Finite Group Databasebeta

← C2×#13080544 #15630224

Subgroups — C2×#13080544 #15630224

Order 2816.

2210 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
1C2×#10623094C211yesmaximal, characteristic, complemented
2#13080544C2116yesmaximal, direct product factor
3C2×#10622694C211yesmaximal, characteristic, complemented
4C2×#10622658C212yesmaximal, complemented
5C2×#10622621C211yesmaximal, characteristic, complemented
6C2×#10622661C212yesmaximal, complemented
7C2×#10622696C211yesmaximal, characteristic, complemented
8C2×#10622851C212yesmaximal, complemented
9C2×C4×#4323C211yesmaximal, characteristic, complemented
10C2×D4×C11⋊C8C211yesmaximal, characteristic, complemented
11C2×#10622846C212yesmaximal, complemented
12C2×C11×#547C211yesmaximal, characteristic, complemented
13C22×C11×#41C2212yes—
14C11×#547—24no—
15C11×#547C2218yescomplemented
16C2×C11×C4⋊Q8C2211yescharacteristic
17C2×C11×#123—22no—
18C2×C11×C22⋊Q8C2212yes—
19C2×C11×#121—21no—
20C2×C4×D4×C11C2211yescharacteristic
21C2×C4×D4×C11C2211yescharacteristic
22C22×C11×C4⋊C4—21no—
23C22×#4325C2212yes—
24C2×#66991—21no—
25C22×#4323C2211yescharacteristic
26C2×#66983C2212yes—
27C2×#66979C2212yes—
28C23×C11⋊C8C2212yes—
29#10623094—24no—
30#10623094C2218yescomplemented
31C2×#66945—21no—
32C2×#66938—22no—
33C2×C4×D44C2211yescharacteristic
34C2×#66922C2211yescharacteristic
35#10622851C22116yescomplemented
36#10622846C22116yescomplemented
37C2×#66899C2212yes—
38C2×#66899C2212yes—
39C2×#66899C2212yes—
40C2×#66896C2212yes—
41C2×#66896C2211yescharacteristic
42C2×C22⋊(C11⋊C8)C2212yes—
43#10622696C2218yescomplemented
44#10622694C2218yescomplemented
45#10622661C22116yescomplemented
46#10622658C22116yescomplemented
47C4×#4323C2218yescomplemented
48D4×C11⋊C8C2218yescomplemented
49#10622621C2218yescomplemented
50C2×#66873C2211yescharacteristic
51C2×#66872C2212yes—
52C2×#66871C2211yescharacteristic
53C2×C4⋊(C11⋊C8)C2211yescharacteristic
54C2×C4×C11⋊C8C2211yescharacteristic
55C2×C11×#41—24no—
56C2×C11×#41—28no—
57C2×C11×#41—44no—
58C2×C11×#41C2318yes—
59C22×Q8×C11C2312yes—
60C22×D4×C11—22no—
61C22×D4×C11D411yescharacteristic
62C22×D4×C11C2311yescharacteristic
63C23×C4×C11—21no—
64C23×C4×C11—22no—
65C23×C4×C11C2312yes—
66C22×C22⋊D11—22no—
67C22×D44C2311yescharacteristic
68C22×C4×D11—21no—
69C22×Dic22—21no—
70C11×C4⋊Q8—22no—
71C11×C4⋊Q8C2314yes—
72C11×#123—216no—
73C11×C22⋊Q8—24no—
74C11×C22⋊Q8C2318yes—
75C11×#121—28no—
76C4×D4×C11—22no—
77C4×D4×C11D414yes—
78C4×D4×C11C2314yes—
79C4×D4×C11C2314yes—
80C2×C11×C4⋊C4—21no—
81C2×C11×C4⋊C4—22no—
82C2×C11×C4⋊C4—21no—
83C2×C11×C4⋊C4—28no—
84C2×C11×C4⋊C4D411yescharacteristic
85C2×C11×C4⋊C4C2311yescharacteristic
86C2×C11×C22⋊C4—22no—
87C2×C11×C22⋊C4—21no—
88C2×C11×C22⋊C4D412yes—
89C2×C42×C11C2311yescharacteristic
90C2×C22⋊Dic11—22no—
91C2×#4325—44no—
92C2×#4325C2318yes—
93#66991—28no—
94C2×#4323—24no—
95C2×#4323C2314yes—
96#66983—24no—
97#66983C2318yes—
98C2×(C2×D11)⋊C4—21no—
99C2×(C2×D11)⋊C4—22no—
100#66979—24no—
101#66979C2318yes—
102C2×C4⋊Dic11—21no—
103C2×C4⋊Dic11C2312yes—
104C2×C4⋊Dic11C2311yescharacteristic
105C2×C4×Dic11—22no—
106C22×C11⋊C8—44no—
107C22×C11⋊C8C2311yescharacteristic
108C22×C11⋊C8C2311yescharacteristic
109C22×C11⋊C8C2312yes—
110C22×C11⋊C8C2318yes—
111#66945—28no—
112#66938—216no—
113C4×D44—22no—
114C4×D44C2314yes—
115#66922—22no—
116#66922C2314yes—
117#66899C2318yes—
118#66899C2318yes—
119#66899C2318yes—
120#66896C2314yes—
121#66896C2318yes—
122C22⋊(C11⋊C8)C2318yes—
123#66873C2314yes—
124#66872C2318yes—
125#66871C2314yes—
126C4⋊(C11⋊C8)C2314yes—
127C4×C11⋊C8C2314yes—
128C2×#3303—111nomaximal, Sylow
129C24×C11—21no—
130C24×C11C2×D412yes—
131C23×D11—21no—
132C11×#41—216no—
133C11×#41—48no—
134C11×#41—416no—
135C2×Q8×C11—22no—
136C2×Q8×C11—24no—
137C2×Q8×C11—24no—
138C2×Q8×C11C2414yes—
139C2×D4×C11—28no—
140C2×D4×C11—24no—
141C2×D4×C11—21no—
142C2×D4×C11—22no—
143C2×D4×C11—48no—
144C2×D4×C11C2×D414yes—
145C2×D4×C11#4114yes—
146C2×D4×C11C2412yes—
147C22×C4×C11—24no—
148C22×C4×C11—24no—
149C22×C4×C11—21no—
150C22×C4×C11—22no—
151C22×C4×C11—24no—
152C22×C4×C11—28no—
153C22×C4×C11—48no—
154C22×C4×C11—44no—
155C22×C4×C11C2×D412yes—
156C22×C4×C11C2×D414yes—
157C22×C4×C11C2×D411yescharacteristic
158C22×C4×C11C2×D411yescharacteristic
159C22×C4×C11C2414yes—
160C22×C4×C11C2411yescharacteristic
161C2×C22⋊D11—28no—
162C2×C22⋊D11—84no—
163C22×Dic11—21no—
164C22×Dic11—22no—
165C2×D44—21no—
166C2×D44—22no—
167C2×D44—22no—
168C2×D44C2412yes—
169C2×C4×D11—24no—
170C2×C4×D11—44no—
171C2×Dic22—24no—
172C2×Dic22—82no—
173C11×C4⋊C4—24no—
174C11×C4⋊C4—28no—
175C11×C4⋊C4—24no—
176C11×C4⋊C4—21no—
177C11×C4⋊C4C2×D414yes—
178C11×C4⋊C4C2412yes—
179C11×C22⋊C4—28no—
180C11×C22⋊C4—24no—
181C11×C22⋊C4C2×D418yes—
182C42×C11C2×D412yes—
183C42×C11C2412yes—
184C22⋊Dic11—28no—
185#4325—48no—
186#4323—28no—
187(C2×D11)⋊C4—24no—
188(C2×D11)⋊C4—28no—
189C4⋊Dic11—22no—
190C4⋊Dic11—24no—
191C4⋊Dic11—21no—
192C4⋊Dic11C2412yes—
193C4⋊Dic11C2414yes—
194C4×Dic11—28no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8—22no—
197C2×C11⋊C8—24no—
198C2×C11⋊C8—48no—
199C2×C11⋊C8C2412yes—
200C2×C11⋊C8C2412yes—
201C2×C11⋊C8C2414yes—
202C2×#555—111no—
203C2×#547D1111yescharacteristic, complemented
204#3303—1116no—
205C2×#487—111no—
206C2×#485—112no—
207C2×#476—111no—
208C2×#465—112no—
209C2×#461—111no—
210C2×#449—112no—
211C2×C4×D8—111no—
212C2×C8×D4—111no—
213C2×#416—112no—
214C23×C11—24no—
215C23×C11—24no—
216C23×C11—24no—
217C23×C11—24no—
218C23×C11—22no—
219C23×C11—42no—
220C23×C11—42no—
221C23×C11C22×D414yes—
222C23×C11C22×D411yescharacteristic
223C22×D11—22no—
224C22×D11—42no—
225C22×D11—42no—
226C22×D11—42no—
227Q8×C11—24no—
228Q8×C11—24no—
229Q8×C11—44no—
230D4×C11—24no—
231D4×C11—28no—
232D4×C11—22no—
233D4×C11—416no—
234D4×C11C2×#4114yes—
235C2×C4×C11—24no—
236C2×C4×C11—22no—
237C2×C4×C11—28no—
238C2×C4×C11—24no—
239C2×C4×C11—24no—
240C2×C4×C11—22no—
241C2×C4×C11—24no—
242C2×C4×C11—416no—
243C2×C4×C11—44no—
244C2×C4×C11—42no—
245C2×C4×C11—48no—
246C2×C4×C11#13514yes—
247C2×C4×C11C22×D414yes—
248C2×C4×C11C22×D412yes—
249C2×C4×C11C22×D411yescharacteristic
250C2×C4×C11C22×D412yes—
251C2×C4×C11C2×#4112yes—
252C2×C4×C11H(5,𝔽2)12yes—
253C2×C4×C11C2511yescharacteristic
254C22⋊D11—88no—
255C2×Dic11—22no—
256C2×Dic11—24no—
257C2×Dic11—42no—
258C2×Dic11—44no—
259D44—22no—
260D44—22no—
261D44—42no—
262C4×D11—48no—
263Dic22—84no—
264C11⋊C8—22no—
265C11⋊C8—22no—
266C11⋊C8—24no—
267C22×#41C2×D1112yes—
268C22×SD16—112no—
269C22×D8—111no—
270C23×C8—112no—
271#555—118no—
272#555—224no—
273#547—24no—
274#547C2×D1118yescomplemented
275C2×C4⋊Q8—111no—
276C2×C4⋊Q8C2×D1111yescharacteristic
277C2×#124—222no—
278C2×#123—22no—
279C2×C22⋊Q8C2×D1112yes—
280C2×#121—21no—
281C2×#121—112no—
282C2×#121—221no—
283C2×C4×D4—111no—
284C2×C4×D4C2×D1111yescharacteristic
285C2×C4×D4C2×D1111yescharacteristic
286C2×#117—112no—
287C2×#117—221no—
288C22×C4⋊C4—21no—
289#487—118no—
290#485—1116no—
291#476—118no—
292#465—1116no—
293#461—118no—
294#449—1116no—
295C4×D8—118no—
296C8×D4—118no—
297C2×#107—111no—
298C2×#106—112no—
299C2×C4⋊C8—111no—
300#416—1116no—
301C2×#103—112no—
302C2×#103—112no—
303C2×#103—112no—
304C2×#102—111no—
305C2×#102—112no—
306C2×#102—111no—
307C2×C22⋊C8—112no—
308C2×C4×C8—111no—
309C22×C11—24no—
310C22×C11—24no—
311C22×C11—24no—
312C22×C11—24no—
313C22×C11—48no—
314C22×C11—48no—
315C22×C11—42no—
316C22×C11—42no—
317C22×C11#54612yes—
318C22×C11C2×#13512yes—
319C22×C11#57712yes—
320C22×C11C23×D411yescharacteristic
321C2×D11—42no—
322C2×D11—42no—
323C2×D11—42no—
324C2×D11—84no—
325C4×C11—24no—
326C4×C11—22no—
327C4×C11—44no—
328C4×C11—42no—
329C4×C11#54612yes—
330C4×C11C2×#13514yes—
331C4×C11C22×#4111yescharacteristic
332C4×C11C2×H(5,𝔽2)11yescharacteristic
333Dic11—42no—
334Dic11—44no—
335C2×#41—24no—
336C2×#41—28no—
337C2×#41—44no—
338C2×#41C22×D1118yes—
339C22×Q8—221no—
340C22×Q8C22×D1112yes—
341C22×D4—22no—
342C22×D4—111no—
343C22×D4—222no—
344C22×D4C22⋊D1111yescharacteristic
345C22×D4C22×D1111yescharacteristic
346C23×C4—21no—
347C23×C4—22no—
348C23×C4—221no—
349C23×C4C22×D1112yes—
350C2×SD16—118no—
351C2×SD16—444no—
352C2×D8—114no—
353C2×D8—224no—
354C22×C8—111no—
355C22×C8—112no—
356C22×C8—111no—
357C22×C8—118no—
358C22×C8—444no—
359C4⋊Q8—22no—
360C4⋊Q8—114no—
361C4⋊Q8—222no—
362C4⋊Q8C22×D1114yes—
363#124—2216no—
364#123—216no—
365C22⋊Q8—24no—
366C22⋊Q8C22×D1118yes—
367#121—28no—
368#121—118no—
369#121—228no—
370#121—224no—
371C4×D4—22no—
372C4×D4—114no—
373C4×D4—222no—
374C4×D4C22⋊D1114yes—
375C4×D4C22×D1114yes—
376C4×D4C22×D1114yes—
377#117—118no—
378#117—224no—
379#117—228no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—28no—
384C2×C4⋊C4—111no—
385C2×C4⋊C4—112no—
386C2×C4⋊C4—221no—
387C2×C4⋊C4C22⋊D1111yescharacteristic
388C2×C4⋊C4C22×D1111yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—222no—
392C2×C22⋊C4—222no—
393C2×C22⋊C4—221no—
394C2×C22⋊C4C22⋊D1112yes—
395C2×C42—222no—
396C2×C42C22×D1111yescharacteristic
397#107—114no—
398#106—118no—
399C4⋊C8—114no—
400#103—118no—
401#103—118no—
402#103—118no—
403#102—114no—
404#102—118no—
405#102—114no—
406C22⋊C8—118no—
407C4×C8—114no—
408C2×C11—44no—
409C2×C11—44no—
410C2×C11#330314yescomplemented
411C2×C11C2×#54611yescharacteristic
412C2×C11C22×#13511yescharacteristic
413C2×C11C2×#57711yescharacteristic
414D11—82no—
415C24—21no—
416C24—221no—
417C24C2×C22⋊D1112yes—
418#41—216no—
419#41—48no—
420#41—416no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—224no—
425C2×Q8—882no—
426C2×Q8C23×D1114yes—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—48no—
432C2×D4—112no—
433C2×D4—228no—
434C2×D4—222no—
435C2×D4—222no—
436C2×D4—221no—
437C2×D4—884no—
438C2×D4#434114yes—
439C2×D4C2×C22⋊D1114yes—
440C2×D4C23×D1112yes—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—221no—
450C22×C4—222no—
451C22×C4—224no—
452C22×C4—444no—
453C22×C4C2×C22⋊D1114yes—
454C22×C4C2×C22⋊D1112yes—
455C22×C4C2×C22⋊D1111yescharacteristic
456C22×C4C2×C22⋊D1111yescharacteristic
457C22×C4C23×D1114yes—
458C22×C4C23×D1111yescharacteristic
459SD16—448no—
460D8—228no—
461C2×C8—112no—
462C2×C8—112no—
463C2×C8—114no—
464C2×C8—222no—
465C2×C8—222no—
466C2×C8—224no—
467C2×C8—448no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—114no—
473C4⋊C4—112no—
474C4⋊C4—221no—
475C4⋊C4—222no—
476C4⋊C4—224no—
477C4⋊C4C2×C22⋊D1114yes—
478C4⋊C4C23×D1112yes—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—228no—
482C22⋊C4—228no—
483C22⋊C4—224no—
484C22⋊C4C2×C22⋊D1118yes—
485C42—228no—
486C42C2×C22⋊D1112yes—
487C42C23×D1112yes—
488C11C2×#330311yescharacteristic, complemented, Sylow
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—222no—
497C23—442no—
498C23—442no—
499C23—442no—
500C23C22×C22⋊D1114yes—
501C23C22×C22⋊D1111yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—44no—
505Q8—884no—
506D4—24no—
507D4—28no—
508D4—22no—
509D4—416no—
510D4—222no—
511D4—222no—
512D4—442no—
513D4—888no—
514D4C2×#434114yes—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—416no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—222no—
527C2×C4—224no—
528C2×C4—442no—
529C2×C4—444no—
530C2×C4—448no—
531C2×C4#6700314yes—
532C2×C4C2×#434112yes—
533C2×C4#6703712yes—
534C2×C4C22×C22⋊D1114yes—
535C2×C4C22×C22⋊D1112yes—
536C2×C4C22×C22⋊D1111yescharacteristic
537C2×C4C22×C22⋊D1112yes—
538C2×C4C24×D1111yescharacteristic
539C8—222no—
540C8—222no—
541C8—224no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—48no—
548C22—48no—
549C22—42no—
550C22—442no—
551C22—442no—
552C22—442no—
553C22—884no—
554C22#1062333712yes—
555C22C2×#6700312yes—
556C22#1062336812yes—
557C22C23×C22⋊D1111yescharacteristic
558C4—24no—
559C4—22no—
560C4—42no—
561C4—44no—
562C4—442no—
563C4—444no—
564C4#1062333712yes—
565C4C2×#6700314yes—
566C4C22×#434111yescharacteristic
567C4C2×#6703711yescharacteristic
568C2—44no—
569C2—44no—
570C2—882no—
571C2#1308054414yesdirect product factor
572C2C2×#1062333711yescharacteristic
573C2C22×#6700311yescharacteristic
574C2C2×#1062336811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingC2×C11×#547C2
socleC23×C11C22×D4
hypercenterC2×#547D11
hypocenterC11C2×#3303
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#10623094C2semidirect
2. #13080544C2direct (N × H)
3. C2×#10622694C2semidirect
4. C2×#10622658C2semidirect
5. C2×#10622621C2semidirect
6. C2×#10622661C2semidirect
7. C2×#10622696C2semidirect
8. C2×#10622851C2semidirect
9. C2×C4×#4323C2semidirect
10. C2×D4×C11⋊C8C2semidirect
11. C2×#10622846C2semidirect
12. C2×C11×#547C2semidirect
15. C11×#547C22semidirect
30. #10623094C22semidirect
35. #10622851C22semidirect
36. #10622846C22semidirect
43. #10622696C22semidirect
44. #10622694C22semidirect
45. #10622661C22semidirect
46. #10622658C22semidirect
47. C4×#4323C22semidirect
48. D4×C11⋊C8C22semidirect
49. #10622621C22semidirect
203. C2×#547D11semidirect
274. #547C2×D11semidirect
410. C2×C11#3303semidirect
488. C11C2×#3303semidirect
571. C2#13080544direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106230942
#130805442
C2×#106226942
C2×#106226582
C2×#106226212
C2×#106226612
C2×#106226962
C2×#106228512
C2×C4×#43232
C2×D4×C11⋊C82
C2×#106228462
C2×C11×#5472
C2×#330311