Finite Group Databasebeta

← C2×#19145786 #25968703

Subgroups — C2×#19145786 #25968703

Order 11008.

2050 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
1C2×#14583450C211yesmaximal, characteristic
2C2×#125×D43C211yesmaximal, characteristic, complemented
3#19145786C2116yesmaximal, direct product factor
4C2×#14583305C211yesmaximal, characteristic, complemented
5C2×#14583265C211yesmaximal, characteristic, complemented
6C2×#14583264C211yesmaximal, characteristic, complemented
7C2×#14583219C212yesmaximal, complemented
8C2×#14583214C212yesmaximal, complemented
9C2×#14583202C212yesmaximal, complemented
10C2×#14583198C212yesmaximal, complemented
11C2×#14582954C211yesmaximal, characteristic, complemented
12C2×C43×#501C211yesmaximal, characteristic, complemented
13C22×#10621902—21no—
14#14583450—24no—
15#14583450C2218yes—
16C2×#13063322C2211yescharacteristic
17C2×#13063317C2211yescharacteristic
18C2×#13063316C2211yescharacteristic
19C2×#13063316C2211yescharacteristic
20C2×#13063315—21no—
21C2×#13063315—21no—
22#125×D43—24no—
23#125×D43C2218yescomplemented
24C2×#13063314—21no—
25C2×#13063300—21no—
26C2×#13063300C2212yes—
27C2×#13063287—21no—
28C2×#13063287C2212yes—
29C2×C4⋊C4×D43—21no—
30C2×C4⋊C4×D43—21no—
31C2×C4⋊C4×D43C2211yescharacteristic
32C2×C4⋊C4×D43C2211yescharacteristic
33C2×C42×D43C2211yescharacteristic
34#14583305C2218yescomplemented
35#14583265C2218yescomplemented
36#14583264C2218yescomplemented
37#14583219C22116yescomplemented
38#14583214C22116yescomplemented
39#14583202C22116yescomplemented
40#14583198C22116yescomplemented
41#14582954C2218yescomplemented
42C2×#13063258C2212yes—
43C2×#13063256C2212yes—
44C2×Dic43⋊C8C2212yes—
45C2×#13063247C2212yes—
46C2×#13063247C2212yes—
47C2×#13063245C2212yes—
48C2×#13063245C2212yes—
49C2×(C2×D43)⋊C8C2212yes—
50C2×#13063236C2211yescharacteristic
51C2×C4⋊Q8×C43C2211yescharacteristic
52C2×#125×C43C2211yescharacteristic
53C43×#501C2218yescomplemented
54C2×#107×C43C2212yes—
55C2×#106×C43C2212yes—
56C2×#97×C43C2211yescharacteristic
57C2×#10621902—28no—
58C2×#10621902—48no—
59C22×D172—22no—
60C22×D172—21no—
61C22×D172—21no—
62C22×D172C2312yes—
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C22×C4×D43—22no—
66C22×C4×D43D411yescharacteristic
67C22×C4×D43D411yescharacteristic
68C22×C4×D43C2311yescharacteristic
69C22×C43⋊C8C2312yes—
70#13063322—22no—
71#13063322C2314yes—
72#13063317—22no—
73#13063317C2314yes—
74#13063316—22no—
75#13063316—22no—
76#13063316C2314yes—
77#13063316C2314yes—
78#13063315—28no—
79#13063315—28no—
80#13063314—28no—
81#13063300—28no—
82#13063300—24no—
83#13063300C2318yes—
84#13063287—28no—
85#13063287—24no—
86#13063287C2318yes—
87C2×C4⋊Dic43—21no—
88C2×C4⋊Dic43—21no—
89C2×C4⋊Dic43C2311yescharacteristic
90C2×C4⋊Dic43C2311yescharacteristic
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C2×Dic43⋊C4—21no—
96C2×Dic43⋊C4—21no—
97C4⋊C4×D43—28no—
98C4⋊C4×D43—28no—
99C4⋊C4×D43—22no—
100C4⋊C4×D43—22no—
101C4⋊C4×D43C2314yes—
102C4⋊C4×D43C2314yes—
103C2×(C2×D43)⋊C4—22no—
104C2×(C2×D43)⋊C4—22no—
105C2×C4×Dic43D411yescharacteristic
106C2×C4×Dic43D411yescharacteristic
107C2×C4×Dic43C2311yescharacteristic
108C42×D43D414yes—
109C42×D43C2314yes—
110#13063258C2318yes—
111#13063256C2318yes—
112Dic43⋊C8C2318yes—
113#13063247C2318yes—
114#13063247C2318yes—
115#13063245C2318yes—
116#13063245C2318yes—
117(C2×D43)⋊C8C2318yes—
118#13063236C2314yes—
119C22×Q8×C43—21no—
120C22×C8×C43C2312yes—
121C4⋊Q8×C43—22no—
122C4⋊Q8×C43C2314yes—
123#125×C43—22no—
124#125×C43C2314yes—
125C2×C4⋊C4×C43—21no—
126C2×C4⋊C4×C43—21no—
127C2×C4⋊C4×C43—21no—
128C2×C4⋊C4×C43C2311yescharacteristic
129C2×C4⋊C4×C43C2311yescharacteristic
130C2×C4⋊C4×C43C2312yes—
131C2×C42×C43C2311yescharacteristic
132#107×C43C2318yes—
133#106×C43C2318yes—
134#97×C43C2314yes—
135C23×D43—22no—
136C23×D43C2×D411yescharacteristic
137C2×Q8×C43—24no—
138C2×Q8×C43—44no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—22no—
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2411yescharacteristic
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43C2×D411yescharacteristic
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150#10621902—432no—
151C2×D172—22no—
152C2×D172—24no—
153C2×D172—24no—
154C2×D172—28no—
155C2×D172—28no—
156C2×D172—44no—
157C2×D172—44no—
158C2×D172—48no—
159C2×D172C2414yes—
160C2×C4×D43—24no—
161C2×C4×D43—28no—
162C2×C4×D43—24no—
163C2×C4×D43—24no—
164C2×C4×D43—24no—
165C2×C4×D43—42no—
166C2×C4×D43—44no—
167C2×C4×D43—48no—
168C2×C4×D43—44no—
169C2×C4×D43C2×D412yes—
170C2×C4×D43C2×D414yes—
171C2×C4×D43C2×D414yes—
172C2×C4×D43C2412yes—
173C2×C8×C43—24no—
174C2×C8×C43C2414yes—
175C4⋊C4×C43—24no—
176C4⋊C4×C43—24no—
177C4⋊C4×C43—24no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—22no—
181C4⋊C4×C43C2412yes—
182C4⋊C4×C43C2414yes—
183C4⋊C4×C43C2412yes—
184C42×C43C2×D412yes—
185C42×C43C2412yes—
186(C2×D43)⋊C4—28no—
187(C2×D43)⋊C4—28no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—24no—
191C4⋊Dic43—24no—
192C4⋊Dic43C2412yes—
193C4⋊Dic43C2412yes—
194Dic43⋊C4—24no—
195Dic43⋊C4—24no—
196Dic43⋊C4—24no—
197Dic43⋊C4—24no—
198Dic43⋊C4—24no—
199Dic43⋊C4—24no—
200C4×Dic43C2×D412yes—
201C4×Dic43C2×D414yes—
202C4×Dic43C2×D414yes—
203C4×Dic43C2412yes—
204C2×C43⋊C8—24no—
205C2×C43⋊C8C2414yes—
206C23×C43C22×D411yescharacteristic
207C22×D43—22no—
208C22×D43—22no—
209C22×D43—24no—
210C22×D43—44no—
211C22×D43—44no—
212C22×D43—44no—
213C22×D43AGL(1,ℤ/8)14yes—
214C22×D43C22×D412yes—
215Q8×C43—48no—
216C2×C4×C43—22no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—22no—
220C2×C4×C43—24no—
221C2×C4×C43—44no—
222C2×C4×C43—42no—
223C2×C4×C43—42no—
224C2×C4×C43C22×D412yes—
225C2×C4×C43C22×D412yes—
226C2×C4×C43C22×D411yescharacteristic
227C2×C4×C432−1+414yes—
228C2×C4×C43C2511yescharacteristic
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43C22×D412yes—
237C2×Dic43C22×D412yes—
238C2×Dic43C22×D412yes—
239D172—28no—
240D172—48no—
241D172—48no—
242D172—416no—
243D172—44no—
244C4×D43—28no—
245C4×D43—24no—
246C4×D43—24no—
247C4×D43—44no—
248C4×D43—416no—
249C4×D43—48no—
250C4×D43—48no—
251C8×C43—24no—
252C43⋊C8—24no—
253C2×#3234—431nomaximal, Sylow
254C22×C43#53612yes—
255C22×C43C2×AGL(1,ℤ/8)12yes—
256C22×C43#57512yes—
257C22×C43C23×D411yescharacteristic
258C2×D43—22no—
259C2×D43—28no—
260C2×D43—22no—
261C2×D43—44no—
262C2×D43—44no—
263C2×D43—44no—
264C2×D43—88no—
265C2×D43C2×AGL(1,ℤ/8)14yes—
266C4×C43—22no—
267C4×C43—22no—
268C4×C43—44no—
269C4×C43—42no—
270C4×C43—42no—
271C4×C43#53612yes—
272C4×C43C2×2−1+412yes—
273Dic43—22no—
274Dic43—22no—
275Dic43—22no—
276Dic43—42no—
277Dic43—42no—
278C2×#532—431no—
279C22×#125—431no—
280#3234—4316no—
281C2×#501—431no—
282C2×#501D4311yescharacteristic, complemented
283C2×#490—431no—
284C2×#490—431no—
285C2×#481—432no—
286C2×#480—432no—
287C2×#478—432no—
288C2×#478—432no—
289C2×#432—431no—
290C2×C43#323414yescomplemented
291C2×C43C2×#53611yescharacteristic
292C2×C43C22×AGL(1,ℤ/8)11yescharacteristic
293C2×C43C2×#57511yescharacteristic
294D43—24no—
295D43—84no—
296C22×#41—861no—
297#532—438no—
298#532—864no—
299C2×C4⋊Q8C2×D4311yescharacteristic
300C2×#127—431no—
301C2×#125—438no—
302C2×#125—431no—
303C2×#125—431no—
304C2×#125—431no—
305C2×#125—861no—
306C2×#125—861no—
307C2×#125—864no—
308C2×#125C2×D4311yescharacteristic
309C2×#124—861no—
310C2×#121—432no—
311C2×#121—861no—
312C2×C4×D4—432no—
313C2×C4×D4—861no—
314C22×C4⋊C4—431no—
315C22×C4⋊C4—431no—
316C22×C4⋊C4—861no—
317C22×C4⋊C4—861no—
318C22×C42—431no—
319#501—438no—
320#501C2×D4318yescomplemented
321#490—438no—
322#490—438no—
323#481—4316no—
324#480—4316no—
325#478—4316no—
326#478—4316no—
327#432—438no—
328C2×#107—432no—
329C2×#107C2×D4312yes—
330C2×#106—432no—
331C2×#106C2×D4312yes—
332C2×C4⋊C8—432no—
333C2×#102—432no—
334C2×#102—432no—
335C2×#102—432no—
336C2×#102—432no—
337C2×C22⋊C8—432no—
338C2×#97—431no—
339C2×#97C2×D4311yescharacteristic
340C43C2×#323411yescharacteristic, complemented, Sylow
341C2×#41—868no—
342C2×#41—1728no—
343C22×Q8—21no—
344C22×D4—432no—
345C22×D4—862no—
346C22×D4—861no—
347C22×D4—861no—
348C23×C4—431no—
349C23×C4—431no—
350C23×C4—431no—
351C23×C4—862no—
352C23×C4—861no—
353C23×C4—861no—
354C22×C8—432no—
355C22×C8C22×D4312yes—
356C4⋊Q8—22no—
357C4⋊Q8C22×D4314yes—
358#127—434no—
359#127—862no—
360#125—22no—
361#125—434no—
362#125—434no—
363#125—434no—
364#125—868no—
365#125—868no—
366#125—862no—
367#125—862no—
368#125—862no—
369#125C22×D4314yes—
370#124—868no—
371#121—438no—
372#121—864no—
373#121—868no—
374C4×D4—438no—
375C4×D4—864no—
376C4×D4—868no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—431no—
381C2×C4⋊C4—431no—
382C2×C4⋊C4—434no—
383C2×C4⋊C4—434no—
384C2×C4⋊C4—862no—
385C2×C4⋊C4—862no—
386C2×C4⋊C4—868no—
387C2×C4⋊C4—868no—
388C2×C4⋊C4—861no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4—861no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—861no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4C22×D4311yescharacteristic
397C2×C4⋊C4C22×D4311yescharacteristic
398C2×C4⋊C4C22×D4312yes—
399C2×C22⋊C4—862no—
400C2×C22⋊C4—862no—
401C2×C42—431no—
402C2×C42—431no—
403C2×C42—431no—
404C2×C42—434no—
405C2×C42—434no—
406C2×C42C22×D4311yescharacteristic
407#107—438no—
408#107C22×D4318yes—
409#106—438no—
410#106C22×D4318yes—
411C4⋊C8—438no—
412#102—438no—
413#102—438no—
414#102—438no—
415#102—438no—
416C22⋊C8—438no—
417#97—434no—
418#97C22×D4314yes—
419C24—431no—
420C24—862no—
421#41—17232no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×D4—434no—
425C2×D4—868no—
426C2×D4—864no—
427C2×D4—862no—
428C2×D4—864no—
429C2×D4—868no—
430C2×D4—1724no—
431C2×D4—1724no—
432C2×D4—1728no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—431no—
437C22×C4—431no—
438C22×C4—431no—
439C22×C4—432no—
440C22×C4—434no—
441C22×C4—432no—
442C22×C4—434no—
443C22×C4—868no—
444C22×C4—864no—
445C22×C4—864no—
446C22×C4—864no—
447C22×C4—864no—
448C22×C4—861no—
449C22×C4—861no—
450C22×C4—1722no—
451C22×C4—1728no—
452C22×C4—1724no—
453C22×C4—1724no—
454C22×C4D4×D4311yescharacteristic
455C22×C4D4×D4311yescharacteristic
456C22×C4C23×D4311yescharacteristic
457C2×C8—24no—
458C2×C8—434no—
459C2×C8—864no—
460C2×C8C23×D4314yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—432no—
468C4⋊C4—432no—
469C4⋊C4—864no—
470C4⋊C4—864no—
471C4⋊C4—864no—
472C4⋊C4—864no—
473C4⋊C4—864no—
474C4⋊C4—864no—
475C4⋊C4—861no—
476C4⋊C4—861no—
477C4⋊C4—864no—
478C4⋊C4—864no—
479C4⋊C4C23×D4312yes—
480C4⋊C4C23×D4312yes—
481C4⋊C4C23×D4314yes—
482C22⋊C4—868no—
483C22⋊C4—868no—
484C42—434no—
485C42—432no—
486C42—434no—
487C42—432no—
488C42D4×D4312yes—
489C42C23×D4312yes—
490C23—434no—
491C23—432no—
492C23—862no—
493C23—862no—
494C23—864no—
495C23—1724no—
496C23—1724no—
497C23—1724no—
498C23C2×D4×D4311yescharacteristic
499Q8—48no—
500D4—868no—
501D4—1728no—
502D4—1728no—
503D4—17216no—
504D4—1724no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—432no—
514C2×C4—432no—
515C2×C4—432no—
516C2×C4—868no—
517C2×C4—862no—
518C2×C4—862no—
519C2×C4—862no—
520C2×C4—864no—
521C2×C4—864no—
522C2×C4—862no—
523C2×C4—862no—
524C2×C4—1724no—
525C2×C4—17216no—
526C2×C4—1728no—
527C2×C4—1728no—
528C2×C4—1722no—
529C2×C4—1722no—
530C2×C4C2×D4×D4312yes—
531C2×C4C2×D4×D4312yes—
532C2×C4C2×D4×D4311yescharacteristic
533C2×C4#1306337314yes—
534C2×C4C24×D4311yescharacteristic
535C8—24no—
536C8—864no—
537C22—434no—
538C22—862no—
539C22—868no—
540C22—862no—
541C22—3448no—
542C22—1724no—
543C22—1724no—
544C22—1724no—
545C22#1458347412yes—
546C22AGL(1,ℤ/8)×D4312yes—
547C22#1458369812yes—
548C22C22×D4×D4311yescharacteristic
549C4—22no—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—862no—
555C4—862no—
556C4—862no—
557C4—1722no—
558C4—1722no—
559C4#1458347412yes—
560C4C2×#1306337312yes—
561C2—864no—
562C2—3444no—
563C2#1914578614yesdirect product factor
564C2C2×#1458347411yescharacteristic
565C2C2×AGL(1,ℤ/8)×D4311yescharacteristic
566C2C2×#1458369811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC2×C4×C43C25
FrattiniC2×C4C24×D43
FittingC2×C43×#501C2
socleC23×C43C22×D4
prime radicalC2×#14583450C2
hypercenterC2×#501D43
hypocenterC43C2×#3234
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#125×D43C2semidirect
3. #19145786C2direct (N × H)
4. C2×#14583305C2semidirect
5. C2×#14583265C2semidirect
6. C2×#14583264C2semidirect
7. C2×#14583219C2semidirect
8. C2×#14583214C2semidirect
9. C2×#14583202C2semidirect
10. C2×#14583198C2semidirect
11. C2×#14582954C2semidirect
12. C2×C43×#501C2semidirect
23. #125×D43C22semidirect
34. #14583305C22semidirect
35. #14583265C22semidirect
36. #14583264C22semidirect
37. #14583219C22semidirect
38. #14583214C22semidirect
39. #14583202C22semidirect
40. #14583198C22semidirect
41. #14582954C22semidirect
53. C43×#501C22semidirect
282. C2×#501D43semidirect
290. C2×C43#3234semidirect
320. #501C2×D43semidirect
340. C43C2×#3234semidirect
563. C2#19145786direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145834502
C2×#125×D432
#191457862
C2×#145833052
C2×#145832652
C2×#145832642
C2×#145832192
C2×#145832142
C2×#145832022
C2×#145831982
C2×#145829542
C2×C43×#5012
C2×#323443