Finite Group Databasebeta

← #26250596

Subgroups — #26250596

Order 12032.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689680C211yesmaximal, characteristic, complemented
2C2×#15689413C211yesmaximal, characteristic, complemented
3C2×#15689328C211yesmaximal, characteristic, complemented
4#20244584C212yesmaximal, complemented
5#20244391C212yesmaximal
6#20244336C211yesmaximal, characteristic, complemented
7#20244311C211yesmaximal, characteristic, complemented
8#20243746C212yesmaximal, complemented
9#20243655C212yesmaximal, complemented
10#20243368C211yesmaximal, characteristic, complemented
11C4×#121×C47C211yesmaximal, characteristic
12C22×#10625335C2212yes—
13#15689680—24no—
14#15689680—24no—
15C2×#13094024C2211yescharacteristic
16C2×#13094024C2211yescharacteristic
17C2×#13094016—21no—
18C2×#13094003—21no—
19C2×#13094003C2212yes—
20C2×C4×Dic94—21no—
21C2×C4×Dic94C2212yes—
22C2×#13093983—21no—
23#15689413—24no—
24#15689328—24no—
25C2×#13093955C2212yes—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093947C2212yes—
29C2×#13093947C2212yes—
30C2×#13093947C2212yes—
31C2×#13093945C2212yes—
32C2×C22⋊(C47⋊C8)C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35#15688878C2212yes—
36C2×#121×C47C2211yescharacteristic
37C2×C4×D4×C47—21no—
38C2×C4×D4×C47C2211yescharacteristic
39C22×C42×C47C2211yescharacteristic
40C47×#390C2211yescharacteristic
41C47×#389C2211yescharacteristic
42C47×#388—21no—
43C47×#386—21no—
44C47×#380C2211yescharacteristic
45C4×C4⋊C4×C47C2211yescharacteristic
46C4×C22⋊C4×C47—21no—
47C22×Dic94—21no—
48C22×Dic94C2312yes—
49C2×#10625335—24no—
50C2×#10625335—24no—
51C2×#10625335—42no—
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2312yes—
54#13094024—22no—
55#13094024—22no—
56#13094024D414yescomplemented
57#13094024D414yescomplemented
58#13094016—28no—
59#13094003—28no—
60#13094003—24no—
61#13094003—24no—
62C4×Dic94—28no—
63C4×Dic94—24no—
64C4×Dic94—24no—
65#13093983—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2312yes—
70C2×Dic47⋊C4—22no—
71C2×Dic47⋊C4—22no—
72C2×C22⋊Dic47—22no—
73C2×C4×Dic47—22no—
74#13093955—24no—
75C4⋊(C47⋊C8)—22no—
76C4⋊(C47⋊C8)—22no—
77#13093947—24no—
78#13093947—24no—
79#13093947—42no—
80#13093945—24no—
81C22⋊(C47⋊C8)—42no—
82C22×D4×C47—21no—
83C22×D4×C47C2311yescharacteristic
84C23×C4×C47—21no—
85C23×C4×C47D411yescharacteristic
86C23×C4×C47D411yescharacteristic
87C23×C4×C47C2311yescharacteristic
88#121×C47—22no—
89#121×C47—22no—
90C4×D4×C47—22no—
91C4×D4×C47—22no—
92C4×D4×C47—44no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47C2311yescharacteristic
95C2×C4⋊C4×C47C2311yescharacteristic
96C2×C4⋊C4×C47C2311yescharacteristic
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×C42×C47—22no—
103C2×C42×C47—22no—
104C2×C42×C47—21no—
105C2×C42×C47D411yescharacteristic
106C2×C42×C47D411yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109#95×C47—21no—
110#95×C47—21no—
111#95×C47—21no—
112C24×C47—21no—
113C24×C47C2×D411yescharacteristic
114C2×D4×C47—24no—
115C2×D4×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—42no—
121C2×D4×C47—42no—
122C2×D4×C47#4112yes—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—41no—
136C22×C4×C47—41no—
137C22×C4×C47—42no—
138C22×C4×C47—42no—
139C22×C4×C47—42no—
140C22×C4×C47SD1614yescomplemented
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2411yescharacteristic
147C22×Dic47—22no—
148C22×Dic47—22no—
149C2×Dic94—22no—
150C2×Dic94—22no—
151C2×Dic94—22no—
152C2×Dic94—22no—
153C2×Dic94—24no—
154C2×Dic94—22no—
155C2×Dic94—42no—
156C2×Dic94—42no—
157C2×Dic94C2×D414yes—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—42no—
164C4⋊C4×C47#4112yes—
165C22⋊C4×C47—24no—
166C22⋊C4×C47—42no—
167C22⋊C4×C47—42no—
168C22⋊C4×C47—42no—
169C22⋊C4×C47—42no—
170C42×C47—22no—
171C42×C47—22no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—42no—
175C42×C47C2×D412yes—
176C42×C47C2×D412yes—
177C22⋊Dic47—28no—
178#10625335—48no—
179C4⋊Dic47—24no—
180C4⋊Dic47—24no—
181C4⋊Dic47—24no—
182C4⋊Dic47—22no—
183C4⋊Dic47C2×D414yes—
184Dic47⋊C4—28no—
185Dic47⋊C4—28no—
186C4×Dic47—28no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—22no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47C2×SD1612yes—
207C23×C47C22×D411yescharacteristic
208D4×C47—22no—
209D4×C47—22no—
210D4×C47—48no—
211D4×C47—41no—
212D4×C47—41no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—28no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—22no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—44no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—84no—
239C2×C4×C47#12012yes—
240C2×C4×C47#12012yes—
241C2×C4×C47C2×SD1612yes—
242C2×C4×C47C2×SD1612yes—
243C2×C4×C47#13712yes—
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47H(5,𝔽2)11yescharacteristic
249C2×C4×C472−1+411yescharacteristic
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256Dic94—24no—
257Dic94—24no—
258Dic94—48no—
259Dic94—42no—
260Dic94—42no—
261C47⋊C8—42no—
262C47⋊C8—42no—
263#24151—471nomaximal, Sylow
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47—81no—
278C22×C47C2×#12011yescharacteristic
279C22×C47#52511yescharacteristic
280C22×C47#53711yescharacteristic
281C22×C47C22×SD1611yescharacteristic
282C22×C47C2×#13711yescharacteristic
283C22×C47#57511yescharacteristic
284C22×C47#57511yescharacteristic
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—42no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—81no—
291C4×C47—81no—
292C4×C47#44814yes—
293C4×C47#54611yescharacteristic
294C4×C47#54711yescharacteristic
295C4×C47#54811yescharacteristic
296C4×C47#55011yescharacteristic
297Dic47—44no—
298Dic47—44no—
299C2×#533—471no—
300C2×#476—471no—
301C2×#461—471no—
302C4×#121D4711yescharacteristic
303#2035—472no—
304#2000—472no—
305#1987—471no—
306#1981—471no—
307#1880—472no—
308#1865—472no—
309#1797—471no—
310C2×C47—42no—
311C2×C47—82no—
312C2×C47#240711yescharacteristic
313C2×C47C2×#44811yescharacteristic
314C2×C47#300511yescharacteristic
315C2×C47#310111yescharacteristic
316C2×C47#311611yescharacteristic
317C2×C47#324611yescharacteristic
318C2×C47#325211yescharacteristic
319C22×SD16—472no—
320#533—944no—
321#533—944no—
322C2×C4⋊Q8—471no—
323C2×C4⋊Q8—471no—
324C2×#125—941no—
325C2×C22⋊Q8—472no—
326C2×C22⋊Q8—941no—
327C2×#121C2×D4711yescharacteristic
328C2×C4×Q8—472no—
329C2×C4×Q8—941no—
330C2×C4×D4—21no—
331C2×C4×D4C2×D4711yescharacteristic
332C2×#117—941no—
333C22×C42C2×D4711yescharacteristic
334#476—944no—
335#461—944no—
336C2×#106—472no—
337C2×C4⋊C8—471no—
338C2×C4⋊C8—471no—
339C2×#103—472no—
340C2×#103—472no—
341C2×#103—472no—
342C2×#102—472no—
343C2×C22⋊C8—472no—
344#390C2×D4711yescharacteristic
345#389C2×D4711yescharacteristic
346#388—21no—
347#386—21no—
348#380C2×D4711yescharacteristic
349C4×C4⋊C4C2×D4711yescharacteristic
350C4×C22⋊C4—21no—
351#340—472no—
352#340—472no—
353#336—472no—
354C47#2415111yescharacteristic, complemented, Sylow
355C22×Q8—472no—
356C22×Q8—941no—
357C22×D4—21no—
358C22×D4C22×D4711yescharacteristic
359C23×C4—21no—
360C23×C4C22⋊D4711yescharacteristic
361C23×C4C22⋊D4711yescharacteristic
362C23×C4C22×D4711yescharacteristic
363C2×SD16—944no—
364C2×SD16—944no—
365C2×SD16—1882no—
366C22×C8—472no—
367C22×C8—472no—
368C4⋊Q8—474no—
369C4⋊Q8—474no—
370C4⋊Q8—942no—
371C4⋊Q8—942no—
372#125—948no—
373C22⋊Q8—944no—
374C22⋊Q8—944no—
375C22⋊Q8—948no—
376#121—22no—
377#121—22no—
378C4×Q8—944no—
379C4×Q8—944no—
380C4×Q8—948no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—44no—
384#117—948no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—472no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C4⋊C4C22×D4711yescharacteristic
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—942no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—942no—
405C2×C42C22⋊D4711yescharacteristic
406C2×C42C22⋊D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4711yescharacteristic
409#106—944no—
410C4⋊C8—942no—
411C4⋊C8—942no—
412#103—944no—
413#103—944no—
414#103—1882no—
415#102—944no—
416C22⋊C8—1882no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—21no—
421C24C2×C22⋊D4711yescharacteristic
422C2×Q8—474no—
423C2×Q8—944no—
424C2×Q8—942no—
425C2×Q8—942no—
426C2×Q8—942no—
427C2×Q8—942no—
428C2×Q8—942no—
429C2×Q8—1882no—
430C2×Q8—1882no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#1062535012yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4#1062533514yescomplemented
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C2×C22⋊D4711yescharacteristic
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C23×D4711yescharacteristic
466SD16—1888no—
467C2×C8—942no—
468C2×C8—942no—
469C2×C8—942no—
470C2×C8—942no—
471C2×C8—942no—
472C2×C8—942no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—474no—
480C4⋊C4—948no—
481C4⋊C4—948no—
482C4⋊C4—944no—
483C4⋊C4—944no—
484C4⋊C4—944no—
485C4⋊C4—942no—
486C4⋊C4#1062535012yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—948no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—948no—
499C42D4×D4712yes—
500C42D4×D4712yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×#1062533512yes—
515C23C22×C22⋊D4711yescharacteristic
516Q8—944no—
517Q8—944no—
518Q8—1888no—
519Q8—1882no—
520Q8—1882no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4#1309399212yes—
559C2×C4#1309399212yes—
560C2×C4C2×#1062533512yes—
561C2×C4C2×#1062533512yes—
562C2×C4#1309405412yes—
563C2×C4C2×D4×D4711yescharacteristic
564C2×C4C2×D4×D4711yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C2×C4C2×#1062535011yescharacteristic
567C2×C4#1309407011yescharacteristic
568C2×C4#1309407111yescharacteristic
569C8—1882no—
570C8—1882no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#1309399211yescharacteristic
586C22#1568963411yescharacteristic
587C22#1568970111yescharacteristic
588C22C22×#1062533511yescharacteristic
589C22C2×#1309405411yescharacteristic
590C22#1568992111yescharacteristic
591C22#1568992111yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—1884no—
600C4—1884no—
601C4#1568926114yes—
602C4#1568976911yescharacteristic
603C4#1568977811yescharacteristic
604C4#1568978211yescharacteristic
605C4#1568979411yescharacteristic
606C2—42no—
607C2—82no—
608C2#2024673311yescharacteristic
609C2C2×#1568926111yescharacteristic
610C2#2025262811yescharacteristic
611C2#2025339611yescharacteristic
612C2#2025355011yescharacteristic
613C2#2025470411yescharacteristic
614C2#2025476111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×#121×C47C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenterC4×#121D47
hypocenterC47#24151
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15689680C2semidirect
2. C2×#15689413C2semidirect
3. C2×#15689328C2semidirect
4. #20244584C2semidirect
6. #20244336C2semidirect
7. #20244311C2semidirect
8. #20243746C2semidirect
9. #20243655C2semidirect
10. #20243368C2semidirect
56. #13094024D4semidirect
57. #13094024D4semidirect
140. C22×C4×C47SD16semidirect
354. C47#24151semidirect
459. C22×C4#10625335semidirect

Maximal subgroups

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

subgroupindex
C2×#156896802
C2×#156894132
C2×#156893282
#202445842
#202443912
#202443362
#202443112
#202437462
#202436552
#202433682
C4×#121×C472
#2415147