—
452C24—224no—
453C24—224no—
454C24—224no—
455C24—224no—
456C24—224no—
457C24—224no—
458C24—222no—
459C24—224no—
460C24—222no—
461C24—224no—
462C24—222no—
463C24C23×D1112yes—
464C24C23×D1111yescharacteristic
465C2×Q8—2216no—
466C2×Q8—228no—
467C2×Q8—4416no—
468C2×D4—2216no—
469C2×D4—2232no—
470C2×D4—2216no—
471C2×D4—2216no—
472C2×D4—2216no—
473C2×D4—228no—
474C2×D4—2216no—
475C2×D4—228no—
476C2×D4—2216no—
477C2×D4—448no—
478C2×D4—4432no—
479C2×D4—4416no—
480C2×D4—448no—
481C2×D4—4432no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—114no—
491C22×C4—118no—
492C22×C4—112no—
493C22×C4—114no—
494C22×C4—118no—
495C22×C4—114no—
496C22×C4—112no—
497C22×C4—112no—
498C22×C4—2216no—
499C22×C4—2216no—
500C22×C4—2216no—
501C22×C4—222no—
502C22×C4—224no—
503C22×C4—222no—
504C22×C4—224no—
505C22×C4—222no—
506C22×C4—224no—
507C22×C4—4432no—
508C22×C4D4×D1114yes—
509C22×C4D4×D1118yes—
510C22×C4C23×D1112yes—
511C22×C4C23×D1114yes—
512C22×C4C23×D1112yes—
513C22×C4C23×D1114yes—
514C4⋊C4—28no—
515C4⋊C4—216no—
516C4⋊C4—1132no—
517C4⋊C4—2216no—
518C22⋊C4—216no—
519C22⋊C4—28no—
520C22⋊C4—2216no—
521C22⋊C4—2216no—
522C22⋊C4—2216no—
523C22⋊C4—228no—
524C22⋊C4—228no—
525C22⋊C4—2216no—
526C22⋊C4—228no—
527C42—1116no—
528C42—448no—
529C11C2×#254011yescharacteristic, complemented, Sylow
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23—44no—
540C23—118no—
541C23—2216no—
542C23—2216no—
543C23—2216no—
544C23—224no—
545C23—224no—
546C23—224no—
547C23—224no—
548C23—224no—
549C23—224no—
550C23—222no—
551C23—224no—
552C23—222no—
553C23—224no—
554C23—222no—
555C23—448no—
556C23—448no—
557C23—448no—
558C23—448no—
559C23—444no—
560C23—444no—
561C23—448no—
562C23—448no—
563C23—444no—
564C23C2×D4×D1112yes—
565C23C2×D4×D1114yes—
566C23#6704012yes—
567C23#41×D1114yes—
568C23#6704212yes—
569C23C24×D1111yescharacteristic
570Q8—4416no—
571D4—2232no—
572D4—4432no—
573D4—4416no—
574D4—4432no—
575C2×C4—216no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—1116no—
588C2×C4—114no—
589C2×C4—118no—
590C2×C4—114no—
591C2×C4—118no—
592C2×C4—228no—
593C2×C4—2216no—
594C2×C4—222no—
595C2×C4—224no—
596C2×C4—222no—
597C2×C4—224no—
598C2×C4—222no—
599C2×C4—224no—
600C2×C4—4432no—
601C2×C4—444no—
602C2×C4C2×D4×D1114yes—
603C2×C4C2×D4×D1118yes—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—88no—
610C22—228no—
611C22—228no—
612C22—228no—
613C22—444no—
614C22—444no—
615C22—4432no—
616C22—444no—
617C22—444no—
618C22—442no—
619C22—442no—
620C22—444no—
621C22—444no—
622C22—442no—
623C22—888no—
624C22#1062316014yes—
625C22#1062316418yes—
626C22#1062318518yes—
627C22#1062318714yes—
628C22D11×#12714yes—
629C22C22×D4×D1111yescharacteristic
630C22C22×D4×D1112yes—
631C22C2×#6704011yescharacteristic
632C22C2×#41×D1112yes—
633C22C2×#6704211yescharacteristic
634C4—28no—
635C4—24no—
636C4—44no—
637C4—42no—
638C4—118no—
639C4—224no—
640C4—228no—
641C4—442no—
642C2—82no—
643C2—448no—
644C2—882no—
645C2#1307355118yesdirect product factor
646C2C2×#1062318711yescharacteristic
647C2C2×D11×#12711yescharacteristic
648C2C2×#1062316011yescharacteristic
649C2C2×#1062318512yes—
650C2C2×#1062316412yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#399C2
socleC24×C11C24
hypercenterC2×#399D11
hypocenterC11C2×#2540
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#66938C2semidirect
2. C22×#66949C2semidirect
3. C22×#66937C2semidirect
4. C22×#66946C2semidirect
5. #13073551C2direct (N × H)
6. C2×#10622605C2semidirect
7. C2×#10622284C2semidirect
8. C2×#10622573C2semidirect
9. C2×#10622601C2semidirect
10. C2×#10622570C2semidirect
11. C2×#10622262C2semidirect
12. C2×C11×#399C2semidirect
35. C2×#66949C22semidirect
37. C2×#66946C22semidirect
39. C2×#66938C22semidirect
41. C2×#66937C22semidirect
42. C11×#399C22semidirect
45. #10622605C22semidirect
46. #10622601C22semidirect
47. #10622573C22semidirect
48. #10622570C22semidirect
49. #10622284C22semidirect
50. #10622262C22semidirect
106. C2×Dic11⋊C4D4semidirect
110. C2×C4×Dic11D4semidirect
202. Dic11⋊C4C2×D4semidirect
204. C4×Dic11C2×D4semidirect
211. C2×#399D11semidirect
321. #399C2×D11semidirect
436. C2×C11#2540semidirect
529. C11C2×#2540semidirect
645. C2#13073551direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669382
C22×#669492
C22×#669372
C22×#669462
#130735512
C2×#106226052
C2×#106222842
C2×#106225732
C2×#106226012
C2×#106225702
C2×#106222622
C2×C11×#3992
C2×#254011