Finite Group Databasebeta

← C2×#24918654 #36276510

Subgroups — C2×#24918654 #36276510

Order 21248.

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×#19128252C211yesmaximal, characteristic
2C2×#125×D83C211yesmaximal, characteristic, complemented
3#24918654C2116yesmaximal, direct product factor
4C2×#19128107C211yesmaximal, characteristic, complemented
5C2×#19128067C211yesmaximal, characteristic, complemented
6C2×#19128066C211yesmaximal, characteristic, complemented
7C2×#19128021C212yesmaximal, complemented
8C2×#19128016C212yesmaximal, complemented
9C2×#19128004C212yesmaximal, complemented
10C2×#19128000C212yesmaximal, complemented
11C2×#19127756C211yesmaximal, characteristic, complemented
12C2×#501×C83C211yesmaximal, characteristic, complemented
13C22×#13050591—21no—
14#19128252—24no—
15#19128252C2218yes—
16C2×#14581789C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581783C2211yescharacteristic
19C2×#14581783C2211yescharacteristic
20C2×#14581782—21no—
21C2×#14581782—21no—
22#125×D83—24no—
23#125×D83C2218yescomplemented
24C2×#14581781—21no—
25C2×#14581767—21no—
26C2×#14581767C2212yes—
27C2×#14581754—21no—
28C2×#14581754C2212yes—
29C2×C4⋊C4×D83—21no—
30C2×C4⋊C4×D83—21no—
31C2×C4⋊C4×D83C2211yescharacteristic
32C2×C4⋊C4×D83C2211yescharacteristic
33C2×C42×D83C2211yescharacteristic
34#19128107C2218yescomplemented
35#19128067C2218yescomplemented
36#19128066C2218yescomplemented
37#19128021C22116yescomplemented
38#19128016C22116yescomplemented
39#19128004C22116yescomplemented
40#19128000C22116yescomplemented
41#19127756C2218yescomplemented
42C2×#14581725C2212yes—
43C2×#14581723C2212yes—
44C2×Dic83⋊C8C2212yes—
45C2×#14581714C2212yes—
46C2×#14581714C2212yes—
47C2×#14581712C2212yes—
48C2×#14581712C2212yes—
49C2×(C2×D83)⋊C8C2212yes—
50C2×#14581703C2211yescharacteristic
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×#125×C83C2211yescharacteristic
53#501×C83C2218yescomplemented
54C2×#107×C83C2212yes—
55C2×#106×C83C2212yes—
56C2×#97×C83C2211yescharacteristic
57C2×#13050591—28no—
58C2×#13050591—48no—
59C22×D332—22no—
60C22×D332—21no—
61C22×D332—21no—
62C22×D332C2312yes—
63C22×C4×D83—21no—
64C22×C4×D83—21no—
65C22×C4×D83—22no—
66C22×C4×D83D411yescharacteristic
67C22×C4×D83D411yescharacteristic
68C22×C4×D83C2311yescharacteristic
69C22×C83⋊C8C2312yes—
70#14581789—22no—
71#14581789C2314yes—
72#14581784—22no—
73#14581784C2314yes—
74#14581783—22no—
75#14581783—22no—
76#14581783C2314yes—
77#14581783C2314yes—
78#14581782—28no—
79#14581782—28no—
80#14581781—28no—
81#14581767—28no—
82#14581767—24no—
83#14581767C2318yes—
84#14581754—28no—
85#14581754—24no—
86#14581754C2318yes—
87C2×C4⋊Dic83—21no—
88C2×C4⋊Dic83—21no—
89C2×C4⋊Dic83C2311yescharacteristic
90C2×C4⋊Dic83C2311yescharacteristic
91C2×Dic83⋊C4—21no—
92C2×Dic83⋊C4—21no—
93C2×Dic83⋊C4—21no—
94C2×Dic83⋊C4—21no—
95C2×Dic83⋊C4—21no—
96C2×Dic83⋊C4—21no—
97C4⋊C4×D83—28no—
98C4⋊C4×D83—28no—
99C4⋊C4×D83—22no—
100C4⋊C4×D83—22no—
101C4⋊C4×D83C2314yes—
102C4⋊C4×D83C2314yes—
103C2×(C2×D83)⋊C4—22no—
104C2×(C2×D83)⋊C4—22no—
105C2×C4×Dic83D411yescharacteristic
106C2×C4×Dic83D411yescharacteristic
107C2×C4×Dic83C2311yescharacteristic
108C42×D83D414yes—
109C42×D83C2314yes—
110#14581725C2318yes—
111#14581723C2318yes—
112Dic83⋊C8C2318yes—
113#14581714C2318yes—
114#14581714C2318yes—
115#14581712C2318yes—
116#14581712C2318yes—
117(C2×D83)⋊C8C2318yes—
118#14581703C2314yes—
119C22×Q8×C83—21no—
120C22×C8×C83C2312yes—
121C4⋊Q8×C83—22no—
122C4⋊Q8×C83C2314yes—
123#125×C83—22no—
124#125×C83C2314yes—
125C2×C4⋊C4×C83—21no—
126C2×C4⋊C4×C83—21no—
127C2×C4⋊C4×C83—21no—
128C2×C4⋊C4×C83C2311yescharacteristic
129C2×C4⋊C4×C83C2311yescharacteristic
130C2×C4⋊C4×C83C2312yes—
131C2×C42×C83C2311yescharacteristic
132#107×C83C2318yes—
133#106×C83C2318yes—
134#97×C83C2314yes—
135C23×D83—22no—
136C23×D83C2×D411yescharacteristic
137#13050591—432no—
138C2×D332—28no—
139C2×D332—28no—
140C2×D332—22no—
141C2×D332—24no—
142C2×D332—24no—
143C2×D332—44no—
144C2×D332—44no—
145C2×D332—48no—
146C2×D332C2414yes—
147C22×Dic83—21no—
148C22×Dic83—21no—
149C22×Dic83C2×D411yescharacteristic
150C22×Dic83C2×D411yescharacteristic
151C22×Dic83C2×D411yescharacteristic
152C2×C4×D83—24no—
153C2×C4×D83—24no—
154C2×C4×D83—24no—
155C2×C4×D83—24no—
156C2×C4×D83—28no—
157C2×C4×D83—42no—
158C2×C4×D83—48no—
159C2×C4×D83—44no—
160C2×C4×D83—44no—
161C2×C4×D83C2×D412yes—
162C2×C4×D83C2×D414yes—
163C2×C4×D83C2×D414yes—
164C2×C4×D83C2412yes—
165C2×C83⋊C8—24no—
166C2×C83⋊C8C2414yes—
167C4⋊Dic83—21no—
168C4⋊Dic83—21no—
169C4⋊Dic83—24no—
170C4⋊Dic83—24no—
171C4⋊Dic83C2412yes—
172C4⋊Dic83C2412yes—
173Dic83⋊C4—24no—
174Dic83⋊C4—24no—
175Dic83⋊C4—24no—
176Dic83⋊C4—24no—
177Dic83⋊C4—24no—
178Dic83⋊C4—24no—
179(C2×D83)⋊C4—28no—
180(C2×D83)⋊C4—28no—
181C4×Dic83C2×D414yes—
182C4×Dic83C2×D414yes—
183C4×Dic83C2×D412yes—
184C4×Dic83C2412yes—
185C2×Q8×C83—24no—
186C2×Q8×C83—44no—
187C22×C4×C83—22no—
188C22×C4×C83—21no—
189C22×C4×C83—21no—
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2411yescharacteristic
193C2×C8×C83—24no—
194C2×C8×C83C2414yes—
195C4⋊C4×C83—24no—
196C4⋊C4×C83—22no—
197C4⋊C4×C83—24no—
198C4⋊C4×C83—24no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83C2414yes—
202C4⋊C4×C83C2412yes—
203C4⋊C4×C83C2412yes—
204C42×C83C2×D412yes—
205C42×C83C2412yes—
206C23×C83C22×D411yescharacteristic
207C22×D83—22no—
208C22×D83—22no—
209C22×D83—24no—
210C22×D83—44no—
211C22×D83—44no—
212C22×D83—44no—
213C22×D83AGL(1,ℤ/8)14yes—
214C22×D83C22×D412yes—
215Q8×C83—48no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—24no—
221C2×C4×C83—44no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83C22×D412yes—
225C2×C4×C83C22×D412yes—
226C2×C4×C83C22×D411yescharacteristic
227C2×C4×C832−1+414yes—
228C2×C4×C83C2511yescharacteristic
229C2×Dic83—22no—
230C2×Dic83—22no—
231C2×Dic83—22no—
232C2×Dic83—22no—
233C2×Dic83—22no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83C22×D412yes—
237C2×Dic83C22×D412yes—
238C2×Dic83C22×D412yes—
239D332—28no—
240D332—48no—
241D332—48no—
242D332—416no—
243D332—44no—
244C4×D83—28no—
245C4×D83—24no—
246C4×D83—24no—
247C4×D83—44no—
248C4×D83—416no—
249C4×D83—48no—
250C4×D83—48no—
251C8×C83—24no—
252C83⋊C8—24no—
253C22×C83#53612yes—
254C22×C83C2×AGL(1,ℤ/8)12yes—
255C22×C83#57512yes—
256C22×C83C23×D411yescharacteristic
257C2×D83—22no—
258C2×D83—28no—
259C2×D83—22no—
260C2×D83—44no—
261C2×D83—44no—
262C2×D83—44no—
263C2×D83—88no—
264C2×D83C2×AGL(1,ℤ/8)14yes—
265C4×C83—22no—
266C4×C83—22no—
267C4×C83—44no—
268C4×C83—42no—
269C4×C83—42no—
270C4×C83#53612yes—
271C4×C83C2×2−1+412yes—
272Dic83—22no—
273Dic83—22no—
274Dic83—22no—
275Dic83—42no—
276Dic83—42no—
277C2×#3234—831nomaximal, Sylow
278C2×C83#323414yescomplemented
279C2×C83C2×#53611yescharacteristic
280C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
281C2×C83C2×#57511yescharacteristic
282D83—24no—
283D83—84no—
284C2×#532—831no—
285C22×#125—831no—
286#3234—8316no—
287C2×#501—831no—
288C2×#501D8311yescharacteristic, complemented
289C2×#490—831no—
290C2×#490—831no—
291C2×#481—832no—
292C2×#480—832no—
293C2×#478—832no—
294C2×#478—832no—
295C2×#432—831no—
296C83C2×#323411yescharacteristic, complemented, Sylow
297C22×#41—1661no—
298#532—838no—
299#532—1664no—
300C2×C4⋊Q8C2×D8311yescharacteristic
301C2×#127—831no—
302C2×#125—838no—
303C2×#125—831no—
304C2×#125—831no—
305C2×#125—831no—
306C2×#125—1661no—
307C2×#125—1661no—
308C2×#125—1664no—
309C2×#125C2×D8311yescharacteristic
310C2×#124—1661no—
311C2×#121—832no—
312C2×#121—1661no—
313C2×C4×D4—832no—
314C2×C4×D4—1661no—
315C22×C4⋊C4—831no—
316C22×C4⋊C4—831no—
317C22×C4⋊C4—1661no—
318C22×C4⋊C4—1661no—
319C22×C42—831no—
320#501—838no—
321#501C2×D8318yescomplemented
322#490—838no—
323#490—838no—
324#481—8316no—
325#480—8316no—
326#478—8316no—
327#478—8316no—
328#432—838no—
329C2×#107—832no—
330C2×#107C2×D8312yes—
331C2×#106—832no—
332C2×#106C2×D8312yes—
333C2×C4⋊C8—832no—
334C2×#102—832no—
335C2×#102—832no—
336C2×#102—832no—
337C2×#102—832no—
338C2×C22⋊C8—832no—
339C2×#97—831no—
340C2×#97C2×D8311yescharacteristic
341C2×#41—3328no—
342C2×#41—1668no—
343C22×Q8—21no—
344C22×D4—832no—
345C22×D4—1662no—
346C22×D4—1661no—
347C22×D4—1661no—
348C23×C4—831no—
349C23×C4—831no—
350C23×C4—831no—
351C23×C4—1662no—
352C23×C4—1661no—
353C23×C4—1661no—
354C22×C8—832no—
355C22×C8C22×D8312yes—
356C4⋊Q8—22no—
357C4⋊Q8C22×D8314yes—
358#127—834no—
359#127—1662no—
360#125—22no—
361#125—834no—
362#125—834no—
363#125—834no—
364#125—1668no—
365#125—1668no—
366#125—1662no—
367#125—1662no—
368#125—1662no—
369#125C22×D8314yes—
370#124—1668no—
371#121—838no—
372#121—1664no—
373#121—1668no—
374C4×D4—838no—
375C4×D4—1664no—
376C4×D4—1668no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—831no—
382C2×C4⋊C4—834no—
383C2×C4⋊C4—834no—
384C2×C4⋊C4—1662no—
385C2×C4⋊C4—1662no—
386C2×C4⋊C4—1668no—
387C2×C4⋊C4—1668no—
388C2×C4⋊C4—1661no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8312yes—
399C2×C22⋊C4—1662no—
400C2×C22⋊C4—1662no—
401C2×C42—831no—
402C2×C42—831no—
403C2×C42—831no—
404C2×C42—834no—
405C2×C42—834no—
406C2×C42C22×D8311yescharacteristic
407#107—838no—
408#107C22×D8318yes—
409#106—838no—
410#106C22×D8318yes—
411C4⋊C8—838no—
412#102—838no—
413#102—838no—
414#102—838no—
415#102—838no—
416C22⋊C8—838no—
417#97—834no—
418#97C22×D8314yes—
419C24—831no—
420C24—1662no—
421#41—33232no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×D4—3324no—
425C2×D4—3324no—
426C2×D4—3328no—
427C2×D4—834no—
428C2×D4—1668no—
429C2×D4—1664no—
430C2×D4—1662no—
431C2×D4—1664no—
432C2×D4—1668no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—3322no—
437C22×C4—3324no—
438C22×C4—3324no—
439C22×C4—3328no—
440C22×C4—834no—
441C22×C4—831no—
442C22×C4—831no—
443C22×C4—831no—
444C22×C4—832no—
445C22×C4—834no—
446C22×C4—832no—
447C22×C4—1668no—
448C22×C4—1664no—
449C22×C4—1664no—
450C22×C4—1664no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4—1664no—
454C22×C4D4×D8311yescharacteristic
455C22×C4D4×D8311yescharacteristic
456C22×C4C23×D8311yescharacteristic
457C2×C8—24no—
458C2×C8—834no—
459C2×C8—1664no—
460C2×C8C23×D8314yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—832no—
468C4⋊C4—832no—
469C4⋊C4—1664no—
470C4⋊C4—1664no—
471C4⋊C4—1664no—
472C4⋊C4—1664no—
473C4⋊C4—1664no—
474C4⋊C4—1664no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1664no—
478C4⋊C4—1664no—
479C4⋊C4C23×D8312yes—
480C4⋊C4C23×D8312yes—
481C4⋊C4C23×D8314yes—
482C22⋊C4—1668no—
483C22⋊C4—1668no—
484C42—834no—
485C42—832no—
486C42—834no—
487C42—832no—
488C42D4×D8312yes—
489C42C23×D8312yes—
490C23—3324no—
491C23—3324no—
492C23—3324no—
493C23—834no—
494C23—832no—
495C23—1664no—
496C23—1662no—
497C23—1662no—
498C23C2×D4×D8311yescharacteristic
499Q8—48no—
500D4—3328no—
501D4—3328no—
502D4—33216no—
503D4—3324no—
504D4—1668no—
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—3324no—
514C2×C4—3328no—
515C2×C4—3328no—
516C2×C4—33216no—
517C2×C4—3322no—
518C2×C4—3322no—
519C2×C4—832no—
520C2×C4—832no—
521C2×C4—832no—
522C2×C4—1668no—
523C2×C4—1662no—
524C2×C4—1662no—
525C2×C4—1662no—
526C2×C4—1662no—
527C2×C4—1662no—
528C2×C4—1664no—
529C2×C4—1664no—
530C2×C4C2×D4×D8312yes—
531C2×C4C2×D4×D8312yes—
532C2×C4C2×D4×D8311yescharacteristic
533C2×C4#1458184014yes—
534C2×C4C24×D8311yescharacteristic
535C8—24no—
536C8—1664no—
537C22—3324no—
538C22—3324no—
539C22—3324no—
540C22—834no—
541C22—6648no—
542C22—1662no—
543C22—1668no—
544C22—1662no—
545C22#1912827612yes—
546C22AGL(1,ℤ/8)×D8312yes—
547C22#1912850012yes—
548C22C22×D4×D8311yescharacteristic
549C4—22no—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—3322no—
555C4—3322no—
556C4—1662no—
557C4—1662no—
558C4—1662no—
559C4#1912827612yes—
560C4C2×#1458184012yes—
561C2—6644no—
562C2—1664no—
563C2#2491865414yesdirect product factor
564C2C2×#1912827611yescharacteristic
565C2C2×AGL(1,ℤ/8)×D8311yescharacteristic
566C2C2×#1912850011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#501×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128252C2
hypercenterC2×#501D83
hypocenterC83C2×#3234
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#125×D83C2semidirect
3. #24918654C2direct (N × H)
4. C2×#19128107C2semidirect
5. C2×#19128067C2semidirect
6. C2×#19128066C2semidirect
7. C2×#19128021C2semidirect
8. C2×#19128016C2semidirect
9. C2×#19128004C2semidirect
10. C2×#19128000C2semidirect
11. C2×#19127756C2semidirect
12. C2×#501×C83C2semidirect
23. #125×D83C22semidirect
34. #19128107C22semidirect
35. #19128067C22semidirect
36. #19128066C22semidirect
37. #19128021C22semidirect
38. #19128016C22semidirect
39. #19128004C22semidirect
40. #19128000C22semidirect
41. #19127756C22semidirect
53. #501×C83C22semidirect
278. C2×C83#3234semidirect
288. C2×#501D83semidirect
296. C83C2×#3234semidirect
321. #501C2×D83semidirect
563. C2#24918654direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282522
C2×#125×D832
#249186542
C2×#191281072
C2×#191280672
C2×#191280662
C2×#191280212
C2×#191280162
C2×#191280042
C2×#191280002
C2×#191277562
C2×#501×C832
C2×#323483