Finite Group Databasebeta

← C2×#20247115 #26384368

Subgroups — C2×#20247115 #26384368

Order 12032.

2942 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094015C211yesmaximal, characteristic, complemented
2C22×C4⋊C4×D47C211yesmaximal, characteristic
3#20247115C2116yesmaximal, direct product factor
4C2×#15689064C212yesmaximal, complemented
5C2×#15689057C212yesmaximal, complemented
6C2×#15689021C211yesmaximal, characteristic, complemented
7C2×#15689019C211yesmaximal, characteristic, complemented
8C2×#15688997C212yesmaximal
9C2×#15688982C212yesmaximal
10C2×#15688980C211yesmaximal, characteristic
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
12C2×C47×#382C211yesmaximal, characteristic, complemented
13C23×C4×D47C2212yes—
14C23×C4×D47C2211yescharacteristic
15C2×#13094015—28no—
16C2×#13094015C2218yescomplemented
17C22×C4⋊Dic47C2212yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C2×C4⋊C4×D47—28no—
25C2×C4⋊C4×D47C2218yes—
26C22×(C2×D47)⋊C4C2211yescharacteristic
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×C4×Dic47C2211yescharacteristic
30#15689064C22116yescomplemented
31#15689057C22116yescomplemented
32#15689021C2218yescomplemented
33#15689019C2218yescomplemented
34#15688997C22116yes—
35#15688982C22116yes—
36#15688980C2218yes—
37C4×(C2×D47)⋊C4C2218yes—
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2211yescharacteristic
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C42×C47C2211yescharacteristic
46C47×#382C2218yescomplemented
47C2×#95×C47C2212yes—
48C2×#95×C47C2212yes—
49C24×D47C2311yescharacteristic
50C23×Dic47C2311yescharacteristic
51C23×Dic47C2312yes—
52C23×Dic47C2311yescharacteristic
53C23×Dic47C2312yes—
54C23×Dic47C2311yescharacteristic
55C22×C4×D47—24no—
56C22×C4×D47—24no—
57C22×C4×D47—28no—
58C22×C4×D47—28no—
59C22×C4×D47—44no—
60C22×C4×D47Q818yes—
61C22×C4×D47C2318yes—
62C22×C4×D47C2314yes—
63#13094015—232no—
64C2×C4⋊Dic47—24no—
65C2×C4⋊Dic47—28no—
66C2×C4⋊Dic47D418yes—
67C2×C4⋊Dic47C2314yes—
68C2×C4⋊Dic47C2318yes—
69C2×C4⋊Dic47C2314yes—
70C2×Dic47⋊C4—28no—
71C2×Dic47⋊C4—28no—
72C2×Dic47⋊C4—44no—
73C2×Dic47⋊C4—44no—
74C2×Dic47⋊C4C2318yes—
75C2×Dic47⋊C4C2318yes—
76C2×Dic47⋊C4C2318yes—
77C2×Dic47⋊C4C2318yes—
78C4⋊C4×D47—232no—
79C2×(C2×D47)⋊C4—24no—
80C2×(C2×D47)⋊C4—28no—
81C2×(C2×D47)⋊C4—24no—
82C2×(C2×D47)⋊C4C2314yes—
83C2×(C2×D47)⋊C4C2314yes—
84C2×(C2×D47)⋊C4C2318yes—
85C2×C4×Dic47—24no—
86C2×C4×Dic47C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2318yes—
90C47⋊#95C2318yes—
91C47⋊#95C2318yes—
92C23×C4×C47C2311yescharacteristic
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2312yes—
95C23×C4×C47C2312yes—
96C23×C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47—24no—
99C2×C4⋊C4×C47C2314yes—
100C2×C4⋊C4×C47C2314yes—
101C2×C42×C47—24no—
102C2×C42×C47C2314yes—
103#95×C47C2318yes—
104#95×C47C2318yes—
105C24×C47C2411yescharacteristic
106C23×D47—22no—
107C23×D47—22no—
108C23×D47—22no—
109C23×D47—22no—
110C23×D47C2×Q814yes—
111C23×D47#4114yes—
112C23×D47#4114yes—
113C23×D47C2412yes—
114C22×C4×C47—24no—
115C22×C4×C47—24no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—24no—
125C22×C4×C47—22no—
126C22×C4×C47C2×D414yes—
127C22×C4×C47C2×Q814yes—
128C22×C4×C47#4114yes—
129C22×C4×C47C2412yes—
130C22×C4×C47C2412yes—
131C22×C4×C47C2414yes—
132C22×C4×C47C2414yes—
133C22×C4×C47C2412yes—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—24no—
137C22×Dic47—24no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—24no—
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—22no—
147C22×Dic47C2×D418yes—
148C22×Dic47C2×Q814yes—
149C22×Dic47C2412yes—
150C22×Dic47C2414yes—
151C22×Dic47C2412yes—
152C22×Dic47C2414yes—
153C22×Dic47C2412yes—
154C2×C4×D47—232no—
155C2×C4×D47—28no—
156C2×C4×D47—28no—
157C2×C4×D47—216no—
158C2×C4×D47—216no—
159C2×C4×D47—48no—
160C2×C4×D47—432no—
161C2×C4×D47C2×Q8116yes—
162C4⋊C4×C47—28no—
163C4⋊C4×C47—28no—
164C42×C47—28no—
165(C2×D47)⋊C4—28no—
166(C2×D47)⋊C4—216no—
167(C2×D47)⋊C4—28no—
168C4⋊Dic47—28no—
169C4⋊Dic47—216no—
170C4⋊Dic47C2×D4116yes—
171Dic47⋊C4—216no—
172Dic47⋊C4—216no—
173Dic47⋊C4—48no—
174Dic47⋊C4—48no—
175C4×Dic47—28no—
176C23×C47C22×D412yes—
177C23×C47C22×Q812yes—
178C23×C47C2×#4112yes—
179C23×C47C2×#4112yes—
180C23×C47C2×#4112yes—
181C23×C47H(5,𝔽2)12yes—
182C23×C472−1+412yes—
183C23×C47C2511yescharacteristic
184C22×D47—28no—
185C22×D47—28no—
186C22×D47—28no—
187C22×D47—28no—
188C22×D47—28no—
189C22×D47—28no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47#125116yescomplemented
195C22×D47C22×Q814yes—
196C22×D47C2×#4114yes—
197C22×D47C2×#4114yes—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—48no—
211C2×C4×C47—48no—
212C2×C4×C47—44no—
213C2×C4×C47—44no—
214C2×C4×C47C22⋊Q8116yes—
215C2×C4×C47C22×D414yes—
216C2×C4×C47C22×Q814yes—
217C2×C4×C47C2×#4114yes—
218C2×Dic47—28no—
219C2×Dic47—216no—
220C2×Dic47—22no—
221C2×Dic47—24no—
222C2×Dic47—22no—
223C2×Dic47—24no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—44no—
234C2×Dic47—44no—
235C2×Dic47—48no—
236C2×Dic47C22×D418yes—
237C2×Dic47C22×Q814yes—
238C4×D47—232no—
239C4×D47—432no—
240C2×#2442—471nomaximal, Sylow
241C22×C47C2×C22⋊Q814yes—
242C22×C47C2×#12514yes—
243C22×C47#54414yes—
244C22×C47#54614yes—
245C22×C47#54814yes—
246C22×C47#55214yes—
247C22×C47#55314yes—
248C22×C47C23×D411yescharacteristic
249C22×C47C23×Q811yescharacteristic
250C22×C47C22×#4111yescharacteristic
251C22×C47C22×#4111yescharacteristic
252C22×C47C22×#4111yescharacteristic
253C22×C47C2×H(5,𝔽2)11yescharacteristic
254C22×C47C2×2−1+411yescharacteristic
255C2×D47—24no—
256C2×D47—24no—
257C2×D47—24no—
258C2×D47—24no—
259C2×D47—232no—
260C2×D47—24no—
261C2×D47—24no—
262C2×D47C2×#12518yescomplemented
263C4×C47—44no—
264C4×C47—44no—
265C4×C47—42no—
266C4×C47—42no—
267C4×C47C2×C22⋊Q818yes—
268Dic47—24no—
269Dic47—28no—
270Dic47—42no—
271Dic47—42no—
272Dic47—44no—
273C22×#125—471no—
274C23×C4⋊C4—471no—
275#2442—4716no—
276C2×#400—472no—
277C2×#398—472no—
278C2×#389—471no—
279C2×#389—471no—
280C2×#385—472no—
281C2×#382D4711yescharacteristic, complemented
282C2×#380—472no—
283C2×#380—471no—
284C2×C4×C22⋊C4—471no—
285C2×C47#244218yescomplemented
286C2×C47C22×C22⋊Q811yescharacteristic
287C2×C47C22×#12511yescharacteristic
288C2×C47C2×#54411yescharacteristic
289C2×C47C2×#54611yescharacteristic
290C2×C47C2×#54811yescharacteristic
291C2×C47C2×#55211yescharacteristic
292C2×C47C2×#55311yescharacteristic
293D47—28no—
294C24×C4—471no—
295C24×C4—472no—
296C2×#125—478no—
297C2×#125—948no—
298C22×C4⋊C4—471no—
299C22×C4⋊C4—472no—
300C22×C4⋊C4—472no—
301C22×C4⋊C4—472no—
302C22×C4⋊C4—471no—
303C22×C4⋊C4—472no—
304C22×C4⋊C4—472no—
305C22×C4⋊C4—478no—
306C22×C4⋊C4—948no—
307C22×C4⋊C4C2×D4711yescharacteristic
308C22×C4⋊C4C2×D4711yescharacteristic
309C22×C22⋊C4—471no—
310C22×C22⋊C4—471no—
311C22×C22⋊C4—472no—
312C22×C42—471no—
313C22×C42C2×D4711yescharacteristic
314#400—4716no—
315#398—4716no—
316#389—478no—
317#389—478no—
318#385—4716no—
319#382C2×D4718yescomplemented
320#380—478no—
321#380—4716no—
322C4×C22⋊C4—478no—
323C2×#95—471no—
324C2×#95—472no—
325C2×#95—471no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95C2×D4712yes—
329C2×#95C2×D4712yes—
330C47C2×#244211yescharacteristic, complemented, Sylow
331C25—471no—
332C23×C4—472no—
333C23×C4—471no—
334C23×C4—478no—
335C23×C4—474no—
336C23×C4—471no—
337C23×C4—472no—
338C23×C4—471no—
339C23×C4—478no—
340C23×C4—948no—
341C23×C4—948no—
342C23×C4—944no—
343C23×C4—944no—
344C23×C4—1884no—
345C23×C4C22×D4712yes—
346C23×C4C22×D4711yescharacteristic
347C23×C4C22×D4711yescharacteristic
348C23×C4C22×D4712yes—
349C23×C4C22×D4711yescharacteristic
350#125—9432no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—478no—
354C2×C4⋊C4—474no—
355C2×C4⋊C4—478no—
356C2×C4⋊C4—478no—
357C2×C4⋊C4—478no—
358C2×C4⋊C4—474no—
359C2×C4⋊C4—478no—
360C2×C4⋊C4—478no—
361C2×C4⋊C4—9432no—
362C2×C4⋊C4—948no—
363C2×C4⋊C4—948no—
364C2×C4⋊C4—948no—
365C2×C4⋊C4—944no—
366C2×C4⋊C4—1884no—
367C2×C4⋊C4—1884no—
368C2×C4⋊C4C22×D4714yes—
369C2×C4⋊C4C22×D4714yes—
370C2×C22⋊C4—474no—
371C2×C22⋊C4—474no—
372C2×C22⋊C4—478no—
373C2×C22⋊C4—944no—
374C2×C22⋊C4—948no—
375C2×C22⋊C4—944no—
376C2×C42—24no—
377C2×C42—474no—
378C2×C42—944no—
379C2×C42C22×D4714yes—
380#95—474no—
381#95—474no—
382#95—478no—
383#95—478no—
384#95—478no—
385#95C22×D4718yes—
386#95C22×D4718yes—
387C24—474no—
388C24—474no—
389C24—474no—
390C24—472no—
391C24—942no—
392C24—942no—
393C24—942no—
394C24—942no—
395C24C23×D4711yescharacteristic
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—478no—
409C22×C4—474no—
410C22×C4—472no—
411C22×C4—474no—
412C22×C4—4716no—
413C22×C4—472no—
414C22×C4—474no—
415C22×C4—472no—
416C22×C4—9416no—
417C22×C4—9416no—
418C22×C4—942no—
419C22×C4—944no—
420C22×C4—942no—
421C22×C4—944no—
422C22×C4—9432no—
423C22×C4—948no—
424C22×C4—948no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—944no—
430C22×C4—944no—
431C22×C4—942no—
432C22×C4—944no—
433C22×C4—942no—
434C22×C4—1888no—
435C22×C4—18832no—
436C22×C4#1062535014yes—
437C22×C4D4×D4714yes—
438C22×C4Q8×D4714yes—
439C22×C4C23×D4712yes—
440C22×C4C23×D4712yes—
441C22×C4C23×D4714yes—
442C22×C4C23×D4714yes—
443C22×C4C23×D4712yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—4716no—
447C4⋊C4—9416no—
448C4⋊C4—9416no—
449C4⋊C4—9416no—
450C4⋊C4—948no—
451C4⋊C4—1888no—
452C4⋊C4—1888no—
453C22⋊C4—948no—
454C22⋊C4—9416no—
455C22⋊C4—948no—
456C42—28no—
457C42—948no—
458C23—4716no—
459C23—474no—
460C23—474no—
461C23—474no—
462C23—948no—
463C23—948no—
464C23—948no—
465C23—948no—
466C23—948no—
467C23—948no—
468C23—942no—
469C23—942no—
470C23—942no—
471C23—942no—
472C23C2×D4×D4712yes—
473C23C2×Q8×D4712yes—
474C23#41×D4712yes—
475C23#41×D4712yes—
476C23C2×#1062535012yes—
477C23#1309407012yes—
478C23#1309407112yes—
479C23C24×D4711yescharacteristic
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—478no—
497C2×C4—474no—
498C2×C4—9416no—
499C2×C4—948no—
500C2×C4—942no—
501C2×C4—944no—
502C2×C4—942no—
503C2×C4—944no—
504C2×C4—9432no—
505C2×C4—942no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—942no—
512C2×C4—944no—
513C2×C4—942no—
514C2×C4—18832no—
515C2×C4—1884no—
516C2×C4—1884no—
517C2×C4—1888no—
518C2×C4#13094002116yes—
519C2×C4C2×D4×D4714yes—
520C2×C4C2×Q8×D4714yes—
521C2×C4C2×#1062535014yes—
522C22—478no—
523C22—944no—
524C22—944no—
525C22—9432no—
526C22—944no—
527C22—944no—
528C22—944no—
529C22—944no—
530C22C2×#1309400214yes—
531C22#125×D4714yes—
532C22#1568975314yes—
533C22#1568976614yes—
534C22#1568978114yes—
535C22#1568981214yes—
536C22#1568981714yes—
537C22C22×D4×D4711yescharacteristic
538C22C22×Q8×D4711yescharacteristic
539C22C2×#41×D4711yescharacteristic
540C22C2×#41×D4711yescharacteristic
541C22C22×#1062535011yescharacteristic
542C22C2×#1309407011yescharacteristic
543C22C2×#1309407111yescharacteristic
544C4—44no—
545C4—44no—
546C4—42no—
547C4—42no—
548C4—948no—
549C4—944no—
550C4—1882no—
551C4—1882no—
552C4—1884no—
553C4C2×#1309400218yes—
554C2—948no—
555C2#2024711518yesdirect product factor
556C2C22×#1309400211yescharacteristic
557C2C2×#125×D4711yescharacteristic
558C2C2×#1568975311yescharacteristic
559C2C2×#1568976611yescharacteristic
560C2C2×#1568978111yescharacteristic
561C2C2×#1568981211yescharacteristic
562C2C2×#1568981711yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#382C2
socleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×#382D47
hypocenterC47C2×#2442
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
1. C22×#13094015C2semidirect
3. #20247115C2direct (N × H)
4. C2×#15689064C2semidirect
5. C2×#15689057C2semidirect
6. C2×#15689021C2semidirect
7. C2×#15689019C2semidirect
12. C2×C47×#382C2semidirect
16. C2×#13094015C22semidirect
30. #15689064C22semidirect
31. #15689057C22semidirect
32. #15689021C22semidirect
33. #15689019C22semidirect
46. C47×#382C22semidirect
194. C22×D47#125semidirect
262. C2×D47C2×#125semidirect
281. C2×#382D47semidirect
285. C2×C47#2442semidirect
319. #382C2×D47semidirect
330. C47C2×#2442semidirect
555. C2#20247115direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940152
C22×C4⋊C4×D472
#202471152
C2×#156890642
C2×#156890572
C2×#156890212
C2×#156890192
C2×#156889972
C2×#156889822
C2×#156889802
C2×C4×(C2×D47)⋊C42
C2×C47×#3822
C2×#244247