Finite Group Databasebeta

← C2×#13074579 #14920889

Subgroups — C2×#13074579 #14920889

Order 2816.

2222 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13074579C2116yesmaximal, direct product factor
2C2×#10622288C212yesmaximal, complemented
3C2×#10622287C212yesmaximal, complemented
4C2×#10622606C212yesmaximal, complemented
5C2×#10622607C212yesmaximal, complemented
6C2×#10622552C211yesmaximal, characteristic
7C2×#10622601C211yesmaximal, characteristic
8C2×#10622601C211yesmaximal, characteristic
9C2×#10622598C211yesmaximal, characteristic, complemented
10C2×#10622586C211yesmaximal, characteristic, complemented
11C2×#10622545C211yesmaximal, characteristic, complemented
12C2×C11×#389C211yesmaximal, characteristic, complemented
13C23×D44C2211yescharacteristic
14C22×C11×C4⋊C4C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C11×C4⋊C4C2212yes—
17C22×C42×C11C2211yescharacteristic
18C22×(C2×D11)⋊C4C2212yes—
19C22×(C2×D11)⋊C4C2212yes—
20C22×(C2×D11)⋊C4C2212yes—
21C22×(C2×D11)⋊C4C2212yes—
22C22×(C2×D11)⋊C4C2212yes—
23C22×(C2×D11)⋊C4C2212yes—
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2212yes—
26C22×Dic11⋊C4C2212yes—
27C22×C4×Dic11C2211yescharacteristic
28C22×C4×Dic11C2211yescharacteristic
29C11×#389C2218yescomplemented
30C2×C11×#95C2212yes—
31C2×C11⋊#95C2211yescharacteristic
32C2×C11⋊#95C2211yescharacteristic
33C2×C11⋊#95C2211yescharacteristic
34C2×C11⋊#95C2211yescharacteristic
35C2×C11⋊#95C2212yes—
36C2×C11⋊#95C2212yes—
37#10622607C22116yescomplemented
38#10622606C22116yescomplemented
39#10622601C2218yes—
40#10622601C2218yes—
41#10622598C2218yescomplemented
42#10622586C2218yescomplemented
43#10622552C2218yes—
44#10622545C2218yescomplemented
45#10622288C22116yescomplemented
46#10622287C22116yescomplemented
47C24×D11C2312yes—
48C23×C4×C11C2312yes—
49C23×C4×C11C2312yes—
50C23×C4×C11C2311yescharacteristic
51C23×C4×C11C2311yescharacteristic
52C23×C4×C11C2311yescharacteristic
53C23×Dic11C2311yescharacteristic
54C23×Dic11C2311yescharacteristic
55C23×Dic11C2311yescharacteristic
56C23×Dic11C2311yescharacteristic
57C23×Dic11C2312yes—
58C22×D44—24no—
59C22×D44—24no—
60C22×D44—24no—
61C22×D44C2314yes—
62C2×C11×C4⋊C4—24no—
63C2×C11×C4⋊C4—24no—
64C2×C11×C4⋊C4—44no—
65C2×C11×C4⋊C4C2314yes—
66C2×C11×C4⋊C4C2314yes—
67C2×C11×C4⋊C4C2318yes—
68C2×C42×C11—24no—
69C2×C42×C11C2314yes—
70C2×(C2×D11)⋊C4—44no—
71C2×(C2×D11)⋊C4—44no—
72C2×(C2×D11)⋊C4—44no—
73C2×(C2×D11)⋊C4—44no—
74C2×(C2×D11)⋊C4—44no—
75C2×(C2×D11)⋊C4—44no—
76C2×(C2×D11)⋊C4C2318yes—
77C2×(C2×D11)⋊C4C2318yes—
78C2×(C2×D11)⋊C4C2318yes—
79C2×(C2×D11)⋊C4C2318yes—
80C2×(C2×D11)⋊C4C2318yes—
81C2×(C2×D11)⋊C4C2318yes—
82C2×C4⋊Dic11—24no—
83C2×C4⋊Dic11—44no—
84C2×C4⋊Dic11C2314yes—
85C2×C4⋊Dic11C2318yes—
86C2×Dic11⋊C4—44no—
87C2×Dic11⋊C4C2318yes—
88C2×C4×Dic11—24no—
89C2×C4×Dic11—24no—
90C2×C4×Dic11C2314yes—
91C2×C4×Dic11C2314yes—
92C11×#95C2318yes—
93C11⋊#95C2314yes—
94C11⋊#95C2318yes—
95C11⋊#95C2314yes—
96C11⋊#95C2318yes—
97C11⋊#95C2314yes—
98C11⋊#95C2314yes—
99C2×#2644—111nomaximal, Sylow
100C24×C11C2411yescharacteristic
101C23×D11—24no—
102C23×D11—24no—
103C23×D11—24no—
104C23×D11—24no—
105C23×D11—24no—
106C23×D11—24no—
107C23×D11—24no—
108C23×D11C2414yes—
109C22×C4×C11—22no—
110C22×C4×C11—22no—
111C22×C4×C11—24no—
112C22×C4×C11—24no—
113C22×C4×C11—24no—
114C22×C4×C11—24no—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—24no—
118C22×C4×C11—24no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11#4114yes—
122C22×C4×C11#4114yes—
123C22×C4×C11#4114yes—
124C22×C4×C11C2412yes—
125C22×C4×C11C2412yes—
126C22×C4×C11C2414yes—
127C22×C4×C11C2414yes—
128C22×C4×C11C2412yes—
129C22×Dic11—22no—
130C22×Dic11—22no—
131C22×Dic11—22no—
132C22×Dic11—22no—
133C22×Dic11—24no—
134C22×Dic11—22no—
135C22×Dic11—24no—
136C22×Dic11—22no—
137C22×Dic11—24no—
138C22×Dic11—22no—
139C22×Dic11—22no—
140C22×Dic11—22no—
141C22×Dic11—22no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11C2412yes—
145C22×Dic11C2412yes—
146C22×Dic11C2412yes—
147C22×Dic11C2412yes—
148C22×Dic11C2414yes—
149C2×D44—28no—
150C2×D44—28no—
151C2×D44—28no—
152C2×D44—416no—
153C11×C4⋊C4—28no—
154C11×C4⋊C4—28no—
155C11×C4⋊C4—48no—
156C42×C11—28no—
157(C2×D11)⋊C4—48no—
158(C2×D11)⋊C4—48no—
159(C2×D11)⋊C4—48no—
160(C2×D11)⋊C4—48no—
161(C2×D11)⋊C4—48no—
162(C2×D11)⋊C4—48no—
163C4⋊Dic11—28no—
164C4⋊Dic11—48no—
165Dic11⋊C4—48no—
166C4×Dic11—28no—
167C4×Dic11—28no—
168#2644—1116no—
169C2×#397—112no—
170C2×#397—112no—
171C2×#397—112no—
172C2×#396—112no—
173C2×#390—111no—
174C2×#390—111no—
175C2×#390—111no—
176C2×#389—111no—
177C2×#389—111no—
178C2×#389D1111yescharacteristic, complemented
179C2×#376—111no—
180C23×C11C2×#4112yes—
181C23×C11C2×#4112yes—
182C23×C11C2×#4112yes—
183C23×C11H(5,𝔽2)12yes—
184C23×C11H(5,𝔽2)12yes—
185C23×C11H(5,𝔽2)12yes—
186C23×C112−1+412yes—
187C23×C11C2511yescharacteristic
188C22×D11—24no—
189C22×D11—24no—
190C22×D11—24no—
191C22×D11—24no—
192C22×D11—24no—
193C22×D11—24no—
194C22×D11—24no—
195C22×D11—48no—
196C22×D11—48no—
197C22×D11—48no—
198C22×D11—48no—
199C22×D11—48no—
200C22×D11—48no—
201C22×D11—48no—
202C2×C4×C11—22no—
203C2×C4×C11—22no—
204C2×C4×C11—24no—
205C2×C4×C11—24no—
206C2×C4×C11—24no—
207C2×C4×C11—24no—
208C2×C4×C11—22no—
209C2×C4×C11—22no—
210C2×C4×C11—24no—
211C2×C4×C11—24no—
212C2×C4×C11—22no—
213C2×C4×C11—22no—
214C2×C4×C11—48no—
215C2×C4×C11—44no—
216C2×C4×C11—48no—
217C2×C4×C11—44no—
218C2×C4×C11#126116yes—
219C2×C4×C11C2×#4114yes—
220C2×C4×C11C2×#4114yes—
221C2×C4×C11C2×#4114yes—
222C2×Dic11—22no—
223C2×Dic11—22no—
224C2×Dic11—22no—
225C2×Dic11—22no—
226C2×Dic11—24no—
227C2×Dic11—22no—
228C2×Dic11—24no—
229C2×Dic11—22no—
230C2×Dic11—24no—
231C2×Dic11—22no—
232C2×Dic11—22no—
233C2×Dic11—22no—
234C2×Dic11—22no—
235C2×Dic11—22no—
236C2×Dic11—22no—
237C2×Dic11—44no—
238C2×Dic11—44no—
239C2×Dic11—48no—
240C2×Dic11—44no—
241C2×Dic11—44no—
242D44—416no—
243C23×D4—111no—
244C22×C4⋊C4—111no—
245C22×C4⋊C4—112no—
246C22×C4⋊C4—112no—
247C22×C4⋊C4C2×D1111yescharacteristic
248C22×C4⋊C4C2×D1111yescharacteristic
249C22×C4⋊C4C2×D1112yes—
250C22×C22⋊C4—112no—
251C22×C22⋊C4—112no—
252C22×C22⋊C4—112no—
253C22×C22⋊C4—112no—
254C22×C22⋊C4—112no—
255C22×C22⋊C4—112no—
256C22×C42—111no—
257C22×C42—111no—
258C22×C42C2×D1111yescharacteristic
259#397—1116no—
260#397—1116no—
261#397—1116no—
262#396—1116no—
263#390—118no—
264#390—118no—
265#390—118no—
266#389—118no—
267#389—118no—
268#389C2×D1118yescomplemented
269#376—118no—
270C2×#95—111no—
271C2×#95—111no—
272C2×#95—111no—
273C2×#95—111no—
274C2×#95—112no—
275C2×#95—112no—
276C2×#95C2×D1112yes—
277C22×C11C2×#12614yes—
278C22×C11#54014yes—
279C22×C11#54014yes—
280C22×C11#54014yes—
281C22×C11#54214yes—
282C22×C11#54214yes—
283C22×C11#54214yes—
284C22×C11C22×#4111yescharacteristic
285C22×C11C22×#4111yescharacteristic
286C22×C11C22×#4111yescharacteristic
287C22×C11C2×H(5,𝔽2)11yescharacteristic
288C22×C11C2×H(5,𝔽2)11yescharacteristic
289C22×C11C2×H(5,𝔽2)11yescharacteristic
290C22×C11C2×2−1+411yescharacteristic
291C2×D11—44no—
292C2×D11—44no—
293C2×D11—44no—
294C2×D11—44no—
295C2×D11—44no—
296C2×D11—44no—
297C2×D11—44no—
298C2×D11—816no—
299C4×C11—44no—
300C4×C11—42no—
301C4×C11—44no—
302C4×C11—42no—
303C4×C11C2×#12618yes—
304Dic11—42no—
305Dic11—42no—
306Dic11—44no—
307Dic11—42no—
308Dic11—42no—
309C25—112no—
310C22×D4—114no—
311C22×D4—224no—
312C22×D4—224no—
313C22×D4—224no—
314C23×C4—112no—
315C23×C4—111no—
316C23×C4—111no—
317C23×C4—111no—
318C23×C4—111no—
319C23×C4C22×D1112yes—
320C23×C4C22×D1112yes—
321C23×C4C22×D1111yescharacteristic
322C23×C4C22×D1111yescharacteristic
323C23×C4C22×D1111yescharacteristic
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—114no—
328C2×C4⋊C4—118no—
329C2×C4⋊C4—118no—
330C2×C4⋊C4—224no—
331C2×C4⋊C4—444no—
332C2×C4⋊C4—444no—
333C2×C4⋊C4C22×D1114yes—
334C2×C4⋊C4C22×D1114yes—
335C2×C4⋊C4C22×D1118yes—
336C2×C22⋊C4—118no—
337C2×C22⋊C4—118no—
338C2×C22⋊C4—118no—
339C2×C22⋊C4—118no—
340C2×C22⋊C4—118no—
341C2×C22⋊C4—118no—
342C2×C22⋊C4—444no—
343C2×C22⋊C4—444no—
344C2×C22⋊C4—444no—
345C2×C22⋊C4—444no—
346C2×C22⋊C4—444no—
347C2×C22⋊C4—444no—
348C2×C42—24no—
349C2×C42—114no—
350C2×C42—114no—
351C2×C42—224no—
352C2×C42—224no—
353C2×C42C22×D1114yes—
354#95—114no—
355#95—114no—
356#95—114no—
357#95—114no—
358#95—118no—
359#95—118no—
360#95C22×D1118yes—
361C2×C11#264418yescomplemented
362C2×C11C22×#12611yescharacteristic
363C2×C11C2×#54011yescharacteristic
364C2×C11C2×#54011yescharacteristic
365C2×C11C2×#54011yescharacteristic
366C2×C11C2×#54211yescharacteristic
367C2×C11C2×#54211yescharacteristic
368C2×C11C2×#54211yescharacteristic
369D11—84no—
370C24—114no—
371C24—224no—
372C24—224no—
373C24—224no—
374C24—224no—
375C24—224no—
376C24—224no—
377C24—224no—
378C24C23×D1111yescharacteristic
379C2×D4—228no—
380C2×D4—228no—
381C2×D4—228no—
382C2×D4—4416no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—112no—
396C22×C4—112no—
397C22×C4—112no—
398C22×C4—112no—
399C22×C4—114no—
400C22×C4—222no—
401C22×C4—222no—
402C22×C4—222no—
403C22×C4—222no—
404C22×C4—222no—
405C22×C4—222no—
406C22×C4—224no—
407C22×C4—222no—
408C22×C4—222no—
409C22×C4—222no—
410C22×C4—222no—
411C22×C4—224no—
412C22×C4—224no—
413C22×C4—222no—
414C22×C4—222no—
415C22×C4#433914yes—
416C22×C4#434114yes—
417C22×C4#434314yes—
418C22×C4C23×D1112yes—
419C22×C4C23×D1112yes—
420C22×C4C23×D1114yes—
421C22×C4C23×D1114yes—
422C22×C4C23×D1112yes—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—48no—
426C4⋊C4—228no—
427C4⋊C4—448no—
428C4⋊C4—448no—
429C22⋊C4—448no—
430C22⋊C4—448no—
431C22⋊C4—448no—
432C22⋊C4—448no—
433C22⋊C4—448no—
434C22⋊C4—448no—
435C42—28no—
436C42—228no—
437C42—228no—
438C11C2×#264411yescharacteristic, complemented, Sylow
439C23—224no—
440C23—224no—
441C23—224no—
442C23—224no—
443C23—224no—
444C23—224no—
445C23—224no—
446C23—448no—
447C23—448no—
448C23—448no—
449C23—448no—
450C23—448no—
451C23—448no—
452C23—448no—
453C23C2×#433912yes—
454C23C2×#434112yes—
455C23#6703712yes—
456C23C2×#434312yes—
457C23#6704012yes—
458C23#6704212yes—
459C23#6704212yes—
460C23C24×D1111yescharacteristic
461D4—4416no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—222no—
479C2×C4—222no—
480C2×C4—222no—
481C2×C4—222no—
482C2×C4—222no—
483C2×C4—222no—
484C2×C4—224no—
485C2×C4—222no—
486C2×C4—222no—
487C2×C4—222no—
488C2×C4—222no—
489C2×C4—224no—
490C2×C4—224no—
491C2×C4—222no—
492C2×C4—222no—
493C2×C4—444no—
494C2×C4—444no—
495C2×C4—444no—
496C2×C4—444no—
497C2×C4—448no—
498C2×C4#66951116yes—
499C2×C4C2×#433914yes—
500C2×C4C2×#434114yes—
501C2×C4C2×#434314yes—
502C22—444no—
503C22—444no—
504C22—444no—
505C22—444no—
506C22—444no—
507C22—444no—
508C22—444no—
509C22—8816no—
510C22C2×#6695114yes—
511C22#1062309314yes—
512C22#1062311314yes—
513C22#1062315014yes—
514C22#1062317414yes—
515C22#1062322914yes—
516C22#1062326414yes—
517C22C22×#433911yescharacteristic
518C22C22×#434111yescharacteristic
519C22C2×#6703711yescharacteristic
520C22C22×#434311yescharacteristic
521C22C2×#6704011yescharacteristic
522C22C2×#6704211yescharacteristic
523C22C2×#6704211yescharacteristic
524C4—44no—
525C4—42no—
526C4—44no—
527C4—42no—
528C4—442no—
529C4—442no—
530C4—444no—
531C4—442no—
532C4—442no—
533C4C2×#6695118yes—
534C2—884no—
535C2#1307457918yesdirect product factor
536C2C22×#6695111yescharacteristic
537C2C2×#1062309311yescharacteristic
538C2C2×#1062315011yescharacteristic
539C2C2×#1062322911yescharacteristic
540C2C2×#1062311311yescharacteristic
541C2C2×#1062317411yescharacteristic
542C2C2×#1062326411yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#389C2
socleC24×C11C24
prime radicalC23×D44C22
hypercenterC2×#389D11
hypocenterC11C2×#2644
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. #13074579C2direct (N × H)
2. C2×#10622288C2semidirect
3. C2×#10622287C2semidirect
4. C2×#10622606C2semidirect
5. C2×#10622607C2semidirect
9. C2×#10622598C2semidirect
10. C2×#10622586C2semidirect
11. C2×#10622545C2semidirect
12. C2×C11×#389C2semidirect
29. C11×#389C22semidirect
37. #10622607C22semidirect
38. #10622606C22semidirect
41. #10622598C22semidirect
42. #10622586C22semidirect
44. #10622545C22semidirect
45. #10622288C22semidirect
46. #10622287C22semidirect
178. C2×#389D11semidirect
268. #389C2×D11semidirect
361. C2×C11#2644semidirect
438. C11C2×#2644semidirect
535. C2#13074579direct (N × H)

Maximal subgroups

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

subgroupindex
#130745792
C2×#106222882
C2×#106222872
C2×#106226062
C2×#106226072
C2×#106225522
C2×#106226012
C2×#106225982
C2×#106225862
C2×#106225452
C2×C11×#3892
C2×#264411