Finite Group Databasebeta

← C2×#13081533 #15634270

Subgroups — C2×#13081533 #15634270

Order 2816.

2006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D44C211yesmaximal, characteristic
2#13081533C2116yesmaximal, direct product factor
3C2×#10622678C212yesmaximal, complemented
4C2×#10622691C211yesmaximal, characteristic, complemented
5C2×#10622634C212yesmaximal, complemented
6C2×#10622633C211yesmaximal, characteristic, complemented
7C2×#10622679C212yesmaximal
8C2×#10622700C211yesmaximal, characteristic
9C2×#10622637C211yesmaximal, characteristic, complemented
10C2×#10622639C212yesmaximal
11C2×C4×#4325C211yesmaximal, characteristic
12C2×C11×#557C211yesmaximal, characteristic, complemented
13C22×Q8×D11—21no—
14C11×#557—24no—
15C11×#557C2218yescomplemented
16C2×C11×C4⋊Q8—21no—
17C2×C11×C4⋊Q8C2211yescharacteristic
18C2×C11×#125—22no—
19C2×C11×#125C2212yes—
20C2×C4×Q8×C11—21no—
21C2×C4×Q8×C11C2211yescharacteristic
22C2×C4×Q8×C11C2212yes—
23C2×C4×Q8×C11C2211yescharacteristic
24C22×#4325C2211yescharacteristic
25Q8×D44—24no—
26Q8×D44C2218yes—
27C2×#66950—21no—
28C2×#66950—22no—
29C2×C4×D44C2211yescharacteristic
30C2×C4×D44C2212yes—
31C2×#66922C2211yescharacteristic
32C2×#66922C2212yes—
33C2×#66899C2212yes—
34C2×#66899C2211yescharacteristic
35#10622700C2218yes—
36#10622691C2218yescomplemented
37#10622679C22116yes—
38#10622678C22116yescomplemented
39#10622639C22116yes—
40C4×#4325C2218yes—
41#10622637C2218yescomplemented
42#10622634C22116yescomplemented
43#10622633C2218yescomplemented
44C2×#66873C2212yes—
45C2×#66873C2211yescharacteristic
46C2×#66872C2211yescharacteristic
47C2×#66872C2212yes—
48C2×#66871C2212yes—
49C2×#66871C2212yes—
50C2×#66871C2212yes—
51C2×C4⋊(C11⋊C8)C2211yescharacteristic
52C2×C4⋊(C11⋊C8)C2212yes—
53C2×#66867C2212yes—
54C2×C4×C11⋊C8C2211yescharacteristic
55C22×Q8×C11—21no—
56C22×Q8×C11D411yescharacteristic
57C22×Q8×C11C2311yescharacteristic
58C2×Q8×D11—28no—
59C2×Q8×D11—48no—
60C22×D44C2311yescharacteristic
61C22×C4×D11—22no—
62C22×C4×D11—21no—
63C22×Dic22—22no—
64C22×Dic22—21no—
65C11×C4⋊Q8—28no—
66C11×C4⋊Q8—22no—
67C11×C4⋊Q8C2314yes—
68C11×#125—216no—
69C11×#125—24no—
70C11×#125C2318yes—
71C4×Q8×C11—28no—
72C4×Q8×C11—24no—
73C4×Q8×C11—22no—
74C4×Q8×C11D414yes—
75C4×Q8×C11C2314yes—
76C4×Q8×C11C2318yes—
77C4×Q8×C11C2314yes—
78C2×C11×C4⋊C4—22no—
79C2×C11×C4⋊C4—21no—
80C2×C11×C4⋊C4—22no—
81C2×C11×C4⋊C4—21no—
82C2×C11×C4⋊C4—22no—
83C2×C11×C4⋊C4D411yescharacteristic
84C2×C11×C4⋊C4D412yes—
85C2×C11×C4⋊C4C2312yes—
86C2×C11×C4⋊C4C2311yescharacteristic
87C2×C42×C11—22no—
88C2×C42×C11—21no—
89C2×C42×C11C2311yescharacteristic
90C2×C42×C11C2312yes—
91C2×#4325Q818yescomplemented
92C2×#4325C2314yes—
93C2×(C2×D11)⋊C4—22no—
94C2×(C2×D11)⋊C4—21no—
95C2×C4⋊Dic11—22no—
96C2×C4⋊Dic11—21no—
97C2×C4⋊Dic11C2312yes—
98C2×C4⋊Dic11C2311yescharacteristic
99C22×C11⋊C8C2311yescharacteristic
100C22×C11⋊C8C2312yes—
101C22×C11⋊C8C2311yescharacteristic
102#66950—28no—
103#66950—216no—
104C4×D44—24no—
105C4×D44—22no—
106C4×D44C2314yes—
107C4×D44C2318yes—
108#66922—24no—
109#66922—22no—
110#66922C2314yes—
111#66922C2318yes—
112#66899C2318yes—
113#66899C2314yes—
114#66873C2314yes—
115#66873C2318yes—
116#66872C2318yes—
117#66872C2314yes—
118#66871C2318yes—
119#66871C2318yes—
120#66871C2318yes—
121C4⋊(C11⋊C8)C2314yes—
122C4⋊(C11⋊C8)C2318yes—
123#66867C2318yes—
124C4×C11⋊C8C2314yes—
125C2×#3386—111nomaximal, Sylow
126C23×D11—21no—
127C2×Q8×C11—24no—
128C2×Q8×C11—21no—
129C2×Q8×C11—24no—
130C2×Q8×C11—22no—
131C2×Q8×C11—44no—
132C2×Q8×C11C2×D414yes—
133C2×Q8×C11C2×Q814yescomplemented
134C2×Q8×C11C2412yes—
135C22×C4×C11—21no—
136C22×C4×C11—21no—
137C22×C4×C11—22no—
138C22×C4×C11C2×D412yes—
139C22×C4×C11C2×D411yescharacteristic
140C22×C4×C11C2×D412yes—
141C22×C4×C11C2×D411yescharacteristic
142C22×C4×C11C2411yescharacteristic
143C22×Dic11—21no—
144C22×Dic11—22no—
145Q8×D11—432no—
146C2×D44—21no—
147C2×D44—22no—
148C2×D44C2×Q814yes—
149C2×D44C2412yes—
150C2×C4×D11—28no—
151C2×C4×D11—24no—
152C2×C4×D11—48no—
153C2×C4×D11—44no—
154C2×Dic22—28no—
155C2×Dic22—24no—
156C2×Dic22—48no—
157C2×Dic22—44no—
158C11×C4⋊C4—28no—
159C11×C4⋊C4—24no—
160C11×C4⋊C4—28no—
161C11×C4⋊C4—24no—
162C11×C4⋊C4—28no—
163C11×C4⋊C4—21no—
164C11×C4⋊C4—22no—
165C11×C4⋊C4C2×D414yes—
166C11×C4⋊C4C2×D418yes—
167C11×C4⋊C4C2414yes—
168C11×C4⋊C4C2412yes—
169C42×C11—28no—
170C42×C11—24no—
171C42×C11C2×D414yes—
172C42×C11C2×D412yes—
173C42×C11C2412yes—
174C42×C11C2414yes—
175#4325C2×Q8116yescomplemented
176(C2×D11)⋊C4—28no—
177(C2×D11)⋊C4—24no—
178C4⋊Dic11—21no—
179C4⋊Dic11—22no—
180C4⋊Dic11—28no—
181C4⋊Dic11—24no—
182C4⋊Dic11C2412yes—
183C4⋊Dic11C2414yes—
184C2×C11⋊C8—22no—
185C2×C11⋊C8—24no—
186C2×C11⋊C8C2×Q814yescomplemented
187C2×C11⋊C8C2412yes—
188C2×C11⋊C8C2414yes—
189C2×C11⋊C8C2412yes—
190C2×#557D1111yescharacteristic, complemented
191C2×D4×Q8—111no—
192#3386—1116no—
193C2×#501—112no—
194C2×#500—111no—
195C2×#477—112no—
196C2×#475—111no—
197C2×#478—112no—
198C2×#474—111no—
199C2×#446—111no—
200C2×#440—112no—
201C2×C4×SD16—111no—
202C23×C11C22×D411yescharacteristic
203C22×D11—24no—
204C22×D11—22no—
205C22×D11—42no—
206C22×D11—42no—
207Q8×C11—24no—
208Q8×C11—28no—
209Q8×C11—22no—
210Q8×C11—48no—
211Q8×C11C22×Q814yescomplemented
212C2×C4×C11—24no—
213C2×C4×C11—24no—
214C2×C4×C11—24no—
215C2×C4×C11—22no—
216C2×C4×C11—22no—
217C2×C4×C11—22no—
218C2×C4×C11—24no—
219C2×C4×C11—44no—
220C2×C4×C11—42no—
221C2×C4×C11AGL(1,ℤ/8)14yes—
222C2×C4×C11C22×D414yes—
223C2×C4×C11C22×D412yes—
224C2×C4×C11C22×D414yes—
225C2×C4×C11C22×D411yescharacteristic
226C2×C4×C11C22×D412yes—
227C2×C4×C11C22×Q812yes—
228C2×C4×C112−1+412yes—
229C2×C4×C11C2511yescharacteristic
230C2×Dic11—22no—
231C2×Dic11—24no—
232C2×Dic11—42no—
233C2×Dic11—44no—
234D44—24no—
235D44—22no—
236D44C22×Q814yes—
237C4×D11—416no—
238C4×D11—48no—
239Dic22—416no—
240Dic22—48no—
241C11⋊C8—22no—
242C11⋊C8—24no—
243C11⋊C8C22×Q814yescomplemented
244C23×Q8—221no—
245C22×SD16—111no—
246#557—24no—
247#557C2×D1118yescomplemented
248D4×Q8—118no—
249D4×Q8—224no—
250C2×C4⋊Q8—21no—
251C2×C4⋊Q8—111no—
252C2×C4⋊Q8—112no—
253C2×C4⋊Q8C2×D1111yescharacteristic
254C2×#125—22no—
255C2×#125C2×D1112yes—
256C2×C22⋊Q8—221no—
257C2×C22⋊Q8—222no—
258C2×C4×Q8—21no—
259C2×C4×Q8C2×D1111yescharacteristic
260C2×C4×Q8C2×D1112yes—
261C2×C4×Q8C2×D1111yescharacteristic
262C2×C4×D4—112no—
263C2×C4×D4—111no—
264#501—1116no—
265#500—118no—
266#478—1116no—
267#477—1116no—
268#475—118no—
269#474—118no—
270#446—118no—
271#440—1116no—
272C4×SD16—118no—
273C2×#107—112no—
274C2×#107—112no—
275C2×#107—112no—
276C2×#106—112no—
277C2×#106—111no—
278C2×C4⋊C8—112no—
279C2×C4⋊C8—111no—
280C2×#103—112no—
281C2×#103—111no—
282C2×#102—111no—
283C2×#102—112no—
284C2×#97—112no—
285C2×C4×C8—111no—
286C22×C11D4×Q812yes—
287C22×C11C2×AGL(1,ℤ/8)12yes—
288C22×C11#57812yes—
289C22×C11C23×D411yescharacteristic
290C2×D11—24no—
291C2×D11—48no—
292C2×D11—42no—
293C2×D11—42no—
294C4×C11—24no—
295C4×C11—24no—
296C4×C11—24no—
297C4×C11—22no—
298C4×C11—44no—
299C4×C11—42no—
300C4×C11D4×Q812yes—
301C4×C11C2×AGL(1,ℤ/8)14yes—
302C4×C11C23×Q811yescharacteristic
303C4×C11C2×2−1+411yescharacteristic
304Dic11—42no—
305Dic11—44no—
306C22×Q8—21no—
307C22×Q8—228no—
308C22×Q8—222no—
309C22×Q8—221no—
310C22×Q8—448no—
311C22×Q8C22⋊D1111yescharacteristic
312C22×Q8C22×D1111yescharacteristic
313C22×D4—111no—
314C23×C4—222no—
315C23×C4—221no—
316C2×SD16—118no—
317C2×SD16—114no—
318C22×C8—111no—
319C22×C8—111no—
320C22×C8—112no—
321C4⋊Q8—28no—
322C4⋊Q8—22no—
323C4⋊Q8—114no—
324C4⋊Q8—118no—
325C4⋊Q8—224no—
326C4⋊Q8—222no—
327C4⋊Q8C22×D1114yes—
328#125—216no—
329#125—24no—
330#125C22×D1118yes—
331C22⋊Q8—228no—
332C22⋊Q8—2216no—
333C4×Q8—28no—
334C4×Q8—24no—
335C4×Q8—22no—
336C4×Q8C22⋊D1114yes—
337C4×Q8C22×D1114yes—
338C4×Q8C22×D1118yes—
339C4×Q8C22×D1114yes—
340C4×D4—114no—
341C4×D4—118no—
342C4×D4—224no—
343C4×D4—222no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—112no—
350C2×C4⋊C4—111no—
351C2×C4⋊C4—222no—
352C2×C4⋊C4—221no—
353C2×C4⋊C4C22⋊D1111yescharacteristic
354C2×C4⋊C4C22⋊D1112yes—
355C2×C4⋊C4C22×D1111yescharacteristic
356C2×C4⋊C4C22×D1112yes—
357C2×C22⋊C4—222no—
358C2×C22⋊C4—221no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42C22×D1112yes—
362C2×C42C22×D1111yescharacteristic
363#107—118no—
364#107—118no—
365#107—118no—
366#106—114no—
367#106—118no—
368C4⋊C8—114no—
369C4⋊C8—118no—
370#103—114no—
371#103—118no—
372#102—114no—
373#102—118no—
374#97—118no—
375C4×C8—114no—
376C2×C11#338614yescomplemented
377C2×C11C2×D4×Q811yescharacteristic
378C2×C11C22×AGL(1,ℤ/8)11yescharacteristic
379C2×C11C2×#57811yescharacteristic
380D11—44no—
381C24—221no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—24no—
386C2×Q8—44no—
387C2×Q8—228no—
388C2×Q8—224no—
389C2×Q8—448no—
390C2×Q8—4432no—
391C2×Q8—444no—
392C2×Q8Q8×D1114yescomplemented
393C2×Q8C2×C22⋊D1114yes—
394C2×Q8C23×D1112yes—
395C2×D4—114no—
396C2×D4—112no—
397C2×D4—222no—
398C2×D4—221no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—222no—
403C22×C4—221no—
404C22×C4—228no—
405C22×C4—224no—
406C22×C4—448no—
407C22×C4—444no—
408C22×C4C2×C22⋊D1112yes—
409C22×C4C2×C22⋊D1111yescharacteristic
410C22×C4C2×C22⋊D1112yes—
411C22×C4C2×C22⋊D1111yescharacteristic
412C22×C4C23×D1111yescharacteristic
413SD16—1116no—
414C2×C8—114no—
415C2×C8—112no—
416C2×C8—112no—
417C2×C8—114no—
418C2×C8—222no—
419C2×C8—224no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—112no—
428C4⋊C4—114no—
429C4⋊C4—222no—
430C4⋊C4—221no—
431C4⋊C4—228no—
432C4⋊C4—224no—
433C4⋊C4C2×C22⋊D1114yes—
434C4⋊C4C2×C22⋊D1118yes—
435C4⋊C4C23×D1112yes—
436C4⋊C4C23×D1114yes—
437C22⋊C4—228no—
438C22⋊C4—224no—
439C42—28no—
440C42—24no—
441C42C2×C22⋊D1114yes—
442C42C2×C22⋊D1112yes—
443C42C23×D1112yes—
444C42C23×D1114yes—
445C11C2×#338611yescharacteristic, complemented, Sylow
446C23—224no—
447C23—222no—
448C23—442no—
449C23—442no—
450C23C22×C22⋊D1111yescharacteristic
451Q8—24no—
452Q8—28no—
453Q8—22no—
454Q8—48no—
455Q8—4416no—
456Q8—448no—
457Q8C2×Q8×D1114yescomplemented
458D4—114no—
459D4—224no—
460D4—222no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—224no—
471C2×C4—222no—
472C2×C4—4416no—
473C2×C4—444no—
474C2×C4—442no—
475C2×C4—448no—
476C2×C4#6700214yes—
477C2×C4C2×Q8×D1112yes—
478C2×C4#6704012yes—
479C2×C4C22×C22⋊D1114yes—
480C2×C4C22×C22⋊D1112yes—
481C2×C4C22×C22⋊D1114yes—
482C2×C4C22×C22⋊D1111yescharacteristic
483C2×C4C22×C22⋊D1112yes—
484C2×C4C24×D1111yescharacteristic
485C8—114no—
486C8—222no—
487C8—224no—
488C22—224no—
489C22—448no—
490C22—442no—
491C22—442no—
492C22Q8×C22⋊D1112yes—
493C22C2×#6700212yes—
494C22#1062336912yes—
495C22C23×C22⋊D1111yescharacteristic
496C4—24no—
497C4—24no—
498C4—24no—
499C4—22no—
500C4—44no—
501C4—42no—
502C4—444no—
503C4—442no—
504C4Q8×C22⋊D1112yes—
505C4C2×#6700214yes—
506C4C22×Q8×D1111yescharacteristic
507C4C2×#6704011yescharacteristic
508C2—444no—
509C2#1308153314yesdirect product factor
510C2C2×Q8×C22⋊D1111yescharacteristic
511C2C22×#6700211yescharacteristic
512C2C2×#1062336911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingC2×C11×#557C2
socleC23×C11C22×D4
prime radicalC22×D44C23
hypercenterC2×#557D11
hypocenterC11C2×#3386
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #13081533C2direct (N × H)
3. C2×#10622678C2semidirect
4. C2×#10622691C2semidirect
5. C2×#10622634C2semidirect
6. C2×#10622633C2semidirect
9. C2×#10622637C2semidirect
12. C2×C11×#557C2semidirect
15. C11×#557C22semidirect
36. #10622691C22semidirect
38. #10622678C22semidirect
41. #10622637C22semidirect
42. #10622634C22semidirect
43. #10622633C22semidirect
91. C2×#4325Q8semidirect
133. C2×Q8×C11C2×Q8semidirect
175. #4325C2×Q8semidirect
186. C2×C11⋊C8C2×Q8semidirect
190. C2×#557D11semidirect
211. Q8×C11C22×Q8semidirect
243. C11⋊C8C22×Q8semidirect
247. #557C2×D11semidirect
376. C2×C11#3386semidirect
392. C2×Q8Q8×D11semidirect
445. C11C2×#3386semidirect
457. Q8C2×Q8×D11semidirect
509. C2#13081533direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D442
#130815332
C2×#106226782
C2×#106226912
C2×#106226342
C2×#106226332
C2×#106226792
C2×#106227002
C2×#106226372
C2×#106226392
C2×C4×#43252
C2×C11×#5572
C2×#338611