Finite Group Databasebeta

← #26227442

Subgroups — #26227442

Order 12032.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689273C211yesmaximal, characteristic
2C2×#15689025C211yesmaximal, characteristic
3#20244245C212yesmaximal, complemented
4#20244070C212yesmaximal, complemented
5#20244015C212yesmaximal, complemented
6#20243743C212yesmaximal, complemented
7#20243696C211yesmaximal, characteristic, complemented
8#20243580C211yesmaximal, characteristic, complemented
9#20243318C211yesmaximal, characteristic
10C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C47C211yesmaximal, characteristic, complemented
12C23×D188C2211yescharacteristic
13C22×#10625336C2212yes—
14C22×C4⋊Dic47C2211yescharacteristic
15C22×(C2×D47)⋊C4—21no—
16C22×(C2×D47)⋊C4—21no—
17#15689273C418yescomplemented
18C2×#13093955C2212yescomplemented
19C2×#13093948C2212yescomplemented
20C2×#13093944C2212yes—
21C2×#13093944C2212yes—
22C2×#13093944C2212yes—
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C4×C47⋊C8C2212yescomplemented
26#15689025—21no—
27#15689025—21no—
28#15689025C2212yes—
29#15689025C2212yes—
30#15688885C2212yescomplemented
31#15688885C2212yescomplemented
32#15688878C2212yescomplemented
33C2×C22⋊Q8×C47C2211yescharacteristic
34C2×C4×Q8×C47C2211yescharacteristic, complemented
35C22×C42×C47C2211yescharacteristic
36C47×#391C2211yescharacteristic, complemented
37C47×#389C2211yescharacteristic, complemented
38C47×#387—21no—
39C47×#385—21no—
40C47×#380C2211yescharacteristic
41C4×C4⋊C4×C47—21no—
42C4×C4⋊C4×C47C2211yescharacteristic, complemented
43C4×C22⋊C4×C47—21no—
44C24×D47—21no—
45C22×D188—21no—
46C22×D188—22no—
47C22×D188—42no—
48C22×D188—42no—
49C22×D188C2×C414yes—
50C22×D188C2312yes—
51C23×Dic47—21no—
52C2×#10625336—42no—
53C2×#10625336—42no—
54C2×#10625336C2×C418yescomplemented
55C22×C47⋊C8C2312yes—
56C22×C47⋊C8C2312yes—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—42no—
59C2×C4⋊Dic47C2312yes—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—22no—
63C2×(C2×D47)⋊C4—22no—
64C2×(C2×D47)⋊C4—44no—
65C2×(C2×D47)⋊C4—44no—
66#13093955—42no—
67#13093948—42no—
68#13093944—42no—
69#13093944—42no—
70#13093944C2×C418yescomplemented
71C22⋊(C47⋊C8)—22no—
72C22⋊(C47⋊C8)C2×C414yes—
73C4×C47⋊C8—42no—
74C22×Q8×C47C2311yescharacteristic
75C23×C4×C47D411yescharacteristic
76C23×C4×C47D411yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C22⋊Q8×C47—22no—
79C22⋊Q8×C47C2×C414yes—
80C4×Q8×C47—42no—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C22⋊C4×C47—21no—
88C2×C22⋊C4×C47—21no—
89C2×C42×C47—21no—
90C2×C42×C47—21no—
91C2×C42×C47—42no—
92C2×C42×C47D411yescharacteristic
93C2×C42×C47D411yescharacteristic
94C2×C42×C47C2311yescharacteristic
95C2×C42×C47C2311yescharacteristic
96#95×C47—21no—
97#95×C47—21no—
98#95×C47—21no—
99#95×C47—21no—
100#95×C47—21no—
101C24×C47C2×D411yescharacteristic
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—42no—
105C23×D47—42no—
106C23×D47—42no—
107C23×D47—42no—
108C23×D47—42no—
109C23×D47—42no—
110C2×Q8×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47C22×C412yes—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—41no—
123C22×C4×C47—41no—
124C22×C4×C47—41no—
125C22×C4×C47SD1614yescomplemented
126C22×C4×C47C22×C412yes—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47#4112yes—
133C22×C4×C47C2411yescharacteristic
134C22×Dic47—22no—
135C22×Dic47—42no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C2×D188—22no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—22no—
142C2×D188—24no—
143C2×D188—22no—
144C2×D188—48no—
145C2×D188—42no—
146C2×D188—42no—
147C2×D188—48no—
148C2×D188C22×C414yes—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—42no—
154C4⋊C4×C47—42no—
155C4⋊C4×C47C22×C412yes—
156C22⋊C4×C47—24no—
157C22⋊C4×C47—42no—
158C42×C47—22no—
159C42×C47—22no—
160C42×C47—41no—
161C42×C47—41no—
162C42×C47—42no—
163C42×C47—42no—
164#10625336—48no—
165(C2×D47)⋊C4—44no—
166(C2×D47)⋊C4—44no—
167(C2×D47)⋊C4—44no—
168(C2×D47)⋊C4—44no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—42no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8C22×C414yes—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47C4×D412yes—
183C23×C47C2×SD1612yes—
184C23×C47#13512yes—
185C23×C47C22×D411yescharacteristic
186C22×D47—24no—
187C22×D47—48no—
188C22×D47—48no—
189C22×D47—42no—
190C22×D47—42no—
191C22×D47—42no—
192C22×D47—42no—
193C22×D47—42no—
194C22×D47—42no—
195C22×D47—48no—
196C22×D47—84no—
197C22×D47—84no—
198C22×D47—84no—
199C22×D47—84no—
200Q8×C47—24no—
201Q8×C47—42no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—44no—
218C2×C4×C47—44no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47C4×D412yes—
230C2×C4×C47C2×SD1612yes—
231C2×C4×C47C2×SD1612yes—
232C2×C4×C47AGL(1,ℤ/8)12yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47H(5,𝔽2)12yes—
236C2×C4×C47H(5,𝔽2)12yes—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—84no—
241D188—24no—
242D188—24no—
243D188—48no—
244D188—42no—
245D188—42no—
246C47⋊C8—24no—
247C47⋊C8—42no—
248#22043—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47C4×SD1614yes—
256C22×C47C2×C4×D411yescharacteristic
257C22×C47#53411yescharacteristic
258C22×C47#53511yescharacteristic
259C22×C47C22×SD1611yescharacteristic
260C22×C47C2×#13511yescharacteristic
261C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C47#57511yescharacteristic
263C2×D47—44no—
264C2×D47—44no—
265C2×D47—44no—
266C2×D47—816no—
267C2×D47—82no—
268C2×D47—82no—
269C2×D47—82no—
270C2×D47—82no—
271C4×C47—22no—
272C4×C47—22no—
273C4×C47—22no—
274C4×C47—22no—
275C4×C47—42no—
276C4×C47—41no—
277C4×C47—42no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47#51812yes—
281C4×C47#53812yes—
282Dic47—82no—
283C2×#450—471no—
284C4×C22⋊Q8D4711yescharacteristic, complemented
285C2×#390—471no—
286#1968—472no—
287#1934—472no—
288#1926—472no—
289#1880—472no—
290#1872—471no—
291#1853—471no—
292#1786—471no—
293C4×C22⋊C8—471no—
294C2×C47—24no—
295C2×C47#232111yescharacteristic
296C2×C47C2×C4×SD1611yescharacteristic
297C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
298C2×C47#318711yescharacteristic
299C2×C47#318811yescharacteristic
300C2×C47#320811yescharacteristic
301C2×C47#321111yescharacteristic
302D47—84no—
303C23×D4—471no—
304C22×SD16—472no—
305C2×C22⋊Q8C2×D4711yescharacteristic
306C2×C4×Q8C2×D4711yescharacteristic, complemented
307C22×C4⋊C4—471no—
308C22×C22⋊C4—941no—
309C22×C22⋊C4—941no—
310C22×C42C2×D4711yescharacteristic
311#450—478no—
312C2×#106—472no—
313C2×#103—472no—
314C2×#102—472no—
315C2×#102—472no—
316C2×#102—472no—
317C2×C22⋊C8—471no—
318C2×C22⋊C8—471no—
319C2×C4×C8—472no—
320#391C2×D4711yescharacteristic, complemented
321#390—472no—
322#390—472no—
323#390—941no—
324#390—941no—
325#389C2×D4711yescharacteristic, complemented
326#387—21no—
327#385—21no—
328#380C2×D4711yescharacteristic
329C4×C4⋊C4—21no—
330C4×C4⋊C4C2×D4711yescharacteristic, complemented
331C4×C22⋊C4—21no—
332#340—472no—
333#340—472no—
334#336—472no—
335C47#2204311yescharacteristic, complemented, Sylow
336C25—941no—
337C22×Q8C22×D4711yescharacteristic
338C22×D4—474no—
339C22×D4—472no—
340C22×D4—942no—
341C22×D4—941no—
342C22×D4—1882no—
343C22×D4—1882no—
344C23×C4—941no—
345C23×C4C22⋊D4711yescharacteristic
346C23×C4C22⋊D4711yescharacteristic
347C23×C4C22×D4711yescharacteristic
348C2×SD16—478no—
349C2×SD16—1882no—
350C2×SD16—1882no—
351C22×C8—472no—
352C22×C8—472no—
353C22⋊Q8—22no—
354C22⋊Q8C4×D4714yes—
355C4×Q8—42no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—472no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4—1882no—
362C2×C4⋊C4C22×D4711yescharacteristic
363C2×C4⋊C4C22×D4711yescharacteristic
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—942no—
370C2×C22⋊C4—942no—
371C2×C22⋊C4—1884no—
372C2×C22⋊C4—1884no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—42no—
376C2×C42C22⋊D4711yescharacteristic
377C2×C42C22⋊D4711yescharacteristic
378C2×C42C22×D4711yescharacteristic
379C2×C42C22×D4711yescharacteristic
380#106—1882no—
381#103—1882no—
382#102—478no—
383#102—1882no—
384#102—1882no—
385C22⋊C8—474no—
386C22⋊C8—942no—
387C4×C8—1882no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393C24—944no—
394C24—942no—
395C24—1882no—
396C24—1882no—
397C24—1882no—
398C24—1882no—
399C24—1882no—
400C24—1882no—
401C24C2×C22⋊D4711yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C2×C4×D4712yes—
406C2×D4—474no—
407C2×D4—944no—
408C2×D4—942no—
409C2×D4—942no—
410C2×D4—942no—
411C2×D4—942no—
412C2×D4—942no—
413C2×D4—1888no—
414C2×D4—1882no—
415C2×D4—1882no—
416C2×D4—1888no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—942no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4—1882no—
432C22×C4#1062533614yescomplemented
433C22×C4C2×C4×D4712yes—
434C22×C4#1062535012yes—
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C2×C22⋊D4711yescharacteristic
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C2×C22⋊D4711yescharacteristic
439C22×C4C2×C22⋊D4711yescharacteristic
440C22×C4C23×D4711yescharacteristic
441SD16—1888no—
442C2×C8—474no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—942no—
446C2×C8—942no—
447C2×C8—942no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—942no—
455C4⋊C4—942no—
456C4⋊C4—1882no—
457C4⋊C4C2×C4×D4712yes—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—1884no—
461C22⋊C4—1884no—
462C22⋊C4—1884no—
463C22⋊C4—1884no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—42no—
469C42—42no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—944no—
475C23—3764no—
476C23—3764no—
477C23—3764no—
478C23—3764no—
479C23—1888no—
480C23—1888no—
481C23—1882no—
482C23—1882no—
483C23—1882no—
484C23—1882no—
485C23—1882no—
486C23—1882no—
487C23—1888no—
488C23C4×C22⋊D4712yes—
489C23C2×#1062533612yes—
490C23#1309404412yes—
491C23C22×C22⋊D4711yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—944no—
495D4—944no—
496D4—1888no—
497D4—1882no—
498D4—1882no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—3764no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4C4×C22⋊D4712yes—
531C2×C4C2×#1062533612yes—
532C2×C4C2×#1062533612yes—
533C2×C4#1309404912yes—
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C2×#1062535011yescharacteristic
536C2×C4#1309406912yes—
537C2×C4#1309407012yes—
538C8—944no—
539C8—1882no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—37616no—
547C22—3762no—
548C22—3762no—
549C22—3762no—
550C22—3762no—
551C22—1884no—
552C22—1884no—
553C22—1884no—
554C22C4×#1062533614yes—
555C22C2×C4×C22⋊D4711yescharacteristic
556C22#1568968211yescharacteristic
557C22#1568968711yescharacteristic
558C22C22×#1062533611yescharacteristic
559C22C2×#1309404411yescharacteristic
560C22C2×#1309404911yescharacteristic
561C22#1568992111yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—3762no—
572C4#1568959512yes—
573C4#1568970612yes—
574C2—24no—
575C2—3764no—
576C2#2024604211yescharacteristic
577C2C2×C4×#1062533611yescharacteristic
578C2C4×#1309404911yescharacteristic
579C2#2025414111yescharacteristic
580C2#2025414911yescharacteristic
581C2#2025432611yescharacteristic
582C2#2025434911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC23×D188C22
cosocleC22×C4×C47C24
hypercenterC4×C22⋊Q8D47
hypocenterC47#22043
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20244245C2semidirect
4. #20244070C2semidirect
5. #20244015C2semidirect
6. #20243743C2semidirect
7. #20243696C2semidirect
8. #20243580C2semidirect
10. C4×C22⋊(C47⋊C8)C2semidirect
11. C4×C22⋊Q8×C47C2semidirect
17. #15689273C4semidirect
18. C2×#13093955C22semidirect
19. C2×#13093948C22semidirect
25. C2×C4×C47⋊C8C22semidirect
30. #15688885C22semidirect
31. #15688885C22semidirect
32. #15688878C22semidirect
34. C2×C4×Q8×C47C22semidirect
36. C47×#391C22semidirect
37. C47×#389C22semidirect
42. C4×C4⋊C4×C47C22semidirect
54. C2×#10625336C2×C4semidirect
70. #13093944C2×C4semidirect
125. C22×C4×C47SD16semidirect
284. C4×C22⋊Q8D47semidirect
306. C2×C4×Q8C2×D47semidirect
320. #391C2×D47semidirect
325. #389C2×D47semidirect
330. C4×C4⋊C4C2×D47semidirect
335. C47#22043semidirect
432. C22×C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156892732
C2×#156890252
#202442452
#202440702
#202440152
#202437432
#202436962
#202435802
#202433182
C4×C22⋊(C47⋊C8)2
C4×C22⋊Q8×C472
#2204347