Finite Group Databasebeta

← C2×#20251421 #26422409

Subgroups — C2×#20251421 #26422409

Order 12032.

2998 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094002C211yesmaximal, characteristic, complemented
2C22×#13093999C211yesmaximal, characteristic
3#20251421C2116yesmaximal, direct product factor
4C2×#15689058C212yesmaximal, complemented
5C2×#15689054C211yesmaximal, characteristic, complemented
6C2×#15689041C212yesmaximal, complemented
7C2×#15689041C211yesmaximal, characteristic, complemented
8C2×#15689040C211yesmaximal, characteristic, complemented
9C2×#15689040C212yesmaximal, complemented
10C2×#15689036C212yesmaximal, complemented
11C2×#15688985C211yesmaximal, characteristic, complemented
12C2×C47×#398C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×D188C2212yes—
15C23×D188C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13094002—28no—
18C2×#13094002C2218yescomplemented
19C2×#13093999—28no—
20C2×#13093999C2218yes—
21C22×Dic47⋊C4C2211yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×(C2×D47)⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2211yescharacteristic
26C22×(C2×D47)⋊C4C2211yescharacteristic
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32#15689058C22116yescomplemented
33#15689054C2218yescomplemented
34#15689041C22116yescomplemented
35#15689041C2218yescomplemented
36#15689040C2218yescomplemented
37#15689040C22116yescomplemented
38#15689036C22116yescomplemented
39#15688985C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2212yes—
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C47×#398C2218yescomplemented
47C2×#95×C47C2211yescharacteristic
48C2×#95×C47C2212yes—
49C24×D47C2312yes—
50C24×D47C2311yescharacteristic
51C22×Dic94—24no—
52C22×Dic94—44no—
53C22×Dic94C2314yes—
54C22×D188—28no—
55C22×D188—24no—
56C22×D188—44no—
57C22×D188—44no—
58C22×D188—44no—
59C22×D188C2318yes—
60C22×D188C2314yes—
61C23×Dic47C2312yes—
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2311yescharacteristic
65C22×C4×D47—44no—
66C22×C4×D47D418yes—
67C22×C4×D47C2314yes—
68#13094002—232no—
69#13093999—232no—
70C2×Dic47⋊C4—24no—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4—44no—
73C2×Dic47⋊C4C2314yes—
74C2×Dic47⋊C4C2314yes—
75C2×Dic47⋊C4C2318yes—
76C2×(C2×D47)⋊C4—24no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—28no—
79C2×(C2×D47)⋊C4—44no—
80C2×(C2×D47)⋊C4—44no—
81C2×(C2×D47)⋊C4—44no—
82C2×(C2×D47)⋊C4—44no—
83C2×(C2×D47)⋊C4—44no—
84C2×(C2×D47)⋊C4C2318yes—
85C2×(C2×D47)⋊C4C2314yes—
86C2×(C2×D47)⋊C4C2314yes—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C47⋊#95C2318yes—
93C47⋊#95C2314yes—
94C47⋊#95C2318yes—
95C23×C4×C47C2312yes—
96C23×C4×C47C2312yes—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47—24no—
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47—44no—
103C2×C4⋊C4×C47C2314yes—
104C2×C4⋊C4×C47C2314yes—
105C2×C4⋊C4×C47C2318yes—
106#95×C47C2314yes—
107#95×C47C2318yes—
108C24×C47C2411yescharacteristic
109C23×D47—24no—
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—24no—
113C23×D47—24no—
114C23×D47—24no—
115C23×D47—24no—
116C23×D47—24no—
117C23×D47—24no—
118C23×D47—24no—
119C23×D47—24no—
120C23×D47C2×D414yes—
121C23×D47C2414yes—
122C23×D47C2412yes—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—22no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2412yes—
137C22×C4×C47C2414yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C22×Dic47—24no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—22no—
149C22×Dic47—24no—
150C22×Dic47C2×D414yes—
151C22×Dic47C2414yes—
152C22×Dic47C2412yes—
153C22×Dic47C2412yes—
154C22×Dic47C2412yes—
155C2×D188—216no—
156C2×D188—28no—
157C2×D188—48no—
158C2×D188—48no—
159C2×D188—48no—
160C2×D188—416no—
161C2×D188—432no—
162C2×C4×D47—416no—
163C2×C4×D47—48no—
164C2×C4×D47C2×D4116yes—
165C2×Dic94—28no—
166C2×Dic94—48no—
167C2×Dic94—416no—
168C4⋊C4×C47—28no—
169C4⋊C4×C47—28no—
170C4⋊C4×C47—48no—
171(C2×D47)⋊C4—28no—
172(C2×D47)⋊C4—28no—
173(C2×D47)⋊C4—216no—
174(C2×D47)⋊C4—48no—
175(C2×D47)⋊C4—48no—
176(C2×D47)⋊C4—48no—
177(C2×D47)⋊C4—48no—
178(C2×D47)⋊C4—48no—
179Dic47⋊C4—28no—
180Dic47⋊C4—28no—
181Dic47⋊C4—48no—
182C23×C47C22×D412yes—
183C23×C47H(5,𝔽2)12yes—
184C23×C47H(5,𝔽2)14yes—
185C23×C47H(5,𝔽2)12yes—
186C23×C472−1+414yes—
187C23×C47C2511yescharacteristic
188C22×D47—28no—
189C22×D47—28no—
190C22×D47—28no—
191C22×D47—24no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—24no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—24no—
202C22×D47—48no—
203C22×D47—48no—
204C22×D47—48no—
205C22×D47—48no—
206C22×D47—48no—
207C22×D47—48no—
208C22×D47—48no—
209C22×D47—48no—
210C22×D47—48no—
211C22×D47C22×D414yes—
212C2×C4×C47—28no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—44no—
226C2×C4×C47—48no—
227C2×C4×C47—44no—
228C2×C4×C47—48no—
229C2×C4×C47C22×D414yes—
230C2×Dic47—28no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—22no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—44no—
241C2×Dic47—44no—
242C2×Dic47—48no—
243C2×Dic47C22×D414yes—
244D188—416no—
245D188—432no—
246C4×D47—416no—
247Dic94—416no—
248C2×#2816—471nomaximal, Sylow
249C22×C47#53414yes—
250C22×C47#53514yes—
251C22×C47#53614yes—
252C22×C47#56218yes—
253C22×C47#56218yes—
254C22×C47C23×D411yescharacteristic
255C22×C47C2×H(5,𝔽2)11yescharacteristic
256C22×C47C2×H(5,𝔽2)12yes—
257C22×C47C2×H(5,𝔽2)11yescharacteristic
258C22×C47C2×2−1+412yes—
259C2×D47—24no—
260C2×D47—24no—
261C2×D47—24no—
262C2×D47—416no—
263C2×D47—44no—
264C2×D47—44no—
265C2×D47—44no—
266C2×D47—44no—
267C2×D47—44no—
268C2×D47—44no—
269C2×D47—44no—
270C2×D47—44no—
271C2×D47—44no—
272C2×D47—816no—
273C4×C47—24no—
274C4×C47—42no—
275C4×C47—44no—
276C4×C47—42no—
277C4×C47—44no—
278Dic47—24no—
279Dic47—42no—
280Dic47—42no—
281Dic47—44no—
282C22×C22⋊Q8—471no—
283C22×#121—471no—
284#2816—4716no—
285C2×#399—472no—
286C2×#398—471no—
287C2×#398D4711yescharacteristic, complemented
288C2×#394—472no—
289C2×#394—471no—
290C2×#394—472no—
291C2×#394—471no—
292C2×#393—472no—
293C2×#381—471no—
294C2×C47#281618yescomplemented
295C2×C47C2×#53411yescharacteristic
296C2×C47C2×#53511yescharacteristic
297C2×C47C2×#53611yescharacteristic
298C2×C47C2×#56212yes—
299C2×C47C2×#56212yes—
300D47—44no—
301D47—84no—
302C23×Q8—471no—
303C23×D4—472no—
304C23×D4—471no—
305C24×C4—471no—
306C2×C22⋊Q8—478no—
307C2×C22⋊Q8—948no—
308C2×#121—478no—
309C2×#121—948no—
310C22×C4⋊C4—471no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—472no—
313C22×C4⋊C4C2×D4711yescharacteristic
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C4⋊C4C2×D4712yes—
316C22×C22⋊C4—472no—
317C22×C22⋊C4—471no—
318C22×C22⋊C4—471no—
319C22×C22⋊C4—472no—
320C22×C22⋊C4—472no—
321C22×C22⋊C4—472no—
322C22×C22⋊C4—472no—
323C22×C22⋊C4—472no—
324#399—4716no—
325#398—478no—
326#398C2×D4718yescomplemented
327#394—478no—
328#394—4716no—
329#394—478no—
330#394—4716no—
331#393—4716no—
332#381—478no—
333C2×#95—472no—
334C2×#95—472no—
335C2×#95—471no—
336C2×#95C2×D4712yes—
337C2×#95C2×D4711yescharacteristic
338C47C2×#281611yescharacteristic, complemented, Sylow
339C25—471no—
340C25—472no—
341C22×Q8—474no—
342C22×Q8—944no—
343C22×Q8—1884no—
344C22×D4—478no—
345C22×D4—474no—
346C22×D4—948no—
347C22×D4—944no—
348C22×D4—1884no—
349C22×D4—1884no—
350C22×D4—1884no—
351C23×C4—471no—
352C23×C4—478no—
353C23×C4—474no—
354C23×C4—471no—
355C23×C4—471no—
356C23×C4—472no—
357C23×C4—1884no—
358C23×C4C22×D4711yescharacteristic
359C23×C4C22×D4711yescharacteristic
360C23×C4C22×D4712yes—
361C23×C4C22×D4712yes—
362C23×C4C22×D4711yescharacteristic
363C22⋊Q8—9432no—
364#121—9432no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—474no—
370C2×C4⋊C4—478no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—1884no—
374C2×C4⋊C4C22×D4714yes—
375C2×C4⋊C4C22×D4714yes—
376C2×C4⋊C4C22×D4718yes—
377C2×C22⋊C4—478no—
378C2×C22⋊C4—474no—
379C2×C22⋊C4—474no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—478no—
382C2×C22⋊C4—478no—
383C2×C22⋊C4—478no—
384C2×C22⋊C4—478no—
385C2×C22⋊C4—948no—
386C2×C22⋊C4—944no—
387C2×C22⋊C4—944no—
388C2×C22⋊C4—1884no—
389C2×C22⋊C4—1884no—
390C2×C22⋊C4—1884no—
391C2×C22⋊C4—1884no—
392C2×C22⋊C4—1884no—
393#95—478no—
394#95—474no—
395#95—478no—
396#95C22×D4714yes—
397#95C22×D4718yes—
398C24—474no—
399C24—472no—
400C24—474no—
401C24—942no—
402C24—944no—
403C24—942no—
404C24—944no—
405C24—944no—
406C24—944no—
407C24—944no—
408C24—944no—
409C24—944no—
410C24—944no—
411C24—944no—
412C24C23×D4711yescharacteristic
413C2×Q8—948no—
414C2×Q8—1888no—
415C2×Q8—18816no—
416C2×D4—9416no—
417C2×D4—948no—
418C2×D4—1888no—
419C2×D4—1888no—
420C2×D4—1888no—
421C2×D4—18816no—
422C2×D4—18832no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—474no—
436C22×C4—472no—
437C22×C4—4716no—
438C22×C4—474no—
439C22×C4—472no—
440C22×C4—472no—
441C22×C4—944no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—944no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—942no—
449C22×C4—944no—
450C22×C4—18816no—
451C22×C4—1888no—
452C22×C4D4×D4714yes—
453C22×C4C23×D4714yes—
454C22×C4C23×D4714yes—
455C22×C4C23×D4712yes—
456C22×C4C23×D4712yes—
457C22×C4C23×D4712yes—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—48no—
461C4⋊C4—948no—
462C4⋊C4—948no—
463C4⋊C4—1888no—
464C22⋊C4—9416no—
465C22⋊C4—948no—
466C22⋊C4—948no—
467C22⋊C4—1888no—
468C22⋊C4—1888no—
469C22⋊C4—1888no—
470C22⋊C4—1888no—
471C22⋊C4—1888no—
472C23—474no—
473C23—948no—
474C23—948no—
475C23—942no—
476C23—948no—
477C23—944no—
478C23—942no—
479C23—944no—
480C23—944no—
481C23—944no—
482C23—944no—
483C23—944no—
484C23—944no—
485C23—944no—
486C23—944no—
487C23—1888no—
488C23—1888no—
489C23—1888no—
490C23—1888no—
491C23—1888no—
492C23—1888no—
493C23—1888no—
494C23—1888no—
495C23—1888no—
496C23C2×D4×D4712yes—
497C23#1309406912yes—
498C23#1309407014yes—
499C23#1309407012yes—
500C23#1309407214yes—
501C23C24×D4711yescharacteristic
502Q8—18816no—
503D4—18816no—
504D4—18832no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—474no—
523C2×C4—948no—
524C2×C4—944no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—944no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—942no—
532C2×C4—944no—
533C2×C4—18816no—
534C2×C4—1888no—
535C2×C4—1884no—
536C2×C4—1884no—
537C2×C4C2×D4×D4714yes—
538C22—944no—
539C22—944no—
540C22—944no—
541C22—37616no—
542C22—1884no—
543C22—1884no—
544C22—1884no—
545C22—1884no—
546C22—1884no—
547C22—1884no—
548C22—1884no—
549C22—18816no—
550C22—1884no—
551C22—1884no—
552C22#1568968414yes—
553C22#1568968914yes—
554C22#1568969814yes—
555C22#1568986918yes—
556C22#1568987118yes—
557C22C22×D4×D4711yescharacteristic
558C22C2×#1309406911yescharacteristic
559C22C2×#1309407012yes—
560C22C2×#1309407011yescharacteristic
561C22C2×#1309407212yes—
562C4—24no—
563C4—42no—
564C4—44no—
565C4—42no—
566C4—44no—
567C4—944no—
568C4—1882no—
569C4—1882no—
570C4—1884no—
571C2—3764no—
572C2—1884no—
573C2#2025142118yesdirect product factor
574C2C2×#1568968411yescharacteristic
575C2C2×#1568968911yescharacteristic
576C2C2×#1568969811yescharacteristic
577C2C2×#1568986912yes—
578C2C2×#1568987112yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#398C2
socleC24×C47C24
prime radicalC22×#13093999C2
hypercenterC2×#398D47
hypocenterC47C2×#2816
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
1. C22×#13094002C2semidirect
3. #20251421C2direct (N × H)
4. C2×#15689058C2semidirect
5. C2×#15689054C2semidirect
6. C2×#15689041C2semidirect
7. C2×#15689041C2semidirect
8. C2×#15689040C2semidirect
9. C2×#15689040C2semidirect
10. C2×#15689036C2semidirect
11. C2×#15688985C2semidirect
12. C2×C47×#398C2semidirect
18. C2×#13094002C22semidirect
32. #15689058C22semidirect
33. #15689054C22semidirect
34. #15689041C22semidirect
35. #15689041C22semidirect
36. #15689040C22semidirect
37. #15689040C22semidirect
38. #15689036C22semidirect
39. #15688985C22semidirect
46. C47×#398C22semidirect
287. C2×#398D47semidirect
294. C2×C47#2816semidirect
326. #398C2×D47semidirect
338. C47C2×#2816semidirect
573. C2#20251421direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940022
C22×#130939992
#202514212
C2×#156890582
C2×#156890542
C2×#156890412
C2×#156890402
C2×#156890362
C2×#156889852
C2×C47×#3982
C2×#281647