Finite Group Databasebeta

← C2×#20242402 #26188520

Subgroups — C2×#20242402 #26188520

Order 12032.

1156 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689451C211yesmaximal, characteristic, complemented
2C2×#15689434C211yesmaximal, characteristic, complemented
3C2×#15689048C211yesmaximal, characteristic, complemented
4#20242402C218yesmaximal, direct product factor
5C2×#15688860C211yesmaximal, characteristic, complemented
6C2×#15688857C211yesmaximal, characteristic, complemented
7C2×#15688847C211yesmaximal, characteristic, complemented
8C2×C47×#538C211yesmaximal, characteristic
9C22×C4⋊Dic47C2211yescharacteristic
10C22×C22⋊Dic47—21no—
11#15689451—22no—
12#15689451C2214yescomplemented
13#15689434—22no—
14#15689434C2214yescomplemented
15C2×#13093957—21no—
16C2×#13093955—21no—
17C2×#13093948—21no—
18C2×#13093945—21no—
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20C2×C22⋊(C47⋊C8)C2211yescharacteristic
21#15689048—22no—
22#15689048C2214yescomplemented
23C2×C47⋊#95—21no—
24C2×C47⋊#95C2211yescharacteristic
25#15688860C2214yescomplemented
26#15688857C2214yescomplemented
27#15688847C2214yescomplemented
28C47×#538—22no—
29C47×#538—22no—
30C47×#538—22no—
31C47×#538C2214yes—
32C2×#126×C47—21no—
33C2×#124×C47—21no—
34C2×#123×C47—21no—
35C2×C22⋊Q8×C47C2211yescharacteristic
36C2×#121×C47—21no—
37C2×#121×C47C2211yescharacteristic
38C2×#120×C47—21no—
39C2×C4×D4×C47—21no—
40C22×C22⋊C4×C47C2211yescharacteristic
41C23×Dic47—21no—
42C23×Dic47—21no—
43C22×C47⋊C8—21no—
44C22×C47⋊C8—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47C2312yes—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47—44no—
53#13093957—24no—
54#13093955—24no—
55#13093948—24no—
56#13093945—24no—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)—21no—
59C22⋊(C47⋊C8)C2312yes—
60C22⋊(C47⋊C8)C2312yes—
61C47⋊#95—24no—
62C47⋊#95—21no—
63C47⋊#95C2312yes—
64C25×C47D411yescharacteristic
65C22×Q8×C47D411yescharacteristic
66C22×D4×C47—21no—
67C22×D4×C47—21no—
68C22×D4×C47—21no—
69C22×D4×C47D411yescharacteristic
70C23×C4×C47—21no—
71C23×C4×C47D411yescharacteristic
72C23×C4×C47C2311yescharacteristic
73#126×C47—24no—
74#126×C47—24no—
75#124×C47—24no—
76#124×C47—24no—
77#123×C47—24no—
78#123×C47—24no—
79C22⋊Q8×C47—21no—
80C22⋊Q8×C47—21no—
81C22⋊Q8×C47D412yes—
82C22⋊Q8×C47C2312yes—
83#121×C47—24no—
84#121×C47—24no—
85#121×C47—21no—
86#121×C47—21no—
87#121×C47D412yes—
88#121×C47C2312yes—
89#120×C47—24no—
90#120×C47—24no—
91C4×D4×C47—24no—
92C4×D4×C47—24no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47D411yescharacteristic
96C2×C4⋊C4×C47D411yescharacteristic
97C2×C22⋊C4×C47—21no—
98C2×C22⋊C4×C47—21no—
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47—21no—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47D412yes—
107C2×C22⋊C4×C47C2312yes—
108C2×C42×C47—21no—
109C24×C47—21no—
110C24×C47—21no—
111C24×C47—22no—
112C24×C47—21no—
113C24×C47—22no—
114C24×C47—42no—
115C24×C47—42no—
116C24×C47C2×D411yescharacteristic
117C24×C47C2×D412yes—
118C2×Q8×C47—21no—
119C2×Q8×C47—42no—
120C2×Q8×C47C2×D412yes—
121C2×D4×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47—24no—
124C2×D4×C47—24no—
125C2×D4×C47—21no—
126C2×D4×C47—42no—
127C2×D4×C47—42no—
128C2×D4×C47—42no—
129C2×D4×C47—82no—
130C2×D4×C47—82no—
131C2×D4×C47C2×D412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—24no—
137C22×C4×C47—21no—
138C22×C4×C47—42no—
139C22×C4×C47—42no—
140C22×C4×C47—82no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D412yes—
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—42no—
151C22×Dic47—42no—
152C22×Dic47—41no—
153C22×Dic47—41no—
154C22×Dic47—42no—
155C22×Dic47—42no—
156C22×Dic47—42no—
157C4⋊C4×C47—24no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47C2×D412yes—
163C4⋊C4×C47C2×D412yes—
164C22⋊C4×C47—24no—
165C22⋊C4×C47—24no—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—22no—
168C22⋊C4×C47—22no—
169C22⋊C4×C47—22no—
170C22⋊C4×C47—22no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C42×C47—24no—
176C22⋊Dic47—48no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47—41no—
180C4⋊Dic47—41no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—42no—
184C2×C47⋊C8—42no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—48no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—81no—
202C23×C47—81no—
203C23×C47—84no—
204C23×C47#12011yescharacteristic
205C23×C47#12011yescharacteristic
206C23×C47#13512yes—
207C23×C47AGL(1,ℤ/8)12yes—
208C23×C47C22×D411yescharacteristic
209Q8×C47—42no—
210Q8×C47—81no—
211D4×C47—42no—
212D4×C47—42no—
213D4×C47—42no—
214D4×C47—44no—
215D4×C47—81no—
216D4×C47—84no—
217D4×C47—84no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—81no—
234C2×C4×C47—84no—
235C2×C4×C47#12011yescharacteristic
236C2×C4×C47#12011yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C22×D411yescharacteristic
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—84no—
246C2×Dic47—82no—
247C2×Dic47—82no—
248C47⋊C8—42no—
249C47⋊C8—42no—
250C2×#1612—471nomaximal, Sylow
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—21no—
254C22×C47—21no—
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—44no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—82no—
265C22×C47—84no—
266C22×C47—84no—
267C22×C47—84no—
268C22×C47#44912yes—
269C22×C47#44912yes—
270C22×C47#45512yes—
271C22×C47#45812yes—
272C22×C47C2×#12011yescharacteristic
273C22×C47C2×#13511yescharacteristic
274C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C47—21no—
276C4×C47—21no—
277C4×C47—21no—
278C4×C47—21no—
279C4×C47—42no—
280C4×C47—42no—
281C4×C47—42no—
282C4×C47—42no—
283Dic47—82no—
284Dic47—82no—
285C2×#538D4711yescharacteristic
286C2×#483—471no—
287C2×#480—471no—
288C2×#397—471no—
289#1612—478no—
290C2×#328—471no—
291C2×#327—471no—
292C2×#323—471no—
293C2×C47—21no—
294C2×C47—21no—
295C2×C47—22no—
296C2×C47—82no—
297C2×C47—82no—
298C2×C47#161214yescomplemented
299C2×C47C2×#44911yescharacteristic
300C2×C47C2×#45511yescharacteristic
301C2×C47C2×#45811yescharacteristic
302#538—22no—
303#538—22no—
304#538—22no—
305#538C2×D4714yes—
306C2×#126—21no—
307C2×#124—21no—
308C2×#123—21no—
309C2×C22⋊Q8C2×D4711yescharacteristic
310C2×#121—21no—
311C2×#121C2×D4711yescharacteristic
312C2×#120—21no—
313C2×C4×D4—21no—
314C22×C4⋊C4—471no—
315C22×C22⋊C4—941no—
316C22×C22⋊C4C2×D4711yescharacteristic
317#483—474no—
318#483—942no—
319#480—474no—
320#480—942no—
321C2×#107—941no—
322C2×#106—941no—
323C2×#103—941no—
324C2×#102—941no—
325C2×C22⋊C8—471no—
326C2×C22⋊C8—471no—
327#397—474no—
328#397—942no—
329C2×#95—471no—
330C2×#95—941no—
331#328—474no—
332#327—474no—
333#323—474no—
334C47C2×#161211yescharacteristic, complemented, Sylow
335C25C22⋊D4711yescharacteristic
336C22×Q8C22⋊D4711yescharacteristic
337C22×D4—21no—
338C22×D4—21no—
339C22×D4—21no—
340C22×D4C22⋊D4711yescharacteristic
341C23×C4—21no—
342C23×C4—941no—
343C23×C4—941no—
344C23×C4C22⋊D4711yescharacteristic
345C23×C4C22×D4711yescharacteristic
346C22×C8—941no—
347C22×C8—941no—
348#126—24no—
349#126—24no—
350#124—24no—
351#124—24no—
352#123—24no—
353#123—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8C22⋊D4712yes—
357C22⋊Q8C22×D4712yes—
358#121—24no—
359#121—24no—
360#121—21no—
361#121—21no—
362#121C22⋊D4712yes—
363#121C22×D4712yes—
364#120—24no—
365#120—24no—
366C4×D4—24no—
367C4×D4—24no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—472no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22⋊D4711yescharacteristic
377C2×C4⋊C4C22⋊D4711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—944no—
388C2×C22⋊C4—1884no—
389C2×C22⋊C4C22⋊D4712yes—
390C2×C22⋊C4C22×D4712yes—
391C2×C42—21no—
392#107—944no—
393#106—944no—
394#103—944no—
395#102—944no—
396C22⋊C8—472no—
397C22⋊C8—472no—
398C22⋊C8—941no—
399C22⋊C8—941no—
400#95—472no—
401#95—941no—
402#95—944no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24C2×C22⋊D4711yescharacteristic
411C24C2×C22⋊D4712yes—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8C2×C22⋊D4712yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×C22⋊D4712yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—1882no—
438C22×C4—1882no—
439C22×C4—1881no—
440C22×C4—1881no—
441C22×C4—1882no—
442C22×C4—1882no—
443C22×C4—1882no—
444C22×C4C2×C22⋊D4711yescharacteristic
445C22×C4C2×C22⋊D4711yescharacteristic
446C22×C4C2×C22⋊D4712yes—
447C22×C4C2×C22⋊D4711yescharacteristic
448C22×C4C2×C22⋊D4711yescharacteristic
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C23×D4711yescharacteristic
451C2×C8—942no—
452C2×C8—942no—
453C2×C8—1882no—
454C2×C8—1882no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—942no—
461C4⋊C4—942no—
462C4⋊C4—1881no—
463C4⋊C4—1881no—
464C4⋊C4C2×C22⋊D4712yes—
465C4⋊C4C2×C22⋊D4712yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—1888no—
478C42—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23—84no—
498C23#1309399311yescharacteristic
499C23#1309399311yescharacteristic
500C23#1309404412yes—
501C23#1309404912yes—
502C23C22×C22⋊D4711yescharacteristic
503Q8—42no—
504Q8—81no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—81no—
510D4—84no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4—3764no—
530C2×C4—3762no—
531C2×C4—3762no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4#1309399311yescharacteristic
539C2×C4#1309399311yescharacteristic
540C2×C4C22×C22⋊D4711yescharacteristic
541C2×C4C22×C22⋊D4711yescharacteristic
542C8—1882no—
543C8—1882no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—24no—
549C22—24no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22#1568926812yes—
562C22#1568926812yes—
563C22#1568930112yes—
564C22#1568931312yes—
565C22C2×#1309399311yescharacteristic
566C22C2×#1309404411yescharacteristic
567C22C2×#1309404911yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—3762no—
577C4—3762no—
578C2—21no—
579C2—22no—
580C2—21no—
581C2—82no—
582C2—82no—
583C2#2024240214yesdirect product factor
584C2C2×#1568926811yescharacteristic
585C2C2×#1568930111yescharacteristic
586C2C2×#1568931311yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#538C2
socleC23×C47#120
hypercenterC2×#538D47
hypocenterC47C2×#1612
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. C2×#15689451C2semidirect
2. C2×#15689434C2semidirect
3. C2×#15689048C2semidirect
4. #20242402C2direct (N × H)
5. C2×#15688860C2semidirect
6. C2×#15688857C2semidirect
7. C2×#15688847C2semidirect
12. #15689451C22semidirect
14. #15689434C22semidirect
22. #15689048C22semidirect
25. #15688860C22semidirect
26. #15688857C22semidirect
27. #15688847C22semidirect
298. C2×C47#1612semidirect
334. C47C2×#1612semidirect
583. C2#20242402direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894512
C2×#156894342
C2×#156890482
#202424022
C2×#156888602
C2×#156888572
C2×#156888472
C2×C47×#5382
C2×#161247