Finite Group Databasebeta

← #26139031

Subgroups — #26139031

Order 12032.

2150 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688997C212yesmaximal
2C2×#15688994C212yesmaximal
3C2×#15688980C211yesmaximal, characteristic
4C2×#15688970C211yesmaximal, characteristic
5C2×#95×D47C211yesmaximal, characteristic
6#20241653C211yesmaximal, characteristic, complemented
7#20241652C211yesmaximal, characteristic, complemented
8#20241651C211yesmaximal, characteristic, complemented
9#20241617C212yesmaximal, complemented
10Dic47⋊#95C212yesmaximal, complemented
11C47×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C23×C4×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C22×C4⋊Dic47C2212yes—
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4C2212yes—
18C22×Dic47⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2212yes—
21C22×C4×Dic47C2212yes—
22#15688997—28no—
23#15688994—28no—
24#15688980—24no—
25#15688970C418yescomplemented
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2211yescharacteristic
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2211yescharacteristic
34#95×D47—24no—
35C22×C4⋊C4×C47C2212yes—
36C22×C4⋊C4×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2211yescharacteristic
40C24×D47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2312yes—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—24no—
53C22×C4×D47—24no—
54C22×C4×D47—42no—
55C22×C4×D47—42no—
56C22×C4×D47—42no—
57C22×C4×D47—42no—
58C22×C4×D47C2×C414yes—
59C22×C4×D47D414yes—
60C22×C4×D47Q814yes—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47—24no—
63C2×C4⋊Dic47—24no—
64C2×C4⋊Dic47—42no—
65C2×C4⋊Dic47—42no—
66C2×Dic47⋊C4—24no—
67C2×Dic47⋊C4—24no—
68C2×Dic47⋊C4—24no—
69C2×Dic47⋊C4—24no—
70C2×Dic47⋊C4—42no—
71C2×Dic47⋊C4—42no—
72C2×(C2×D47)⋊C4—24no—
73C2×(C2×D47)⋊C4—24no—
74C2×(C2×D47)⋊C4—24no—
75C2×(C2×D47)⋊C4C2×C418yes—
76C2×(C2×D47)⋊C4D418yes—
77C2×(C2×D47)⋊C4Q818yes—
78C2×C4×Dic47—42no—
79C2×C4×Dic47—42no—
80C2×C4×Dic47C2×C418yescomplemented
81C47⋊#95—22no—
82C47⋊#95—24no—
83C47⋊#95—22no—
84C47⋊#95—22no—
85C47⋊#95—24no—
86C47⋊#95—24no—
87C47⋊#95—24no—
88C47⋊#95C2×C414yescomplemented
89C23×C4×C47C2312yes—
90C23×C4×C47C2311yescharacteristic
91C23×C4×C47C2311yescharacteristic
92C23×C4×C47C2311yescharacteristic
93C23×C4×C47C2312yes—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—24no—
96C2×C4⋊C4×C47—24no—
97C2×C4⋊C4×C47—42no—
98C2×C4⋊C4×C47—42no—
99#95×C47—22no—
100#95×C47—24no—
101#95×C47C2×C414yescomplemented
102C24×C47C2411yescharacteristic
103C23×D47—21no—
104C23×D47—21no—
105C23×D47—21no—
106C23×D47—21no—
107C23×D47—22no—
108C23×D47—22no—
109C23×D47C22×C412yes—
110C23×D47C2×D414yes—
111C23×D47C2×D412yes—
112C23×D47C2×Q814yes—
113C23×D47C2×Q812yes—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C22×C414yes—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×Q814yes—
140C22×C4×C47C2×Q812yes—
141C22×C4×C47#4112yes—
142C22×C4×C47#4112yes—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47C22×C414yes—
165C22×Dic47C22×C412yes—
166C22×Dic47C2×D412yes—
167C22×Dic47C2×D414yes—
168C22×Dic47C2×Q814yes—
169C22×Dic47C2×Q812yes—
170C22×Dic47#4112yes—
171C22×Dic47#4112yes—
172C2×C4×D47—28no—
173C2×C4×D47—28no—
174C2×C4×D47—48no—
175C2×C4×D47—44no—
176C2×C4×D47—44no—
177C2×C4×D47—48no—
178C2×C4×D47—48no—
179C2×C4×D47—48no—
180C2×C4×D47—48no—
181C2×C4×D47—48no—
182C2×C4×D47—44no—
183C2×C4×D47—44no—
184C2×C4×D47—44no—
185C2×C4×D47—44no—
186C2×C4×D47C4⋊C4116yescomplemented
187C4⋊C4×C47—48no—
188C4⋊C4×C47—44no—
189(C2×D47)⋊C4—48no—
190(C2×D47)⋊C4C4⋊C4132yescomplemented
191C4⋊Dic47—48no—
192C4⋊Dic47—44no—
193Dic47⋊C4—48no—
194Dic47⋊C4—48no—
195C4×Dic47—48no—
196C23×C47C23×C411yescharacteristic
197C23×C47C22×D411yescharacteristic
198C23×C47C22×D412yes—
199C23×C47C22×Q811yescharacteristic
200C23×C47C22×Q812yes—
201C23×C47C2×#4111yescharacteristic
202C23×C47C2×#4111yescharacteristic
203C23×C47C2×#4111yescharacteristic
204C23×C47C2×#4111yescharacteristic
205C23×C47H(5,𝔽2)12yes—
206C23×C47H(5,𝔽2)12yes—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—24no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47—24no—
221C22×D47—24no—
222C22×D47—24no—
223C22×D47—24no—
224C22×D47—24no—
225C22×D47C2×C4⋊C418yes—
226C22×D47C2×C4⋊C414yes—
227C22×D47#12714yes—
228C22×D47C4⋊Q814yes—
229C22×D47C4⋊Q818yes—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47C2×C4⋊C418yes—
258C2×C4×C47C2×C4⋊C414yes—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47—42no—
285C2×Dic47—42no—
286C2×Dic47C2×C4⋊C418yes—
287C2×Dic47C2×C4⋊C414yes—
288C4×D47—416no—
289C4×D47—416no—
290C4×D47—416no—
291#13965—471nomaximal, Sylow
292C22×C47C22×C4⋊C411yescharacteristic
293C22×C47C22×C4⋊C412yes—
294C22×C47C4×#4111yescharacteristic
295C22×C47C4×#4111yescharacteristic
296C22×C47#51812yes—
297C22×C47#52511yescharacteristic
298C22×C47#52511yescharacteristic
299C22×C47C2×#12711yescharacteristic
300C22×C47C2×C4⋊Q811yescharacteristic
301C22×C47C2×C4⋊Q812yes—
302C22×C47#53311yescharacteristic
303C22×C47#53311yescharacteristic
304C22×C47#53411yescharacteristic
305C22×C47#53411yescharacteristic
306C22×C47#54312yes—
307C22×C47#54612yes—
308C22×C47#54612yes—
309C22×C47#54612yes—
310C22×C47#54612yes—
311C22×C47#55412yes—
312C22×C47#55412yes—
313C22×C47#55412yes—
314C22×C47#55412yes—
315C2×D47—24no—
316C2×D47—24no—
317C2×D47—24no—
318C2×D47—28no—
319C2×D47—28no—
320C2×D47—24no—
321C2×D47—24no—
322C2×D47—28no—
323C2×D47—28no—
324C2×D47—24no—
325C2×D47#38418yes—
326C4×C47—24no—
327C4×C47—28no—
328C4×C47—42no—
329C4×C47—42no—
330C4×C47—82no—
331Dic47—24no—
332Dic47—28no—
333Dic47—42no—
334Dic47—42no—
335Dic47—82no—
336C2×#385—472no—
337C2×#385—472no—
338C2×#380—471no—
339C2×C4×C22⋊C4—471no—
340C22×#95—471no—
341#1444—471no—
342#1444—471no—
343#1444—471no—
344#1444D4711yescharacteristic, complemented
345C4⋊#95—472no—
346C4⋊#95—472no—
347C2×C47C2×#38411yescharacteristic
348C2×C47#229611yescharacteristic
349C2×C47#231511yescharacteristic
350C2×C47#234912yes—
351C2×C47#234912yes—
352C2×C47#253911yescharacteristic
353C2×C47#253911yescharacteristic
354C2×C47#254912yes—
355C2×C47#254912yes—
356C2×C47#258311yescharacteristic
357C2×C47#258311yescharacteristic
358D47—28no—
359C24×C4—471no—
360C24×C4—471no—
361C24×C4—471no—
362C22×C4⋊C4—472no—
363C22×C4⋊C4—472no—
364C22×C4⋊C4—472no—
365C22×C4⋊C4—471no—
366C22×C4⋊C4C2×D4712yes—
367C22×C4⋊C4C2×D4711yescharacteristic
368C22×C22⋊C4—472no—
369C22×C22⋊C4—472no—
370C22×C42—472no—
371#385—948no—
372#385—948no—
373#380—944no—
374C4×C22⋊C4—478no—
375C2×#95—471no—
376C2×#95—471no—
377C2×#95—471no—
378C2×#95—472no—
379C2×#95—472no—
380C2×#95—472no—
381C2×#95—472no—
382C2×#95—471no—
383C2×#95—944no—
384C2×#95C2×D4712yes—
385C2×#95C2×D4711yescharacteristic
386C2×#95C2×D4711yescharacteristic
387C47#1396511yescharacteristic, complemented, Sylow
388C25—471no—
389C23×C4—472no—
390C23×C4—471no—
391C23×C4—474no—
392C23×C4—474no—
393C23×C4—474no—
394C23×C4—471no—
395C23×C4—471no—
396C23×C4—472no—
397C23×C4—942no—
398C23×C4—942no—
399C23×C4—942no—
400C23×C4—942no—
401C23×C4—944no—
402C23×C4—944no—
403C23×C4—942no—
404C23×C4—942no—
405C23×C4—1882no—
406C23×C4—1882no—
407C23×C4—1882no—
408C23×C4—1882no—
409C23×C4C22×D4712yes—
410C23×C4C22×D4711yescharacteristic
411C23×C4C22×D4711yescharacteristic
412C23×C4C22×D4712yes—
413C23×C4C22×D4711yescharacteristic
414C2×C4⋊C4—22no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—42no—
418C2×C4⋊C4—42no—
419C2×C4⋊C4—944no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—944no—
422C2×C4⋊C4—944no—
423C2×C4⋊C4—942no—
424C2×C4⋊C4—944no—
425C2×C4⋊C4—944no—
426C2×C4⋊C4—1882no—
427C2×C4⋊C4—1882no—
428C2×C4⋊C4—1882no—
429C2×C4⋊C4—1882no—
430C2×C22⋊C4—478no—
431C2×C22⋊C4—478no—
432C2×C22⋊C4—478no—
433C2×C22⋊C4—944no—
434C2×C22⋊C4—944no—
435C2×C22⋊C4—944no—
436C2×C42—478no—
437C2×C42—1882no—
438C2×C42—1882no—
439#95—22no—
440#95—24no—
441#95—474no—
442#95—942no—
443#95—942no—
444#95—944no—
445#95—944no—
446#95—944no—
447#95—942no—
448#95—944no—
449#95C4×D4714yescomplemented
450C24—472no—
451C24—474no—
452C24—472no—
453C24—474no—
454C24—472no—
455C24—941no—
456C24—941no—
457C24—941no—
458C24—941no—
459C24—942no—
460C24—942no—
461C24C23×D4711yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—474no—
484C22×C4—472no—
485C22×C4—472no—
486C22×C4—474no—
487C22×C4—474no—
488C22×C4—472no—
489C22×C4—4716no—
490C22×C4—472no—
491C22×C4—472no—
492C22×C4—948no—
493C22×C4—948no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—941no—
505C22×C4—941no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—1888no—
516C22×C4—1888no—
517C22×C4—1884no—
518C22×C4—1884no—
519C22×C4—1888no—
520C22×C4—1888no—
521C22×C4—1888no—
522C22×C4—1888no—
523C22×C4—1884no—
524C22×C4—1884no—
525C22×C4—1884no—
526C22×C4—1884no—
527C22×C4C2×C4×D4712yes—
528C22×C4C2×C4×D4714yes—
529C22×C4D4×D4714yes—
530C22×C4D4×D4712yes—
531C22×C4#1062535212yes—
532C22×C4Q8×D4714yes—
533C22×C4Q8×D4712yes—
534C22×C4#1062535412yes—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—1888no—
538C4⋊C4—1888no—
539C4⋊C4—1884no—
540C4⋊C4—1888no—
541C22⋊C4—4732no—
542C22⋊C4—1888no—
543C42—1888no—
544C23—478no—
545C23—474no—
546C23—474no—
547C23—478no—
548C23—474no—
549C23—944no—
550C23—944no—
551C23—944no—
552C23—944no—
553C23—944no—
554C23—944no—
555C23—944no—
556C23—944no—
557C23—944no—
558C23—942no—
559C23—942no—
560C23—944no—
561C23—942no—
562C23—942no—
563C23—942no—
564C23—942no—
565C23—942no—
566C23—942no—
567C23C22×C4×D4711yescharacteristic
568C23C2×D4×D4712yes—
569C23C2×D4×D4711yescharacteristic
570C23C2×Q8×D4711yescharacteristic
571C23C2×Q8×D4712yes—
572C23#41×D4711yescharacteristic
573C23#41×D4711yescharacteristic
574C23C2×#1062535211yescharacteristic
575C23C2×#1062535411yescharacteristic
576C23#1309406912yes—
577C23#1309407012yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—478no—
606C2×C4—474no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—944no—
612C2×C4—944no—
613C2×C4—944no—
614C2×C4—944no—
615C2×C4—942no—
616C2×C4—942no—
617C2×C4—944no—
618C2×C4—944no—
619C2×C4—942no—
620C2×C4—942no—
621C2×C4—18816no—
622C2×C4—18816no—
623C2×C4—18816no—
624C2×C4—1882no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4—1882no—
629C2×C4—1882no—
630C2×C4—1882no—
631C2×C4—1882no—
632C2×C4—1882no—
633C2×C4—1881no—
634C2×C4—1881no—
635C2×C4—1881no—
636C2×C4—1881no—
637C2×C4C4⋊C4×D4718yes—
638C2×C4C4⋊C4×D4714yes—
639C22—478no—
640C22—948no—
641C22—948no—
642C22—948no—
643C22—948no—
644C22—944no—
645C22—944no—
646C22—944no—
647C22—944no—
648C22—944no—
649C22—944no—
650C22C2×C4⋊C4×D4712yes—
651C22C2×C4⋊C4×D4711yescharacteristic
652C22C4×#1062535211yescharacteristic
653C22C4×#1062535411yescharacteristic
654C22#1568959512yes—
655C22#1568963311yescharacteristic
656C22#1568963611yescharacteristic
657C22#127×D4711yescharacteristic
658C22C4⋊Q8×D4712yes—
659C22C4⋊Q8×D4711yescharacteristic
660C22#1568967811yescharacteristic
661C22#1568967911yescharacteristic
662C22#1568968311yescharacteristic
663C22#1568968411yescharacteristic
664C22#1568974812yes—
665C22#1568976012yes—
666C22#1568976412yes—
667C22#1568976512yes—
668C22#1568977012yes—
669C22#1568982712yes—
670C22#1568982912yes—
671C22#1568983112yes—
672C22#1568983412yes—
673C4—28no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—948no—
679C4—944no—
680C4—3762no—
681C4—1882no—
682C4—1882no—
683C2—948no—
684C2#384×D4711yescharacteristic
685C2#2024586911yescharacteristic
686C2#2024599911yescharacteristic
687C2#2024629412yes—
688C2#2024629712yes—
689C2#2024831311yescharacteristic
690C2#2024831511yescharacteristic
691C2#2024839312yes—
692C2#2024839512yes—
693C2#2024876211yescharacteristic
694C2#2024876411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1444C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1444D47
hypocenterC47#13965
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
6. #20241653C2semidirect
7. #20241652C2semidirect
8. #20241651C2semidirect
9. #20241617C2semidirect
10. Dic47⋊#95C2semidirect
11. C47×#1444C2semidirect
25. #15688970C4semidirect
80. C2×C4×Dic47C2×C4semidirect
88. C47⋊#95C2×C4semidirect
101. #95×C47C2×C4semidirect
186. C2×C4×D47C4⋊C4semidirect
190. (C2×D47)⋊C4C4⋊C4semidirect
344. #1444D47semidirect
387. C47#13965semidirect
449. #95C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889972
C2×#156889942
C2×#156889802
C2×#156889702
C2×#95×D472
#202416532
#202416522
#202416512
#202416172
Dic47⋊#952
C47×#14442
#1396547