Finite Group Databasebeta

← C2×#20241923 #26183517

Subgroups — C2×#20241923 #26183517

Order 12032.

1098 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689654C211yesmaximal, characteristic
2C2×#15689177C211yesmaximal, characteristic, complemented
3C2×#15688987C211yesmaximal, characteristic, complemented
4#20241923C218yesmaximal, direct product factor
5C2×#15688861C211yesmaximal, characteristic, complemented
6C2×#15688859C211yesmaximal, characteristic, complemented
7C2×#15688857C211yesmaximal, characteristic
8C2×C47×#327C211yesmaximal, characteristic
9#15689654—22no—
10#15689654—22no—
11#15689654C414yescomplemented
12#15689654C2214yes—
13C2×#13094017—21no—
14C2×#13094015C411yescharacteristic, complemented
15C2×#13094011C411yescharacteristic, complemented
16C2×#13094005—21no—
17C2×#13094003C2211yescharacteristic
18C2×#13093989C411yescharacteristic, complemented
19C2×D4×Dic47—21no—
20C2×D4×Dic47C411yescharacteristic, complemented
21C2×#13093980—21no—
22C22×Dic47⋊C4—21no—
23C22×C4×Dic47C2211yescharacteristic
24#15689177—22no—
25#15689177C2214yescomplemented
26C2×Dic47⋊C8—21no—
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093934—21no—
29#15688987—22no—
30#15688987C2214yescomplemented
31C2×C47⋊#95—21no—
32C2×C47⋊#95C2211yescharacteristic
33#15688861C2214yescomplemented
34#15688859C2214yescomplemented
35#15688857C2214yes—
36C2×#121×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C47×#327C2214yes—
40C22×Dic94C2×C411yescharacteristic
41C23×Dic47—21no—
42C23×Dic47—21no—
43C23×Dic47D411yescharacteristic
44C23×Dic47D411yescharacteristic
45C22×C47⋊C8—21no—
46#13094017—24no—
47#13094017—24no—
48#13094015—22no—
49#13094015C2×C414yescomplemented
50#13094011—22no—
51#13094011C2×C414yescomplemented
52#13094005—24no—
53#13094005—24no—
54#13094003—21no—
55#13094003—21no—
56#13094003C2×C412yes—
57#13094003C2312yes—
58#13093989—22no—
59#13093989C2×C414yescomplemented
60D4×Dic47—24no—
61D4×Dic47—24no—
62D4×Dic47—22no—
63D4×Dic47C2×C414yescomplemented
64#13093980—24no—
65#13093980—24no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2×C411yescharacteristic
68C2×Dic47⋊C4—21no—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4—44no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47C2×C411yescharacteristic
79C2×C4×Dic47C2×C411yescharacteristic
80C2×C4×Dic47C2×C412yes—
81C2×C4×Dic47D412yes—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D412yes—
85C2×C4×Dic47C2312yes—
86Dic47⋊C8—24no—
87C22⋊(C47⋊C8)—21no—
88C22⋊(C47⋊C8)C2312yes—
89#13093934—24no—
90C47⋊#95—21no—
91C47⋊#95—24no—
92C47⋊#95C2312yes—
93C22×D4×C47—21no—
94C22×D4×C47C2×C411yescharacteristic
95C23×C4×C47—21no—
96C23×C4×C47C2311yescharacteristic
97C22×C8×C47—21no—
98#121×C47—21no—
99#121×C47—21no—
100#121×C47C2×C412yes—
101#121×C47C2312yes—
102C2×C4⋊C4×C47C2×C411yescharacteristic
103C2×C22⋊C4×C47—21no—
104C22⋊C8×C47—21no—
105C22⋊C8×C47C2312yes—
106#95×C47—21no—
107#95×C47C2312yes—
108C24×C47—21no—
109C24×C47C2×D411yescharacteristic
110C2×D4×C47—22no—
111C2×D4×C47—22no—
112C2×D4×C47—21no—
113C2×D4×C47—42no—
114C2×D4×C47—42no—
115C2×D4×C47—42no—
116C2×D4×C47C22×C412yes—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—42no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47—42no—
123C22×C4×C47C22×C411yescharacteristic
124C22×C4×C47C22×C411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—24no—
134C22×Dic47—22no—
135C22×Dic47—42no—
136C22×Dic47—44no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47C22⋊C411yescharacteristic
141C22×Dic47C22⋊C411yescharacteristic
142C22×Dic47C22⋊C411yescharacteristic
143C22×Dic47C22⋊C411yescharacteristic
144C22×Dic47C22⋊C412yes—
145C22×Dic47C22⋊C412yes—
146C22×Dic47C2×D412yes—
147C22×Dic47C2×D412yes—
148C2×Dic94—21no—
149C2×Dic94—42no—
150C2×Dic94C22×C412yes—
151C2×C8×C47—22no—
152C2×C8×C47—42no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47C22×C412yes—
155C22⋊C4×C47—22no—
156C22⋊C4×C47—22no—
157C22⋊Dic47—22no—
158C22⋊Dic47—22no—
159C22⋊Dic47—24no—
160C22⋊Dic47—24no—
161C22⋊Dic47—24no—
162C4⋊Dic47—21no—
163C4⋊Dic47—24no—
164C4⋊Dic47C22×C412yes—
165Dic47⋊C4—22no—
166Dic47⋊C4—22no—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—48no—
170C4×Dic47—24no—
171C4×Dic47C22⋊C412yes—
172C4×Dic47C22⋊C412yes—
173C4×Dic47C22⋊C412yes—
174C4×Dic47C22⋊C412yes—
175C4×Dic47C22×C412yes—
176C4×Dic47C22×C412yes—
177C4×Dic47C2×D412yes—
178C4×Dic47C2×D412yes—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—42no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47C2×C22⋊C411yescharacteristic
188C23×C47C2×C22⋊C411yescharacteristic
189C23×C47AGL(1,ℤ/8)12yes—
190C23×C47#13712yes—
191C23×C47C22×D411yescharacteristic
192D4×C47—42no—
193D4×C47—42no—
194D4×C47—42no—
195D4×C47—42no—
196D4×C47—44no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—84no—
209C2×C4×C47C2×C22⋊C411yescharacteristic
210C2×C4×C47C2×C22⋊C411yescharacteristic
211C2×C4×C47C23×C411yescharacteristic
212C2×C4×C47C22×D411yescharacteristic
213C2×Dic47—22no—
214C2×Dic47—22no—
215C2×Dic47—24no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—48no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—84no—
231C2×Dic47C2×C22⋊C412yes—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C2×C22⋊C412yes—
234C2×Dic47C2×C22⋊C412yes—
235Dic94—42no—
236Dic94—42no—
237C8×C47—42no—
238C47⋊C8—42no—
239C2×#1512—471nomaximal, Sylow
240C22×C47—21no—
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—21no—
244C22×C47—24no—
245C22×C47—24no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47—84no—
250C22×C47#41012yes—
251C22×C47#41712yes—
252C22×C47#41712yes—
253C22×C47#42112yes—
254C22×C47C22×C22⋊C411yescharacteristic
255C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C47C2×#13711yescharacteristic
257C4×C47—21no—
258C4×C47—21no—
259C4×C47—21no—
260C4×C47—21no—
261C4×C47—42no—
262C4×C47—82no—
263Dic47—22no—
264Dic47—22no—
265Dic47—22no—
266Dic47—22no—
267Dic47—42no—
268Dic47—42no—
269Dic47—82no—
270C2×#529—471no—
271C2×#432—471no—
272C2×#382—471no—
273#1512—478no—
274C2×#328—471no—
275C2×#328—471no—
276C2×#327—471no—
277C2×#327D4711yescharacteristic
278C2×C47—21no—
279C2×C47—21no—
280C2×C47—22no—
281C2×C47—82no—
282C2×C47#151214yescomplemented
283C2×C47C2×#41011yescharacteristic
284C2×C47C2×#41711yescharacteristic
285C2×C47C2×#42111yescharacteristic
286#529—474no—
287#529—474no—
288#529—942no—
289#529—942no—
290C2×#126—941no—
291C2×#125—471no—
292C2×#124—471no—
293C2×#123—941no—
294C2×C22⋊Q8—471no—
295C2×#121C2×D4711yescharacteristic
296C2×C4×Q8—471no—
297C2×C4×D4—471no—
298C2×C4×D4—941no—
299C2×#117—941no—
300C22×C4⋊C4—941no—
301C22×C42—471no—
302#432—474no—
303#432—942no—
304C2×C4⋊C8—941no—
305C2×C22⋊C8—471no—
306C2×C22⋊C8C2×D4711yescharacteristic
307C2×#97—941no—
308#382—474no—
309#382—942no—
310C2×#95—471no—
311C2×#95—941no—
312C2×#95C2×D4711yescharacteristic
313#328—474no—
314#328—474no—
315#327—474no—
316#327C2×D4714yes—
317C47C2×#151211yescharacteristic, complemented, Sylow
318C22×Q8—471no—
319C22×D4—21no—
320C22×D4C4×D4711yescharacteristic
321C23×C4—21no—
322C23×C4—471no—
323C23×C4—471no—
324C23×C4—941no—
325C23×C4—941no—
326C23×C4C22×D4711yescharacteristic
327C22×C8—21no—
328C22×C8—941no—
329#126—944no—
330#126—944no—
331#125—474no—
332#125—942no—
333#124—474no—
334#124—942no—
335#123—944no—
336#123—944no—
337C22⋊Q8—472no—
338C22⋊Q8—472no—
339C22⋊Q8—941no—
340C22⋊Q8—941no—
341#121—21no—
342#121—21no—
343#121C4×D4712yes—
344#121C22×D4712yes—
345C4×Q8—474no—
346C4×Q8—942no—
347C4×D4—474no—
348C4×D4—942no—
349C4×D4—944no—
350C4×D4—944no—
351#117—944no—
352#117—944no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—941no—
355C2×C4⋊C4—941no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—1884no—
360C2×C4⋊C4C4×D4711yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—941no—
363C2×C22⋊C4—941no—
364C2×C22⋊C4—941no—
365C2×C22⋊C4—941no—
366C2×C42—471no—
367C2×C42—471no—
368C2×C42—472no—
369C2×C42—471no—
370C2×C42—471no—
371C2×C42—472no—
372C2×C42—472no—
373C2×C42—472no—
374C2×C42—941no—
375C4⋊C8—944no—
376C22⋊C8—21no—
377C22⋊C8—472no—
378C22⋊C8—941no—
379C22⋊C8C22×D4712yes—
380#97—944no—
381#95—21no—
382#95—472no—
383#95—944no—
384#95—941no—
385#95C22×D4712yes—
386C24—21no—
387C24D4×D4711yescharacteristic
388C2×Q8—472no—
389C2×Q8—941no—
390C2×Q8—1882no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C2×C4×D4712yes—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—472no—
405C22×C4—471no—
406C22×C4—471no—
407C22×C4—471no—
408C22×C4—471no—
409C22×C4—472no—
410C22×C4—472no—
411C22×C4—472no—
412C22×C4—944no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—942no—
418C22×C4—1882no—
419C22×C4—1884no—
420C22×C4—1882no—
421C22×C4—1882no—
422C22×C4—1882no—
423C22×C4C2×C4×D4711yescharacteristic
424C22×C4C2×C4×D4711yescharacteristic
425C22×C4D4×D4711yescharacteristic
426C22×C4D4×D4711yescharacteristic
427C22×C4D4×D4711yescharacteristic
428C22×C4C23×D4711yescharacteristic
429C2×C8—22no—
430C2×C8—42no—
431C2×C8—942no—
432C2×C8—1882no—
433C4⋊C4—21no—
434C4⋊C4—472no—
435C4⋊C4—944no—
436C4⋊C4—944no—
437C4⋊C4—944no—
438C4⋊C4—942no—
439C4⋊C4—941no—
440C4⋊C4—942no—
441C4⋊C4—1888no—
442C4⋊C4C2×C4×D4712yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—944no—
446C22⋊C4—944no—
447C22⋊C4—944no—
448C22⋊C4—942no—
449C22⋊C4—942no—
450C42—472no—
451C42—472no—
452C42—472no—
453C42—472no—
454C42—472no—
455C42—472no—
456C42—472no—
457C42—472no—
458C42—944no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22⋊C4×D4711yescharacteristic
466C23C22⋊C4×D4711yescharacteristic
467C23#1309404812yes—
468C23#1309405312yes—
469C23C2×D4×D4711yescharacteristic
470Q8—1882no—
471Q8—1882no—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—42no—
476D4—42no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—472no—
490C2×C4—472no—
491C2×C4—472no—
492C2×C4—472no—
493C2×C4—944no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—942no—
502C2×C4—3764no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1888no—
506C2×C4—1882no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4C22⋊C4×D4711yescharacteristic
512C2×C4C22⋊C4×D4711yescharacteristic
513C2×C4C22×C4×D4711yescharacteristic
514C2×C4C2×D4×D4711yescharacteristic
515C8—42no—
516C8—1882no—
517C22—24no—
518C22—24no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—84no—
527C22#1568909512yes—
528C22#1568912112yes—
529C22#1568912112yes—
530C22#1568913812yes—
531C22C2×C22⋊C4×D4711yescharacteristic
532C22C2×#1309404811yescharacteristic
533C22C2×#1309405311yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—82no—
540C4—942no—
541C4—942no—
542C4—942no—
543C4—942no—
544C4—3762no—
545C4—1882no—
546C4—1882no—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—82no—
551C2#2024192314yesdirect product factor
552C2C2×#1568909511yescharacteristic
553C2C2×#1568912111yescharacteristic
554C2C2×#1568913811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#327C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#327D47
hypocenterC47C2×#1512
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#15689177C2semidirect
3. C2×#15688987C2semidirect
4. #20241923C2direct (N × H)
5. C2×#15688861C2semidirect
6. C2×#15688859C2semidirect
11. #15689654C4semidirect
14. C2×#13094015C4semidirect
15. C2×#13094011C4semidirect
18. C2×#13093989C4semidirect
20. C2×D4×Dic47C4semidirect
25. #15689177C22semidirect
30. #15688987C22semidirect
33. #15688861C22semidirect
34. #15688859C22semidirect
49. #13094015C2×C4semidirect
51. #13094011C2×C4semidirect
59. #13093989C2×C4semidirect
63. D4×Dic47C2×C4semidirect
282. C2×C47#1512semidirect
317. C47C2×#1512semidirect
551. C2#20241923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896542
C2×#156891772
C2×#156889872
#202419232
C2×#156888612
C2×#156888592
C2×#156888572
C2×C47×#3272
C2×#151247