Finite Group Databasebeta

← #26139161

Subgroups — #26139161

Order 12032.

2558 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D188C211yesmaximal, characteristic
2C2×#15689024C211yesmaximal, characteristic
3C2×#15689024C211yesmaximal, characteristic
4C2×#15688994C214yesmaximal, complemented
5C2×#15688992C211yesmaximal, characteristic, complemented
6#20241632C214yesmaximal, complemented
7#20241617C211yesmaximal, characteristic, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C2×C47×#389C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C23×C4×D47C2212yes—
12C2×C4×D188—24no—
13C2×C4×D188—24no—
14C2×C4×D188—24no—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2214yes—
17C22×(C2×D47)⋊C4C2212yes—
18C22×(C2×D47)⋊C4C2214yes—
19C22×(C2×D47)⋊C4C2214yes—
20C22×C4×Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22#15689024—24no—
23#15689024C418yescomplemented
24#15688994—216no—
25#15688992—24no—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2212yes—
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C42×C47C2211yescharacteristic
33C47×#389—24no—
34C2×#95×C47C2212yes—
35C24×D47C2312yes—
36C22×D188—22no—
37C22×D188—22no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188C2×C414yes—
41C22×D188D414yes—
42C22×D188Q814yes—
43C23×Dic47C2312yes—
44C23×Dic47C2312yes—
45C23×Dic47C2312yes—
46C22×C4×D47—24no—
47C22×C4×D47—24no—
48C22×C4×D47—24no—
49C22×C4×D47—28no—
50C22×C4×D47—44no—
51C4×D188—232no—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47D414yes—
55C2×Dic47⋊C4—28no—
56C2×Dic47⋊C4—28no—
57C2×Dic47⋊C4—44no—
58C2×(C2×D47)⋊C4—24no—
59C2×(C2×D47)⋊C4—24no—
60C2×(C2×D47)⋊C4—28no—
61C2×(C2×D47)⋊C4—24no—
62C2×(C2×D47)⋊C4—28no—
63C2×(C2×D47)⋊C4—28no—
64C2×(C2×D47)⋊C4—28no—
65C2×(C2×D47)⋊C4—44no—
66C2×(C2×D47)⋊C4C2×C4116yescomplemented
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—22no—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47C2×C414yescomplemented
73C47⋊#95—28no—
74C47⋊#95—24no—
75C47⋊#95D418yescomplemented
76C23×C4×C47C2312yes—
77C23×C4×C47C2312yes—
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2311yescharacteristic
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—28no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—24no—
88C2×C4⋊C4×C47C2×C414yescomplemented
89C2×C42×C47—22no—
90C2×C42×C47—22no—
91C2×C42×C47D414yes—
92#95×C47—24no—
93C24×C47C2411yescharacteristic
94C23×D47—24no—
95C23×D47—22no—
96C23×D47—22no—
97C23×D47—24no—
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—24no—
101C23×D47—24no—
102C23×D47C22×C414yes—
103C23×D47C2×D414yes—
104C23×D47C2×Q814yes—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—24no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—24no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47C22×C414yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D414yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×Q812yes—
130C22×C4×C47#4112yes—
131C22×C4×C47#4112yes—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—24no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—22no—
146C22×Dic47C22×C414yes—
147C22×Dic47C2×D414yes—
148C22×Dic47C2×D414yes—
149C22×Dic47C2×D414yes—
150C22×Dic47#4114yes—
151C2×D188—28no—
152C2×D188—28no—
153C2×D188—28no—
154C2×D188—28no—
155C2×D188—28no—
156C2×D188—28no—
157C2×D188C4⋊C4116yescomplemented
158C2×C4×D47—216no—
159C2×C4×D47—416no—
160C2×C4×D47—416no—
161C2×C4×D47—48no—
162C2×C4×D47—48no—
163C4⋊C4×C47—28no—
164C4⋊C4×C47—44no—
165C4⋊C4×C47—48no—
166C42×C47—28no—
167(C2×D47)⋊C4—216no—
168(C2×D47)⋊C4—232no—
169(C2×D47)⋊C4—416no—
170C4⋊Dic47—28no—
171Dic47⋊C4—416no—
172C4×Dic47—28no—
173C4×Dic47—44no—
174C23×C47C23×C411yescharacteristic
175C23×C47C22×D411yescharacteristic
176C23×C47C22×D411yescharacteristic
177C23×C47C22×D412yes—
178C23×C47C22×D411yescharacteristic
179C23×C47C22×Q811yescharacteristic
180C23×C47C2×#4112yes—
181C23×C47C2×#4111yescharacteristic
182C23×C47C2×#4111yescharacteristic
183C23×C47H(5,𝔽2)12yes—
184C23×C472−1+412yes—
185C22×D47—28no—
186C22×D47—24no—
187C22×D47—24no—
188C22×D47—28no—
189C22×D47—24no—
190C22×D47—24no—
191C22×D47—24no—
192C22×D47—24no—
193C22×D47—28no—
194C22×D47—28no—
195C22×D47—28no—
196C22×D47—28no—
197C22×D47—42no—
198C22×D47—42no—
199C22×D47—44no—
200C22×D47—42no—
201C22×D47—42no—
202C22×D47—44no—
203C22×D47—44no—
204C22×D47—44no—
205C22×D47—44no—
206C22×D47—44no—
207C22×D47C2×C4⋊C418yes—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—28no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—44no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—44no—
226C2×C4×C47—44no—
227C2×C4×C47C2×C4⋊C414yes—
228C2×C4×C47C4×D414yes—
229C2×C4×C47C4×D414yes—
230C2×C4×C47#12014yes—
231C2×C4×C47C22⋊Q814yes—
232C2×C4×C47C22⋊Q814yes—
233C2×C4×C47#12314yes—
234C2×C4×C47#12714yes—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—28no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—44no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47C4×D418yescomplemented
254D188—232no—
255C4×D47—432no—
256#13977—471nomaximal, Sylow
257C22×C47C22×C4⋊C411yescharacteristic
258C22×C47C2×C4×D412yes—
259C22×C47C2×C4×D411yescharacteristic
260C22×C47C2×C4×D411yescharacteristic
261C22×C47#52012yes—
262C22×C47C2×#12011yescharacteristic
263C22×C47C2×C22⋊Q811yescharacteristic
264C22×C47C2×C22⋊Q811yescharacteristic
265C22×C47C2×#12311yescharacteristic
266C22×C47C2×#12711yescharacteristic
267C22×C47#53211yescharacteristic
268C22×C47#53212yes—
269C22×C47#53511yescharacteristic
270C22×C47#53611yescharacteristic
271C22×C47#54212yes—
272C22×C47D4212yes—
273C22×C47D4212yes—
274C22×C47#54712yes—
275C22×C47#54712yes—
276C22×C47#54812yes—
277C22×C47D4×Q812yes—
278C22×C47#55312yes—
279C22×C47#55412yes—
280C2×D47—216no—
281C2×D47—28no—
282C2×D47—44no—
283C2×D47—44no—
284C2×D47—44no—
285C2×D47—48no—
286C2×D47—48no—
287C2×D47—44no—
288C2×D47—44no—
289C2×D47—48no—
290C2×D47—44no—
291C4×C47—24no—
292C4×C47—44no—
293C4×C47—42no—
294C4×C47—82no—
295C4×C47#38518yescomplemented
296Dic47—28no—
297Dic47—44no—
298Dic47—44no—
299C22×C4×D4—471no—
300C2×#390—471no—
301C2×#390—471no—
302C2×#389D4711yescharacteristic, complemented
303C2×#385—474no—
304C2×#384—471no—
305#1439—474no—
306C4⋊#95—471no—
307C4×#95—471no—
308C2×C47C2×#38511yescharacteristic
309C2×C47C4×#12711yescharacteristic
310C2×C47#232611yescharacteristic
311C2×C47D4×C4⋊C412yes—
312C2×C47#236112yes—
313C2×C47#242212yes—
314C2×C47#243211yescharacteristic
315C2×C47#243311yescharacteristic
316C2×C47#245212yes—
317C2×C47#254211yescharacteristic
318C2×C47#256611yescharacteristic
319D47—48no—
320C23×D4—471no—
321C24×C4—472no—
322C2×C4×D4—944no—
323C2×C4×D4—944no—
324C2×C4×D4—944no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4—474no—
327C22×C4⋊C4C2×D4711yescharacteristic
328C22×C4⋊C4C2×D4712yes—
329C22×C4⋊C4C2×D4711yescharacteristic
330C22×C22⋊C4—474no—
331C22×C22⋊C4—472no—
332C22×C22⋊C4—474no—
333C22×C42—471no—
334C22×C42—471no—
335C22×C42C2×D4711yescharacteristic
336#390—478no—
337#390—944no—
338#389—24no—
339#385—9416no—
340#384—944no—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—474no—
344C2×#95C2×D4712yes—
345C47#1397711yescharacteristic, complemented, Sylow
346C25—472no—
347C22×D4—474no—
348C22×D4—474no—
349C22×D4—474no—
350C22×D4—942no—
351C22×D4—942no—
352C22×D4—942no—
353C22×D4—942no—
354C23×C4—472no—
355C23×C4—472no—
356C23×C4—472no—
357C23×C4—944no—
358C23×C4—944no—
359C23×C4—944no—
360C23×C4—948no—
361C23×C4—1884no—
362C23×C4C22×D4712yes—
363C23×C4C22×D4712yes—
364C23×C4C22×D4711yescharacteristic
365C23×C4C22×D4711yescharacteristic
366C23×C4C22×D4711yescharacteristic
367C4×D4—9432no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—1884no—
381C2×C4⋊C4C4×D4714yescomplemented
382C2×C22⋊C4—4716no—
383C2×C22⋊C4—944no—
384C2×C22⋊C4—948no—
385C2×C22⋊C4—948no—
386C2×C22⋊C4—944no—
387C2×C22⋊C4—944no—
388C2×C22⋊C4—948no—
389C2×C22⋊C4—948no—
390C2×C22⋊C4—1884no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—474no—
394C2×C42—942no—
395C2×C42—942no—
396C2×C42—942no—
397C2×C42—942no—
398C2×C42—942no—
399C2×C42C22⋊D4714yes—
400#95—24no—
401#95—478no—
402#95—944no—
403#95—948no—
404C24—474no—
405C24—474no—
406C24—474no—
407C24—944no—
408C24—942no—
409C24—942no—
410C24—944no—
411C24—942no—
412C24—942no—
413C24—944no—
414C24—944no—
415C24C23×D4711yescharacteristic
416C2×D4—4716no—
417C2×D4—948no—
418C2×D4—948no—
419C2×D4—948no—
420C2×D4—948no—
421C2×D4—948no—
422C2×D4—948no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—474no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—9416no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—944no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—944no—
458C22×C4—942no—
459C22×C4—18816no—
460C22×C4—18816no—
461C22×C4—1888no—
462C22×C4—1888no—
463C22×C4C2×C4×D4714yes—
464C22×C4C2×C4×D4712yes—
465C22×C4#1062535012yes—
466C22×C4D4×D4714yes—
467C22×C4D4×D4712yes—
468C22×C4D4×D4712yes—
469C22×C4Q8×D4712yes—
470C22×C4#1062535412yes—
471C22×C4C2×C22⋊D4712yes—
472C22×C4C2×C22⋊D4712yes—
473C22×C4C2×C22⋊D4712yes—
474C4⋊C4—28no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—948no—
478C4⋊C4—18816no—
479C22⋊C4—9432no—
480C22⋊C4—9416no—
481C22⋊C4—18816no—
482C42—28no—
483C42—948no—
484C42—1884no—
485C23—478no—
486C23—948no—
487C23—948no—
488C23—948no—
489C23—948no—
490C23—948no—
491C23—944no—
492C23—944no—
493C23—948no—
494C23—944no—
495C23—944no—
496C23—944no—
497C23—944no—
498C23—1884no—
499C23—1884no—
500C23—1884no—
501C23—1884no—
502C23—1884no—
503C23—1882no—
504C23—1882no—
505C23—1884no—
506C23—1882no—
507C23—1882no—
508C23C22×C4×D4711yescharacteristic
509C23C2×D4×D4711yescharacteristic
510C23C2×D4×D4712yes—
511C23C2×D4×D4711yescharacteristic
512C23C22×C22⋊D4711yescharacteristic
513C23C2×Q8×D4711yescharacteristic
514C23#41×D4712yes—
515C23C2×#1062535011yescharacteristic
516C23C2×#1062535411yescharacteristic
517C23#1309407012yes—
518C23#1309407212yes—
519D4—9432no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—478no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—944no—
543C2×C4—944no—
544C2×C4—944no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—948no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—18832no—
552C2×C4—1882no—
553C2×C4—1884no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4C4⋊C4×D4714yes—
560C2×C4C4×C22⋊D4714yes—
561C2×C4#1309398614yes—
562C2×C4#1309399214yes—
563C2×C4#1309400014yes—
564C2×C4#1309400214yes—
565C2×C4#1309400914yes—
566C2×C4#1309402014yes—
567C22—9416no—
568C22—948no—
569C22—1884no—
570C22—1884no—
571C22—1884no—
572C22—1888no—
573C22—1888no—
574C22—1884no—
575C22—1888no—
576C22—1884no—
577C22—1884no—
578C22C2×C4⋊C4×D4711yescharacteristic
579C22C4×D4×D4712yes—
580C22C2×C4×C22⋊D4711yescharacteristic
581C22C2×#1309398611yescharacteristic
582C22#1568960512yes—
583C22C2×#1309399211yescharacteristic
584C22C2×#1309400011yescharacteristic
585C22C2×#1309400211yescharacteristic
586C22C2×#1309400911yescharacteristic
587C22C2×#1309402011yescharacteristic
588C22#1568966911yescharacteristic
589C22#1568967012yes—
590C22#1568968711yescharacteristic
591C22#1568969711yescharacteristic
592C22#1568974512yes—
593C22#1568975712yes—
594C22#1568975712yes—
595C22#1568977512yes—
596C22#1568977612yes—
597C22#1568978312yes—
598C22#1568978812yes—
599C22#1568982612yes—
600C22#1568982912yes—
601C4—24no—
602C4—44no—
603C4—42no—
604C4—82no—
605C4—948no—
606C4—1884no—
607C4—1884no—
608C4#1568899418yescomplemented
609C2—1888no—
610C2C2×#1568899411yescharacteristic
611C2C4×#1309402011yescharacteristic
612C2#2024609511yescharacteristic
613C2#2024628912yes—
614C2#2024640312yes—
615C2#2024690312yes—
616C2#2024701711yescharacteristic
617C2#2024702611yescharacteristic
618C2#2024723912yes—
619C2#2024834211yescharacteristic
620C2#2024858911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#389C2
socle, cosocleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#389D47
hypocenterC47#13977
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. C2×#15688994C2semidirect
5. C2×#15688992C2semidirect
6. #20241632C2semidirect
7. #20241617C2semidirect
8. C4×C47⋊#95C2semidirect
9. C2×C47×#389C2semidirect
23. #15689024C4semidirect
66. C2×(C2×D47)⋊C4C2×C4semidirect
72. C2×C4×Dic47C2×C4semidirect
75. C47⋊#95D4semidirect
88. C2×C4⋊C4×C47C2×C4semidirect
157. C2×D188C4⋊C4semidirect
253. C2×Dic47C4×D4semidirect
295. C4×C47#385semidirect
302. C2×#389D47semidirect
345. C47#13977semidirect
381. C2×C4⋊C4C4×D47semidirect
608. C4#15688994semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D1882
C2×#156890242
C2×#156889942
C2×#156889922
#202416322
#202416172
C4×C47⋊#952
C2×C47×#3892
#1397747