Finite Group Databasebeta

← C2×#20253925 #27085909

Subgroups — C2×#20253925 #27085909

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689753C212yesmaximal
2#20253925C2116yesmaximal, direct product factor
3C2×#15689533C212yesmaximal, complemented
4C2×#15689530C212yesmaximal, complemented
5C2×#15689528C211yesmaximal, characteristic, complemented
6C2×#15689527C211yesmaximal, characteristic, complemented
7C2×#15689524C211yesmaximal, characteristic, complemented
8C2×#15689162C212yesmaximal
9C2×#15689160C211yesmaximal, characteristic
10C2×#107×D47C211yesmaximal, characteristic
11C2×#15689077C211yesmaximal, characteristic
12C2×C47×#499C211yesmaximal, characteristic, complemented
13C22×#10625324C2211yescharacteristic
14C22×C8×D47C2211yescharacteristic
15#15689753—28no—
16#15689753C22116yes—
17C2×#13094024C2212yes—
18C2×#13094023—22no—
19C2×#13094023C2212yes—
20C2×#13094015C2212yes—
21C2×#13094014—22no—
22C2×#13094004—22no—
23C2×#13094002—22no—
24C2×#13093982C2212yes—
25C2×#13093981C2211yescharacteristic
26C2×C4⋊C4×D47C2212yes—
27#15689533C22116yescomplemented
28#15689530C22116yescomplemented
29#15689528C2218yescomplemented
30#15689527C2218yescomplemented
31#15689524C2218yescomplemented
32#15689162C22116yes—
33#15689160C2218yes—
34C2×#13093958C2211yescharacteristic
35C2×#13093958C2211yescharacteristic
36C2×#13093958C2212yes—
37C2×#13093957C2212yes—
38C2×#13093957C2212yes—
39#107×D47C2218yes—
40C2×#13093955C2212yes—
41C2×#13093955C2212yes—
42#15689077C2218yes—
43C2×#13093935C2211yescharacteristic
44C2×#13093934C2211yescharacteristic
45C2×C8×Dic47C2211yescharacteristic
46C2×#125×C47C2212yes—
47C47×#499C2218yescomplemented
48C2×#107×C47C2211yescharacteristic
49C2×#107×C47C2211yescharacteristic
50C2×#106×C47C2212yes—
51C2×C4×C8×C47C2211yescharacteristic
52C22×Dic94—22no—
53C22×Dic94—22no—
54C22×C4×D47—22no—
55C22×C4×D47C2311yescharacteristic
56C2×#10625324Q818yescomplemented
57C2×#10625324C2314yes—
58C22×C47⋊C8C2311yescharacteristic
59C22×C47⋊C8C2311yescharacteristic
60C2×C8×D47Q818yescomplemented
61C2×C8×D47C2314yes—
62#13094024—24no—
63#13094024C2318yes—
64#13094023—216no—
65#13094023—24no—
66#13094023C2318yes—
67#13094015—24no—
68#13094015C2318yes—
69#13094014—216no—
70#13094004—216no—
71#13094002—216no—
72#13093982—24no—
73#13093982C2318yes—
74#13093981D414yes—
75#13093981C2314yes—
76C2×C4⋊Dic47—22no—
77C2×C4⋊Dic47—22no—
78C2×C4⋊Dic47C2312yes—
79C2×C4⋊Dic47C2312yes—
80C2×Dic47⋊C4—22no—
81C2×Dic47⋊C4—22no—
82C2×Dic47⋊C4—22no—
83C2×Dic47⋊C4D411yescharacteristic
84C2×Dic47⋊C4D411yescharacteristic
85C4⋊C4×D47—24no—
86C4⋊C4×D47C2318yes—
87C2×(C2×D47)⋊C4—22no—
88C2×(C2×D47)⋊C4D411yescharacteristic
89C2×(C2×D47)⋊C4D411yescharacteristic
90C2×C4×Dic47—22no—
91C2×C4×Dic47C2311yescharacteristic
92#13093958C2314yes—
93#13093958C2314yes—
94#13093958C2318yes—
95#13093957C2318yes—
96#13093957C2318yes—
97#13093955C2318yes—
98#13093955C2318yes—
99#13093935C2314yes—
100#13093934C2314yes—
101C8×Dic47C2314yes—
102C22×C8×C47C2311yescharacteristic
103C22×C8×C47C2311yescharacteristic
104#125×C47—24no—
105#125×C47C2318yes—
106C2×C4⋊C4×C47—22no—
107C2×C4⋊C4×C47—22no—
108C2×C4⋊C4×C47C2312yes—
109C2×C4⋊C4×C47C2312yes—
110C2×C42×C47C2311yescharacteristic
111#107×C47C2314yes—
112#107×C47C2314yes—
113#106×C47C2318yes—
114C4×C8×C47C2314yes—
115C23×D47C2×D411yescharacteristic
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2411yescharacteristic
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47C2×D411yescharacteristic
124C22×Dic47C2×D411yescharacteristic
125C22×Dic47C2×D411yescharacteristic
126C2×C4×D47—28no—
127C2×C4×D47—84no—
128C2×C4×D47C2×D412yes—
129C2×C4×D47C2×Q814yes—
130C2×C4×D47C2×Q814yes—
131C2×C4×D47C2412yes—
132C2×Dic94—28no—
133C2×Dic94—28no—
134C2×Dic94—84no—
135C2×Dic94—84no—
136C2×C8×C47C2×Q814yes—
137C2×C8×C47C2×Q814yes—
138C2×C8×C47C2412yes—
139C2×C8×C47C2412yes—
140C4⋊C4×C47—28no—
141C4⋊C4×C47—28no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47C2414yes—
145C4⋊C4×C47C2414yes—
146C42×C47C2×D412yes—
147C42×C47C2412yes—
148(C2×D47)⋊C4—28no—
149(C2×D47)⋊C4C2×D414yes—
150(C2×D47)⋊C4C2×D414yes—
151C4⋊Dic47—22no—
152C4⋊Dic47—28no—
153C4⋊Dic47—22no—
154C4⋊Dic47—28no—
155C4⋊Dic47C2414yes—
156C4⋊Dic47C2414yes—
157Dic47⋊C4—28no—
158Dic47⋊C4—28no—
159Dic47⋊C4—28no—
160Dic47⋊C4C2×D414yes—
161Dic47⋊C4C2×D414yes—
162C4×Dic47—28no—
163C4×Dic47C2×D412yes—
164C4×Dic47C2412yes—
165C2×C47⋊C8C2×Q814yes—
166C2×C47⋊C8C2×Q814yes—
167C2×C47⋊C8C2412yes—
168C2×C47⋊C8C2412yes—
169#10625324C2×Q8116yescomplemented
170C8×D47C2×Q8116yescomplemented
171C23×C47C22×D411yescharacteristic
172C22×D47—22no—
173C22×D47—22no—
174C22×D47C4⋊Q814yes—
175C22×D47C22×D412yes—
176C2×C4×C47—22no—
177C2×C4×C47—22no—
178C2×C4×C47—24no—
179C2×C4×C47—24no—
180C2×C4×C47—44no—
181C2×C4×C47—44no—
182C2×C4×C47C22×D412yes—
183C2×C4×C47C22×D412yes—
184C2×C4×C47C22×D411yescharacteristic
185C2×C4×C47C22×Q812yes—
186C2×C4×C47C22×Q812yes—
187C2×C4×C47C2511yescharacteristic
188C2×Dic47—22no—
189C2×Dic47—22no—
190C2×Dic47—24no—
191C2×Dic47—24no—
192C2×Dic47—44no—
193C2×Dic47—44no—
194C2×Dic47C4⋊Q814yes—
195C2×Dic47C22×D412yes—
196C2×Dic47C22×D412yes—
197C2×Dic47C22×D412yes—
198C4×D47—88no—
199C4×D47C4⋊Q818yes—
200C4×D47C22×Q814yes—
201C4×D47C22×Q814yes—
202Dic94—88no—
203Dic94—88no—
204C8×C47C22×Q814yes—
205C8×C47C22×Q814yes—
206C47⋊C8C22×Q814yes—
207C47⋊C8C22×Q814yes—
208C2×#3164—471nomaximal, Sylow
209C22×C47C2×C4⋊Q812yes—
210C22×C47#57612yes—
211C22×C47#57812yes—
212C22×C47C23×D411yescharacteristic
213C2×D47—28no—
214C2×D47—22no—
215C2×D47—22no—
216C2×D47C2×C4⋊Q814yes—
217C4×C47—22no—
218C4×C47—22no—
219C4×C47—44no—
220C4×C47—44no—
221C4×C47C2×C4⋊Q812yes—
222C4×C47C23×Q811yescharacteristic
223C4×C47C23×Q811yescharacteristic
224Dic47—22no—
225Dic47—22no—
226Dic47—44no—
227Dic47—44no—
228Dic47C2×C4⋊Q814yes—
229C2×#544—472no—
230#3164—4716no—
231C2×#501—472no—
232C2×#501—472no—
233C2×#500—471no—
234C2×#500—471no—
235C2×#499—471no—
236C2×#499D4711yescharacteristic, complemented
237C2×#428—472no—
238C2×#427—471no—
239C22×#107—471no—
240C2×#405—471no—
241C2×C47#316414yescomplemented
242C2×C47C22×C4⋊Q811yescharacteristic
243C2×C47C2×#57611yescharacteristic
244C2×C47C2×#57811yescharacteristic
245D47—24no—
246C22×M16—471no—
247C23×C8—471no—
248#544—4716no—
249#544—948no—
250C2×C4⋊Q8—472no—
251C2×C4⋊Q8—472no—
252C2×C4⋊Q8—942no—
253C2×#125—472no—
254C2×#125—942no—
255C2×#125C2×D4712yes—
256C2×C22⋊Q8—942no—
257C2×C22⋊Q8—942no—
258C2×#117—471no—
259C2×#117—472no—
260C22×C4⋊C4—472no—
261#501—4716no—
262#501—4716no—
263#500—478no—
264#500—478no—
265#499—478no—
266#499C2×D4718yescomplemented
267#428—4716no—
268#427—478no—
269C2×#107—471no—
270C2×#107—471no—
271C2×#107—472no—
272C2×#107—472no—
273C2×#107—472no—
274C2×#107—478no—
275C2×#107C2×D4711yescharacteristic
276C2×#107C2×D4711yescharacteristic
277C2×#106—472no—
278C2×#106—472no—
279C2×#106C2×D4712yes—
280#405—478no—
281C2×#97—471no—
282C2×#97—471no—
283C2×C4×C8—471no—
284C2×C4×C8C2×D4711yescharacteristic
285C47C2×#316411yescharacteristic, complemented, Sylow
286C22×Q8—942no—
287C22×Q8—942no—
288C23×C4—471no—
289C23×C4—942no—
290C2×M16—478no—
291C2×M16—474no—
292C22×C8—471no—
293C22×C8—471no—
294C22×C8—478no—
295C22×C8—474no—
296C22×C8C22×D4711yescharacteristic
297C22×C8C22×D4711yescharacteristic
298C4⋊Q8—478no—
299C4⋊Q8—478no—
300C4⋊Q8—944no—
301C4⋊Q8—9416no—
302C4⋊Q8—944no—
303#125—24no—
304#125—478no—
305#125—944no—
306#125—9416no—
307#125C22×D4718yes—
308C22⋊Q8—9416no—
309C22⋊Q8—9416no—
310#117—474no—
311#117—474no—
312#117—478no—
313#117—944no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—471no—
317C2×C4⋊C4—471no—
318C2×C4⋊C4—472no—
319C2×C4⋊C4—472no—
320C2×C4⋊C4—478no—
321C2×C4⋊C4—944no—
322C2×C4⋊C4—942no—
323C2×C4⋊C4—942no—
324C2×C4⋊C4—942no—
325C2×C4⋊C4—942no—
326C2×C4⋊C4—942no—
327C2×C4⋊C4C22×D4712yes—
328C2×C4⋊C4C22×D4712yes—
329C2×C22⋊C4—471no—
330C2×C22⋊C4—471no—
331C2×C22⋊C4—942no—
332C2×C42—471no—
333C2×C42—942no—
334C2×C42C22×D4711yescharacteristic
335#107—474no—
336#107—474no—
337#107—478no—
338#107—478no—
339#107—478no—
340#107C22×D4714yes—
341#107C22×D4714yes—
342#106—478no—
343#106—478no—
344#106C22×D4718yes—
345#97—474no—
346#97—474no—
347C4×C8—474no—
348C4×C8C22×D4714yes—
349C24—471no—
350C2×Q8—948no—
351C2×Q8—948no—
352C2×Q8—3764no—
353C2×Q8—3764no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—471no—
357C22×C4—472no—
358C22×C4—474no—
359C22×C4—474no—
360C22×C4—472no—
361C22×C4—471no—
362C22×C4—471no—
363C22×C4—948no—
364C22×C4—942no—
365C22×C4—942no—
366C22×C4—3764no—
367C22×C4D4×D4711yescharacteristic
368C22×C4D4×D4711yescharacteristic
369C22×C4C23×D4711yescharacteristic
370M16—4716no—
371C2×C8—474no—
372C2×C8—474no—
373C2×C8—472no—
374C2×C8—472no—
375C2×C8—4716no—
376C2×C8Q8×D4714yes—
377C2×C8Q8×D4714yes—
378C2×C8C23×D4712yes—
379C2×C8C23×D4712yes—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—474no—
385C4⋊C4—474no—
386C4⋊C4—474no—
387C4⋊C4—474no—
388C4⋊C4—948no—
389C4⋊C4—948no—
390C4⋊C4—948no—
391C4⋊C4—948no—
392C4⋊C4—942no—
393C4⋊C4—942no—
394C4⋊C4—948no—
395C4⋊C4C23×D4714yes—
396C4⋊C4C23×D4714yes—
397C22⋊C4—474no—
398C22⋊C4—474no—
399C22⋊C4—948no—
400C42—472no—
401C42—472no—
402C42—948no—
403C42D4×D4712yes—
404C42C23×D4712yes—
405C23—474no—
406C23—472no—
407C23—942no—
408C23—942no—
409C23C2×D4×D4711yescharacteristic
410Q8—3768no—
411Q8—3768no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—474no—
419C2×C4—472no—
420C2×C4—478no—
421C2×C4—474no—
422C2×C4—474no—
423C2×C4—472no—
424C2×C4—472no—
425C2×C4—942no—
426C2×C4—942no—
427C2×C4—944no—
428C2×C4—944no—
429C2×C4—3768no—
430C2×C4—1884no—
431C2×C4—1884no—
432C2×C4C2×D4×D4712yes—
433C2×C4C2×D4×D4712yes—
434C2×C4C2×D4×D4711yescharacteristic
435C2×C4C2×Q8×D4712yes—
436C2×C4C2×Q8×D4712yes—
437C2×C4C24×D4711yescharacteristic
438C8—474no—
439C8—474no—
440C8C2×Q8×D4714yes—
441C8C2×Q8×D4714yes—
442C22—474no—
443C22—942no—
444C22—942no—
445C22—948no—
446C22C4⋊Q8×D4712yes—
447C22#1568992512yes—
448C22#1568993712yes—
449C22C22×D4×D4711yescharacteristic
450C4—22no—
451C4—22no—
452C4—44no—
453C4—44no—
454C4—474no—
455C4—942no—
456C4—942no—
457C4—1884no—
458C4—1884no—
459C4C4⋊Q8×D4712yes—
460C4C22×Q8×D4711yescharacteristic
461C4C22×Q8×D4711yescharacteristic
462C2—944no—
463C2#2025392514yesdirect product factor
464C2C2×C4⋊Q8×D4711yescharacteristic
465C2C2×#1568992511yescharacteristic
466C2C2×#1568993711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#499C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC2×#499D47
hypocenterC47C2×#3164
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
2. #20253925C2direct (N × H)
3. C2×#15689533C2semidirect
4. C2×#15689530C2semidirect
5. C2×#15689528C2semidirect
6. C2×#15689527C2semidirect
7. C2×#15689524C2semidirect
12. C2×C47×#499C2semidirect
27. #15689533C22semidirect
28. #15689530C22semidirect
29. #15689528C22semidirect
30. #15689527C22semidirect
31. #15689524C22semidirect
47. C47×#499C22semidirect
56. C2×#10625324Q8semidirect
60. C2×C8×D47Q8semidirect
169. #10625324C2×Q8semidirect
170. C8×D47C2×Q8semidirect
236. C2×#499D47semidirect
241. C2×C47#3164semidirect
266. #499C2×D47semidirect
285. C47C2×#3164semidirect
463. C2#20253925direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897532
#202539252
C2×#156895332
C2×#156895302
C2×#156895282
C2×#156895272
C2×#156895242
C2×#156891622
C2×#156891602
C2×#107×D472
C2×#156890772
C2×C47×#4992
C2×#316447