Finite Group Databasebeta

← #26232379

Subgroups — #26232379

Order 12032.

1412 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625350C211yesmaximal, characteristic
2C2×#15689199C212yesmaximal, complemented
3#20244322C212yesmaximal, complemented
4#20243685C212yesmaximal, complemented
5#20243684C212yesmaximal, complemented
6#20243456C211yesmaximal, characteristic, complemented
7#20243455C211yesmaximal, characteristic, complemented
8#20243367C211yesmaximal, characteristic, complemented
9#20243145C211yesmaximal, characteristic, complemented
10#20243080C211yesmaximal, characteristic, complemented
11C47×#1728C211yesmaximal, characteristic, complemented
12C22×#10625350C411yescharacteristic
13C22×#10625324C2212yes—
14C4×#10625350—28no—
15C2×C4×Dic94C411yescharacteristic
16C2×C4×Dic94C2211yescharacteristic
17C2×C4×D188C411yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×C4×C22⋊D47C411yescharacteristic
20C2×C4×C22⋊D47C411yescharacteristic
21C2×C4×C22⋊D47C2211yescharacteristic
22C2×C4×C22⋊D47C2211yescharacteristic
23C2×#13093983C411yescharacteristic
24C2×#13093981C411yescharacteristic
25C2×#13093981C411yescharacteristic
26C2×C42×D47C2211yescharacteristic
27C2×C42×D47C2211yescharacteristic
28#15689199—28no—
29C2×Dic47⋊C8C2212yes—
30C2×C22⋊(C47⋊C8)C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32C2×(C2×D47)⋊C8C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34C2×C8×Dic47C2212yes—
35#15688878C2212yes—
36#15688878C2211yescharacteristic
37#15688878C2211yescharacteristic
38#15688877C2212yes—
39#15688877C2212yes—
40C22×M16×C47C2212yes—
41C22×C42×C47C2211yescharacteristic
42C47×#336C2212yes—
43C47×#336C2211yescharacteristic
44C47×#336C2211yescharacteristic
45C2×#10625350—24no—
46C2×#10625350—24no—
47C2×#10625350—24no—
48C22×Dic94C2×C411yescharacteristic
49C22×D188C2×C411yescharacteristic
50C22×C22⋊D47C2×C411yescharacteristic
51C22×C22⋊D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2×C411yescharacteristic
54C22×C4×D47C2×C411yescharacteristic
55C22×C4×D47C2×C411yescharacteristic
56C22×C4×D47C2311yescharacteristic
57C22×C4×D47C2311yescharacteristic
58C2×#10625324—24no—
59C2×#10625324—24no—
60C2×#10625324—42no—
61C22×C47⋊C8C2312yes—
62C22×C47⋊C8C2312yes—
63C4×Dic94—24no—
64C4×Dic94—24no—
65C4×D188—24no—
66C4×D188—24no—
67C4×C22⋊D47—24no—
68C4×C22⋊D47—24no—
69C4×C22⋊D47—24no—
70C4×C22⋊D47D418yes—
71#13093983—24no—
72#13093981—24no—
73#13093981—24no—
74C2×C4⋊Dic47C2×C411yescharacteristic
75C2×C4⋊Dic47C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×Dic47⋊C4C2×C411yescharacteristic
79C2×Dic47⋊C4C2×C411yescharacteristic
80C2×C22⋊Dic47C2×C411yescharacteristic
81C2×C22⋊Dic47C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2×C411yescharacteristic
85C2×(C2×D47)⋊C4C2×C411yescharacteristic
86C2×C4×Dic47C2×C411yescharacteristic
87C2×C4×Dic47C2×C411yescharacteristic
88C2×C4×Dic47C2×C411yescharacteristic
89C2×C4×Dic47C2×C411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92C42×D47—24no—
93C42×D47—24no—
94Dic47⋊C8—24no—
95C22⋊(C47⋊C8)—24no—
96(C2×D47)⋊C8—24no—
97(C2×D47)⋊C8—24no—
98(C2×D47)⋊C8—42no—
99C8×Dic47—24no—
100C23×C4×C47C2×C411yescharacteristic
101C23×C4×C47C2×C411yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C2×M16×C47—24no—
104C2×M16×C47—24no—
105C2×M16×C47—42no—
106C22×C8×C47C2312yes—
107C22×C8×C47C2312yes—
108C2×C42×C47—24no—
109C2×C42×C47C2×C411yescharacteristic
110C2×C42×C47C2×C411yescharacteristic
111C2×C42×C47C2311yescharacteristic
112C2×C42×C47C2311yescharacteristic
113C24×C47C22×C411yescharacteristic
114C23×D47C22×C411yescharacteristic
115C23×D47C22×C411yescharacteristic
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—41no—
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C22×C411yescharacteristic
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C2×D414yes—
131C22×C4×C47C2411yescharacteristic
132C2×C22⋊D47—22no—
133C2×C22⋊D47—22no—
134C2×C22⋊D47—22no—
135C2×C22⋊D47—22no—
136C2×C22⋊D47—22no—
137C2×C22⋊D47C22⋊C414yes—
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144#10625350—232no—
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—21no—
162C2×C4×D47—21no—
163C2×C4×D47—22no—
164C2×C4×D47—41no—
165C2×C4×D47—41no—
166C2×C4×D47M1614yescomplemented
167C2×C4×D47C2×D414yes—
168C2×Dic94—22no—
169C2×Dic94—22no—
170C2×Dic94M1614yescomplemented
171M16×C47—48no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—22no—
177C2×C8×C47#4114yescomplemented
178C42×C47—22no—
179C42×C47—22no—
180C42×C47—22no—
181C42×C47—22no—
182C22⋊Dic47—22no—
183C22⋊Dic47C22⋊C414yes—
184(C2×D47)⋊C4—22no—
185(C2×D47)⋊C4—22no—
186(C2×D47)⋊C4—22no—
187(C2×D47)⋊C4C22⋊C414yes—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190Dic47⋊C4—22no—
191Dic47⋊C4—22no—
192Dic47⋊C4—22no—
193Dic47⋊C4C22⋊C414yes—
194C4×Dic47—22no—
195C4×Dic47—22no—
196C4×Dic47—22no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C4×Dic47C2×D414yes—
200C2×C47⋊C8—22no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204C2×C47⋊C8—22no—
205C2×C47⋊C8—22no—
206#10625324—48no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47C2×C22⋊C412yes—
214C23×C47C23×C411yescharacteristic
215C22×D47—21no—
216C22×D47—21no—
217C22×D47—21no—
218C22×D47—21no—
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—21no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—21no—
227C22×D47C2×C22⋊C412yes—
228C22×D47C2×M1612yes—
229C2×C4×C47—28no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—44no—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×M1612yes—
246C2×C4×C47C2×M1612yes—
247C2×C4×C47#13112yes—
248C2×C4×C47C22×D412yes—
249C2×C4×C47C2×#4112yes—
250C2×C4×C47H(5,𝔽2)11yescharacteristic
251C2×C4×C472−1+411yescharacteristic
252C22⋊D47—28no—
253C22⋊D47—28no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47C2×C22⋊C412yes—
267C2×Dic47C2×C22⋊C412yes—
268C2×Dic47C2×C22⋊C412yes—
269C2×Dic47C2×M1612yes—
270C2×Dic47C2×M1612yes—
271C2×Dic47C2×M1612yes—
272D188—28no—
273C4×D47—28no—
274C4×D47—24no—
275C4×D47—24no—
276C4×D47—28no—
277C4×D47—44no—
278C4×D47—44no—
279C4×D47—42no—
280C4×D47—42no—
281Dic94—28no—
282C8×C47—24no—
283C8×C47—42no—
284C47⋊C8—42no—
285C47⋊C8—42no—
286#22473—471nomaximal, Sylow
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47—41no—
294C22×C47C22×C22⋊C411yescharacteristic
295C22×C47C4×#4111yescharacteristic
296C22×C47#52011yescharacteristic
297C22×C47C22×M1611yescharacteristic
298C22×C47C22×M1611yescharacteristic
299C22×C47C2×#13111yescharacteristic
300C22×C47#56811yescharacteristic
301C2×D47—22no—
302C2×D47—22no—
303C2×D47—22no—
304C2×D47—22no—
305C2×D47—22no—
306C2×D47—22no—
307C2×D47—41no—
308C2×D47—41no—
309C2×D47—41no—
310C2×D47—41no—
311C2×D47—41no—
312C2×D47—41no—
313C2×D47—41no—
314C2×D47—41no—
315C4×C47—22no—
316C4×C47—22no—
317C4×C47—22no—
318C4×C47—22no—
319C4×C47—41no—
320C4×C47#54612yes—
321C4×C47#54712yes—
322Dic47—22no—
323Dic47—22no—
324Dic47—22no—
325Dic47—41no—
326Dic47—41no—
327Dic47#40714yes—
328C2×C4×#41—471no—
329C2×#436—472no—
330#1984—472no—
331#1870—472no—
332#1870—472no—
333#1817—471no—
334#1817—471no—
335#1797—471no—
336C4×C22⋊C8—471no—
337#1728—471no—
338#1728D4711yescharacteristic, complemented
339C2×C47—42no—
340C2×C47#233911yescharacteristic
341C2×C47C2×#40711yescharacteristic
342C2×C47#288011yescharacteristic
343C2×C47#296511yescharacteristic
344C2×C47#297111yescharacteristic
345C2×C47#298911yescharacteristic
346C2×C47#299111yescharacteristic
347D47—42no—
348D47—42no—
349C22×#41—471no—
350C22×M16—472no—
351C22×M16C2×D4712yes—
352C4×#41—948no—
353C2×C4×Q8—471no—
354C2×C4×Q8—471no—
355C2×C4×D4—471no—
356C2×C4×D4—471no—
357C2×C4×D4—471no—
358C2×C4×D4—471no—
359C2×C4×D4—471no—
360C2×C4×D4—471no—
361C2×#117—471no—
362C2×#117—471no—
363C2×#117—471no—
364C22×C42—471no—
365C22×C42—471no—
366C22×C42C2×D4711yescharacteristic
367#436—948no—
368C2×C4⋊C8—472no—
369C2×C22⋊C8—472no—
370C2×C22⋊C8—472no—
371C2×C22⋊C8—472no—
372C2×C22⋊C8—472no—
373C2×C4×C8—472no—
374#336—472no—
375#336—472no—
376#336—471no—
377#336—471no—
378#336—472no—
379#336C2×D4711yescharacteristic
380#336C2×D4712yes—
381#336C2×D4711yescharacteristic
382C47#2247311yescharacteristic, complemented, Sylow
383C2×#41—944no—
384C2×#41—944no—
385C2×#41—944no—
386C22×Q8—471no—
387C22×D4—471no—
388C22×D4—471no—
389C22×D4—471no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—471no—
395C23×C4—471no—
396C23×C4C4×D4711yescharacteristic
397C23×C4C4×D4711yescharacteristic
398C23×C4C22×D4711yescharacteristic
399C2×M16—24no—
400C2×M16—24no—
401C2×M16—42no—
402C2×M16—944no—
403C2×M16—944no—
404C2×M16—1882no—
405C22×C8—472no—
406C22×C8—472no—
407C22×C8C22×D4712yes—
408C22×C8C22×D4712yes—
409C4×Q8—944no—
410C4×Q8—944no—
411C4×D4—478no—
412C4×D4—944no—
413C4×D4—944no—
414C4×D4—944no—
415C4×D4—944no—
416C4×D4—944no—
417#117—944no—
418#117—944no—
419#117—944no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C22⋊C4—471no—
427C2×C22⋊C4—471no—
428C2×C22⋊C4—471no—
429C2×C22⋊C4—471no—
430C2×C22⋊C4—471no—
431C2×C22⋊C4—471no—
432C2×C42—24no—
433C2×C42—471no—
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—944no—
440C2×C42—944no—
441C2×C42C4×D4711yescharacteristic
442C2×C42C4×D4711yescharacteristic
443C2×C42C22×D4711yescharacteristic
444C2×C42C22×D4711yescharacteristic
445C4⋊C8—944no—
446C22⋊C8—944no—
447C22⋊C8—944no—
448C22⋊C8—944no—
449C22⋊C8—1882no—
450C4×C8—944no—
451C24—471no—
452C24—471no—
453C24C2×C4×D4711yescharacteristic
454#41—9432no—
455C2×Q8—474no—
456C2×Q8—942no—
457C2×Q8—942no—
458C2×D4—474no—
459C2×D4—942no—
460C2×D4—942no—
461C2×D4—942no—
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—942no—
465C2×D4—942no—
466C2×D4—942no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—1881no—
501C22×C4—1881no—
502C22×C4C2×C4×D4711yescharacteristic
503C22×C4C2×C4×D4711yescharacteristic
504C22×C4C2×C4×D4711yescharacteristic
505C22×C4C2×C4×D4711yescharacteristic
506C22×C4C2×C4×D4711yescharacteristic
507C22×C4D4×D4714yes—
508C22×C4C23×D4711yescharacteristic
509M16—48no—
510M16—1888no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—942no—
517C2×C8—942no—
518C2×C8—942no—
519C2×C8—942no—
520C2×C8—942no—
521C2×C8—942no—
522C2×C8#1062535014yescomplemented
523C4⋊C4—474no—
524C4⋊C4—942no—
525C4⋊C4—942no—
526C4⋊C4—942no—
527C4⋊C4—942no—
528C4⋊C4—942no—
529C22⋊C4—474no—
530C22⋊C4—474no—
531C22⋊C4—942no—
532C22⋊C4—942no—
533C22⋊C4—942no—
534C22⋊C4—942no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—474no—
540C42—942no—
541C42—942no—
542C42—942no—
543C42—942no—
544C42—942no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—472no—
552C23—472no—
553C23—941no—
554C23—941no—
555C23—941no—
556C23—941no—
557C23—941no—
558C23—941no—
559C23—941no—
560C23—941no—
561C23—941no—
562C23—941no—
563C23—941no—
564C23—941no—
565C23C22⋊C4×D4712yes—
566C23C22×C4×D4711yescharacteristic
567Q8—948no—
568D4—948no—
569D4—948no—
570D4—948no—
571C2×C4—28no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—44no—
586C2×C4—472no—
587C2×C4—472no—
588C2×C4—472no—
589C2×C4—472no—
590C2×C4—472no—
591C2×C4—472no—
592C2×C4—948no—
593C2×C4—948no—
594C2×C4—941no—
595C2×C4—941no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—941no—
603C2×C4—941no—
604C2×C4—941no—
605C2×C4—941no—
606C2×C4—941no—
607C2×C4—941no—
608C2×C4—1884no—
609C2×C4—1884no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4C22⋊C4×D4712yes—
613C2×C4M16×D4712yes—
614C2×C4M16×D4712yes—
615C2×C4#1309403212yes—
616C2×C4C2×D4×D4712yes—
617C2×C4C2×#1062535012yes—
618C2×C4#1309406911yescharacteristic
619C2×C4#1309407211yescharacteristic
620C8—24no—
621C8—42no—
622C8—1882no—
623C8—1882no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—942no—
632C22—942no—
633C22—942no—
634C22—942no—
635C22—942no—
636C22—942no—
637C22—1881no—
638C22—1881no—
639C22—1881no—
640C22—1881no—
641C22—1881no—
642C22—1881no—
643C22—1881no—
644C22—1881no—
645C22C2×C22⋊C4×D4711yescharacteristic
646C22C4×#1062535011yescharacteristic
647C22#1568960411yescharacteristic
648C22C2×M16×D4711yescharacteristic
649C22C2×M16×D4711yescharacteristic
650C22C2×#1309403211yescharacteristic
651C22#1568988911yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—41no—
657C4—474no—
658C4—942no—
659C4—942no—
660C4—942no—
661C4—1881no—
662C4—1881no—
663C4#1568976312yes—
664C4#1568977412yes—
665C2—42no—
666C2—1882no—
667C2—1882no—
668C2#2024621511yescharacteristic
669C2#407×D4711yescharacteristic
670C2#2025183811yescharacteristic
671C2#2025235011yescharacteristic
672C2#2025240611yescharacteristic
673C2#2025250211yescharacteristic
674C2#2025251811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1728C2
prime radicalC22×#10625350C4
cosocleC22×C4×C47C24
hypercenter#1728D47
hypocenterC47#22473
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1728

Complements

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

NH ≅ G/Ntype
2. C2×#15689199C2semidirect
3. #20244322C2semidirect
4. #20243685C2semidirect
5. #20243684C2semidirect
6. #20243456C2semidirect
7. #20243455C2semidirect
8. #20243367C2semidirect
9. #20243145C2semidirect
10. #20243080C2semidirect
11. C47×#1728C2semidirect
166. C2×C4×D47M16semidirect
170. C2×Dic94M16semidirect
177. C2×C8×C47#41semidirect
338. #1728D47semidirect
382. C47#22473semidirect
522. C2×C8#10625350semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253502
C2×#156891992
#202443222
#202436852
#202436842
#202434562
#202434552
#202433672
#202431452
#202430802
C47×#17282
#2247347