Finite Group Databasebeta

← #26217908

Subgroups — #26217908

Order 12032.

2126 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689682C211yesmaximal, characteristic
2C2×#15689598C211yesmaximal, characteristic, complemented
3#20245963C212yesmaximal, complemented
4#20243799C214yesmaximal, complemented
5#20243794C214yesmaximal, complemented
6C2×#15688889C212yesmaximal, complemented
7C22×#99×C47C211yesmaximal, characteristic, complemented
8C23×C22⋊D47C2211yescharacteristic
9#15689682—24no—
10#15689682C418yescomplemented
11C2×#13094007—21no—
12C2×#13094005C2212yes—
13C2×#13093996—22no—
14C2×#13093993—21no—
15C2×#13093992C2212yes—
16#15689598—24no—
17#15689598—22no—
18#15689598C2214yes—
19C2×D4×Dic47—24no—
20C2×#13093983—21no—
21C22×C22⋊Dic47—21no—
22C22×C22⋊Dic47C2212yes—
23C22×C22⋊Dic47C2211yescharacteristic
24#15689002—24no—
25#15689000—24no—
26#15688984C2214yescomplemented
27#15688978C2214yescomplemented
28#15688889C2218yescomplemented
29C23×D4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2212yes—
31C2×#99×C47—24no—
32C2×#99×C47C2214yes—
33C24×D47C2311yescharacteristic
34C22×C22⋊D47—22no—
35C22×C22⋊D47—22no—
36C22×C22⋊D47—21no—
37C22×C22⋊D47—21no—
38C22×C22⋊D47—22no—
39C22×C22⋊D47—24no—
40C22×C22⋊D47—22no—
41C22×C22⋊D47C2×C414yescomplemented
42C23×Dic47—22no—
43C23×Dic47—21no—
44C23×Dic47D414yescomplemented
45C23×Dic47C2311yescharacteristic
46#13094007—28no—
47#13094005—24no—
48#13094005C2×C418yescomplemented
49#13093996—216no—
50#13093993—28no—
51#13093992—24no—
52#13093992C2×C418yescomplemented
53D4×Dic47—216no—
54D4×Dic47—216no—
55#13093983—24no—
56#13093983—24no—
57C2×C4⋊Dic47—22no—
58C2×Dic47⋊C4—22no—
59C2×C22⋊Dic47—22no—
60C2×C22⋊Dic47—24no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—24no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—22no—
65C2×C22⋊Dic47—22no—
66C2×C22⋊Dic47—21no—
67C2×C22⋊Dic47—22no—
68C2×C22⋊Dic47—22no—
69C2×C22⋊Dic47C2×C414yescomplemented
70C2×C22⋊Dic47D414yescomplemented
71C2×C22⋊Dic47C2314yes—
72C2×C22⋊Dic47C2312yes—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4—28no—
75C2×C4×Dic47—22no—
76C47⋊#95—28no—
77C25×C47D411yescharacteristic
78C25×C47C2311yescharacteristic
79C22×D4×C47—22no—
80C22×D4×C47—22no—
81C22×D4×C47—24no—
82C22×D4×C47C2×C414yes—
83C22×D4×C47D414yes—
84C22×D4×C47D412yes—
85C22×D4×C47C2312yes—
86C23×C4×C47—22no—
87C23×C4×C47D411yescharacteristic
88C2×C22⋊C4×C47—24no—
89C2×C22⋊C4×C47—24no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47C2314yes—
92#99×C47—28no—
93C24×C47—21no—
94C24×C47—21no—
95C24×C47—22no—
96C24×C47—82no—
97C24×C47—82no—
98C24×C47C22⋊C414yescomplemented
99C24×C47C22×C412yes—
100C24×C47C2×D412yes—
101C24×C47C2×D411yescharacteristic
102C24×C47C2×D411yescharacteristic
103C24×C47C2×D412yes—
104C24×C47C2×D412yes—
105C24×C47C2×D412yes—
106C24×C47#4114yes—
107C24×C47C2411yescharacteristic
108C23×D47—22no—
109C23×D47—21no—
110C23×D47—21no—
111C23×D47—22no—
112C23×D47—82no—
113C23×D47C22×C412yescomplemented
114C2×D4×C47—28no—
115C2×D4×C47—28no—
116C2×D4×C47—24no—
117C2×D4×C47—28no—
118C2×D4×C47—28no—
119C2×D4×C47—24no—
120C2×D4×C47—24no—
121C2×D4×C47—24no—
122C2×D4×C47C22⋊C418yescomplemented
123C2×D4×C47C22×C414yes—
124C2×D4×C47C2×D414yes—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—44no—
130C22×C4×C47—84no—
131C22×C4×C47C22⋊C414yescomplemented
132C22×C4×C47C2×D412yes—
133C2×C22⋊D47—24no—
134C2×C22⋊D47—24no—
135C2×C22⋊D47—28no—
136C2×C22⋊D47—28no—
137C2×C22⋊D47—28no—
138C2×C22⋊D47—41no—
139C2×C22⋊D47—42no—
140C2×C22⋊D47—41no—
141C2×C22⋊D47—44no—
142C2×C22⋊D47—44no—
143C2×C22⋊D47—42no—
144C2×C22⋊D47—42no—
145C2×C22⋊D47—84no—
146C22×Dic47—24no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—22no—
152C22×Dic47—28no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—84no—
156C22×Dic47—84no—
157C22×Dic47—82no—
158C22×Dic47C22×C412yescomplemented
159C22×Dic47C2×D418yes—
160C22⋊C4×C47—24no—
161C22⋊C4×C47—24no—
162C22⋊C4×C47—44no—
163C22⋊Dic47—28no—
164C22⋊Dic47—24no—
165C22⋊Dic47—28no—
166C22⋊Dic47—24no—
167C22⋊Dic47—24no—
168C22⋊Dic47—24no—
169C22⋊Dic47—24no—
170C22⋊Dic47—22no—
171C22⋊Dic47C22×C414yescomplemented
172(C2×D47)⋊C4—28no—
173(C2×D47)⋊C4—88no—
174C4⋊Dic47—28no—
175Dic47⋊C4—28no—
176C4×Dic47—28no—
177C23×C47—22no—
178C23×C47—24no—
179C23×C47—22no—
180C23×C47—24no—
181C23×C47—42no—
182C23×C47—44no—
183C23×C47—44no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—44no—
187C23×C47—44no—
188C23×C47—82no—
189C23×C47—82no—
190C23×C47—82no—
191C23×C47—84no—
192C23×C47—84no—
193C23×C47—84no—
194C23×C47C2×C22⋊C414yes—
195C23×C47C2×C22⋊C412yes—
196C23×C47C2×C22⋊C412yes—
197C23×C47C4×D414yes—
198C23×C47C4×D414yes—
199C23×C47#12011yescharacteristic
200C23×C47#12012yes—
201C23×C47#12011yescharacteristic
202C23×C47#12114yes—
203C23×C47#12114yes—
204C23×C47#12312yes—
205C23×C47#12312yes—
206C23×C47C23×C411yescharacteristic
207C23×C47C22×D412yes—
208C23×C47C22×D411yescharacteristic
209C23×C47C22×D411yescharacteristic
210C23×C47C2×#4112yes—
211C22×D47—22no—
212C22×D47—24no—
213C22×D47—48no—
214C22×D47—44no—
215C22×D47—41no—
216C22×D47—42no—
217C22×D47—41no—
218C22×D47—82no—
219C22×D47—82no—
220C22×D47—84no—
221D4×C47—28no—
222D4×C47—28no—
223D4×C47—216no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—48no—
229C2×C4×C47—44no—
230C2×C4×C47—88no—
231C2×C4×C47C2×C22⋊C414yes—
232C22⋊D47—44no—
233C22⋊D47—44no—
234C22⋊D47—48no—
235C22⋊D47—88no—
236C2×Dic47—22no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—28no—
240C2×Dic47—48no—
241C2×Dic47—44no—
242C2×Dic47—41no—
243C2×Dic47—42no—
244C2×Dic47—41no—
245C2×Dic47—48no—
246C2×Dic47—84no—
247C2×Dic47—84no—
248C2×Dic47—88no—
249#20960—471nomaximal, Sylow
250C22×C47—22no—
251C22×C47—24no—
252C22×C47—22no—
253C22×C47—24no—
254C22×C47—22no—
255C22×C47—24no—
256C22×C47—42no—
257C22×C47—44no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—44no—
264C22×C47—44no—
265C22×C47—44no—
266C22×C47—42no—
267C22×C47—44no—
268C22×C47—48no—
269C22×C47—44no—
270C22×C47—82no—
271C22×C47—84no—
272C22×C47#38612yes—
273C22×C47#38612yes—
274C22×C47#38614yes—
275C22×C47C22×C22⋊C411yescharacteristic
276C22×C47C2×C4×D412yes—
277C22×C47C2×#12011yescharacteristic
278C22×C47C2×#12112yes—
279C22×C47C2×#12311yescharacteristic
280C2×D47—44no—
281C2×D47—48no—
282C2×D47—42no—
283C2×D47—84no—
284C2×D47—82no—
285C2×D47—82no—
286C4×C47—24no—
287C4×C47—84no—
288Dic47—44no—
289Dic47—48no—
290Dic47—42no—
291C2×#534—471no—
292C2×#518—471no—
293C22×#99D4711yescharacteristic, complemented
294#2308—472no—
295#1890—474no—
296#1889—474no—
297C2×#342—472no—
298C2×C47—21no—
299C2×C47—22no—
300C2×C47—22no—
301C2×C47—21no—
302C2×C47—22no—
303C2×C47—22no—
304C2×C47—24no—
305C2×C47—44no—
306C2×C47C2×#38611yescharacteristic
307C2×C47#302612yes—
308D47—82no—
309C23×D4—471no—
310C23×D4C2×D4711yescharacteristic
311#534—478no—
312#534—944no—
313C2×#123—472no—
314C2×#123—941no—
315C2×#121—942no—
316C2×#120—472no—
317C2×#120—941no—
318#518—474no—
319#518—944no—
320#518—942no—
321C2×C4×D4—944no—
322C2×#117—941no—
323C22×C22⋊C4—471no—
324C22×C22⋊C4—472no—
325C22×C22⋊C4—941no—
326C22×C22⋊C4C2×D4712yes—
327C2×#99—24no—
328C2×#99C2×D4714yes—
329#386—944no—
330#386—944no—
331#381—474no—
332#379—474no—
333#342—478no—
334C47#2096011yescharacteristic, complemented, Sylow
335C25—471no—
336C25D18811yescharacteristic
337C25C22×D4711yescharacteristic
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—24no—
341C22×D4—474no—
342C22×D4—942no—
343C22×D4—941no—
344C22×D4—941no—
345C22×D4—942no—
346C22×D4—942no—
347C22×D4—944no—
348C22×D4—942no—
349C22×D4C4×D4714yes—
350C22×D4D18812yes—
351C22×D4C22⋊D4714yes—
352C22×D4C22×D4712yes—
353C23×C4—22no—
354C23×C4—471no—
355C23×C4—474no—
356C23×C4—942no—
357C23×C4—941no—
358C23×C4D18811yescharacteristic
359#123—478no—
360#123—944no—
361#123—948no—
362#121—9416no—
363#120—478no—
364#120—948no—
365#120—944no—
366C4×D4—9416no—
367C4×D4—9416no—
368#117—944no—
369#117—944no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—942no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—474no—
376C2×C22⋊C4—472no—
377C2×C22⋊C4—474no—
378C2×C22⋊C4—474no—
379C2×C22⋊C4—942no—
380C2×C22⋊C4—944no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4—944no—
385C2×C22⋊C4—942no—
386C2×C22⋊C4—942no—
387C2×C22⋊C4—942no—
388C2×C22⋊C4—942no—
389C2×C22⋊C4—942no—
390C2×C22⋊C4—948no—
391C2×C22⋊C4C22×D4714yes—
392C2×C42—942no—
393#99—28no—
394#95—948no—
395C24—21no—
396C24—21no—
397C24—22no—
398C24—82no—
399C24—82no—
400C24—472no—
401C24—941no—
402C24—942no—
403C24—942no—
404C24—941no—
405C24—3762no—
406C24(C2×D47)⋊C414yescomplemented
407C24C2×C4×D4712yes—
408C24C2×D18812yes—
409C24C2×D18811yescharacteristic
410C24C2×D18811yescharacteristic
411C24D4×D4712yes—
412C24D4×D4712yes—
413C24#1062535214yes—
414C24C2×C22⋊D4712yes—
415C24C23×D4711yescharacteristic
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—948no—
425C2×D4—944no—
426C2×D4—944no—
427C2×D4—948no—
428C2×D4—948no—
429C2×D4—3764no—
430C2×D4—1881no—
431C2×D4—1882no—
432C2×D4—1881no—
433C2×D4—1884no—
434C2×D4—1884no—
435C2×D4—1882no—
436C2×D4—1882no—
437C2×D4(C2×D47)⋊C418yescomplemented
438C2×D4C2×C4×D4714yes—
439C2×D4C2×C22⋊D4714yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4—472no—
447C22×C4—478no—
448C22×C4—948no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—942no—
453C22×C4—941no—
454C22×C4—942no—
455C22×C4—941no—
456C22×C4—942no—
457C22×C4—3764no—
458C22×C4—3762no—
459C22×C4—3764no—
460C22×C4(C2×D47)⋊C414yescomplemented
461C22×C4C2×D18812yes—
462C4⋊C4—948no—
463C4⋊C4—948no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—44no—
467C22⋊C4—474no—
468C22⋊C4—948no—
469C22⋊C4—944no—
470C22⋊C4—948no—
471C22⋊C4—944no—
472C22⋊C4—944no—
473C22⋊C4—944no—
474C22⋊C4—942no—
475C22⋊C4—944no—
476C22⋊C4—948no—
477C22⋊C4—3768no—
478C42—948no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23—84no—
493C23—82no—
494C23—84no—
495C23—84no—
496C23—942no—
497C23—944no—
498C23—3762no—
499C23—3764no—
500C23—3762no—
501C23—1888no—
502C23—1884no—
503C23—1881no—
504C23—1882no—
505C23—1881no—
506C23C2×(C2×D47)⋊C414yes—
507C23C2×(C2×D47)⋊C412yes—
508C23C2×(C2×D47)⋊C412yes—
509C23(C22⋊D47)⋊C414yes—
510C23(C22⋊D47)⋊C414yes—
511C23#1309399411yescharacteristic
512C23#1309399412yes—
513C23#1309399411yescharacteristic
514C23#1309399614yes—
515C23#1309399614yes—
516C23#1309400812yes—
517C23#1309400812yes—
518C23C22×C4×D4711yescharacteristic
519C23C2×D4×D4712yes—
520C23C22×C22⋊D4711yescharacteristic
521C23C22×D18811yescharacteristic
522C23C2×#1062535212yes—
523D4—216no—
524D4—28no—
525D4—28no—
526D4—3768no—
527D4—1884no—
528D4—1884no—
529D4—1888no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—88no—
537C2×C4—944no—
538C2×C4—942no—
539C2×C4—944no—
540C2×C4—948no—
541C2×C4—3764no—
542C2×C4—3764no—
543C2×C4—3768no—
544C2×C4—1888no—
545C2×C4—1884no—
546C2×C4—1881no—
547C2×C4—1882no—
548C2×C4—1881no—
549C2×C4—1888no—
550C2×C4C2×(C2×D47)⋊C414yes—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22—48no—
570C22—44no—
571C22—82no—
572C22—84no—
573C22—3764no—
574C22—3762no—
575C22—3762no—
576C22—1884no—
577C22—1888no—
578C22—1882no—
579C22#1568900012yes—
580C22#1568900012yes—
581C22#1568900014yes—
582C22C22×(C2×D47)⋊C411yescharacteristic
583C22C2×(C22⋊D47)⋊C412yes—
584C22C2×#1309399411yescharacteristic
585C22C2×#1309399612yes—
586C22C2×#1309400811yescharacteristic
587C4—24no—
588C4—84no—
589C4—1884no—
590C4—1882no—
591C4—1888no—
592C2—22no—
593C2—21no—
594C2—22no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—24no—
599C2—44no—
600C2—3762no—
601C2C2×#1568900011yescharacteristic
602C2#2025278712yes—
subgroupG/H
centerC22C2×#13093994
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC22×#99×C47C2
socleC22×C47C2×#120
prime radicalC2×#15689682C2
cosocleC24×C47C24
hypercenterC22×#99D47
hypocenterC47#20960
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
2. C2×#15689598C2semidirect
3. #20245963C2semidirect
4. #20243799C2semidirect
5. #20243794C2semidirect
6. C2×#15688889C2semidirect
7. C22×#99×C47C2semidirect
10. #15689682C4semidirect
26. #15688984C22semidirect
27. #15688978C22semidirect
28. #15688889C22semidirect
41. C22×C22⋊D47C2×C4semidirect
44. C23×Dic47D4semidirect
48. #13094005C2×C4semidirect
52. #13093992C2×C4semidirect
69. C2×C22⋊Dic47C2×C4semidirect
70. C2×C22⋊Dic47D4semidirect
98. C24×C47C22⋊C4semidirect
113. C23×D47C22×C4semidirect
122. C2×D4×C47C22⋊C4semidirect
131. C22×C4×C47C22⋊C4semidirect
158. C22×Dic47C22×C4semidirect
171. C22⋊Dic47C22×C4semidirect
293. C22×#99D47semidirect
334. C47#20960semidirect
406. C24(C2×D47)⋊C4semidirect
437. C2×D4(C2×D47)⋊C4semidirect
460. C22×C4(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156896822
C2×#156895982
#202459632
#202437992
#202437942
C2×#156888892
C22×#99×C472
#2096047