Finite Group Databasebeta

← #26226558

Subgroups — #26226558

Order 12032.

1234 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689087C211yesmaximal, characteristic
2C2×#15688983C211yesmaximal, characteristic
3#20243581C212yesmaximal, complemented
4#20243580C212yesmaximal, complemented
5#20243576C212yesmaximal, complemented
6#20243575C212yesmaximal, complemented
7#20243419C214yesmaximal, complemented
8C2×C47×#380C211yesmaximal, characteristic
9C22×#10625329C2212yes—
10C22×C4⋊Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×C22⋊Dic47—21no—
13C2×#13093957C2214yescomplemented
14C2×#13093955C2214yescomplemented
15#15689087C418yescomplemented
16C2×C22⋊(C47⋊C8)C2212yes—
17C2×C22⋊(C47⋊C8)C2212yes—
18#15688983—21no—
19#15688983—21no—
20#15688983C2212yes—
21#15688983C2212yes—
22C2×C47⋊#95—21no—
23#15688885C2214yescomplemented
24#15688885C2214yescomplemented
25C24×C4×C47C2211yescharacteristic
26C22×C4⋊C4×C47C2211yescharacteristic
27C22×C4⋊C4×C47C2211yescharacteristic
28C22×C22⋊C4×C47—21no—
29C47×#380—21no—
30C47×#380—21no—
31C47×#380C2212yes—
32C47×#380C2212yes—
33C2×#95×C47—21no—
34C23×Dic47—21no—
35C23×Dic47—21no—
36C2×#10625329—42no—
37C2×#10625329—42no—
38C2×#10625329C2×C418yescomplemented
39C22×C47⋊C8C2314yes—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—42no—
43C2×C4⋊Dic47—42no—
44C2×C4⋊Dic47C2312yes—
45C2×C4⋊Dic47C2312yes—
46C2×C22⋊Dic47—22no—
47C2×C22⋊Dic47—22no—
48C2×C22⋊Dic47—44no—
49#13093957—44no—
50#13093955—44no—
51C22⋊(C47⋊C8)C2×C418yescomplemented
52C22⋊(C47⋊C8)C2×C418yescomplemented
53C47⋊#95—22no—
54C47⋊#95—22no—
55C25×C47D411yescharacteristic
56C23×C4×C47—21no—
57C23×C4×C47—21no—
58C23×C4×C47—42no—
59C23×C4×C47—42no—
60C23×C4×C47C2×C414yes—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47D411yescharacteristic
64C23×C4×C47Q814yes—
65C23×C4×C47C2311yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C23×C4×C47C2311yescharacteristic
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—42no—
71C2×C4⋊C4×C47—42no—
72C2×C4⋊C4×C47C2312yes—
73C2×C4⋊C4×C47C2312yes—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47—22no—
76C2×C22⋊C4×C47—44no—
77#95×C47—22no—
78#95×C47—22no—
79C24×C47—22no—
80C24×C47—22no—
81C24×C47—22no—
82C24×C47—22no—
83C24×C47—41no—
84C24×C47—41no—
85C24×C47C4⋊C414yes—
86C24×C47C2×D411yescharacteristic
87C24×C47C2×D411yescharacteristic
88C24×C47C2×D411yescharacteristic
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—48no—
98C22×C4×C47—48no—
99C22×C4×C47—41no—
100C22×C4×C47—41no—
101C22×C4×C47—41no—
102C22×C4×C47—41no—
103C22×C4×C47—41no—
104C22×C4×C47—41no—
105C22×C4×C47—42no—
106C22×C4×C47—42no—
107C22×C4×C47—42no—
108C22×C4×C47—42no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47C4⋊C414yes—
112C22×C4×C47C4⋊C414yes—
113C22×C4×C47C4⋊C414yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×Q812yes—
121C22×C4×C47C2×Q812yes—
122C22×C4×C47C2×Q812yes—
123C22×C4×C47C2411yescharacteristic
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—42no—
127C22×Dic47—42no—
128C22×Dic47—42no—
129C22×Dic47—42no—
130C22×Dic47—42no—
131C22×Dic47—42no—
132C4⋊C4×C47—24no—
133C4⋊C4×C47—24no—
134C4⋊C4×C47—42no—
135C4⋊C4×C47—42no—
136C22⋊C4×C47—44no—
137C22⋊C4×C47—44no—
138C22⋊Dic47—44no—
139C22⋊Dic47—44no—
140C4⋊Dic47—24no—
141C4⋊Dic47—24no—
142C4⋊Dic47—42no—
143C4⋊Dic47—42no—
144#10625329—48no—
145C2×C47⋊C8—24no—
146C2×C47⋊C8—24no—
147C2×C47⋊C8C22×C418yes—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—21no—
151C23×C47—21no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—21no—
159C23×C47—21no—
160C23×C47—21no—
161C23×C47—21no—
162C23×C47—28no—
163C23×C47—28no—
164C23×C47—44no—
165C23×C47—44no—
166C23×C47—44no—
167C23×C47—44no—
168C23×C47—44no—
169C23×C47C2×C4⋊C412yes—
170C23×C47C2×C4⋊C412yes—
171C23×C47C2×C4⋊C412yes—
172C23×C47AGL(1,ℤ/8)12yes—
173C23×C47AGL(1,ℤ/8)12yes—
174C23×C47#13712yes—
175C23×C47#13712yes—
176C23×C47C22×D411yescharacteristic
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—21no—
183C2×C4×C47—24no—
184C2×C4×C47—24no—
185C2×C4×C47—24no—
186C2×C4×C47—24no—
187C2×C4×C47—24no—
188C2×C4×C47—24no—
189C2×C4×C47—44no—
190C2×C4×C47—44no—
191C2×C4×C47—44no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—84no—
199C2×C4×C47—84no—
200C2×C4×C47C2×C4⋊C412yes—
201C2×C4×C47C2×C4⋊C412yes—
202C2×C4×C47C2×C4⋊C412yes—
203C2×C4×C47C23×C411yescharacteristic
204C2×C4×C47C22×Q811yescharacteristic
205C2×C4×C47H(5,𝔽2)12yes—
206C2×C4×C47H(5,𝔽2)12yes—
207C2×Dic47—42no—
208C2×Dic47—42no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—84no—
214C2×Dic47—84no—
215C47⋊C8—28no—
216#21942—471nomaximal, Sylow
217C22×C47—22no—
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—22no—
221C22×C47—22no—
222C22×C47—22no—
223C22×C47—22no—
224C22×C47—22no—
225C22×C47—22no—
226C22×C47—22no—
227C22×C47—22no—
228C22×C47—22no—
229C22×C47—22no—
230C22×C47—22no—
231C22×C47—22no—
232C22×C47—416no—
233C22×C47—41no—
234C22×C47—41no—
235C22×C47—41no—
236C22×C47—41no—
237C22×C47#42814yes—
238C22×C47#42814yes—
239C22×C47C22×C4⋊C411yescharacteristic
240C22×C47#53411yescharacteristic
241C22×C47#53411yescharacteristic
242C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C47C2×#13711yescharacteristic
245C22×C47C2×#13711yescharacteristic
246C4×C47—22no—
247C4×C47—22no—
248C4×C47—22no—
249C4×C47—82no—
250C4×C47—82no—
251C4×C47#51812yes—
252C4×C47#54312yes—
253Dic47—82no—
254Dic47—82no—
255C2×#407—471no—
256C2×#380—471no—
257C2×#380D4711yescharacteristic
258#1853—472no—
259#1853—472no—
260#1852—472no—
261#1852—472no—
262#1809—474no—
263C2×C47—24no—
264C2×C47—24no—
265C2×C47—42no—
266C2×C47#231511yescharacteristic
267C2×C47C2×#42811yescharacteristic
268C2×C47C2×#42811yescharacteristic
269C2×C47#319211yescharacteristic
270C2×C47#319311yescharacteristic
271C2×C47#319311yescharacteristic
272C2×C47#319511yescharacteristic
273C24×C4C2×D4711yescharacteristic
274C22×M16—472no—
275C22×C4⋊C4—471no—
276C22×C4⋊C4—471no—
277C22×C4⋊C4C2×D4711yescharacteristic
278C22×C4⋊C4C2×D4711yescharacteristic
279C22×C22⋊C4—21no—
280C22×C22⋊C4—941no—
281C2×#107—474no—
282C2×#106—474no—
283#407—478no—
284C2×C22⋊C8—472no—
285C2×C22⋊C8—472no—
286#380—21no—
287#380—21no—
288#380—472no—
289#380—472no—
290#380—941no—
291#380—941no—
292#380C2×D4712yes—
293#380C2×D4712yes—
294C2×#95—21no—
295C2×#95—941no—
296#340—474no—
297#340—474no—
298C47#2194211yescharacteristic, complemented, Sylow
299C25C22⋊D4711yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—42no—
303C23×C4—42no—
304C23×C4—941no—
305C23×C4—941no—
306C23×C4Dic9414yes—
307C23×C4C4×D4714yes—
308C23×C4C22⋊D4711yescharacteristic
309C23×C4C22⋊D4711yescharacteristic
310C23×C4C22⋊D4711yescharacteristic
311C23×C4C22×D4711yescharacteristic
312C23×C4C22×D4711yescharacteristic
313C23×C4C22×D4711yescharacteristic
314C2×M16—478no—
315C2×M16—1882no—
316C2×M16—1882no—
317C22×C8—474no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—472no—
323C2×C4⋊C4—472no—
324C2×C4⋊C4—941no—
325C2×C4⋊C4—941no—
326C2×C4⋊C4—1882no—
327C2×C4⋊C4—1882no—
328C2×C4⋊C4C22×D4712yes—
329C2×C4⋊C4C22×D4712yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—942no—
334C2×C22⋊C4—942no—
335C2×C22⋊C4—1884no—
336#107—1884no—
337#106—1884no—
338C22⋊C8—478no—
339C22⋊C8—478no—
340#95—22no—
341#95—22no—
342#95—942no—
343#95—942no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—41no—
349C24—41no—
350C24Dic47⋊C414yes—
351C24C2×C22⋊D4711yescharacteristic
352C24C2×C22⋊D4711yescharacteristic
353C24C2×C22⋊D4711yescharacteristic
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—942no—
377C22×C4—942no—
378C22×C4—1882no—
379C22×C4—1882no—
380C22×C4—1882no—
381C22×C4—1882no—
382C22×C4—1882no—
383C22×C4—1882no—
384C22×C4Dic47⋊C414yes—
385C22×C4Dic47⋊C414yes—
386C22×C4Dic47⋊C414yes—
387C22×C4C2×Dic9412yes—
388C22×C4C2×Dic9412yes—
389C22×C4C2×Dic9412yes—
390C22×C4C2×C4×D4712yes—
391C22×C4C2×C4×D4712yes—
392C22×C4C2×C4×D4712yes—
393C22×C4C2×C22⋊D4711yescharacteristic
394C22×C4C2×C22⋊D4711yescharacteristic
395C22×C4C2×C22⋊D4711yescharacteristic
396C22×C4C23×D4711yescharacteristic
397M16—1888no—
398C2×C8—478no—
399C2×C8—944no—
400C2×C8—944no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—944no—
406C4⋊C4—944no—
407C4⋊C4—1882no—
408C4⋊C4—1882no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—1884no—
412C22⋊C4—1884no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—28no—
426C23—28no—
427C23—21no—
428C23—21no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23C2×Dic47⋊C412yes—
435C23C2×Dic47⋊C412yes—
436C23C2×Dic47⋊C412yes—
437C23#1309404912yes—
438C23#1309405012yes—
439C23#1309405412yes—
440C23#1309405512yes—
441C23C22×C22⋊D4711yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—3764no—
466C2×C4—3764no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1882no—
470C2×C4—1882no—
471C2×C4—1882no—
472C2×C4—1882no—
473C2×C4C2×Dic47⋊C412yes—
474C2×C4C2×Dic47⋊C412yes—
475C2×C4C2×Dic47⋊C412yes—
476C2×C4C22×C4×D4711yescharacteristic
477C2×C4C22×Dic9411yescharacteristic
478C2×C4#1309406912yes—
479C2×C4#1309406912yes—
480C8—948no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—416no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22#1568916314yes—
502C22#1568916614yes—
503C22C22×Dic47⋊C411yescharacteristic
504C22#1568968211yescharacteristic
505C22#1568968211yescharacteristic
506C22C2×#1309404911yescharacteristic
507C22C2×#1309405011yescharacteristic
508C22C2×#1309405411yescharacteristic
509C22C2×#1309405511yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—82no—
514C4—82no—
515C4—3762no—
516C4—3762no—
517C4#1568959612yes—
518C4#1568974912yes—
519C2—24no—
520C2—24no—
521C2—42no—
522C2#2024600011yescharacteristic
523C2C2×#1568916311yescharacteristic
524C2C2×#1568916611yescharacteristic
525C2#2025418511yescharacteristic
526C2#2025419011yescharacteristic
527C2#2025419411yescharacteristic
528C2#2025420911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#380C2
socleC23×C47C22×D4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC2×#380D47
hypocenterC47#21942
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. #20243581C2semidirect
4. #20243580C2semidirect
5. #20243576C2semidirect
6. #20243575C2semidirect
7. #20243419C2semidirect
13. C2×#13093957C22semidirect
14. C2×#13093955C22semidirect
15. #15689087C4semidirect
23. #15688885C22semidirect
24. #15688885C22semidirect
38. C2×#10625329C2×C4semidirect
51. C22⋊(C47⋊C8)C2×C4semidirect
52. C22⋊(C47⋊C8)C2×C4semidirect
298. C47#21942semidirect

Maximal subgroups

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

subgroupindex
C2×#156890872
C2×#156889832
#202435812
#202435802
#202435762
#202435752
#202434192
C2×C47×#3802
#2194247