Finite Group Databasebeta

← C2×#20253909 #27085891

Subgroups — C2×#20253909 #27085891

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689679C211yesmaximal, characteristic
2C2×#15689678C211yesmaximal, characteristic
3#20253909C2116yesmaximal, direct product factor
4C2×#15689533C211yesmaximal, characteristic, complemented
5C2×#15689532C211yesmaximal, characteristic, complemented
6C2×#15689531C212yesmaximal, complemented
7C2×#15689530C212yesmaximal, complemented
8C2×#15689529C211yesmaximal, characteristic, complemented
9C2×#15689160C212yesmaximal
10C2×#15689158C212yesmaximal
11C2×#97×D47C211yesmaximal, characteristic
12C2×C47×#501C211yesmaximal, characteristic, complemented
13C22×C8×D47C2212yes—
14#15689679—24no—
15#15689679C2218yes—
16#15689678—24no—
17#15689678C2218yes—
18C2×#13094024C2211yescharacteristic
19C2×#13094023C2211yescharacteristic
20C2×#13094023C2211yescharacteristic
21C2×#13094016C2211yescharacteristic
22C2×#13094015C2212yes—
23C2×#13094004—21no—
24C2×#13094002—21no—
25C2×#13094002—21no—
26C2×#13094000—21no—
27C2×Q8×Dic47—21no—
28C2×#13093989—21no—
29C2×#13093989—21no—
30C2×#13093989—21no—
31C2×#13093982C2211yescharacteristic
32C2×#13093982C2212yes—
33C2×#13093982C2211yescharacteristic
34C2×C42×D47C2211yescharacteristic
35#15689533C2218yescomplemented
36#15689532C2218yescomplemented
37#15689531C22116yescomplemented
38#15689530C22116yescomplemented
39#15689529C2218yescomplemented
40#15689160C22116yes—
41#15689158C22116yes—
42C2×#13093958C2212yes—
43C2×#13093957C2212yes—
44C2×#13093957C2212yes—
45C2×#13093956C2212yes—
46C2×#13093955C2212yes—
47C2×#13093955C2212yes—
48C2×#13093935C2211yescharacteristic
49C2×#13093934C2212yes—
50#97×D47C2218yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#501C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×#106×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×Dic94—21no—
59C22×Dic94—21no—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2312yes—
64C2×C8×D47Q8116yescomplemented
65C2×C8×D47C2318yes—
66#13094024—22no—
67#13094024C2314yes—
68#13094023—22no—
69#13094023—22no—
70#13094023C2314yes—
71#13094023C2314yes—
72#13094016—22no—
73#13094016C2314yes—
74#13094015—24no—
75#13094015C2318yes—
76#13094004—28no—
77#13094002—28no—
78#13094002—28no—
79#13094000—28no—
80Q8×Dic47—28no—
81#13093989—28no—
82#13093989—28no—
83#13093989—28no—
84#13093982—24no—
85#13093982—22no—
86#13093982—22no—
87#13093982C2314yes—
88#13093982C2318yes—
89#13093982C2314yes—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—22no—
92C2×C4⋊Dic47C2312yes—
93C2×C4⋊Dic47C2311yescharacteristic
94C2×C4⋊Dic47C2311yescharacteristic
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—22no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×(C2×D47)⋊C4—22no—
101C2×(C2×D47)⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—22no—
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47D411yescharacteristic
108C2×C4×Dic47C2311yescharacteristic
109C42×D47D414yes—
110C42×D47C2314yes—
111#13093958C2318yes—
112#13093957C2318yes—
113#13093957C2318yes—
114#13093956C2318yes—
115#13093955C2318yes—
116#13093955C2318yes—
117#13093935C2314yes—
118#13093934C2318yes—
119C22×Q8×C47—21no—
120C22×C8×C47C2312yes—
121C4⋊Q8×C47—22no—
122C4⋊Q8×C47C2314yes—
123#125×C47—22no—
124#125×C47C2314yes—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2312yes—
131C2×C42×C47C2311yescharacteristic
132#107×C47C2318yes—
133#106×C47C2318yes—
134#97×C47C2314yes—
135C23×D47C2×D411yescharacteristic
136C2×Q8×C47—24no—
137C2×Q8×C47—44no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—22no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—22no—
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47C2×D414yes—
153C2×C4×D47C2×D414yes—
154C2×C4×D47C2×D412yes—
155C2×C4×D47C2×Q818yes—
156C2×C4×D47C2412yes—
157C2×Dic94—24no—
158C2×Dic94—24no—
159C2×Dic94—24no—
160C2×Dic94—44no—
161C2×Dic94—44no—
162C2×Dic94—44no—
163C2×C8×C47C2×Q818yes—
164C2×C8×C47C2414yes—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—24no—
167C4⋊C4×C47—24no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47C2412yes—
172C4⋊C4×C47C2414yes—
173C4⋊C4×C47C2412yes—
174C42×C47C2×D412yes—
175C42×C47C2412yes—
176(C2×D47)⋊C4—28no—
177(C2×D47)⋊C4—24no—
178(C2×D47)⋊C4—24no—
179C4⋊Dic47—24no—
180C4⋊Dic47—22no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—28no—
184C4⋊Dic47C2412yes—
185C4⋊Dic47C2414yes—
186C4⋊Dic47C2412yes—
187Dic47⋊C4—24no—
188Dic47⋊C4—24no—
189Dic47⋊C4—28no—
190Dic47⋊C4—24no—
191Dic47⋊C4—24no—
192C4×Dic47—24no—
193C4×Dic47—24no—
194C4×Dic47—28no—
195C4×Dic47C2×D412yes—
196C4×Dic47C2×D414yes—
197C4×Dic47C2×D414yes—
198C4×Dic47C2412yes—
199C2×C47⋊C8C2×Q818yes—
200C2×C47⋊C8C2414yes—
201C8×D47C2×Q8132yescomplemented
202C23×C47C22×D411yescharacteristic
203C22×D47—22no—
204C22×D47—22no—
205C22×D47C4⋊Q814yes—
206C22×D47C22×D412yes—
207Q8×C47—48no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—44no—
216C2×C4×C47C22×D412yes—
217C2×C4×C47C22×D412yes—
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×Q814yes—
220C2×C4×C47C2511yescharacteristic
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—24no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—44no—
229C2×Dic47C4⋊Q814yes—
230C2×Dic47C22×D412yes—
231C2×Dic47C22×D412yes—
232C2×Dic47C22×D412yes—
233C4×D47—44no—
234C4×D47—44no—
235C4×D47C4⋊Q818yes—
236C4×D47C22×Q818yes—
237Dic94—48no—
238Dic94—48no—
239Dic94—48no—
240C8×C47C22×Q818yes—
241C47⋊C8C22×Q818yes—
242C2×#3162—471nomaximal, Sylow
243C22×C47C2×C4⋊Q812yes—
244C22×C47#57512yes—
245C22×C47#57512yes—
246C22×C47C23×D411yescharacteristic
247C2×D47—28no—
248C2×D47—22no—
249C2×D47—22no—
250C2×D47C2×C4⋊Q814yes—
251C4×C47—22no—
252C4×C47—22no—
253C4×C47—42no—
254C4×C47—42no—
255C4×C47—44no—
256C4×C47C2×C4⋊Q812yes—
257C4×C47C23×Q812yes—
258Dic47—22no—
259Dic47—22no—
260Dic47—42no—
261Dic47—42no—
262Dic47—44no—
263Dic47C2×C4⋊Q814yes—
264C2×#533—471no—
265C2×#533—471no—
266#3162—4716no—
267C2×#501—472no—
268C2×#501—472no—
269C2×#501—471no—
270C2×#501—471no—
271C2×#501—471no—
272C2×#501D4711yescharacteristic, complemented
273C2×#427—472no—
274C2×#427—472no—
275C22×#97—471no—
276C2×C47#316214yescomplemented
277C2×C47C22×C4⋊Q811yescharacteristic
278C2×C47C2×#57511yescharacteristic
279C2×C47C2×#57511yescharacteristic
280D47—24no—
281C23×C8—472no—
282#533—478no—
283#533—478no—
284#533—944no—
285#533—944no—
286C2×C4⋊Q8—471no—
287C2×C4⋊Q8—471no—
288C2×C4⋊Q8—471no—
289C2×C4⋊Q8C2×D4711yescharacteristic
290C2×#125—472no—
291C2×#125—471no—
292C2×#125C2×D4711yescharacteristic
293C2×C22⋊Q8—941no—
294C2×C22⋊Q8—941no—
295C2×C22⋊Q8—941no—
296C2×C22⋊Q8—941no—
297C2×C4×Q8—941no—
298C2×C4×Q8—941no—
299C2×C4×Q8—941no—
300C2×C4×Q8—941no—
301C2×#117—472no—
302C2×#117—471no—
303C2×#117—471no—
304C22×C42—471no—
305#501—478no—
306#501—478no—
307#501—4716no—
308#501—4716no—
309#501—478no—
310#501C2×D4718yescomplemented
311#427—4716no—
312#427—4716no—
313C2×#107—472no—
314C2×#107—472no—
315C2×#107—472no—
316C2×#107C2×D4712yes—
317C2×#106—472no—
318C2×#106—472no—
319C2×#106—472no—
320C2×#106C2×D4712yes—
321C2×#97—472no—
322C2×#97—471no—
323C2×#97—478no—
324C2×#97C2×D4711yescharacteristic
325C47C2×#316211yescharacteristic, complemented, Sylow
326C22×Q8—21no—
327C22×Q8—941no—
328C22×Q8—941no—
329C22×Q8—941no—
330C23×C4—471no—
331C23×C4—471no—
332C23×C4—471no—
333C22×C8—472no—
334C22×C8—4716no—
335C22×C8—478no—
336C22×C8C22×D4712yes—
337C4⋊Q8—22no—
338C4⋊Q8—474no—
339C4⋊Q8—474no—
340C4⋊Q8—474no—
341C4⋊Q8—942no—
342C4⋊Q8—942no—
343C4⋊Q8—942no—
344C4⋊Q8C22×D4714yes—
345#125—22no—
346#125—478no—
347#125—474no—
348#125—944no—
349#125—942no—
350#125C22×D4714yes—
351C22⋊Q8—948no—
352C22⋊Q8—948no—
353C22⋊Q8—948no—
354C22⋊Q8—948no—
355C4×Q8—948no—
356C4×Q8—948no—
357C4×Q8—948no—
358C4×Q8—948no—
359#117—474no—
360#117—478no—
361#117—474no—
362#117—942no—
363#117—942no—
364#117—944no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4C22×D4712yes—
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—942no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—471no—
387C2×C42—474no—
388C2×C42—474no—
389C2×C42—941no—
390C2×C42—941no—
391C2×C42—942no—
392C2×C42C22×D4711yescharacteristic
393#107—478no—
394#107—478no—
395#107—478no—
396#107C22×D4718yes—
397#106—478no—
398#106—478no—
399#106—478no—
400#106C22×D4718yes—
401#97—478no—
402#97—474no—
403#97C22×D4714yes—
404C24—471no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—944no—
408C2×Q8—944no—
409C2×Q8—944no—
410C2×Q8—1884no—
411C2×Q8—1884no—
412C2×Q8—1884no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—471no—
417C22×C4—472no—
418C22×C4—478no—
419C22×C4—472no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—474no—
423C22×C4—474no—
424C22×C4—941no—
425C22×C4—941no—
426C22×C4—942no—
427C22×C4—1882no—
428C22×C4—1882no—
429C22×C4D4×D4711yescharacteristic
430C22×C4D4×D4711yescharacteristic
431C22×C4C23×D4711yescharacteristic
432C2×C8—478no—
433C2×C8—474no—
434C2×C8—4732no—
435C2×C8Q8×D4718yes—
436C2×C8C23×D4714yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—472no—
444C4⋊C4—474no—
445C4⋊C4—472no—
446C4⋊C4—948no—
447C4⋊C4—944no—
448C4⋊C4—944no—
449C4⋊C4—948no—
450C4⋊C4—944no—
451C4⋊C4—944no—
452C4⋊C4—942no—
453C4⋊C4—941no—
454C4⋊C4—941no—
455C4⋊C4—944no—
456C4⋊C4C23×D4712yes—
457C4⋊C4C23×D4714yes—
458C4⋊C4C23×D4712yes—
459C22⋊C4—944no—
460C22⋊C4—944no—
461C22⋊C4—948no—
462C42—474no—
463C42—474no—
464C42—472no—
465C42—472no—
466C42—944no—
467C42—944no—
468C42—948no—
469C42D4×D4712yes—
470C42C23×D4712yes—
471C23—474no—
472C23—472no—
473C23—942no—
474C23—942no—
475C23C2×D4×D4711yescharacteristic
476Q8—48no—
477Q8—1888no—
478Q8—1888no—
479Q8—1888no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—474no—
489C2×C4—472no—
490C2×C4—478no—
491C2×C4—478no—
492C2×C4—472no—
493C2×C4—472no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—944no—
499C2×C4—1884no—
500C2×C4—1884no—
501C2×C4—1882no—
502C2×C4—1884no—
503C2×C4—1882no—
504C2×C4C2×D4×D4712yes—
505C2×C4C2×D4×D4712yes—
506C2×C4C2×D4×D4711yescharacteristic
507C2×C4C2×Q8×D4714yes—
508C2×C4C24×D4711yescharacteristic
509C8—478no—
510C8C2×Q8×D4718yes—
511C22—474no—
512C22—942no—
513C22—942no—
514C22—948no—
515C22C4⋊Q8×D4712yes—
516C22#1568991912yes—
517C22#1568992212yes—
518C22C22×D4×D4711yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—474no—
525C4—942no—
526C4—942no—
527C4—1882no—
528C4—1884no—
529C4—1882no—
530C4C4⋊Q8×D4712yes—
531C4C22×Q8×D4712yes—
532C2—944no—
533C2#2025390914yesdirect product factor
534C2C2×C4⋊Q8×D4711yescharacteristic
535C2C2×#1568991911yescharacteristic
536C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#501C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC2×#501D47
hypocenterC47C2×#3162
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #20253909C2direct (N × H)
4. C2×#15689533C2semidirect
5. C2×#15689532C2semidirect
6. C2×#15689531C2semidirect
7. C2×#15689530C2semidirect
8. C2×#15689529C2semidirect
12. C2×C47×#501C2semidirect
35. #15689533C22semidirect
36. #15689532C22semidirect
37. #15689531C22semidirect
38. #15689530C22semidirect
39. #15689529C22semidirect
53. C47×#501C22semidirect
64. C2×C8×D47Q8semidirect
201. C8×D47C2×Q8semidirect
272. C2×#501D47semidirect
276. C2×C47#3162semidirect
310. #501C2×D47semidirect
325. C47C2×#3162semidirect
533. C2#20253909direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896792
C2×#156896782
#202539092
C2×#156895332
C2×#156895322
C2×#156895312
C2×#156895302
C2×#156895292
C2×#156891602
C2×#156891582
C2×#97×D472
C2×C47×#5012
C2×#316247