Finite Group Databasebeta

← C2×#20243725 #26217597

Subgroups — C2×#20243725 #26217597

Order 12032.

2254 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625336C211yesmaximal, characteristic
2C22×C4×D188C211yesmaximal, characteristic
3C22×#13093948C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic
5#20243725C218yesmaximal, direct product factor
6C2×#15688885C211yesmaximal, characteristic, complemented
7C2×#15688878C211yesmaximal, characteristic, complemented
8C2×C47×#391C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C23×C4×D47—21no—
11C22×#10625336—24no—
12C22×#10625336—24no—
13C22×#10625336C418yescomplemented
14C22×#10625336C2214yes—
15C23×C47⋊C8C2211yescharacteristic
16C23×C47⋊C8C2211yescharacteristic
17C2×C4×D188—24no—
18C2×C4×D188—24no—
19C2×C4×D188—22no—
20C2×C4×D188C2214yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×(C2×D47)⋊C4—21no—
23C2×#13093948—24no—
24C2×#13093948C2214yescomplemented
25C2×#13093944—24no—
26C2×#13093944C2214yes—
27#15688885C2214yescomplemented
28#15688878C2214yescomplemented
29C23×Q8×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C42×C47C2211yescharacteristic
32C47×#391—22no—
33C47×#391C2214yescomplemented
34C2×#95×C47—21no—
35C2×#95×C47—21no—
36C24×D47—21no—
37C22×D188—21no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188—22no—
41C22×D188C2×C414yes—
42C22×D188D414yes—
43C22×D188D414yes—
44C22×D188C2312yes—
45C23×Dic47—21no—
46C22×C4×D47—28no—
47C22×C4×D47—24no—
48C22×C4×D47—44no—
49C22×C4×D47—44no—
50C2×#10625336—232no—
51C2×#10625336—28no—
52C2×#10625336—28no—
53C2×#10625336C2×C4116yescomplemented
54C22×C47⋊C8—22no—
55C22×C47⋊C8—22no—
56C22×C47⋊C8—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8—22no—
59C22×C47⋊C8C2×C414yescomplemented
60C22×C47⋊C8C2312yes—
61C22×C47⋊C8C2312yes—
62C4×D188—216no—
63C4×D188—28no—
64C4×D188—28no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47D414yes—
68C2×C4⋊Dic47C2312yes—
69C2×(C2×D47)⋊C4—28no—
70C2×(C2×D47)⋊C4—24no—
71#13093948—28no—
72#13093944—28no—
73C22×Q8×C47—21no—
74C22×Q8×C47—22no—
75C22×Q8×C47—22no—
76C22×Q8×C47—22no—
77C22×Q8×C47—22no—
78C22×Q8×C47—22no—
79C22×Q8×C47C2×C414yescomplemented
80C22×Q8×C47C2312yes—
81C23×C4×C47—21no—
82C23×C4×C47—21no—
83C23×C4×C47D411yescharacteristic
84C23×C4×C47D411yescharacteristic
85C23×C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47C2312yes—
90C2×C42×C47—22no—
91C2×C42×C47—22no—
92C2×C42×C47D412yes—
93C2×C42×C47C2312yes—
94#95×C47—24no—
95#95×C47—24no—
96C24×C47C2×D411yescharacteristic
97C23×D47—24no—
98C23×D47—24no—
99C23×D47—24no—
100C23×D47—22no—
101C23×D47—42no—
102C23×D47—42no—
103C23×D47—42no—
104C23×D47—42no—
105C2×Q8×C47—24no—
106C2×Q8×C47—22no—
107C2×Q8×C47—28no—
108C2×Q8×C47—28no—
109C2×Q8×C47—22no—
110C2×Q8×C47—22no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—42no—
114C2×Q8×C47—42no—
115C2×Q8×C47—44no—
116C2×Q8×C47—44no—
117C2×Q8×C47C22×C414yescomplemented
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47SD1614yescomplemented
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47#4112yes—
140C22×C4×C47#4112yes—
141C22×C4×C47C2411yescharacteristic
142C22×Dic47—24no—
143C22×Dic47—22no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C2×D188—28no—
147C2×D188—24no—
148C2×D188—24no—
149C2×D188—28no—
150C2×D188—24no—
151C2×D188—28no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188C22⋊C4116yes—
156C2×D188C22×C414yes—
157C2×D188C2×D414yes—
158C2×D188C2×D414yes—
159C2×C4×D47—216no—
160C2×C4×D47—48no—
161C2×C4×D47—432no—
162C2×C4×D47—48no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—42no—
166C42×C47—24no—
167C42×C47—22no—
168C42×C47—22no—
169#10625336—232no—
170(C2×D47)⋊C4—216no—
171C4⋊Dic47—24no—
172C4⋊Dic47—22no—
173C4⋊Dic47C2×D414yes—
174C2×C47⋊C8—28no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—44no—
181C2×C47⋊C8C22×C414yescomplemented
182C23×C47C4×D412yes—
183C23×C47#12012yes—
184C23×C47#12112yes—
185C23×C47C2×SD1612yes—
186C23×C47#13512yes—
187C23×C47AGL(1,ℤ/8)12yes—
188C23×C47#13712yes—
189C23×C47C22×D411yescharacteristic
190C22×D47—24no—
191C22×D47—24no—
192C22×D47—24no—
193C22×D47—216no—
194C22×D47—42no—
195C22×D47—42no—
196C22×D47—42no—
197C22×D47—42no—
198C22×D47—48no—
199C22×D47—48no—
200C22×D47—48no—
201C22×D47—48no—
202C22×D47—48no—
203C22×D47—48no—
204Q8×C47—24no—
205Q8×C47—24no—
206Q8×C47—42no—
207Q8×C47—42no—
208Q8×C47—48no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—28no—
212C2×C4×C47—24no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—48no—
219C2×C4×C47—44no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—84no—
224C2×C4×C47C2×C22⋊C414yes—
225C2×C4×C47C4×D414yes—
226C2×C4×C47C4×D412yes—
227C2×C4×C47#12012yes—
228C2×C4×C47#12112yes—
229C2×C4×C47#12114yes—
230C2×C4×C47#12314yes—
231C2×C4×C47C2×SD1614yescomplemented
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×Dic47—24no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—48no—
242D188—24no—
243D188—24no—
244D188—24no—
245D188—216no—
246D188C2×C22⋊C418yes—
247C4×D47—432no—
248C47⋊C8—24no—
249C47⋊C8—42no—
250C2×#1877—471nomaximal, Sylow
251C22×C47#38614yes—
252C22×C47C4×SD1614yes—
253C22×C47#44014yes—
254C22×C47#45014yes—
255C22×C47#45214yes—
256C22×C47#46114yes—
257C22×C47#46314yes—
258C22×C47C2×C4×D411yescharacteristic
259C22×C47C2×#12011yescharacteristic
260C22×C47C2×#12111yescharacteristic
261C22×C47C22×SD1611yescharacteristic
262C22×C47C2×#13511yescharacteristic
263C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C47C2×#13711yescharacteristic
265C2×D47—28no—
266C2×D47—432no—
267C2×D47—44no—
268C2×D47—44no—
269C2×D47—44no—
270C2×D47—44no—
271C2×D47—44no—
272C2×D47—44no—
273C4×C47—24no—
274C4×C47—42no—
275C4×C47—44no—
276C4×C47—82no—
277C4×C47#38614yes—
278C4×C47C22×C22⋊C411yescharacteristic
279C4×C47C2×C4×D411yescharacteristic
280C4×C47C2×#12111yescharacteristic
281C4×C47C2×#12311yescharacteristic
282Dic47—44no—
283C23×SD16—471no—
284C22×C4×D4—471no—
285C22×#103—471no—
286C22×#102—471no—
287C2×#391D4711yescharacteristic, complemented
288#1877—478no—
289C2×#340—471no—
290C2×#336—471no—
291C2×C47#187718yescomplemented
292C2×C47C2×#38611yescharacteristic
293C2×C47C2×C4×SD1611yescharacteristic
294C2×C47C2×#44011yescharacteristic
295C2×C47C2×#45011yescharacteristic
296C2×C47C2×#45211yescharacteristic
297C2×C47C2×#46111yescharacteristic
298C2×C47C2×#46311yescharacteristic
299D47—48no—
300C23×Q8C2×D4711yescharacteristic
301C23×D4—471no—
302C24×C4—941no—
303C22×SD16—478no—
304C22×SD16—474no—
305C22×SD16—944no—
306C22×SD16—944no—
307C23×C8—471no—
308C23×C8—471no—
309C2×C4×D4—474no—
310C2×C4×D4—944no—
311C2×C4×D4—944no—
312C2×C4×D4—942no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C22⋊C4—941no—
316C22×C42C2×D4711yescharacteristic
317C2×#103—474no—
318C2×#103—944no—
319C2×#102—474no—
320C2×#102—944no—
321#391—22no—
322#391C2×D4714yescomplemented
323C2×#95—21no—
324C2×#95—21no—
325#340—474no—
326#336—474no—
327C47C2×#187711yescharacteristic, complemented, Sylow
328C25—941no—
329C22×Q8—22no—
330C22×Q8—21no—
331C22×Q8—22no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8C4×D4714yescomplemented
336C22×Q8C22×D4712yes—
337C22×D4—474no—
338C22×D4—474no—
339C22×D4—474no—
340C22×D4—472no—
341C22×D4—942no—
342C22×D4—942no—
343C22×D4—942no—
344C22×D4—941no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—948no—
348C23×C4—941no—
349C23×C4—944no—
350C23×C4—1884no—
351C23×C4—1884no—
352C23×C4C22⋊D4711yescharacteristic
353C23×C4C22⋊D4711yescharacteristic
354C23×C4C22×D4711yescharacteristic
355C2×SD16—4716no—
356C2×SD16—9432no—
357C2×SD16—948no—
358C2×SD16—948no—
359C22×C8—474no—
360C22×C8—472no—
361C22×C8—472no—
362C22×C8—942no—
363C22×C8—942no—
364C22×C8—942no—
365C22×C8—942no—
366C22×C8—942no—
367C4×D4—948no—
368C4×D4—948no—
369C4×D4—9416no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4C22×D4712yes—
378C2×C22⋊C4—948no—
379C2×C22⋊C4—944no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42C22⋊D4712yes—
383C2×C42C22×D4712yes—
384#103—948no—
385#102—948no—
386#95—24no—
387#95—24no—
388C24—944no—
389C24—944no—
390C24—944no—
391C24—942no—
392C24—1882no—
393C24—1882no—
394C24—1882no—
395C24—1882no—
396C24C2×C22⋊D4711yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8C2×C4×D4714yescomplemented
410C2×D4—4716no—
411C2×D4—474no—
412C2×D4—474no—
413C2×D4—474no—
414C2×D4—948no—
415C2×D4—944no—
416C2×D4—944no—
417C2×D4—944no—
418C2×D4—948no—
419C2×D4—948no—
420C2×D4—942no—
421C2×D4—942no—
422C2×D4—942no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—9416no—
437C22×C4—944no—
438C22×C4—942no—
439C22×C4—18832no—
440C22×C4—1888no—
441C22×C4—1882no—
442C22×C4—1882no—
443C22×C4—1888no—
444C22×C4#1062533614yescomplemented
445C22×C4C2×C4×D4712yes—
446C22×C4#1062535012yes—
447C22×C4D4×D4712yes—
448C22×C4D4×D4712yes—
449C22×C4D4×D4712yes—
450C22×C4#1062535212yes—
451C22×C4C2×C22⋊D4712yes—
452C22×C4C2×C22⋊D4712yes—
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C23×D4711yescharacteristic
455SD16—9432no—
456C2×C8—474no—
457C2×C8—948no—
458C2×C8—942no—
459C2×C8—942no—
460C2×C8—942no—
461C2×C8—942no—
462C2×C8—942no—
463C2×C8—1884no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—474no—
468C4⋊C4—944no—
469C4⋊C4—942no—
470C22⋊C4—9416no—
471C42—24no—
472C42—22no—
473C42—22no—
474C23—9416no—
475C23—944no—
476C23—944no—
477C23—944no—
478C23—1888no—
479C23—1888no—
480C23—1888no—
481C23—1888no—
482C23—1888no—
483C23—1882no—
484C23—1882no—
485C23—1882no—
486C23—1882no—
487C23—1888no—
488C23C4×C22⋊D4712yes—
489C23#1309399212yes—
490C23#1309399512yes—
491C23C2×#1062533612yes—
492C23#1309404412yes—
493C23#1309404912yes—
494C23#1309405512yes—
495C23C22×C22⋊D4711yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—48no—
499Q8—42no—
500Q8—42no—
501D4—478no—
502D4—9416no—
503D4—944no—
504D4—944no—
505D4—944no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—944no—
522C2×C4—18832no—
523C2×C4—1888no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4C22⋊C4×D4714yes—
527C2×C4(C22⋊D47)⋊C414yes—
528C2×C4C4×C22⋊D4712yes—
529C2×C4#1309399212yes—
530C2×C4#1309399512yes—
531C2×C4#1309399714yes—
532C2×C4#1309400614yes—
533C2×C4C2×#1062533614yescomplemented
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C2×D4×D4711yescharacteristic
536C2×C4C2×D4×D4711yescharacteristic
537C2×C4C2×D4×D4711yescharacteristic
538C2×C4C2×#1062535211yescharacteristic
539C2×C4C2×#1062535011yescharacteristic
540C8—944no—
541C8—1882no—
542C22—948no—
543C22—1884no—
544C22—18832no—
545C22—1884no—
546C22—1884no—
547C22—1884no—
548C22—1884no—
549C22—1884no—
550C22#1568899914yes—
551C22C4×#1062533614yes—
552C22#1568922114yes—
553C22#1568927314yes—
554C22#1568928314yes—
555C22#1568932514yes—
556C22#1568933514yes—
557C22C2×C4×C22⋊D4711yescharacteristic
558C22C2×#1309399211yescharacteristic
559C22C2×#1309399511yescharacteristic
560C22C22×#1062533611yescharacteristic
561C22C2×#1309404411yescharacteristic
562C22C2×#1309404911yescharacteristic
563C22C2×#1309405511yescharacteristic
564C4—24no—
565C4—44no—
566C4—42no—
567C4—82no—
568C4—1884no—
569C4#1568899914yes—
570C4C2×C22⋊C4×D4711yescharacteristic
571C4C2×(C22⋊D47)⋊C411yescharacteristic
572C4C2×#1309399711yescharacteristic
573C4C2×#1309400611yescharacteristic
574C2—1888no—
575C2#2024372518yesdirect product factor
576C2C2×#1568899911yescharacteristic
577C2C2×C4×#1062533611yescharacteristic
578C2C2×#1568922111yescharacteristic
579C2C2×#1568927311yescharacteristic
580C2C2×#1568928311yescharacteristic
581C2C2×#1568932511yescharacteristic
582C2C2×#1568933511yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#391C2
socleC24×C47C2×D4
prime radicalC23×D188C22
cosocleC22×C4×C47C24
hypercenterC2×#391D47
hypocenterC47C2×#1877
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C22×#13093948C2semidirect
5. #20243725C2direct (N × H)
6. C2×#15688885C2semidirect
7. C2×#15688878C2semidirect
8. C2×C47×#391C2semidirect
13. C22×#10625336C4semidirect
24. C2×#13093948C22semidirect
27. #15688885C22semidirect
28. #15688878C22semidirect
33. C47×#391C22semidirect
53. C2×#10625336C2×C4semidirect
59. C22×C47⋊C8C2×C4semidirect
79. C22×Q8×C47C2×C4semidirect
117. C2×Q8×C47C22×C4semidirect
131. C22×C4×C47SD16semidirect
181. C2×C47⋊C8C22×C4semidirect
231. C2×C4×C47C2×SD16semidirect
287. C2×#391D47semidirect
291. C2×C47#1877semidirect
322. #391C2×D47semidirect
327. C47C2×#1877semidirect
335. C22×Q8C4×D47semidirect
409. C2×Q8C2×C4×D47semidirect
444. C22×C4#10625336semidirect
533. C2×C4C2×#10625336semidirect
575. C2#20243725direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253362
C22×C4×D1882
C22×#130939482
C22×#130939442
#202437252
C2×#156888852
C2×#156888782
C2×C47×#3912
C2×#187747