Finite Group Databasebeta

← C2×#20241984 #26184178

Subgroups — C2×#20241984 #26184178

Order 12032.

1394 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#124×D47C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic, complemented
3C2×#15689075C211yesmaximal, characteristic, complemented
4#20241984C218yesmaximal, direct product factor
5C2×#15688863C211yesmaximal, characteristic, complemented
6C2×#15688862C211yesmaximal, characteristic, complemented
7C2×#15688862C211yesmaximal, characteristic, complemented
8C2×C47×#329C211yesmaximal, characteristic, complemented
9C22×Q8×D47C411yescharacteristic
10C22×D4×D47C411yescharacteristic
11C22×#10625324—21no—
12C2×#13094013C411yescharacteristic
13C2×#13094012C2211yescharacteristic
14C2×#13094011C411yescharacteristic
15C2×#13094010—21no—
16C2×#13094010—21no—
17#124×D47—22no—
18#124×D47—22no—
19#124×D47C414yes—
20#124×D47C2214yes—
21C2×C22⋊C4×D47—21no—
22C2×C22⋊C4×D47—21no—
23C2×C42×D47C2211yescharacteristic
24#15689178—22no—
25#15689178C2214yescomplemented
26C2×Dic47⋊C8—21no—
27C2×(C2×D47)⋊C8—21no—
28#15689075—22no—
29#15689075C2214yescomplemented
30C2×#13093935C2211yescharacteristic, complemented
31C2×#13093935C2211yescharacteristic, complemented
32C2×C8×Dic47—21no—
33#15688863C2214yescomplemented
34#15688862C2214yescomplemented
35#15688862C2214yescomplemented
36C2×#124×C47C2211yescharacteristic
37C2×#97×C47C2211yescharacteristic, complemented
38C2×#97×C47C2211yescharacteristic, complemented
39C47×#329C2214yescomplemented
40C24×D47—21no—
41C22×Dic94—21no—
42C22×Dic94C2×C411yescharacteristic
43C2×Q8×D47—22no—
44C2×Q8×D47—44no—
45C2×Q8×D47C2×C414yes—
46C22×D188C2×C411yescharacteristic
47C22×C22⋊D47—21no—
48C2×D4×D47—22no—
49C2×D4×D47—44no—
50C2×D4×D47C2×C414yes—
51C22×C4×D47—21no—
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47D411yescharacteristic
54C22×C4×D47D411yescharacteristic
55C2×#10625324—28no—
56C2×#10625324—24no—
57C22×C47⋊C8—21no—
58C22×C47⋊C8—21no—
59#13094013—22no—
60#13094013C2×C414yes—
61#13094012—21no—
62#13094012—21no—
63#13094012C2×C412yes—
64#13094012C2312yes—
65#13094011—22no—
66#13094011C2×C414yes—
67#13094010—24no—
68#13094010—24no—
69#13094010—24no—
70#13094010—24no—
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—21no—
73C2×(C2×D47)⋊C4—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C22⋊C4×D47—24no—
78C22⋊C4×D47—24no—
79C22⋊C4×D47—24no—
80C22⋊C4×D47—24no—
81C2×C4×Dic47C2×C411yescharacteristic
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C42×D47C2×C412yes—
85C42×D47D412yes—
86C42×D47D412yes—
87C42×D47C2312yes—
88Dic47⋊C8—24no—
89(C2×D47)⋊C8—24no—
90#13093935—21no—
91#13093935—21no—
92#13093935C2312yescomplemented
93#13093935C2312yescomplemented
94C8×Dic47—24no—
95C22×Q8×C47C2×C411yescharacteristic
96C22×D4×C47C2×C411yescharacteristic
97C22×C8×C47—21no—
98C22×C8×C47—21no—
99#124×C47—21no—
100#124×C47—21no—
101#124×C47C2×C412yes—
102#124×C47C2312yes—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C42×C47C2311yescharacteristic
106#97×C47—21no—
107#97×C47—21no—
108#97×C47C2312yescomplemented
109#97×C47C2312yescomplemented
110C24×C47—21no—
111C23×D47—21no—
112C23×D47—24no—
113C23×D47—44no—
114C23×D47—44no—
115C23×D47—44no—
116C23×D47C22⋊C411yescharacteristic
117C2×Q8×C47—21no—
118C2×Q8×C47—42no—
119C2×Q8×C47C22×C412yes—
120C2×D4×C47—21no—
121C2×D4×C47—42no—
122C2×D4×C47C22×C412yes—
123C22×C4×C47—21no—
124C22×C4×C47C22×C411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C2×C22⋊D47—24no—
128C2×C22⋊D47—44no—
129C22×Dic47—21no—
130C22×Dic47C22⋊C411yescharacteristic
131C22×Dic47C22⋊C411yescharacteristic
132C22×Dic47C22⋊C411yescharacteristic
133Q8×D47—48no—
134Q8×D47—48no—
135D4×D47—48no—
136D4×D47—48no—
137C2×D188—21no—
138C2×D188—42no—
139C2×D188C22×C412yes—
140C2×C4×D47—24no—
141C2×C4×D47—24no—
142C2×C4×D47—24no—
143C2×C4×D47—42no—
144C2×C4×D47—44no—
145C2×C4×D47C22⋊C412yes—
146C2×C4×D47C22⋊C412yes—
147C2×C4×D47C22⋊C412yes—
148C2×C4×D47C22×C412yes—
149C2×C4×D47C2×D412yes—
150C2×C4×D47C2×D412yes—
151C2×Dic94—21no—
152C2×Dic94—24no—
153C2×Dic94—44no—
154C2×Dic94—42no—
155C2×Dic94C22×C412yes—
156C2×C8×C47—24no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C2×C8×C47—42no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C22⋊C4×C47—22no—
164C42×C47C22×C411yescharacteristic
165C42×C47C2×D411yescharacteristic
166C42×C47C2×D411yescharacteristic
167C42×C47C2411yescharacteristic
168C22⋊Dic47—24no—
169C22⋊Dic47—24no—
170(C2×D47)⋊C4—22no—
171(C2×D47)⋊C4—22no—
172(C2×D47)⋊C4—22no—
173(C2×D47)⋊C4—22no—
174(C2×D47)⋊C4—24no—
175(C2×D47)⋊C4—24no—
176C4×Dic47C22⋊C412yes—
177C4×Dic47C22⋊C412yes—
178C4×Dic47C22⋊C412yes—
179C4×Dic47C22×C412yes—
180C4×Dic47C2×D412yes—
181C4×Dic47C2×D412yes—
182C2×C47⋊C8—24no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—42no—
186#10625324—216no—
187C23×C47—22no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47C2×C22⋊C411yescharacteristic
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—48no—
196C22×D47—48no—
197C22×D47—42no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—48no—
201C22×D47—816no—
202C22×D47#10414yes—
203C22×D47C2×C22⋊C412yes—
204Q8×C47—42no—
205Q8×C47—42no—
206D4×C47—42no—
207D4×C47—42no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—42no—
213C2×C4×C47C2×C22⋊C411yescharacteristic
214C2×C4×C47C2×C22⋊C411yescharacteristic
215C2×C4×C47C2×C22⋊C411yescharacteristic
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C22⋊D47—48no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—42no—
224C2×Dic47#10014yes—
225C2×Dic47C2×C22⋊C412yes—
226C2×Dic47C2×C22⋊C412yes—
227C2×Dic47C2×C22⋊C412yes—
228D188—42no—
229D188—42no—
230C4×D47—24no—
231C4×D47—24no—
232C4×D47—24no—
233C4×D47—24no—
234C4×D47—42no—
235C4×D47—42no—
236C4×D47—48no—
237Dic94—42no—
238Dic94—48no—
239Dic94—42no—
240C8×C47—24no—
241C8×C47—42no—
242C47⋊C8—24no—
243C47⋊C8—42no—
244C2×#1527—471nomaximal, Sylow
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—84no—
249C22×C47C2×#10012yes—
250C22×C47C2×#10412yes—
251C22×C47#42112yes—
252C22×C47C22×C22⋊C411yescharacteristic
253C2×D47—22no—
254C2×D47—22no—
255C2×D47—28no—
256C2×D47—42no—
257C2×D47—42no—
258C2×D47—42no—
259C2×D47—816no—
260C2×D47—84no—
261C2×D47C2×#10414yes—
262C4×C47—21no—
263C4×C47—21no—
264C4×C47—21no—
265C4×C47—21no—
266C4×C47—21no—
267C4×C47—21no—
268C4×C47—42no—
269Dic47—22no—
270Dic47—22no—
271Dic47—42no—
272Dic47C2×#10014yes—
273C22×#124—471no—
274C2×#432—471no—
275C2×C4×M16—471no—
276#1527—478no—
277C2×#329—471no—
278C2×#329—471no—
279C2×#329—471no—
280C2×#329D4711yescharacteristic, complemented
281C2×C47—82no—
282C2×C47#152714yescomplemented
283C2×C47C22×#10011yescharacteristic
284C2×C47C22×#10411yescharacteristic
285C2×C47C2×#42111yescharacteristic
286D47—24no—
287D47—82no—
288C23×Q8—471no—
289C23×D4—471no—
290C22×M16—941no—
291C2×#124—471no—
292C2×#124—471no—
293C2×#124—474no—
294C2×#124—474no—
295C2×#124—471no—
296C2×#124—941no—
297C2×#124—941no—
298C2×#124—942no—
299C2×#124—942no—
300C2×#124C2×D4711yescharacteristic
301C22×C22⋊C4—941no—
302C22×C22⋊C4—941no—
303C22×C42—471no—
304#432—474no—
305#432—942no—
306C2×C4⋊C8—941no—
307C2×C22⋊C8—941no—
308C4×M16—474no—
309C4×M16—942no—
310C2×#97—471no—
311C2×#97—471no—
312C2×#97C2×D4711yescharacteristic, complemented
313C2×#97C2×D4711yescharacteristic, complemented
314C2×C4×C8—941no—
315#329—474no—
316#329—474no—
317#329—474no—
318#329C2×D4714yescomplemented
319C47C2×#152711yescharacteristic, complemented, Sylow
320C25—941no—
321C22×Q8—474no—
322C22×Q8—471no—
323C22×Q8—941no—
324C22×Q8—942no—
325C22×Q8—1884no—
326C22×Q8C4×D4711yescharacteristic
327C22×D4—474no—
328C22×D4—471no—
329C22×D4—942no—
330C22×D4—941no—
331C22×D4—1884no—
332C22×D4C4×D4711yescharacteristic
333C23×C4—471no—
334C23×C4—471no—
335C23×C4—471no—
336C23×C4—941no—
337C2×M16—944no—
338C2×M16—948no—
339C22×C8—21no—
340C22×C8—21no—
341C22×C8—941no—
342C22×C8—941no—
343#124—21no—
344#124—21no—
345#124—474no—
346#124—474no—
347#124—472no—
348#124—472no—
349#124—944no—
350#124—944no—
351#124—944no—
352#124—944no—
353#124—942no—
354#124—942no—
355#124—941no—
356#124—941no—
357#124C4×D4712yes—
358#124C22×D4712yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—944no—
362C2×C22⋊C4—944no—
363C2×C22⋊C4—944no—
364C2×C22⋊C4—944no—
365C2×C22⋊C4—941no—
366C2×C22⋊C4—941no—
367C2×C22⋊C4—941no—
368C2×C22⋊C4—941no—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—941no—
371C2×C42—471no—
372C2×C42—471no—
373C2×C42—471no—
374C2×C42—472no—
375C2×C42—472no—
376C2×C42—472no—
377C2×C42—472no—
378C2×C42C22×D4711yescharacteristic
379C4⋊C8—944no—
380C22⋊C8—944no—
381#97—21no—
382#97—21no—
383#97—472no—
384#97—472no—
385#97—941no—
386#97—941no—
387#97C22×D4712yescomplemented
388#97C22×D4712yescomplemented
389C4×C8—944no—
390C24—21no—
391C24—471no—
392C24—944no—
393C24—941no—
394C24—1884no—
395C24—1884no—
396C24—1884no—
397C2×Q8—21no—
398C2×Q8—42no—
399C2×Q8—472no—
400C2×Q8—944no—
401C2×Q8—941no—
402C2×Q8—1882no—
403C2×Q8—1884no—
404C2×Q8—1888no—
405C2×Q8—1888no—
406C2×Q8C2×C4×D4712yes—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—472no—
410C2×D4—944no—
411C2×D4—941no—
412C2×D4—1882no—
413C2×D4—1884no—
414C2×D4—1888no—
415C2×D4—1888no—
416C2×D4C2×C4×D4712yes—
417C22×C4—21no—
418C22×C4—471no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—472no—
422C22×C4—472no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—472no—
427C22×C4—944no—
428C22×C4—944no—
429C22×C4—944no—
430C22×C4—941no—
431C22×C4—1882no—
432C22×C4—1884no—
433C22×C4C2×C4×D4711yescharacteristic
434C22×C4D4×D4711yescharacteristic
435C22×C4D4×D4711yescharacteristic
436M16—9416no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—42no—
441C2×C8—944no—
442C2×C8—942no—
443C2×C8—942no—
444C2×C8—1882no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—944no—
450C22⋊C4—944no—
451C22⋊C4—944no—
452C22⋊C4—944no—
453C22⋊C4—942no—
454C22⋊C4—942no—
455C22⋊C4—942no—
456C22⋊C4—942no—
457C42—472no—
458C42—472no—
459C42—472no—
460C42—472no—
461C42—472no—
462C42—472no—
463C42C2×C4×D4711yescharacteristic
464C42D4×D4711yescharacteristic
465C42D4×D4711yescharacteristic
466C42C23×D4711yescharacteristic
467C23—22no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—472no—
472C23—474no—
473C23—942no—
474C23—942no—
475C23—942no—
476C23—37616no—
477C23—1888no—
478C23—1888no—
479C23—1888no—
480C23—1882no—
481C23—1882no—
482C23—1882no—
483C23C22⋊C4×D4711yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—1882no—
487Q8—1882no—
488Q8—1888no—
489D4—42no—
490D4—42no—
491D4—1882no—
492D4—1882no—
493D4—1888no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—472no—
500C2×C4—474no—
501C2×C4—472no—
502C2×C4—472no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—944no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—1882no—
511C2×C4—1882no—
512C2×C4—1888no—
513C2×C4—1882no—
514C2×C4C22⋊C4×D4711yescharacteristic
515C2×C4C22⋊C4×D4711yescharacteristic
516C2×C4C22⋊C4×D4711yescharacteristic
517C2×C4C22×C4×D4711yescharacteristic
518C2×C4C2×D4×D4711yescharacteristic
519C2×C4C2×D4×D4711yescharacteristic
520C8—24no—
521C8—42no—
522C8—944no—
523C8—1882no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—84no—
528C22—474no—
529C22—942no—
530C22—948no—
531C22—942no—
532C22—37616no—
533C22—3764no—
534C22—1882no—
535C22—1882no—
536C22—1882no—
537C22#100×D4712yes—
538C22#104×D4712yes—
539C22#1568913812yes—
540C22C2×C22⋊C4×D4711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—42no—
548C4—474no—
549C4—942no—
550C4—942no—
551C4—1882no—
552C2—82no—
553C2—944no—
554C2—3762no—
555C2#2024198414yesdirect product factor
556C2C2×#100×D4711yescharacteristic
557C2C2×#104×D4711yescharacteristic
558C2C2×#1568913811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC2×C47×#329C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×D47C4
cosocleC42×C47C24
hypercenterC2×#329D47
hypocenterC47C2×#1527
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. C2×#15689178C2semidirect
3. C2×#15689075C2semidirect
4. #20241984C2direct (N × H)
5. C2×#15688863C2semidirect
6. C2×#15688862C2semidirect
7. C2×#15688862C2semidirect
8. C2×C47×#329C2semidirect
25. #15689178C22semidirect
29. #15689075C22semidirect
30. C2×#13093935C22semidirect
31. C2×#13093935C22semidirect
33. #15688863C22semidirect
34. #15688862C22semidirect
35. #15688862C22semidirect
37. C2×#97×C47C22semidirect
38. C2×#97×C47C22semidirect
39. C47×#329C22semidirect
92. #13093935C23semidirect
93. #13093935C23semidirect
108. #97×C47C23semidirect
109. #97×C47C23semidirect
280. C2×#329D47semidirect
282. C2×C47#1527semidirect
312. C2×#97C2×D47semidirect
313. C2×#97C2×D47semidirect
318. #329C2×D47semidirect
319. C47C2×#1527semidirect
387. #97C22×D47semidirect
388. #97C22×D47semidirect
555. C2#20241984direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#124×D472
C2×#156891782
C2×#156890752
#202419842
C2×#156888632
C2×#156888622
C2×C47×#3292
C2×#152747