Finite Group Databasebeta

← C2×#13074441 #14920789

Subgroups — C2×#13074441 #14920789

Order 2816.

2742 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66941C211yesmaximal, characteristic, complemented
2C22×#66979C211yesmaximal, characteristic
3#13074441C2116yesmaximal, direct product factor
4C2×#10622273C212yesmaximal, complemented
5C2×#10622562C212yesmaximal
6C2×#10622269C212yesmaximal, complemented
7C2×#10622560C212yesmaximal
8C2×#10622598C211yesmaximal, characteristic, complemented
9C2×#10622586C211yesmaximal, characteristic, complemented
10C2×#10622590C211yesmaximal, characteristic, complemented
11C2×#10622825C211yesmaximal, characteristic
12C2×C11×#380C211yesmaximal, characteristic
13C24×C4×C11C2211yescharacteristic
14C23×Dic22C2212yes—
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C22⋊C4C2212yes—
18C22×C22⋊Dic11C2211yescharacteristic
19C22×C22⋊Dic11C2211yescharacteristic
20C22×C22⋊Dic11C2212yes—
21C2×#66979—28no—
22C2×#66979C2218yes—
23C22×C4⋊Dic11C2211yescharacteristic
24C22×C4⋊Dic11C2211yescharacteristic
25C22×C4⋊Dic11C2211yescharacteristic
26C22×C4⋊Dic11C2211yescharacteristic
27C22×C4⋊Dic11C2212yes—
28C22×Dic11⋊C4C2212yes—
29C22×Dic11⋊C4C2212yes—
30C22×C4×Dic11C2211yescharacteristic
31C22×C4×Dic11C2211yescharacteristic
32C2×#66941—28no—
33C2×#66941C2218yescomplemented
34C11×#380C2218yes—
35C2×C11×#95C2212yes—
36#10622825C2218yes—
37C2×C11⋊#95C2212yes—
38C2×C11⋊#95C2212yes—
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42#10622598C2218yescomplemented
43#10622590C2218yescomplemented
44#10622586C2218yescomplemented
45#10622562C22116yes—
46#10622560C22116yes—
47#10622273C22116yescomplemented
48#10622269C22116yescomplemented
49C25×C11C2311yescharacteristic
50C23×C4×C11—24no—
51C23×C4×C11—24no—
52C23×C4×C11Q818yescomplemented
53C23×C4×C11C2312yes—
54C23×C4×C11C2312yes—
55C23×C4×C11C2311yescharacteristic
56C23×C4×C11C2314yes—
57C23×C4×C11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C23×Dic11C2311yescharacteristic
60C23×Dic11C2312yes—
61C23×Dic11C2312yes—
62C23×Dic11C2311yescharacteristic
63C23×Dic11C2311yescharacteristic
64C22×Dic22—28no—
65C22×Dic22—44no—
66C22×Dic22—44no—
67C22×Dic22C2318yes—
68C2×C11×C4⋊C4—24no—
69C2×C11×C4⋊C4—24no—
70C2×C11×C4⋊C4C2314yes—
71C2×C11×C4⋊C4C2314yes—
72C2×C11×C22⋊C4—44no—
73C2×C11×C22⋊C4C2318yes—
74C2×C22⋊Dic11—24no—
75C2×C22⋊Dic11—24no—
76C2×C22⋊Dic11—44no—
77C2×C22⋊Dic11C2314yes—
78C2×C22⋊Dic11C2314yes—
79C2×C22⋊Dic11C2318yes—
80#66979—232no—
81C2×C4⋊Dic11—24no—
82C2×C4⋊Dic11—24no—
83C2×C4⋊Dic11—24no—
84C2×C4⋊Dic11—24no—
85C2×C4⋊Dic11—44no—
86C2×C4⋊Dic11C2314yes—
87C2×C4⋊Dic11C2314yes—
88C2×C4⋊Dic11C2314yes—
89C2×C4⋊Dic11C2314yes—
90C2×C4⋊Dic11C2318yes—
91C2×Dic11⋊C4—28no—
92C2×Dic11⋊C4—44no—
93C2×Dic11⋊C4C2318yes—
94C2×Dic11⋊C4C2318yes—
95C2×C4×Dic11—24no—
96C2×C4×Dic11D418yes—
97C2×C4×Dic11C2314yes—
98C2×C4×Dic11C2314yes—
99#66941—232no—
100C11×#95C2318yes—
101C11⋊#95C2318yes—
102C11⋊#95C2318yes—
103C11⋊#95C2318yes—
104C11⋊#95C2318yes—
105C11⋊#95C2318yes—
106C2×#2628—111nomaximal, Sylow
107C24×C11—22no—
108C24×C11—22no—
109C24×C11—22no—
110C24×C11—22no—
111C24×C11—22no—
112C24×C11—22no—
113C24×C11C2×Q814yes—
114C24×C11C2412yes—
115C24×C11C2411yescharacteristic
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11—232no—
119C22×C4×C11—28no—
120C22×C4×C11—28no—
121C22×C4×C11—24no—
122C22×C4×C11—24no—
123C22×C4×C11—24no—
124C22×C4×C11—24no—
125C22×C4×C11—24no—
126C22×C4×C11—24no—
127C22×C4×C11C2×D414yes—
128C22×C4×C11C2×Q814yes—
129C22×C4×C11C2×Q814yes—
130C22×C4×C11C2×Q8116yescomplemented
131C22×C4×C11#4114yes—
132C22×C4×C11C2414yes—
133C22×C4×C11C2414yes—
134C22×C4×C11C2412yes—
135C22×C4×C11C2412yes—
136C22×Dic11—22no—
137C22×Dic11—22no—
138C22×Dic11—24no—
139C22×Dic11—24no—
140C22×Dic11—22no—
141C22×Dic11—22no—
142C22×Dic11—22no—
143C22×Dic11—24no—
144C22×Dic11—22no—
145C22×Dic11—24no—
146C22×Dic11—24no—
147C22×Dic11—22no—
148C22×Dic11—22no—
149C22×Dic11—22no—
150C22×Dic11—24no—
151C22×Dic11—22no—
152C22×Dic11C2×D414yes—
153C22×Dic11C2×D414yes—
154C22×Dic11C2412yes—
155C22×Dic11C2414yes—
156C22×Dic11C2414yes—
157C22×Dic11C2412yes—
158C22×Dic11C2412yes—
159C22×Dic11C2412yes—
160C2×Dic22—216no—
161C2×Dic22—48no—
162C2×Dic22—48no—
163C2×Dic22—432no—
164C11×C4⋊C4—28no—
165C11×C4⋊C4—28no—
166C11×C22⋊C4—48no—
167C22⋊Dic11—28no—
168C22⋊Dic11—28no—
169C22⋊Dic11—48no—
170C4⋊Dic11—28no—
171C4⋊Dic11—28no—
172C4⋊Dic11—28no—
173C4⋊Dic11—28no—
174C4⋊Dic11—48no—
175Dic11⋊C4—216no—
176Dic11⋊C4—48no—
177C4×Dic11—28no—
178C4×Dic11C2×D4116yes—
179C22×C4⋊Q8—111no—
180C22×#117—111no—
181#2628—1116no—
182C2×#400—112no—
183C2×#399—112no—
184C2×#395—112no—
185C2×#393—112no—
186C2×#389—111no—
187C2×#389—111no—
188C2×#382—111no—
189C2×#380—111no—
190C2×#380D1111yescharacteristic
191C23×C11—28no—
192C23×C11—28no—
193C23×C11—28no—
194C23×C11—22no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11—22no—
198C23×C11—22no—
199C23×C11—22no—
200C23×C11—44no—
201C23×C11—44no—
202C23×C11—44no—
203C23×C11—44no—
204C23×C11C22×D412yes—
205C23×C11C22×Q814yes—
206C23×C11C22×Q812yes—
207C23×C11C2×#4112yes—
208C23×C11H(5,𝔽2)12yes—
209C23×C11H(5,𝔽2)12yes—
210C23×C112−1+412yes—
211C23×C112−1+412yes—
212C23×C11C2511yescharacteristic
213C2×C4×C11—28no—
214C2×C4×C11—22no—
215C2×C4×C11—22no—
216C2×C4×C11—232no—
217C2×C4×C11—24no—
218C2×C4×C11—24no—
219C2×C4×C11—24no—
220C2×C4×C11—24no—
221C2×C4×C11—24no—
222C2×C4×C11—24no—
223C2×C4×C11—48no—
224C2×C4×C11—48no—
225C2×C4×C11C22⋊Q8116yes—
226C2×C4×C11C22×D414yes—
227C2×C4×C11C22×Q814yes—
228C2×C4×C11C22×Q814yes—
229C2×C4×C11C2×#4114yes—
230C2×Dic11—28no—
231C2×Dic11—28no—
232C2×Dic11—22no—
233C2×Dic11—22no—
234C2×Dic11—22no—
235C2×Dic11—22no—
236C2×Dic11—22no—
237C2×Dic11—22no—
238C2×Dic11—24no—
239C2×Dic11—24no—
240C2×Dic11—22no—
241C2×Dic11—24no—
242C2×Dic11—22no—
243C2×Dic11—24no—
244C2×Dic11—24no—
245C2×Dic11—22no—
246C2×Dic11—24no—
247C2×Dic11—22no—
248C2×Dic11—48no—
249C2×Dic11—44no—
250C2×Dic11—44no—
251C2×Dic11—48no—
252C2×Dic11C22×D414yes—
253C2×Dic11C22×D414yes—
254Dic22—432no—
255C23×Q8—112no—
256C24×C4C2×D1111yescharacteristic
257C2×C4⋊Q8—118no—
258C2×C4⋊Q8—228no—
259C2×#117—118no—
260C2×#117—228no—
261C22×C4⋊C4—112no—
262C22×C4⋊C4—111no—
263C22×C4⋊C4—111no—
264C22×C4⋊C4—111no—
265C22×C4⋊C4—111no—
266C22×C4⋊C4—112no—
267C22×C4⋊C4—112no—
268C22×C4⋊C4C2×D1111yescharacteristic
269C22×C4⋊C4C2×D1111yescharacteristic
270C22×C22⋊C4—111no—
271C22×C22⋊C4—111no—
272C22×C22⋊C4—112no—
273C22×C22⋊C4C2×D1112yes—
274C22×C42—111no—
275C22×C42—111no—
276#400—1116no—
277#399—1116no—
278#395—1116no—
279#393—1116no—
280#389—118no—
281#389—118no—
282#382—118no—
283#380—118no—
284#380C2×D1118yes—
285C2×#95—112no—
286C2×#95—112no—
287C2×#95—112no—
288C2×#95—112no—
289C2×#95—112no—
290C2×#95C2×D1112yes—
291C22×C11—24no—
292C22×C11—24no—
293C22×C11—24no—
294C22×C11—416no—
295C22×C11—42no—
296C22×C11—42no—
297C22×C11—42no—
298C22×C11—42no—
299C22×C11C2×C22⋊Q814yes—
300C22×C11#53514yes—
301C22×C11#53614yes—
302C22×C11#54214yes—
303C22×C11#54214yes—
304C22×C11#54414yes—
305C22×C11#54414yes—
306C22×C11C23×D411yescharacteristic
307C22×C11C23×Q811yescharacteristic
308C22×C11C22×#4111yescharacteristic
309C22×C11C2×H(5,𝔽2)11yescharacteristic
310C22×C11C2×H(5,𝔽2)11yescharacteristic
311C22×C11C2×2−1+411yescharacteristic
312C22×C11C2×2−1+411yescharacteristic
313C4×C11—24no—
314C4×C11—44no—
315C4×C11—44no—
316C4×C11C2×C22⋊Q818yes—
317Dic11—24no—
318Dic11—24no—
319Dic11—44no—
320Dic11—42no—
321Dic11—42no—
322Dic11—44no—
323C25C22×D1111yescharacteristic
324C22×Q8—118no—
325C22×Q8—228no—
326C22×Q8—444no—
327C22×Q8—444no—
328C23×C4—24no—
329C23×C4—24no—
330C23×C4—111no—
331C23×C4—111no—
332C23×C4—111no—
333C23×C4—112no—
334C23×C4—112no—
335C23×C4—111no—
336C23×C4Dic2218yescomplemented
337C23×C4C22×D1112yes—
338C23×C4C22×D1112yes—
339C23×C4C22×D1111yescharacteristic
340C23×C4C22×D1114yes—
341C23×C4C22×D1111yescharacteristic
342C4⋊Q8—2232no—
343#117—2232no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—114no—
347C2×C4⋊C4—114no—
348C2×C4⋊C4—114no—
349C2×C4⋊C4—118no—
350C2×C4⋊C4—114no—
351C2×C4⋊C4—118no—
352C2×C4⋊C4—118no—
353C2×C4⋊C4—224no—
354C2×C4⋊C4—224no—
355C2×C4⋊C4—228no—
356C2×C4⋊C4—224no—
357C2×C4⋊C4—224no—
358C2×C4⋊C4—444no—
359C2×C4⋊C4—444no—
360C2×C4⋊C4C22×D1114yes—
361C2×C4⋊C4C22×D1114yes—
362C2×C22⋊C4—44no—
363C2×C22⋊C4—114no—
364C2×C22⋊C4—114no—
365C2×C22⋊C4—118no—
366C2×C22⋊C4—224no—
367C2×C22⋊C4—224no—
368C2×C22⋊C4—444no—
369C2×C22⋊C4C22×D1118yes—
370C2×C42—118no—
371C2×C42—114no—
372C2×C42—114no—
373C2×C42—224no—
374#95—118no—
375#95—118no—
376#95—118no—
377#95—118no—
378#95—118no—
379#95C22×D1118yes—
380C2×C11—44no—
381C2×C11#262818yescomplemented
382C2×C11C22×C22⋊Q811yescharacteristic
383C2×C11C2×#53511yescharacteristic
384C2×C11C2×#53611yescharacteristic
385C2×C11C2×#54211yescharacteristic
386C2×C11C2×#54211yescharacteristic
387C2×C11C2×#54411yescharacteristic
388C2×C11C2×#54411yescharacteristic
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24C2×Dic2214yes—
396C24C23×D1112yes—
397C24C23×D1111yescharacteristic
398C2×Q8—2216no—
399C2×Q8—448no—
400C2×Q8—448no—
401C2×Q8—4432no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—232no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—114no—
414C22×C4—114no—
415C22×C4—112no—
416C22×C4—112no—
417C22×C4—112no—
418C22×C4—114no—
419C22×C4—114no—
420C22×C4—112no—
421C22×C4—222no—
422C22×C4—222no—
423C22×C4—222no—
424C22×C4—222no—
425C22×C4—222no—
426C22×C4—222no—
427C22×C4—224no—
428C22×C4—224no—
429C22×C4—222no—
430C22×C4—224no—
431C22×C4—222no—
432C22×C4—224no—
433C22×C4—224no—
434C22×C4—224no—
435C22×C4—222no—
436C22×C4—222no—
437C22×C4C2×Dic2214yes—
438C22×C4C2×Dic2214yes—
439C22×C4C2×Dic22116yescomplemented
440C22×C4D4×D1114yes—
441C22×C4#434114yes—
442C22×C4C23×D1114yes—
443C22×C4C23×D1114yes—
444C22×C4C23×D1112yes—
445C22×C4C23×D1112yes—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—228no—
449C4⋊C4—228no—
450C4⋊C4—2216no—
451C4⋊C4—228no—
452C4⋊C4—228no—
453C4⋊C4—448no—
454C4⋊C4—448no—
455C22⋊C4—48no—
456C22⋊C4—228no—
457C22⋊C4—228no—
458C22⋊C4—448no—
459C42—1116no—
460C42—228no—
461C11C2×#262811yescharacteristic, complemented, Sylow
462C23—28no—
463C23—28no—
464C23—28no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23C22×Dic2214yes—
476C23C22×Dic2212yes—
477C23C2×D4×D1112yes—
478C23C2×#434112yes—
479C23#6703712yes—
480C23#6704012yes—
481C23#6704212yes—
482C23#6704312yes—
483C23C24×D1111yescharacteristic
484Q8—4432no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—232no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—114no—
498C2×C4—114no—
499C2×C4—228no—
500C2×C4—228no—
501C2×C4—222no—
502C2×C4—222no—
503C2×C4—222no—
504C2×C4—222no—
505C2×C4—222no—
506C2×C4—222no—
507C2×C4—224no—
508C2×C4—224no—
509C2×C4—222no—
510C2×C4—224no—
511C2×C4—222no—
512C2×C4—224no—
513C2×C4—224no—
514C2×C4—224no—
515C2×C4—222no—
516C2×C4—222no—
517C2×C4—448no—
518C2×C4—444no—
519C2×C4—444no—
520C2×C4—448no—
521C2×C4#66931116yes—
522C2×C4C22×Dic2214yes—
523C2×C4C22×Dic2214yes—
524C2×C4C2×D4×D1114yes—
525C2×C4C2×#434114yes—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—416no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22C2×#6693114yes—
535C22#1062303114yes—
536C22#1062305014yes—
537C22#1062313414yes—
538C22#1062313814yes—
539C22#1062315914yes—
540C22#1062317414yes—
541C22C23×Dic2211yescharacteristic
542C22C22×D4×D1111yescharacteristic
543C22C22×#434111yescharacteristic
544C22C2×#6703711yescharacteristic
545C22C2×#6704011yescharacteristic
546C22C2×#6704211yescharacteristic
547C22C2×#6704311yescharacteristic
548C4—24no—
549C4—44no—
550C4—44no—
551C4—224no—
552C4—224no—
553C4—444no—
554C4—442no—
555C4—442no—
556C4—444no—
557C4C2×#6693118yes—
558C2—44no—
559C2#1307444118yesdirect product factor
560C2C22×#6693111yescharacteristic
561C2C2×#1062313811yescharacteristic
562C2C2×#1062315911yescharacteristic
563C2C2×#1062313411yescharacteristic
564C2C2×#1062317411yescharacteristic
565C2C2×#1062305011yescharacteristic
566C2C2×#1062303111yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#380C2
socleC24×C11C24
prime radicalC25×C11C23
hypercenterC2×#380D11
hypocenterC11C2×#2628
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#66941C2semidirect
3. #13074441C2direct (N × H)
4. C2×#10622273C2semidirect
6. C2×#10622269C2semidirect
8. C2×#10622598C2semidirect
9. C2×#10622586C2semidirect
10. C2×#10622590C2semidirect
33. C2×#66941C22semidirect
42. #10622598C22semidirect
43. #10622590C22semidirect
44. #10622586C22semidirect
47. #10622273C22semidirect
48. #10622269C22semidirect
52. C23×C4×C11Q8semidirect
130. C22×C4×C11C2×Q8semidirect
336. C23×C4Dic22semidirect
381. C2×C11#2628semidirect
439. C22×C4C2×Dic22semidirect
461. C11C2×#2628semidirect
559. C2#13074441direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669412
C22×#669792
#130744412
C2×#106222732
C2×#106225622
C2×#106222692
C2×#106225602
C2×#106225982
C2×#106225862
C2×#106225902
C2×#106228252
C2×C11×#3802
C2×#262811