Finite Group Databasebeta

← C2×#13074243 #14924413

Subgroups — C2×#13074243 #14924413

Order 2816.

3330 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66929C211yesmaximal, characteristic, complemented
2C22×C4×D44C211yesmaximal, characteristic, complemented
3#13074243C2116yesmaximal, direct product factor
4C2×#10622274C211yesmaximal, characteristic, complemented
5C2×#10622287C211yesmaximal, characteristic, complemented
6C2×#10622285C211yesmaximal, characteristic, complemented
7C2×#10622284C211yesmaximal, characteristic
8C2×#10622552C212yesmaximal, complemented
9C2×#10622553C212yesmaximal, complemented
10C2×#10622278C212yesmaximal, complemented
11C2×#10622281C212yesmaximal, complemented
12C2×C11×#376C211yesmaximal, characteristic, complemented
13C23×D44C2211yescharacteristic
14C23×D44C2212yes—
15C23×C4×D11C2212yes—
16C22×C42×C11C2212yes—
17C22×C42×C11C2211yescharacteristic
18C22×(C2×D11)⋊C4C2211yescharacteristic
19C22×(C2×D11)⋊C4C2211yescharacteristic
20C22×(C2×D11)⋊C4C2211yescharacteristic
21C22×(C2×D11)⋊C4C2212yes—
22C22×(C2×D11)⋊C4C2212yes—
23C22×(C2×D11)⋊C4C2212yes—
24C22×(C2×D11)⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2212yes—
26C22×C4⋊Dic11C2211yescharacteristic
27C22×C4⋊Dic11C2212yes—
28C22×Dic11⋊C4C2211yescharacteristic
29C22×Dic11⋊C4C2211yescharacteristic
30C22×Dic11⋊C4C2211yescharacteristic
31C2×#66929—28no—
32C2×#66929C2218yescomplemented
33C2×C4×D44—28no—
34C2×C4×D44C2218yescomplemented
35C11×#376C2218yescomplemented
36C2×C11×#95C2212yes—
37C2×C11×#95C2211yescharacteristic
38C2×C11×#95C2211yescharacteristic
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41#10622553C22116yescomplemented
42#10622552C22116yescomplemented
43#10622287C2218yescomplemented
44#10622285C2218yescomplemented
45#10622284C2218yes—
46#10622281C22116yescomplemented
47#10622278C22116yescomplemented
48#10622274C2218yescomplemented
49C24×D11C2311yescharacteristic
50C24×D11C2312yes—
51C23×C4×C11C2312yes—
52C23×C4×C11C2311yescharacteristic
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×C4×C11C2312yes—
56C23×Dic11C2311yescharacteristic
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2312yes—
60C22×D44—28no—
61C22×D44—24no—
62C22×D44—28no—
63C22×D44—28no—
64C22×D44—44no—
65C22×D44C2314yes—
66C22×D44C2318yes—
67C22×C4×D11—28no—
68C22×C4×D11—44no—
69C22×C4×D11—44no—
70C22×C4×D11C2318yes—
71C2×C42×C11—28no—
72C2×C42×C11D418yescomplemented
73C2×C42×C11C2318yes—
74C2×C42×C11C2314yes—
75C2×(C2×D11)⋊C4—28no—
76C2×(C2×D11)⋊C4—28no—
77C2×(C2×D11)⋊C4—24no—
78C2×(C2×D11)⋊C4—24no—
79C2×(C2×D11)⋊C4—24no—
80C2×(C2×D11)⋊C4—28no—
81C2×(C2×D11)⋊C4—44no—
82C2×(C2×D11)⋊C4—44no—
83C2×(C2×D11)⋊C4C2318yes—
84C2×(C2×D11)⋊C4C2314yes—
85C2×(C2×D11)⋊C4C2314yes—
86C2×(C2×D11)⋊C4C2314yes—
87C2×(C2×D11)⋊C4C2318yes—
88C2×(C2×D11)⋊C4C2318yes—
89C2×(C2×D11)⋊C4C2318yes—
90C2×(C2×D11)⋊C4C2318yes—
91C2×C4⋊Dic11—24no—
92C2×C4⋊Dic11—44no—
93C2×C4⋊Dic11C2314yes—
94C2×C4⋊Dic11C2318yes—
95C2×Dic11⋊C4—24no—
96C2×Dic11⋊C4—24no—
97C2×Dic11⋊C4—24no—
98C2×Dic11⋊C4C2314yes—
99C2×Dic11⋊C4C2314yes—
100C2×Dic11⋊C4C2314yes—
101#66929—232no—
102C4×D44—232no—
103C11×#95C2314yes—
104C11×#95C2318yes—
105C11×#95C2314yes—
106C11⋊#95C2318yes—
107C11⋊#95C2318yes—
108C2×#2603—111nomaximal, Sylow
109C24×C11C2411yescharacteristic
110C23×D11—24no—
111C23×D11—24no—
112C23×D11—24no—
113C23×D11—24no—
114C23×D11—24no—
115C23×D11—24no—
116C23×D11—22no—
117C23×D11—24no—
118C23×D11—24no—
119C23×D11—22no—
120C23×D11—22no—
121C23×D11#4118yes—
122C23×D11C2412yes—
123C23×D11C2414yes—
124C22×C4×C11—24no—
125C22×C4×C11—24no—
126C22×C4×C11—24no—
127C22×C4×C11—24no—
128C22×C4×C11—22no—
129C22×C4×C11—22no—
130C22×C4×C11—24no—
131C22×C4×C11—24no—
132C22×C4×C11—24no—
133C22×C4×C11C2×D414yes—
134C22×C4×C11C2×D414yes—
135C22×C4×C11C2×D414yes—
136C22×C4×C11#4118yes—
137C22×C4×C11C2414yes—
138C22×C4×C11C2412yes—
139C22×C4×C11C2412yes—
140C22×C4×C11C2412yes—
141C22×C4×C11C2414yes—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—22no—
145C22×Dic11—24no—
146C22×Dic11—24no—
147C22×Dic11—24no—
148C22×Dic11—22no—
149C22×Dic11—24no—
150C22×Dic11—22no—
151C22×Dic11—24no—
152C22×Dic11C2412yes—
153C22×Dic11C2412yes—
154C22×Dic11C2414yes—
155C22×Dic11C2412yes—
156C2×D44—232no—
157C2×D44—216no—
158C2×D44—216no—
159C2×D44—28no—
160C2×D44—216no—
161C2×D44—48no—
162C2×D44—432no—
163C2×C4×D11—216no—
164C2×C4×D11—48no—
165C2×C4×D11—48no—
166C2×C4×D11—432no—
167C42×C11—216no—
168C42×C11C2×D4116yescomplemented
169(C2×D11)⋊C4—216no—
170(C2×D11)⋊C4—216no—
171(C2×D11)⋊C4—28no—
172(C2×D11)⋊C4—28no—
173(C2×D11)⋊C4—28no—
174(C2×D11)⋊C4—216no—
175(C2×D11)⋊C4—48no—
176(C2×D11)⋊C4—48no—
177C4⋊Dic11—28no—
178C4⋊Dic11—48no—
179Dic11⋊C4—28no—
180Dic11⋊C4—28no—
181Dic11⋊C4—28no—
182C22×#126—111no—
183C22×C4×D4—111no—
184#2603—1116no—
185C2×#398—111no—
186C2×#397—111no—
187C2×#394—111no—
188C2×#392—111no—
189C2×#390—112no—
190C2×#388—112no—
191C2×#386—112no—
192C2×#385—112no—
193C2×#376D1111yescharacteristic, complemented
194C23×C11C22×D412yes—
195C23×C11C2×#4114yes—
196C23×C11C2×#4114yes—
197C23×C11H(5,𝔽2)12yes—
198C23×C11H(5,𝔽2)12yes—
199C23×C11C2511yescharacteristic
200C22×D11—216no—
201C22×D11—216no—
202C22×D11—216no—
203C22×D11—24no—
204C22×D11—24no—
205C22×D11—24no—
206C22×D11—24no—
207C22×D11—24no—
208C22×D11—24no—
209C22×D11—22no—
210C22×D11—24no—
211C22×D11—24no—
212C22×D11—22no—
213C22×D11—22no—
214C22×D11—48no—
215C22×D11—48no—
216C22×D11—48no—
217C22×D11—48no—
218C22×D11—44no—
219C22×D11—44no—
220C22×D11—44no—
221C22×D11—48no—
222C22×D11—48no—
223C22×D11C2×#4118yes—
224C2×C4×C11—28no—
225C2×C4×C11—28no—
226C2×C4×C11—28no—
227C2×C4×C11—216no—
228C2×C4×C11—24no—
229C2×C4×C11—24no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—22no—
233C2×C4×C11—22no—
234C2×C4×C11—24no—
235C2×C4×C11—24no—
236C2×C4×C11—24no—
237C2×C4×C11—48no—
238C2×C4×C11C22×D414yes—
239C2×C4×C11C22×D414yes—
240C2×C4×C11C22×D414yes—
241C2×C4×C11C2×#4118yes—
242C2×Dic11—22no—
243C2×Dic11—22no—
244C2×Dic11—22no—
245C2×Dic11—24no—
246C2×Dic11—24no—
247C2×Dic11—24no—
248C2×Dic11—22no—
249C2×Dic11—24no—
250C2×Dic11—22no—
251C2×Dic11—24no—
252C2×Dic11—44no—
253C2×Dic11—44no—
254C2×Dic11—48no—
255C2×Dic11—44no—
256D44—232no—
257D44—432no—
258C4×D11—432no—
259C23×D4—111no—
260C23×D4—112no—
261C24×C4—112no—
262C2×#126—118no—
263C2×#126—228no—
264C2×C4×D4—118no—
265C2×C4×D4—228no—
266C22×C4⋊C4—111no—
267C22×C4⋊C4—111no—
268C22×C4⋊C4—111no—
269C22×C4⋊C4—111no—
270C22×C4⋊C4—112no—
271C22×C22⋊C4—111no—
272C22×C22⋊C4—112no—
273C22×C22⋊C4—112no—
274C22×C22⋊C4—112no—
275C22×C22⋊C4—111no—
276C22×C22⋊C4—111no—
277C22×C22⋊C4—112no—
278C22×C22⋊C4—112no—
279C22×C42C2×D1112yes—
280C22×C42C2×D1111yescharacteristic
281#398—118no—
282#397—118no—
283#394—118no—
284#392—118no—
285#390—1116no—
286#388—1116no—
287#386—1116no—
288#385—1116no—
289#376C2×D1118yescomplemented
290C2×#95—112no—
291C2×#95—112no—
292C2×#95C2×D1111yescharacteristic
293C2×#95C2×D1111yescharacteristic
294C2×#95C2×D1112yes—
295C22×C11#52514yes—
296C22×C11#53214yes—
297C22×C11#53514yes—
298C22×C11#55118yes—
299C22×C11#55318yes—
300C22×C11C23×D411yescharacteristic
301C22×C11C22×#4112yes—
302C22×C11C22×#4112yes—
303C22×C11C2×H(5,𝔽2)11yescharacteristic
304C22×C11C2×H(5,𝔽2)11yescharacteristic
305C2×D11—28no—
306C2×D11—28no—
307C2×D11—28no—
308C2×D11—432no—
309C2×D11—44no—
310C2×D11—44no—
311C2×D11—44no—
312C2×D11—44no—
313C2×D11—42no—
314C2×D11—42no—
315C2×D11—42no—
316C2×D11—44no—
317C2×D11—44no—
318C2×D11—88no—
319C4×C11—24no—
320C4×C11—24no—
321C4×C11—24no—
322C4×C11—28no—
323C4×C11—44no—
324Dic11—42no—
325Dic11—42no—
326Dic11—44no—
327Dic11—42no—
328C25—112no—
329C25—111no—
330C22×D4—114no—
331C22×D4—118no—
332C22×D4—228no—
333C22×D4—228no—
334C22×D4—224no—
335C22×D4—228no—
336C22×D4—444no—
337C23×C4—111no—
338C23×C4—111no—
339C23×C4—111no—
340C23×C4—112no—
341C23×C4—118no—
342C23×C4—228no—
343C23×C4—444no—
344C23×C4—444no—
345C23×C4C22×D1112yes—
346C23×C4C22×D1111yescharacteristic
347C23×C4C22×D1111yescharacteristic
348C23×C4C22×D1111yescharacteristic
349C23×C4C22×D1112yes—
350#126—2232no—
351C4×D4—2232no—
352C2×C4⋊C4—114no—
353C2×C4⋊C4—114no—
354C2×C4⋊C4—114no—
355C2×C4⋊C4—114no—
356C2×C4⋊C4—118no—
357C2×C4⋊C4—224no—
358C2×C4⋊C4—224no—
359C2×C4⋊C4—224no—
360C2×C4⋊C4—224no—
361C2×C4⋊C4—444no—
362C2×C22⋊C4—118no—
363C2×C22⋊C4—114no—
364C2×C22⋊C4—114no—
365C2×C22⋊C4—114no—
366C2×C22⋊C4—118no—
367C2×C22⋊C4—118no—
368C2×C22⋊C4—118no—
369C2×C22⋊C4—118no—
370C2×C22⋊C4—228no—
371C2×C22⋊C4—228no—
372C2×C22⋊C4—224no—
373C2×C22⋊C4—224no—
374C2×C22⋊C4—224no—
375C2×C22⋊C4—228no—
376C2×C22⋊C4—444no—
377C2×C22⋊C4—444no—
378C2×C42—28no—
379C2×C42D4418yescomplemented
380C2×C42C22×D1118yes—
381C2×C42C22×D1114yes—
382#95—118no—
383#95—118no—
384#95C22×D1118yes—
385#95C22×D1114yes—
386#95C22×D1114yes—
387C2×C11#260318yescomplemented
388C2×C11C2×#52511yescharacteristic
389C2×C11C2×#53211yescharacteristic
390C2×C11C2×#53511yescharacteristic
391C2×C11C2×#55112yes—
392C2×C11C2×#55312yes—
393D11—48no—
394D11—82no—
395C24—118no—
396C24—114no—
397C24—112no—
398C24—224no—
399C24—224no—
400C24—224no—
401C24—224no—
402C24—224no—
403C24—224no—
404C24—222no—
405C24—224no—
406C24—224no—
407C24—222no—
408C24—222no—
409C24C23×D1111yescharacteristic
410C2×D4—2216no—
411C2×D4—2232no—
412C2×D4—2216no—
413C2×D4—228no—
414C2×D4—2216no—
415C2×D4—448no—
416C2×D4—4432no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—112no—
427C22×C4—112no—
428C22×C4—114no—
429C22×C4—112no—
430C22×C4—222no—
431C22×C4—222no—
432C22×C4—222no—
433C22×C4—224no—
434C22×C4—224no—
435C22×C4—224no—
436C22×C4—222no—
437C22×C4—224no—
438C22×C4—222no—
439C22×C4—224no—
440C22×C4—2216no—
441C22×C4—448no—
442C22×C4—448no—
443C22×C4—4432no—
444C22×C4C2×D4414yes—
445C22×C4C2×D4414yes—
446C22×C4C2×D4414yes—
447C22×C4#433918yes—
448C22×C4C23×D1114yes—
449C22×C4C23×D1112yes—
450C22×C4C23×D1112yes—
451C22×C4C23×D1112yes—
452C22×C4C23×D1114yes—
453C4⋊C4—228no—
454C4⋊C4—228no—
455C4⋊C4—228no—
456C4⋊C4—228no—
457C4⋊C4—448no—
458C22⋊C4—2216no—
459C22⋊C4—2216no—
460C22⋊C4—228no—
461C22⋊C4—228no—
462C22⋊C4—228no—
463C22⋊C4—2216no—
464C22⋊C4—448no—
465C22⋊C4—448no—
466C42—216no—
467C42C2×D44116yescomplemented
468C11C2×#260311yescharacteristic, complemented, Sylow
469C23—118no—
470C23—2216no—
471C23—2216no—
472C23—2216no—
473C23—224no—
474C23—224no—
475C23—224no—
476C23—224no—
477C23—224no—
478C23—224no—
479C23—222no—
480C23—224no—
481C23—224no—
482C23—222no—
483C23—222no—
484C23—448no—
485C23—448no—
486C23—448no—
487C23—448no—
488C23—444no—
489C23—444no—
490C23—444no—
491C23—448no—
492C23—448no—
493C23C22×D4412yes—
494C23C2×#433914yes—
495C23#41×D1114yes—
496C23#6704212yes—
497C23#6704212yes—
498C23C24×D1111yescharacteristic
499D4—2232no—
500D4—4432no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—216no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—48no—
515C2×C4—222no—
516C2×C4—222no—
517C2×C4—222no—
518C2×C4—224no—
519C2×C4—224no—
520C2×C4—224no—
521C2×C4—222no—
522C2×C4—224no—
523C2×C4—222no—
524C2×C4—224no—
525C2×C4—4432no—
526C2×C4—444no—
527C2×C4—444no—
528C2×C4—448no—
529C2×C4—444no—
530C2×C4C22×D4414yes—
531C2×C4C22×D4414yes—
532C2×C4C22×D4414yes—
533C2×C4C2×#433918yes—
534C22—228no—
535C22—228no—
536C22—228no—
537C22—444no—
538C22—444no—
539C22—4432no—
540C22—444no—
541C22—444no—
542C22—442no—
543C22—444no—
544C22—442no—
545C22—442no—
546C22—444no—
547C22—888no—
548C22#1062300814yes—
549C22#1062305514yes—
550C22#1062305614yes—
551C22#1062305818yes—
552C22#1062306118yes—
553C22C23×D4411yescharacteristic
554C22C22×#433912yes—
555C22C2×#41×D1112yes—
556C22C2×#6704211yescharacteristic
557C22C2×#6704211yescharacteristic
558C4—24no—
559C4—24no—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—442no—
564C4—442no—
565C4—444no—
566C4—442no—
567C2—448no—
568C2—882no—
569C2#1307424318yesdirect product factor
570C2C2×#1062305511yescharacteristic
571C2C2×#1062300811yescharacteristic
572C2C2×#1062305611yescharacteristic
573C2C2×#1062305812yes—
574C2C2×#1062306112yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#376C2
socleC24×C11C24
prime radicalC2×#10622284C2
hypercenterC2×#376D11
hypocenterC11C2×#2603
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
1. C22×#66929C2semidirect
2. C22×C4×D44C2semidirect
3. #13074243C2direct (N × H)
4. C2×#10622274C2semidirect
5. C2×#10622287C2semidirect
6. C2×#10622285C2semidirect
8. C2×#10622552C2semidirect
9. C2×#10622553C2semidirect
10. C2×#10622278C2semidirect
11. C2×#10622281C2semidirect
12. C2×C11×#376C2semidirect
32. C2×#66929C22semidirect
34. C2×C4×D44C22semidirect
35. C11×#376C22semidirect
41. #10622553C22semidirect
42. #10622552C22semidirect
43. #10622287C22semidirect
44. #10622285C22semidirect
46. #10622281C22semidirect
47. #10622278C22semidirect
48. #10622274C22semidirect
72. C2×C42×C11D4semidirect
168. C42×C11C2×D4semidirect
193. C2×#376D11semidirect
289. #376C2×D11semidirect
379. C2×C42D44semidirect
387. C2×C11#2603semidirect
467. C42C2×D44semidirect
468. C11C2×#2603semidirect
569. C2#13074243direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669292
C22×C4×D442
#130742432
C2×#106222742
C2×#106222872
C2×#106222852
C2×#106222842
C2×#106225522
C2×#106225532
C2×#106222782
C2×#106222812
C2×C11×#3762
C2×#260311