Finite Group Databasebeta

← C2×#13079488 #15624448

Subgroups — C2×#13079488 #15624448

Order 2816.

2314 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623356C211yesmaximal, characteristic, complemented
2#13079488C2116yesmaximal, direct product factor
3C2×#10622609C212yesmaximal, complemented
4C2×#10622648C212yesmaximal, complemented
5C2×#10622655C212yesmaximal, complemented
6C2×#10622654C212yesmaximal, complemented
7C2×#10622653C212yesmaximal, complemented
8C2×#10622831C212yesmaximal, complemented
9C2×#10622847C211yesmaximal, characteristic, complemented
10C2×C11×#535C211yesmaximal, characteristic
11C22×C11×#41C2211yescharacteristic
12C23×D4×C11C2211yescharacteristic
13C23×Dic22C2211yescharacteristic, complemented
14C11×#535—24no—
15C11×#535C2218yes—
16C2×C11×#127—21no—
17C2×C11×#124—21no—
18C2×C11×#121C2212yes—
19C2×C11×#121C2212yes—
20C2×C11×#120—21no—
21C2×C11×#120—21no—
22C2×C11×#117C2211yescharacteristic
23#10623356—24no—
24#10623356C2218yescomplemented
25C2×#66997—21no—
26C2×#66989—21no—
27C2×#66988—21no—
28C2×#66988—21no—
29C22×#4324C2212yes—
30C22×#4324C2212yes—
31C2×#66979C2211yescharacteristic, complemented
32C2×#66977C2212yes—
33C2×#66977C2212yescomplemented
34C22×#4318C2211yescharacteristic, complemented
35C23×C11⋊C8C2211yescharacteristic, complemented
36#10622847C2218yescomplemented
37#10622831C22116yescomplemented
38C2×#66896C2212yes—
39C2×#66896C2212yes—
40C2×C22⋊(C11⋊C8)C2212yes—
41C2×C22⋊(C11⋊C8)C2212yescomplemented
42#10622655C22116yescomplemented
43#10622654C22116yescomplemented
44#10622653C22116yescomplemented
45#10622648C22116yescomplemented
46#10622609C22116yescomplemented
47C2×#66874C2212yescomplemented
48C2×#66874C2212yescomplemented
49C2×#66872C2212yescomplemented
50C2×#66871C2212yescomplemented
51C25×C11—21no—
52C2×C11×#41—42no—
53C2×C11×#41—42no—
54C2×C11×#41D414yes—
55C2×C11×#41C2314yes—
56C22×Q8×C11D411yescharacteristic
57C22×D4×C11—22no—
58C22×D4×C11—22no—
59C22×D4×C11—21no—
60C22×D4×C11—22no—
61C22×D4×C11—42no—
62C22×D4×C11—42no—
63C22×D4×C11D411yescharacteristic
64C22×D4×C11D411yescharacteristic
65C22×D4×C11D411yescharacteristic
66C22×D4×C11C2312yes—
67C22×D4×C11C2314yes—
68C23×C4×C11C2311yescharacteristic
69C23×C4×C11C2311yescharacteristic
70C23×C4×C11C2311yescharacteristic
71C23×Dic11—21no—
72C22×Dic22—21no—
73C22×Dic22—22no—
74C22×Dic22—42no—
75C22×Dic22—42no—
76C22×Dic22C2312yes—
77C22×Dic22C2314yescomplemented
78C11×#127—28no—
79C11×#124—28no—
80C11×#121—24no—
81C11×#121—24no—
82C11×#121C2318yes—
83C11×#121C2318yes—
84C11×#120—28no—
85C11×#120—28no—
86C11×#117—22no—
87C11×#117C2314yes—
88C2×C11×C4⋊C4C2312yes—
89C2×C11×C22⋊C4—21no—
90C2×C11×C22⋊C4—22no—
91C2×C11×C22⋊C4—22no—
92C2×C42×C11—21no—
93C2×C22⋊Dic11—22no—
94C2×C22⋊Dic11—22no—
95C2×C22⋊Dic11—21no—
96#66997—28no—
97#66989—28no—
98#66988—28no—
99#66988—28no—
100C2×#4324—44no—
101C2×#4324—44no—
102C2×#4324C2318yes—
103C2×#4324C2318yes—
104#66979—22no—
105#66979C2314yescomplemented
106C2×C4⋊Dic11C2312yes—
107#66977—24no—
108#66977—24no—
109#66977C2318yes—
110#66977C2318yescomplemented
111C2×Dic11⋊C4—22no—
112C2×Dic11⋊C4—22no—
113C2×C4×Dic11—21no—
114C2×#4318—42no—
115C2×#4318C2314yescomplemented
116C22×C11⋊C8—42no—
117C22×C11⋊C8C2312yes—
118C22×C11⋊C8C2312yes—
119C22×C11⋊C8C2314yescomplemented
120#66896C2318yes—
121#66896C2318yes—
122C22⋊(C11⋊C8)C2318yes—
123C22⋊(C11⋊C8)C2318yescomplemented
124#66874C2318yescomplemented
125#66874C2318yescomplemented
126#66872C2318yescomplemented
127#66871C2318yescomplemented
128C2×#3200—111nomaximal, Sylow
129C24×C11—22no—
130C24×C11—24no—
131C24×C11—44no—
132C24×C11—84no—
133C24×C11C2×D411yescharacteristic
134C24×C11C2×D411yescharacteristic
135C24×C11C2×D411yescharacteristic
136C11×#41—48no—
137C11×#41—44no—
138C11×#41—44no—
139C2×Q8×C11—42no—
140C2×Q8×C11C2×D414yes—
141C2×D4×C11—28no—
142C2×D4×C11—28no—
143C2×D4×C11—22no—
144C2×D4×C11—24no—
145C2×D4×C11—24no—
146C2×D4×C11—24no—
147C2×D4×C11—44no—
148C2×D4×C11—44no—
149C2×D4×C11—42no—
150C2×D4×C11—42no—
151C2×D4×C11—42no—
152C2×D4×C11—44no—
153C2×D4×C11—48no—
154C2×D4×C11—84no—
155C2×D4×C11—84no—
156C2×D4×C11C2×D414yes—
157C2×D4×C11C2×D414yes—
158C2×D4×C11C2×D414yes—
159C2×D4×C11C2414yes—
160C22×C4×C11—22no—
161C22×C4×C11—22no—
162C22×C4×C11—22no—
163C22×C4×C11—22no—
164C22×C4×C11—22no—
165C22×C4×C11—22no—
166C22×C4×C11—22no—
167C22×C4×C11C2×D411yescharacteristic
168C22×C4×C11C2×D411yescharacteristic
169C22×C4×C11C2×D411yescharacteristic
170C22×C4×C11C2×D412yes—
171C22×C4×C11C2×D412yes—
172C22×C4×C11C2×D412yes—
173C22×C4×C11C2412yes—
174C22×C4×C11C2412yes—
175C22×C4×C11C2412yes—
176C22×C4×C11C2411yescharacteristic
177C22×Dic11—22no—
178C22×Dic11—22no—
179C22×Dic11—24no—
180C22×Dic11—44no—
181C2×Dic22—22no—
182C2×Dic22—24no—
183C2×Dic22—24no—
184C2×Dic22—24no—
185C2×Dic22—44no—
186C2×Dic22—44no—
187C2×Dic22—44no—
188C2×Dic22—48no—
189C2×Dic22C2414yes—
190C11×C4⋊C4—22no—
191C11×C4⋊C4C2414yes—
192C11×C22⋊C4—24no—
193C11×C22⋊C4—28no—
194C11×C22⋊C4—28no—
195C42×C11—24no—
196C22⋊Dic11—28no—
197C22⋊Dic11—28no—
198C22⋊Dic11—24no—
199#4324—48no—
200#4324—48no—
201C4⋊Dic11—22no—
202C4⋊Dic11C2414yes—
203Dic11⋊C4—28no—
204Dic11⋊C4—28no—
205C4×Dic11—24no—
206#4318—44no—
207C2×C11⋊C8—24no—
208C2×C11⋊C8—24no—
209C2×C11⋊C8—44no—
210C2×C11⋊C8C2414yes—
211C2×C11⋊C8C2414yes—
212C2×#536—111no—
213C2×#535D1111yescharacteristic
214#3200—1116no—
215C2×#485—112no—
216C2×#482—112no—
217C2×#468—112no—
218C2×#465—112no—
219C2×#448—112no—
220C2×#450—112no—
221C2×#408—111no—
222C23×C11—22no—
223C23×C11—22no—
224C23×C11—22no—
225C23×C11—24no—
226C23×C11—24no—
227C23×C11—24no—
228C23×C11—24no—
229C23×C11—48no—
230C23×C11—44no—
231C23×C11—44no—
232C23×C11—44no—
233C23×C11—88no—
234C23×C11—816no—
235C23×C11C22×D412yes—
236C23×C11C22×D412yes—
237C23×C11C22×D412yes—
238C23×C11C22×D411yescharacteristic
239Q8×C11—44no—
240D4×C11—24no—
241D4×C11—24no—
242D4×C11—44no—
243D4×C11—44no—
244D4×C11—44no—
245D4×C11—48no—
246D4×C11—44no—
247D4×C11—88no—
248D4×C11—88no—
249C2×C4×C11—22no—
250C2×C4×C11—22no—
251C2×C4×C11—22no—
252C2×C4×C11—24no—
253C2×C4×C11—24no—
254C2×C4×C11—24no—
255C2×C4×C11—22no—
256C2×C4×C11—22no—
257C2×C4×C11—22no—
258C2×C4×C11—22no—
259C2×C4×C11—22no—
260C2×C4×C11—22no—
261C2×C4×C11—24no—
262C2×C4×C11—44no—
263C2×C4×C11C22×D412yes—
264C2×C4×C11C22×D412yes—
265C2×C4×C11C22×D412yes—
266C2×C4×C11C22×D411yescharacteristic
267C2×C4×C11H(5,𝔽2)12yes—
268C2×C4×C11H(5,𝔽2)12yes—
269C2×C4×C11C2511yescharacteristic
270C2×Dic11—24no—
271C2×Dic11—24no—
272C2×Dic11—48no—
273C2×Dic11—44no—
274C2×Dic11—44no—
275Dic22—24no—
276Dic22—24no—
277Dic22—48no—
278Dic22—44no—
279C11⋊C8—24no—
280C11⋊C8—24no—
281C22×#41C2×D1111yescharacteristic
282C23×Q8—111no—
283C23×D4C2×D1111yescharacteristic
284C22×SD16—112no—
285C22×SD16—112no—
286C22×M16—111no—
287C23×C8—111no—
288#536—118no—
289#536—224no—
290#535—24no—
291#535C2×D1118yes—
292C2×C4⋊Q8—221no—
293C2×#127—21no—
294C2×#124—21no—
295C2×#124—221no—
296C2×#123—221no—
297C2×#123—221no—
298C2×C22⋊Q8—112no—
299C2×C22⋊Q8—112no—
300C2×#121C2×D1112yes—
301C2×#121C2×D1112yes—
302C2×#120—21no—
303C2×#120—21no—
304C2×#117—111no—
305C2×#117C2×D1111yescharacteristic
306#485—1116no—
307#482—1116no—
308#468—1116no—
309#465—1116no—
310#450—1116no—
311#448—1116no—
312C2×#107—112no—
313C2×#106—112no—
314C2×#103—112no—
315C2×#103—112no—
316C2×#102—112no—
317C2×#102—112no—
318#408—118no—
319C2×C22⋊C8—112no—
320C2×C22⋊C8—112no—
321C22×C11—22no—
322C22×C11—22no—
323C22×C11—22no—
324C22×C11—24no—
325C22×C11—24no—
326C22×C11—24no—
327C22×C11—44no—
328C22×C11—44no—
329C22×C11—44no—
330C22×C11—44no—
331C22×C11—44no—
332C22×C11—44no—
333C22×C11—816no—
334C22×C11—88no—
335C22×C11#53412yes—
336C22×C11#57714yes—
337C22×C11C23×D411yescharacteristic
338C4×C11—22no—
339C4×C11—22no—
340C4×C11—22no—
341C4×C11—44no—
342C4×C11#53412yes—
343C4×C11C2×H(5,𝔽2)11yescharacteristic
344C4×C11C2×H(5,𝔽2)11yescharacteristic
345Dic11—44no—
346Dic11—44no—
347C25—21no—
348C2×#41—42no—
349C2×#41—42no—
350C2×#41C22⋊D1114yes—
351C2×#41C22×D1114yes—
352C22×Q8—112no—
353C22×Q8—114no—
354C22×Q8—221no—
355C22×Q8—222no—
356C22×Q8—442no—
357C22×Q8—442no—
358C22×Q8C22⋊D1111yescharacteristic
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—21no—
362C22×D4—22no—
363C22×D4—42no—
364C22×D4—42no—
365C22×D4C22⋊D1111yescharacteristic
366C22×D4C22⋊D1111yescharacteristic
367C22×D4C22⋊D1111yescharacteristic
368C22×D4C22×D1112yes—
369C22×D4C22×D1114yes—
370C23×C4—221no—
371C23×C4C22×D1111yescharacteristic
372C23×C4C22×D1111yescharacteristic
373C23×C4C22×D1111yescharacteristic
374C2×SD16—118no—
375C2×SD16—118no—
376C2×SD16—444no—
377C2×SD16—444no—
378C2×M16—114no—
379C2×M16—442no—
380C22×C8—112no—
381C22×C8—112no—
382C22×C8—114no—
383C22×C8—442no—
384C4⋊Q8—228no—
385#127—28no—
386#124—28no—
387#124—228no—
388#123—228no—
389#123—228no—
390C22⋊Q8—118no—
391C22⋊Q8—118no—
392C22⋊Q8—224no—
393C22⋊Q8—224no—
394#121—24no—
395#121—24no—
396#121C22×D1118yes—
397#121C22×D1118yes—
398#120—28no—
399#120—28no—
400#117—22no—
401#117—114no—
402#117—222no—
403#117C22×D1114yes—
404C2×C4⋊C4—112no—
405C2×C4⋊C4—222no—
406C2×C4⋊C4—222no—
407C2×C4⋊C4C22×D1112yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—221no—
412C2×C22⋊C4—222no—
413C2×C22⋊C4—222no—
414C2×C42—21no—
415C2×C42—221no—
416#107—118no—
417#106—118no—
418#103—118no—
419#103—118no—
420#102—118no—
421#102—118no—
422C22⋊C8—118no—
423C22⋊C8—118no—
424C2×C11—42no—
425C2×C11—42no—
426C2×C11—42no—
427C2×C11—84no—
428C2×C11#320014yescomplemented
429C2×C11C2×#53411yescharacteristic
430C2×C11C2×#57712yes—
431C24—22no—
432C24—24no—
433C24—44no—
434C24—84no—
435C24C2×C22⋊D1111yescharacteristic
436C24C2×C22⋊D1111yescharacteristic
437C24C2×C22⋊D1111yescharacteristic
438#41—44no—
439#41—44no—
440#41—48no—
441C2×Q8—42no—
442C2×Q8—114no—
443C2×Q8—222no—
444C2×Q8—224no—
445C2×Q8—224no—
446C2×Q8—224no—
447C2×Q8—444no—
448C2×Q8—444no—
449C2×Q8—444no—
450C2×Q8—448no—
451C2×Q8C2×C22⋊D1114yes—
452C2×D4—28no—
453C2×D4—28no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—48no—
465C2×D4—84no—
466C2×D4—84no—
467C2×D4C2×C22⋊D1114yes—
468C2×D4C2×C22⋊D1114yes—
469C2×D4C2×C22⋊D1114yes—
470C2×D4C23×D1114yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—224no—
479C22×C4—222no—
480C22×C4—222no—
481C22×C4—444no—
482C22×C4C2×C22⋊D1112yes—
483C22×C4C2×C22⋊D1112yes—
484C22×C4C2×C22⋊D1112yes—
485C22×C4C2×C22⋊D1111yescharacteristic
486C22×C4C2×C22⋊D1111yescharacteristic
487C22×C4C2×C22⋊D1111yescharacteristic
488C22×C4C23×D1112yes—
489C22×C4C23×D1112yes—
490C22×C4C23×D1112yes—
491C22×C4C23×D1111yescharacteristic
492SD16—448no—
493SD16—448no—
494M16—444no—
495C2×C8—114no—
496C2×C8—114no—
497C2×C8—224no—
498C2×C8—224no—
499C2×C8—444no—
500C4⋊C4—22no—
501C4⋊C4—114no—
502C4⋊C4—222no—
503C4⋊C4—228no—
504C4⋊C4—228no—
505C4⋊C4C23×D1114yes—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—224no—
510C22⋊C4—228no—
511C22⋊C4—228no—
512C42—24no—
513C42—224no—
514C11C2×#320011yescharacteristic, complemented, Sylow
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—48no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—88no—
527C23—816no—
528C23C22×C22⋊D1112yes—
529C23C22×C22⋊D1112yes—
530C23C22×C22⋊D1112yes—
531C23C22×C22⋊D1111yescharacteristic
532Q8—44no—
533Q8—224no—
534Q8—224no—
535Q8—448no—
536Q8—444no—
537D4—24no—
538D4—24no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—48no—
543D4—44no—
544D4—88no—
545D4—88no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—44no—
560C2×C4—224no—
561C2×C4—224no—
562C2×C4—448no—
563C2×C4—444no—
564C2×C4—444no—
565C2×C4#6703712yes—
566C2×C4#6703712yes—
567C2×C4C22×C22⋊D1112yes—
568C2×C4C22×C22⋊D1112yes—
569C2×C4C22×C22⋊D1112yes—
570C2×C4C22×C22⋊D1111yescharacteristic
571C2×C4C24×D1111yescharacteristic
572C8—224no—
573C8—224no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—816no—
587C22—88no—
588C22#1062333512yes—
589C22#1062336714yes—
590C22C23×C22⋊D1111yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—44no—
595C4—444no—
596C4—444no—
597C4#1062333512yes—
598C4C2×#6703711yescharacteristic
599C4C2×#6703711yescharacteristic
600C2—42no—
601C2—42no—
602C2—42no—
603C2—84no—
604C2#1307948814yesdirect product factor
605C2C2×#1062333511yescharacteristic
606C2C2×#1062336712yes—
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
Fitting, prime radicalC2×C11×#535C2
socleC23×C11C22×D4
hypercenterC2×#535D11
hypocenterC11C2×#3200
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#10623356C2semidirect
2. #13079488C2direct (N × H)
3. C2×#10622609C2semidirect
4. C2×#10622648C2semidirect
5. C2×#10622655C2semidirect
6. C2×#10622654C2semidirect
7. C2×#10622653C2semidirect
8. C2×#10622831C2semidirect
9. C2×#10622847C2semidirect
13. C23×Dic22C22semidirect
24. #10623356C22semidirect
31. C2×#66979C22semidirect
33. C2×#66977C22semidirect
34. C22×#4318C22semidirect
35. C23×C11⋊C8C22semidirect
36. #10622847C22semidirect
37. #10622831C22semidirect
41. C2×C22⋊(C11⋊C8)C22semidirect
42. #10622655C22semidirect
43. #10622654C22semidirect
44. #10622653C22semidirect
45. #10622648C22semidirect
46. #10622609C22semidirect
47. C2×#66874C22semidirect
48. C2×#66874C22semidirect
49. C2×#66872C22semidirect
50. C2×#66871C22semidirect
77. C22×Dic22C23semidirect
105. #66979C23semidirect
110. #66977C23semidirect
115. C2×#4318C23semidirect
119. C22×C11⋊C8C23semidirect
123. C22⋊(C11⋊C8)C23semidirect
124. #66874C23semidirect
125. #66874C23semidirect
126. #66872C23semidirect
127. #66871C23semidirect
428. C2×C11#3200semidirect
514. C11C2×#3200semidirect
604. C2#13079488direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106233562
#130794882
C2×#106226092
C2×#106226482
C2×#106226552
C2×#106226542
C2×#106226532
C2×#106228312
C2×#106228472
C2×C11×#5352
C2×#320011