Finite Group Databasebeta

← C2×#19133910 #25077729

Subgroups — C2×#19133910 #25077729

Order 11008.

1022 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583647C211yesmaximal, characteristic
2C2×#14583291C211yesmaximal, characteristic, complemented
3C2×#14583261C211yesmaximal, characteristic, complemented
4#19133910C218yesmaximal, direct product factor
5C2×#14582641C211yesmaximal, characteristic, complemented
6C2×#14582638C211yesmaximal, characteristic, complemented
7C2×#14582638C211yesmaximal, characteristic, complemented
8C2×C43×#501C211yesmaximal, characteristic, complemented
9C22×#10621885—21no—
10C22×#10621884—21no—
11#14583647—22no—
12#14583647—22no—
13#14583647—22no—
14#14583647C2214yes—
15C2×#13063313C2211yescharacteristic
16C2×#13063313C2211yescharacteristic
17C2×#13063310—21no—
18C2×#13063310—21no—
19C2×#13063303—21no—
20C2×#13063303—21no—
21C2×#13063300—21no—
22C2×#13063300—21no—
23#14583291—22no—
24#14583291C2214yescomplemented
25#14583261—22no—
26#14583261C2214yescomplemented
27C2×#13063248—21no—
28C2×#13063245—21no—
29C2×#13063236C2211yescharacteristic
30C2×#13063236C2211yescharacteristic
31#14582641C2214yescomplemented
32#14582638C2214yescomplemented
33#14582638C2214yescomplemented
34C2×C4⋊Q8×C43C2211yescharacteristic
35C2×#125×C43C2211yescharacteristic
36C43×#501—22no—
37C43×#501C2214yescomplemented
38C2×#107×C43—21no—
39C2×#106×C43—21no—
40C2×#97×C43C2211yescharacteristic
41C22×Dic86D411yescharacteristic
42C22×Dic86D411yescharacteristic
43C22×D172—21no—
44C22×D172—21no—
45C22×D172D411yescharacteristic
46C22×D172D411yescharacteristic
47C22×C4×D43—21no—
48C22×C4×D43—21no—
49C2×#10621885—24no—
50C2×#10621885—44no—
51C2×#10621884—24no—
52C2×#10621884—44no—
53C22×C43⋊C8—21no—
54C22×C43⋊C8—21no—
55#13063313—21no—
56#13063313—21no—
57#13063313—21no—
58#13063313—21no—
59#13063313D412yes—
60#13063313D412yes—
61#13063313C2312yes—
62#13063313C2312yes—
63#13063310—24no—
64#13063310—24no—
65#13063310—24no—
66#13063310—24no—
67#13063303—24no—
68#13063303—24no—
69#13063303—24no—
70#13063303—24no—
71#13063300—24no—
72#13063300—24no—
73#13063300—24no—
74#13063300—24no—
75C2×Dic43⋊C4—21no—
76C2×Dic43⋊C4—21no—
77C2×(C2×D43)⋊C4—21no—
78C2×(C2×D43)⋊C4—21no—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83#13063248—24no—
84#13063245—24no—
85#13063236—21no—
86#13063236—21no—
87#13063236C2312yes—
88#13063236C2312yes—
89C22×Q8×C43—21no—
90C22×C8×C43—21no—
91C4⋊Q8×C43—21no—
92C4⋊Q8×C43—21no—
93C4⋊Q8×C43—21no—
94C4⋊Q8×C43C2312yes—
95#125×C43—21no—
96#125×C43—21no—
97#125×C43D412yes—
98#125×C43C2312yes—
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43D411yescharacteristic
104C2×C4⋊C4×C43D411yescharacteristic
105C2×C42×C43C2311yescharacteristic
106#107×C43—24no—
107#106×C43—24no—
108#97×C43—21no—
109#97×C43C2312yes—
110C23×D43—21no—
111C23×D43—21no—
112C2×Q8×C43—22no—
113C2×Q8×C43—42no—
114C2×Q8×C43—42no—
115C2×Q8×C43—41no—
116C22×C4×C43—21no—
117C22×C4×C43—21no—
118C22×C4×C43—41no—
119C22×C4×C43C2×D411yescharacteristic
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43C2×D411yescharacteristic
122C22×Dic43—21no—
123C22×Dic43—21no—
124C2×D172—24no—
125C2×D172—24no—
126C2×D172—21no—
127C2×D172—21no—
128C2×D172—42no—
129C2×D172—42no—
130C2×D172—82no—
131C2×D172—82no—
132C2×D172C2×D412yes—
133C2×D172C2×D412yes—
134C2×C4×D43—24no—
135C2×C4×D43—24no—
136C2×C4×D43—82no—
137C2×C4×D43—82no—
138C2×Dic86—21no—
139C2×Dic86—21no—
140C2×Dic86—42no—
141C2×Dic86—42no—
142C2×Dic86C2×D412yes—
143C2×Dic86C2×D412yes—
144C2×C8×C43—24no—
145C2×C8×C43—22no—
146C4⋊C4×C43—22no—
147C4⋊C4×C43—22no—
148C4⋊C4×C43—21no—
149C4⋊C4×C43—21no—
150C4⋊C4×C43—22no—
151C4⋊C4×C43—22no—
152C4⋊C4×C43—22no—
153C4⋊C4×C43—22no—
154C4⋊C4×C43—41no—
155C4⋊C4×C43—41no—
156C4⋊C4×C43C2×D412yes—
157C4⋊C4×C43C2×D412yes—
158C42×C43C2×D411yescharacteristic
159C42×C43C2×D411yescharacteristic
160C42×C43C2×D411yescharacteristic
161C42×C43C2411yescharacteristic
162#10621885—48no—
163#10621884—48no—
164(C2×D43)⋊C4—24no—
165(C2×D43)⋊C4—24no—
166(C2×D43)⋊C4—22no—
167(C2×D43)⋊C4—22no—
168(C2×D43)⋊C4—22no—
169(C2×D43)⋊C4—22no—
170(C2×D43)⋊C4—22no—
171(C2×D43)⋊C4—22no—
172(C2×D43)⋊C4—22no—
173(C2×D43)⋊C4—22no—
174Dic43⋊C4—24no—
175Dic43⋊C4—24no—
176C2×C43⋊C8—24no—
177C2×C43⋊C8—22no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8—42no—
180C23×C43#12011yescharacteristic
181C22×D43—22no—
182C22×D43—22no—
183C22×D43—42no—
184C22×D43—42no—
185C22×D43—42no—
186C22×D43—42no—
187C22×D43—42no—
188C22×D43—42no—
189Q8×C43—42no—
190Q8×C43—42no—
191Q8×C43—82no—
192C2×C4×C43—22no—
193C2×C4×C43—22no—
194C2×C4×C43—22no—
195C2×C4×C43—22no—
196C2×C4×C43—22no—
197C2×C4×C43—42no—
198C2×C4×C43—42no—
199C2×C4×C43—42no—
200C2×C4×C43—82no—
201C2×C4×C43#12011yescharacteristic
202C2×C4×C43#12011yescharacteristic
203C2×C4×C43#12011yescharacteristic
204C2×C4×C43C22×D411yescharacteristic
205C2×C4×C43C22×D411yescharacteristic
206C2×C4×C43C22×D411yescharacteristic
207C2×Dic43—22no—
208C2×Dic43—22no—
209C2×Dic43—42no—
210C2×Dic43—42no—
211D172—42no—
212D172—42no—
213D172—81no—
214D172—81no—
215D172—84no—
216D172—84no—
217C4×D43—84no—
218C4×D43—84no—
219Dic86—42no—
220Dic86—42no—
221Dic86—81no—
222Dic86—81no—
223C8×C43—24no—
224C43⋊C8—24no—
225C43⋊C8—42no—
226C2×#1661—431nomaximal, Sylow
227C22×C43#45312yes—
228C22×C43#45512yes—
229C22×C43#45612yes—
230C22×C43C2×#12011yescharacteristic
231C2×D43—42no—
232C2×D43—42no—
233C2×D43—42no—
234C2×D43—42no—
235C2×D43—42no—
236C2×D43—42no—
237C2×D43—84no—
238C2×D43—84no—
239C4×C43—21no—
240C4×C43—21no—
241C4×C43—21no—
242C4×C43—21no—
243C4×C43—21no—
244C4×C43—21no—
245C4×C43—42no—
246C4×C43—42no—
247C4×C43—82no—
248Dic43—42no—
249Dic43—42no—
250C2×#562—431no—
251C2×#501D4311yescharacteristic, complemented
252C2×#497—431no—
253C2×#489—431no—
254#1661—438no—
255C2×#330—431no—
256C2×#329—431no—
257C2×#329—431no—
258C2×C43#166114yescomplemented
259C2×C43C2×#45311yescharacteristic
260C2×C43C2×#45511yescharacteristic
261C2×C43C2×#45611yescharacteristic
262D43—82no—
263D43—82no—
264C22×Q16—861no—
265C22×SD16—861no—
266#562—434no—
267#562—862no—
268#562—862no—
269#562—862no—
270C2×C4⋊Q8C2×D4311yescharacteristic
271C2×#125C2×D4311yescharacteristic
272C2×#124—431no—
273C2×#124—431no—
274C2×#123—861no—
275C2×#123—861no—
276C2×C22⋊Q8—861no—
277C2×C22⋊Q8—861no—
278C2×#121—861no—
279C2×#121—861no—
280#501—22no—
281#501C2×D4314yescomplemented
282#497—434no—
283#497—862no—
284#489—434no—
285#489—862no—
286C2×#107—21no—
287C2×#106—21no—
288C2×#103—861no—
289C2×#102—861no—
290C2×#97—431no—
291C2×#97—431no—
292C2×#97C2×D4311yescharacteristic
293#330—434no—
294#329—434no—
295#329—434no—
296C43C2×#166111yescharacteristic, complemented, Sylow
297C22×Q8—21no—
298C22×Q8—431no—
299C22×Q8—431no—
300C22×D4—431no—
301C22×D4—431no—
302C22×D4—861no—
303C22×D4—861no—
304C23×C4—861no—
305C23×C4—861no—
306C2×Q16—864no—
307C2×Q16—1724no—
308C2×SD16—864no—
309C2×SD16—1724no—
310C22×C8—21no—
311C22×C8—861no—
312C22×C8—861no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8C22×D4312yes—
317#125—21no—
318#125—21no—
319#125C22⋊D4312yes—
320#125C22×D4312yes—
321#124—432no—
322#124—432no—
323#124—432no—
324#124—432no—
325#124—861no—
326#124—861no—
327#124—861no—
328#124—861no—
329#123—864no—
330#123—864no—
331#123—864no—
332#123—864no—
333C22⋊Q8—864no—
334C22⋊Q8—864no—
335C22⋊Q8—864no—
336C22⋊Q8—864no—
337#121—864no—
338#121—864no—
339#121—864no—
340#121—864no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—861no—
346C2×C4⋊C4—861no—
347C2×C4⋊C4C22⋊D4311yescharacteristic
348C2×C4⋊C4C22⋊D4311yescharacteristic
349C2×C22⋊C4—861no—
350C2×C22⋊C4—861no—
351C2×C22⋊C4—861no—
352C2×C22⋊C4—861no—
353C2×C22⋊C4—861no—
354C2×C22⋊C4—861no—
355C2×C42C22×D4311yescharacteristic
356#107—24no—
357#106—24no—
358#103—864no—
359#102—864no—
360#97—21no—
361#97—432no—
362#97—432no—
363#97—861no—
364#97—861no—
365#97C22×D4312yes—
366C24—861no—
367C24—861no—
368C2×Q8—22no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—41no—
372C2×Q8—432no—
373C2×Q8—432no—
374C2×Q8—861no—
375C2×Q8—861no—
376C2×Q8—1722no—
377C2×Q8—1722no—
378C2×D4—432no—
379C2×D4—432no—
380C2×D4—864no—
381C2×D4—864no—
382C2×D4—861no—
383C2×D4—861no—
384C2×D4—3442no—
385C2×D4—3442no—
386C2×D4—1722no—
387C2×D4—1722no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—864no—
392C22×C4—864no—
393C22×C4—861no—
394C22×C4—861no—
395C22×C4—3442no—
396C22×C4—3442no—
397C22×C4D4×D4311yescharacteristic
398C22×C4D4×D4311yescharacteristic
399C22×C4C2×C22⋊D4311yescharacteristic
400Q16—1728no—
401SD16—1728no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—864no—
405C2×C8—862no—
406C2×C8—862no—
407C2×C8—1722no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—864no—
419C4⋊C4—864no—
420C4⋊C4C2×C22⋊D4312yes—
421C4⋊C4C2×C22⋊D4312yes—
422C22⋊C4—864no—
423C22⋊C4—864no—
424C22⋊C4—862no—
425C22⋊C4—862no—
426C22⋊C4—862no—
427C22⋊C4—862no—
428C22⋊C4—862no—
429C22⋊C4—862no—
430C22⋊C4—862no—
431C22⋊C4—862no—
432C42D4×D4311yescharacteristic
433C42D4×D4311yescharacteristic
434C42C2×C22⋊D4311yescharacteristic
435C42C23×D4311yescharacteristic
436C23—862no—
437C23—862no—
438C23—1722no—
439C23—1722no—
440C23—1722no—
441C23—1722no—
442C23—1722no—
443C23—1722no—
444C23#1306329311yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—82no—
448Q8—3441no—
449Q8—3441no—
450Q8—1722no—
451Q8—1722no—
452D4—3441no—
453D4—3441no—
454D4—3444no—
455D4—3444no—
456D4—1722no—
457D4—1722no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—862no—
468C2×C4—862no—
469C2×C4—3444no—
470C2×C4—3444no—
471C2×C4—1722no—
472C2×C4—1722no—
473C2×C4#1306329311yescharacteristic
474C2×C4#1306329311yescharacteristic
475C2×C4#1306329311yescharacteristic
476C2×C4C2×D4×D4311yescharacteristic
477C2×C4C2×D4×D4311yescharacteristic
478C2×C4C22×C22⋊D4311yescharacteristic
479C8—24no—
480C8—864no—
481C8—1722no—
482C22—3444no—
483C22—3444no—
484C22—1722no—
485C22—1722no—
486C22—1722no—
487C22—1722no—
488C22—1722no—
489C22—1722no—
490C22#1458306512yes—
491C22#1458307512yes—
492C22#1458308112yes—
493C22C2×#1306329311yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—1722no—
504C4—1722no—
505C2—3442no—
506C2—3442no—
507C2#1913391014yesdirect product factor
508C2C2×#1458306511yescharacteristic
509C2C2×#1458307511yescharacteristic
510C2C2×#1458308111yescharacteristic
subgroupG/H
centerC23#13063293
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×C43×#501C2
socleC23×C43#120
prime radicalC2×#14583647C2
hypercenterC2×#501D43
hypocenterC43C2×#1661
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#14583291C2semidirect
3. C2×#14583261C2semidirect
4. #19133910C2direct (N × H)
5. C2×#14582641C2semidirect
6. C2×#14582638C2semidirect
7. C2×#14582638C2semidirect
8. C2×C43×#501C2semidirect
24. #14583291C22semidirect
26. #14583261C22semidirect
31. #14582641C22semidirect
32. #14582638C22semidirect
33. #14582638C22semidirect
37. C43×#501C22semidirect
251. C2×#501D43semidirect
258. C2×C43#1661semidirect
281. #501C2×D43semidirect
296. C43C2×#1661semidirect
507. C2#19133910direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145836472
C2×#145832912
C2×#145832612
#191339102
C2×#145826412
C2×#145826382
C2×C43×#5012
C2×#166143