Finite Group Databasebeta

← C2×#13072334 #14921856

Subgroups — C2×#13072334 #14921856

Order 2816.

2942 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66943C211yesmaximal, characteristic, complemented
2#13072334C2116yesmaximal, direct product factor
3C2×#10622599C212yesmaximal, complemented
4C2×#10622597C212yesmaximal, complemented
5C2×#10622598C211yesmaximal, characteristic, complemented
6C2×#10622586C211yesmaximal, characteristic, complemented
7C2×#10622823C212yesmaximal
8C2×#10622590C211yesmaximal, characteristic, complemented
9C2×#10622825C211yesmaximal, characteristic
10C2×#10622825C212yesmaximal
11C2×C4×C22⋊Dic11C211yesmaximal, characteristic
12C23×C11×C4⋊C4C211yesmaximal, characteristic
13C24×C4×C11C2212yes—
14C24×C4×C11C2211yescharacteristic
15C22×C11×C4⋊C4—28no—
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C4⋊C4C2218yes—
18C22×C11×C4⋊C4C2212yes—
19C22×C11×C4⋊C4C2211yescharacteristic
20C22×C22⋊Dic11C2211yescharacteristic
21C22×C22⋊Dic11C2211yescharacteristic
22C22×C22⋊Dic11C2212yes—
23C22×C4⋊Dic11C2211yescharacteristic
24C22×C4⋊Dic11C2212yes—
25C22×C4⋊Dic11C2211yescharacteristic
26C22×C4⋊Dic11C2212yes—
27C22×Dic11⋊C4C2212yes—
28C22×Dic11⋊C4C2212yes—
29C22×C4×Dic11C2211yescharacteristic
30C22×C4×Dic11C2211yescharacteristic
31C2×#66943—28no—
32C2×#66943C2218yescomplemented
33#10622825C22116yes—
34#10622825C2218yes—
35#10622823C22116yes—
36C4×C22⋊Dic11C2218yes—
37C2×C11⋊#95C2211yescharacteristic
38C2×C11⋊#95C2211yescharacteristic
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2212yes—
44#10622599C22116yescomplemented
45#10622598C2218yescomplemented
46#10622597C22116yescomplemented
47#10622590C2218yescomplemented
48#10622586C2218yescomplemented
49C25×C11C2311yescharacteristic
50C23×C4×C11—24no—
51C23×C4×C11—24no—
52C23×C4×C11—28no—
53C23×C4×C11—28no—
54C23×C4×C11—44no—
55C23×C4×C11Q818yes—
56C23×C4×C11C2312yes—
57C23×C4×C11C2318yes—
58C23×C4×C11C2312yes—
59C23×C4×C11C2311yescharacteristic
60C23×C4×C11C2314yes—
61C23×C4×C11C2311yescharacteristic
62C23×Dic11C2311yescharacteristic
63C23×Dic11C2311yescharacteristic
64C23×Dic11C2311yescharacteristic
65C23×Dic11C2311yescharacteristic
66C23×Dic11C2312yes—
67C23×Dic11C2312yes—
68C2×C11×C4⋊C4—28no—
69C2×C11×C4⋊C4—24no—
70C2×C11×C4⋊C4—232no—
71C2×C11×C4⋊C4D418yes—
72C2×C11×C4⋊C4C2314yes—
73C2×C11×C4⋊C4C2318yes—
74C2×C11×C4⋊C4C2314yes—
75C2×C22⋊Dic11—24no—
76C2×C22⋊Dic11—24no—
77C2×C22⋊Dic11—28no—
78C2×C22⋊Dic11C2314yes—
79C2×C22⋊Dic11C2314yes—
80C2×C22⋊Dic11C2318yes—
81C2×C4⋊Dic11—28no—
82C2×C4⋊Dic11—24no—
83C2×C4⋊Dic11—24no—
84C2×C4⋊Dic11—44no—
85C2×C4⋊Dic11C2314yes—
86C2×C4⋊Dic11C2318yes—
87C2×C4⋊Dic11C2314yes—
88C2×C4⋊Dic11C2318yes—
89C2×Dic11⋊C4—28no—
90C2×Dic11⋊C4—44no—
91C2×Dic11⋊C4C2318yes—
92C2×Dic11⋊C4C2318yes—
93C2×C4×Dic11—24no—
94C2×C4×Dic11—24no—
95C2×C4×Dic11C2314yes—
96C2×C4×Dic11C2314yes—
97#66943—232no—
98C11⋊#95C2314yes—
99C11⋊#95C2314yes—
100C11⋊#95C2318yes—
101C11⋊#95C2318yes—
102C11⋊#95C2318yes—
103C11⋊#95C2318yes—
104C11⋊#95C2318yes—
105C2×#2442—111nomaximal, Sylow
106C24×C11—22no—
107C24×C11—22no—
108C24×C11—22no—
109C24×C11—22no—
110C24×C11C2×Q814yes—
111C24×C11#4114yes—
112C24×C11#4114yes—
113C24×C11C2412yes—
114C24×C11C2411yescharacteristic
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—24no—
118C22×C4×C11—24no—
119C22×C4×C11—232no—
120C22×C4×C11—28no—
121C22×C4×C11—28no—
122C22×C4×C11—24no—
123C22×C4×C11—216no—
124C22×C4×C11—24no—
125C22×C4×C11—24no—
126C22×C4×C11—216no—
127C22×C4×C11—48no—
128C22×C4×C11—432no—
129C22×C4×C11C2×D418yes—
130C22×C4×C11C2×D414yes—
131C22×C4×C11C2×Q814yes—
132C22×C4×C11C2×Q8116yes—
133C22×C4×C11C2×Q814yes—
134C22×C4×C11#4114yes—
135C22×C4×C11C2414yes—
136C22×C4×C11C2414yes—
137C22×C4×C11C2412yes—
138C22×C4×C11C2412yes—
139C22×Dic11—22no—
140C22×Dic11—22no—
141C22×Dic11—24no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—24no—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—22no—
148C22×Dic11—24no—
149C22×Dic11—22no—
150C22×Dic11—24no—
151C22×Dic11—22no—
152C22×Dic11—24no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—24no—
156C22×Dic11—22no—
157C22×Dic11C2412yes—
158C22×Dic11C2412yes—
159C22×Dic11C2412yes—
160C22×Dic11C2414yes—
161C22×Dic11C2414yes—
162C22×Dic11C2412yes—
163C11×C4⋊C4—216no—
164C11×C4⋊C4—28no—
165C11×C4⋊C4C2×D4116yes—
166C22⋊Dic11—28no—
167C22⋊Dic11—28no—
168C22⋊Dic11—216no—
169C4⋊Dic11—216no—
170C4⋊Dic11—28no—
171C4⋊Dic11—28no—
172C4⋊Dic11—48no—
173Dic11⋊C4—216no—
174Dic11⋊C4—48no—
175C4×Dic11—28no—
176C4×Dic11—28no—
177C22×#125—111no—
178C23×C4⋊C4D1111yescharacteristic
179#2442—1116no—
180C2×#400—112no—
181C2×#398—112no—
182C2×#389—111no—
183C2×#389—111no—
184C2×#385—112no—
185C2×#382—111no—
186C2×#380—112no—
187C2×#380—111no—
188C2×C4×C22⋊C4—111no—
189C23×C11—28no—
190C23×C11—28no—
191C23×C11—28no—
192C23×C11—28no—
193C23×C11—28no—
194C23×C11—28no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11—22no—
198C23×C11—22no—
199C23×C11#125116yescomplemented
200C23×C11C22×D412yes—
201C23×C11C22×Q814yes—
202C23×C11C22×Q812yes—
203C23×C11C2×#4112yes—
204C23×C11C2×#4114yes—
205C23×C11C2×#4112yes—
206C23×C11C2×#4114yes—
207C23×C11C2×#4112yes—
208C23×C11H(5,𝔽2)12yes—
209C23×C112−1+412yes—
210C23×C11C2511yescharacteristic
211C2×C4×C11—28no—
212C2×C4×C11—216no—
213C2×C4×C11—22no—
214C2×C4×C11—22no—
215C2×C4×C11—24no—
216C2×C4×C11—24no—
217C2×C4×C11—232no—
218C2×C4×C11—24no—
219C2×C4×C11—24no—
220C2×C4×C11—24no—
221C2×C4×C11—432no—
222C2×C4×C11—48no—
223C2×C4×C11C22⋊Q8116yes—
224C2×C4×C11C22×D418yes—
225C2×C4×C11C22×D414yes—
226C2×C4×C11C22×Q814yes—
227C2×C4×C11C22×Q814yes—
228C2×C4×C11C2×#4114yes—
229C2×Dic11—22no—
230C2×Dic11—22no—
231C2×Dic11—24no—
232C2×Dic11—22no—
233C2×Dic11—22no—
234C2×Dic11—24no—
235C2×Dic11—22no—
236C2×Dic11—22no—
237C2×Dic11—22no—
238C2×Dic11—24no—
239C2×Dic11—22no—
240C2×Dic11—24no—
241C2×Dic11—22no—
242C2×Dic11—24no—
243C2×Dic11—22no—
244C2×Dic11—22no—
245C2×Dic11—24no—
246C2×Dic11—22no—
247C2×Dic11—44no—
248C2×Dic11—44no—
249C2×Dic11—48no—
250C2×Dic11—44no—
251C2×Dic11—44no—
252C2×Dic11—48no—
253C24×C4C2×D1112yes—
254C24×C4C2×D1111yescharacteristic
255C2×#125—118no—
256C2×#125—228no—
257C22×C4⋊C4—28no—
258C22×C4⋊C4—112no—
259C22×C4⋊C4—112no—
260C22×C4⋊C4—111no—
261C22×C4⋊C4—111no—
262C22×C4⋊C4—112no—
263C22×C4⋊C4—112no—
264C22×C4⋊C4C2×D1112yes—
265C22×C4⋊C4C2×D1118yes—
266C22×C4⋊C4C2×D1111yescharacteristic
267C22×C4⋊C4C2×D1111yescharacteristic
268C22×C22⋊C4—111no—
269C22×C22⋊C4—111no—
270C22×C22⋊C4—112no—
271C22×C42—111no—
272C22×C42—111no—
273#400—1116no—
274#398—1116no—
275#389—118no—
276#389—118no—
277#385—1116no—
278#382—118no—
279#380—1116no—
280#380—118no—
281C4×C22⋊C4—118no—
282C2×#95—111no—
283C2×#95—112no—
284C2×#95—112no—
285C2×#95—111no—
286C2×#95—112no—
287C2×#95—112no—
288C2×#95—112no—
289C22×C11—24no—
290C22×C11—24no—
291C22×C11—24no—
292C22×C11—24no—
293C22×C11—232no—
294C22×C11—24no—
295C22×C11—24no—
296C22×C11C2×C22⋊Q814yes—
297C22×C11C2×#12514yes—
298C22×C11C2×#12518yescomplemented
299C22×C11#54414yes—
300C22×C11#54614yes—
301C22×C11#54814yes—
302C22×C11#55214yes—
303C22×C11#55314yes—
304C22×C11C23×D411yescharacteristic
305C22×C11C23×Q811yescharacteristic
306C22×C11C22×#4111yescharacteristic
307C22×C11C22×#4111yescharacteristic
308C22×C11C22×#4111yescharacteristic
309C22×C11C2×H(5,𝔽2)11yescharacteristic
310C22×C11C2×2−1+411yescharacteristic
311C4×C11—24no—
312C4×C11—28no—
313C4×C11—44no—
314C4×C11C2×C22⋊Q818yes—
315Dic11—42no—
316Dic11—42no—
317Dic11—44no—
318Dic11—42no—
319Dic11—42no—
320Dic11—44no—
321C25C22×D1111yescharacteristic
322C23×C4—24no—
323C23×C4—24no—
324C23×C4—28no—
325C23×C4—28no—
326C23×C4—44no—
327C23×C4—111no—
328C23×C4—112no—
329C23×C4—112no—
330C23×C4—111no—
331C23×C4—111no—
332C23×C4—111no—
333C23×C4Dic2218yes—
334C23×C4C22×D1112yes—
335C23×C4C22×D1118yes—
336C23×C4C22×D1114yes—
337C23×C4C22×D1111yescharacteristic
338C23×C4C22×D1112yes—
339C23×C4C22×D1111yescharacteristic
340#125—2232no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—232no—
344C2×C4⋊C4—114no—
345C2×C4⋊C4—118no—
346C2×C4⋊C4—118no—
347C2×C4⋊C4—114no—
348C2×C4⋊C4—118no—
349C2×C4⋊C4—118no—
350C2×C4⋊C4—224no—
351C2×C4⋊C4—228no—
352C2×C4⋊C4—228no—
353C2×C4⋊C4—224no—
354C2×C4⋊C4—444no—
355C2×C4⋊C4—444no—
356C2×C4⋊C4C22⋊D1118yes—
357C2×C4⋊C4C22×D1114yes—
358C2×C4⋊C4C22×D1118yes—
359C2×C4⋊C4C22×D1114yes—
360C2×C22⋊C4—114no—
361C2×C22⋊C4—114no—
362C2×C22⋊C4—118no—
363C2×C22⋊C4—224no—
364C2×C22⋊C4—224no—
365C2×C22⋊C4—228no—
366C2×C42—114no—
367C2×C42—114no—
368C2×C42—224no—
369C2×C42—224no—
370#95—114no—
371#95—114no—
372#95—118no—
373#95—118no—
374#95—118no—
375#95—118no—
376#95—118no—
377C2×C11—28no—
378C2×C11#244218yescomplemented
379C2×C11C22×C22⋊Q811yescharacteristic
380C2×C11C22×#12511yescharacteristic
381C2×C11C2×#54411yescharacteristic
382C2×C11C2×#54611yescharacteristic
383C2×C11C2×#54811yescharacteristic
384C2×C11C2×#55211yescharacteristic
385C2×C11C2×#55311yescharacteristic
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24C2×Dic2214yes—
391C24#434114yes—
392C24#434314yes—
393C24C23×D1112yes—
394C24C23×D1111yescharacteristic
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—232no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—216no—
407C22×C4—48no—
408C22×C4—432no—
409C22×C4—112no—
410C22×C4—112no—
411C22×C4—112no—
412C22×C4—114no—
413C22×C4—114no—
414C22×C4—112no—
415C22×C4—222no—
416C22×C4—222no—
417C22×C4—224no—
418C22×C4—222no—
419C22×C4—222no—
420C22×C4—224no—
421C22×C4—222no—
422C22×C4—222no—
423C22×C4—224no—
424C22×C4—222no—
425C22×C4—224no—
426C22×C4—222no—
427C22×C4—222no—
428C22×C4—224no—
429C22×C4—222no—
430C22×C4—222no—
431C22×C4—222no—
432C22×C4—224no—
433C22×C4C2×Dic2214yes—
434C22×C4C2×Dic22116yes—
435C22×C4C2×Dic2214yes—
436C22×C4#433914yes—
437C22×C4C2×C22⋊D1118yes—
438C22×C4C2×C22⋊D1114yes—
439C22×C4C23×D1114yes—
440C22×C4C23×D1112yes—
441C22×C4C23×D1114yes—
442C22×C4C23×D1112yes—
443C4⋊C4—216no—
444C4⋊C4—28no—
445C4⋊C4—228no—
446C4⋊C4—2216no—
447C4⋊C4—2216no—
448C4⋊C4—228no—
449C4⋊C4—448no—
450C4⋊C4—448no—
451C4⋊C4C2×C22⋊D11116yes—
452C22⋊C4—228no—
453C22⋊C4—228no—
454C22⋊C4—2216no—
455C42—228no—
456C42—228no—
457C11C2×#244211yescharacteristic, complemented, Sylow
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23#66943116yescomplemented
469C23C22×Dic2214yes—
470C23C22×Dic2212yes—
471C23C2×#433912yes—
472C23C2×#434114yes—
473C23C2×#434112yes—
474C23#6703712yes—
475C23C2×#434314yes—
476C23C2×#434312yes—
477C23#6704012yes—
478C23C22×C22⋊D1112yes—
479C23C24×D1111yescharacteristic
480C2×C4—28no—
481C2×C4—216no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—232no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—432no—
491C2×C4—48no—
492C2×C4—222no—
493C2×C4—222no—
494C2×C4—224no—
495C2×C4—222no—
496C2×C4—222no—
497C2×C4—224no—
498C2×C4—222no—
499C2×C4—222no—
500C2×C4—222no—
501C2×C4—224no—
502C2×C4—222no—
503C2×C4—224no—
504C2×C4—222no—
505C2×C4—222no—
506C2×C4—224no—
507C2×C4—222no—
508C2×C4—222no—
509C2×C4—224no—
510C2×C4—444no—
511C2×C4—444no—
512C2×C4—448no—
513C2×C4—444no—
514C2×C4—444no—
515C2×C4—448no—
516C2×C4#66977116yes—
517C2×C4C22×Dic2214yes—
518C2×C4C22×Dic2214yes—
519C2×C4C2×#433914yes—
520C2×C4C22×C22⋊D1118yes—
521C2×C4C22×C22⋊D1114yes—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—232no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22C2×#6694314yes—
530C22C2×#6694318yescomplemented
531C22#1062303114yes—
532C22#1062304414yes—
533C22#1062304514yes—
534C22C2×#6697714yes—
535C22#1062333714yes—
536C22#1062334814yes—
537C22C23×Dic2211yescharacteristic
538C22C22×#433911yescharacteristic
539C22C22×#434111yescharacteristic
540C22C2×#6703711yescharacteristic
541C22C22×#434311yescharacteristic
542C22C2×#6704011yescharacteristic
543C22C23×C22⋊D1111yescharacteristic
544C4—28no—
545C4—24no—
546C4—44no—
547C4—442no—
548C4—442no—
549C4—444no—
550C4—442no—
551C4—442no—
552C4—444no—
553C4C2×#6697718yes—
554C2—28no—
555C2#1307233418yesdirect product factor
556C2C22×#6697711yescharacteristic
557C2C22×#6694311yescharacteristic
558C2C2×#1062303111yescharacteristic
559C2C2×#1062333711yescharacteristic
560C2C2×#1062334811yescharacteristic
561C2C2×#1062304411yescharacteristic
562C2C2×#1062304511yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC23×C11×C4⋊C4C2
socleC24×C11C24
prime radicalC25×C11C23
hypercenterC23×C4⋊C4D11
hypocenterC11C2×#2442
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#66943C2semidirect
2. #13072334C2direct (N × H)
3. C2×#10622599C2semidirect
4. C2×#10622597C2semidirect
5. C2×#10622598C2semidirect
6. C2×#10622586C2semidirect
8. C2×#10622590C2semidirect
32. C2×#66943C22semidirect
44. #10622599C22semidirect
45. #10622598C22semidirect
46. #10622597C22semidirect
47. #10622590C22semidirect
48. #10622586C22semidirect
199. C23×C11#125semidirect
298. C22×C11C2×#125semidirect
378. C2×C11#2442semidirect
457. C11C2×#2442semidirect
468. C23#66943semidirect
530. C22C2×#66943semidirect
555. C2#13072334direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669432
#130723342
C2×#106225992
C2×#106225972
C2×#106225982
C2×#106225862
C2×#106228232
C2×#106225902
C2×#106228252
C2×C4×C22⋊Dic112
C23×C11×C4⋊C42
C2×#244211