Finite Group Databasebeta

← #26226831

Subgroups — #26226831

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689178C211yesmaximal, characteristic
2#20245793C211yesmaximal, characteristic
3#382×D47C211yesmaximal, characteristic
4#20244157C212yesmaximal, complemented
5#20244153C212yesmaximal, complemented
6#20243579C212yesmaximal
7#20243574C212yesmaximal
8#20243554C211yesmaximal, characteristic, complemented
9#20243522C211yesmaximal, characteristic, complemented
10#20243521C211yesmaximal, characteristic, complemented
11C47×#1847C211yesmaximal, characteristic, complemented
12C2×#13093982—21no—
13C2×C4⋊C4×D47C2211yescharacteristic
14C2×C4⋊C4×D47C2211yescharacteristic
15C2×C42×D47C2211yescharacteristic
16#15689178C418yescomplemented
17C2×#13093957C2212yes—
18C2×#13093955C2212yes—
19C2×Dic47⋊C8C2212yes—
20C2×(C2×D47)⋊C8C2212yes—
21C2×#13093935C2211yescharacteristic
22#15688993C2211yescharacteristic
23#15688989C2211yescharacteristic
24#15688988—21no—
25#15688988C2211yescharacteristic
26#15688988C2211yescharacteristic
27#15688987—21no—
28#15688987—21no—
29#15688980—21no—
30#15688980C2212yes—
31C4⋊C4×Dic47—21no—
32C4⋊C4×Dic47C2212yes—
33#95×D47—21no—
34#95×D47—21no—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×#107×C47C2212yes—
40C2×#106×C47C2212yes—
41C2×#97×C47C2211yescharacteristic
42C47×#384C2211yescharacteristic
43C47×#382C2211yescharacteristic
44C22×C4×D47—21no—
45C22×C4×D47—21no—
46C22×C4×D47D411yescharacteristic
47C22×C4×D47D411yescharacteristic
48C22×C4×D47C2311yescharacteristic
49C22×C47⋊C8C2312yes—
50#13093982—44no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47C2312yes—
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2311yescharacteristic
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C4⋊C4×D47—42no—
59C4⋊C4×D47—42no—
60C2×(C2×D47)⋊C4—22no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47D411yescharacteristic
63C2×C4×Dic47D411yescharacteristic
64C2×C4×Dic47C2311yescharacteristic
65C42×D47C2×C414yes—
66C42×D47Q814yes—
67#13093957—42no—
68#13093955—42no—
69Dic47⋊C8C2×C418yescomplemented
70(C2×D47)⋊C8C2×C418yes—
71#13093935C2×C414yescomplemented
72C47⋊#95—21no—
73C47⋊#95—21no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—22no—
79C47⋊#95—22no—
80C22×C8×C47C2312yes—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47C2312yes—
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C42×C47C2311yescharacteristic
87#107×C47—42no—
88#106×C47—42no—
89#97×C47C2×C414yescomplemented
90#95×C47—21no—
91#95×C47—21no—
92C23×D47C2×D411yescharacteristic
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2411yescharacteristic
99C22×Dic47—21no—
100C22×Dic47—21no—
101C22×Dic47—22no—
102C22×Dic47C2×D411yescharacteristic
103C22×Dic47C2×D411yescharacteristic
104C22×Dic47C2×D411yescharacteristic
105C2×C4×D47—22no—
106C2×C4×D47—22no—
107C2×C4×D47—22no—
108C2×C4×D47—22no—
109C2×C4×D47—41no—
110C2×C4×D47—41no—
111C2×C4×D47—42no—
112C2×C4×D47—42no—
113C2×C4×D47—42no—
114C2×C4×D47—42no—
115C2×C4×D47—42no—
116C2×C4×D47—42no—
117C2×C4×D47C4⋊C414yes—
118C2×C4×D47C4⋊C414yes—
119C2×C4×D47C22×C412yes—
120C2×C4×D47C2×Q812yes—
121C2×C8×C47—22no—
122C2×C8×C47—22no—
123C2×C8×C47C22×C414yes—
124C4⋊C4×C47—22no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—22no—
127C4⋊C4×C47—22no—
128C4⋊C4×C47—42no—
129C4⋊C4×C47—42no—
130C42×C47C22×C412yes—
131C42×C47C2×Q812yes—
132(C2×D47)⋊C4—44no—
133C4⋊Dic47—22no—
134C4⋊Dic47—22no—
135C4⋊Dic47—24no—
136C4⋊Dic47—42no—
137C4⋊Dic47—42no—
138Dic47⋊C4—42no—
139Dic47⋊C4—42no—
140C4×Dic47—44no—
141C4×Dic47C4⋊C414yes—
142C4×Dic47C4⋊C414yes—
143C4×Dic47C22×C412yes—
144C4×Dic47C2×Q812yes—
145C2×C47⋊C8—22no—
146C2×C47⋊C8—22no—
147C2×C47⋊C8C22×C414yes—
148C23×C47C22×D411yescharacteristic
149C22×D47—21no—
150C22×D47—21no—
151C22×D47—21no—
152C22×D47—21no—
153C22×D47C2×C4⋊C412yes—
154C22×D47AGL(1,ℤ/8)12yes—
155C22×D47#13712yes—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—42no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47C2×C4⋊C412yes—
170C2×C4×C47C2×C4⋊C412yes—
171C2×C4×C47C23×C411yescharacteristic
172C2×C4×C47C22×Q811yescharacteristic
173C2×C4×C47H(5,𝔽2)12yes—
174C2×C4×C472−1+412yes—
175C2×Dic47—21no—
176C2×Dic47—21no—
177C2×Dic47—21no—
178C2×Dic47—21no—
179C2×Dic47—21no—
180C2×Dic47—21no—
181C2×Dic47—41no—
182C2×Dic47—41no—
183C2×Dic47—42no—
184C2×Dic47—42no—
185C2×Dic47—41no—
186C2×Dic47—41no—
187C2×Dic47—42no—
188C2×Dic47—41no—
189C2×Dic47—41no—
190C2×Dic47C2×C4⋊C412yes—
191C2×Dic47C2×C4⋊C412yes—
192C2×Dic47C2×C4⋊C412yes—
193C4×D47—28no—
194C4×D47—24no—
195C4×D47—24no—
196C4×D47—44no—
197C4×D47—84no—
198C4×D47—84no—
199C8×C47—24no—
200C47⋊C8—24no—
201#21986—471nomaximal, Sylow
202C22×C47C22×C4⋊C411yescharacteristic
203C22×C47#53511yescharacteristic
204C22×C47#53611yescharacteristic
205C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C47C2×#13711yescharacteristic
207C22×C47#57511yescharacteristic
208C22×C47#57511yescharacteristic
209C2×D47—22no—
210C2×D47—22no—
211C2×D47—22no—
212C2×D47—22no—
213C2×D47—22no—
214C2×D47—22no—
215C2×D47#42814yes—
216C4×C47—22no—
217C4×C47—22no—
218C4×C47—81no—
219C4×C47—82no—
220C4×C47—81no—
221C4×C47#52012yes—
222C4×C47#54412yes—
223Dic47—22no—
224Dic47—22no—
225Dic47—22no—
226Dic47—81no—
227Dic47—81no—
228Dic47—82no—
229C2×#432—471no—
230#2285—471no—
231C2×#382—471no—
232#1949—472no—
233#1949—472no—
234#1853—472no—
235#1852—472no—
236#1847—471no—
237#1847D4711yescharacteristic, complemented
238#1836—471no—
239#1836—471no—
240C2×C47#231711yescharacteristic
241C2×C47C2×#42811yescharacteristic
242C2×C47#291111yescharacteristic
243C2×C47#321311yescharacteristic
244C2×C47#321511yescharacteristic
245C2×C47#323411yescharacteristic
246C2×C47#323611yescharacteristic
247D47—24no—
248C2×#117—941no—
249C22×C4⋊C4—471no—
250C22×C4⋊C4—471no—
251C22×C42—471no—
252#432—478no—
253C2×#107—472no—
254C2×#107C2×D4712yes—
255C2×#106—472no—
256C2×#106C2×D4712yes—
257C2×C4⋊C8—472no—
258C2×C22⋊C8—472no—
259C2×#97—471no—
260C2×#97C2×D4711yescharacteristic
261#384—471no—
262#384C2×D4711yescharacteristic
263#382—471no—
264#382—471no—
265#382—471no—
266#382—941no—
267#382—941no—
268#382—941no—
269#382C2×D4711yescharacteristic
270#380—472no—
271#380—941no—
272C4×C4⋊C4—472no—
273C4×C4⋊C4—941no—
274C2×#95—941no—
275C2×#95—941no—
276#340—472no—
277#340—472no—
278#340—472no—
279#340—472no—
280C47#2198611yescharacteristic, complemented, Sylow
281C23×C4—471no—
282C23×C4—471no—
283C23×C4—471no—
284C23×C4—941no—
285C23×C4—941no—
286C22×C8—472no—
287C22×C8C22×D4712yes—
288#117—1884no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—471no—
292C2×C4⋊C4—471no—
293C2×C4⋊C4—472no—
294C2×C4⋊C4—941no—
295C2×C4⋊C4—941no—
296C2×C4⋊C4—941no—
297C2×C4⋊C4—941no—
298C2×C4⋊C4—1882no—
299C2×C4⋊C4—1882no—
300C2×C4⋊C4C22×D4711yescharacteristic
301C2×C4⋊C4C22×D4711yescharacteristic
302C2×C4⋊C4C22×D4712yes—
303C2×C22⋊C4—942no—
304C2×C42—471no—
305C2×C42—471no—
306C2×C42—471no—
307C2×C42—474no—
308C2×C42—474no—
309C2×C42—942no—
310C2×C42C22×D4711yescharacteristic
311#107—42no—
312#107—1882no—
313#106—42no—
314#106—1882no—
315C4⋊C8—478no—
316C22⋊C8—478no—
317#97—474no—
318#97C4×D4714yescomplemented
319#95—21no—
320#95—21no—
321#95—941no—
322#95—941no—
323#95—941no—
324#95—941no—
325#95—942no—
326#95—942no—
327#95—941no—
328#95—941no—
329C24—471no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—471no—
334C22×C4—471no—
335C22×C4—471no—
336C22×C4—474no—
337C22×C4—472no—
338C22×C4—472no—
339C22×C4—474no—
340C22×C4—942no—
341C22×C4—942no—
342C22×C4—942no—
343C22×C4—942no—
344C22×C4—941no—
345C22×C4—941no—
346C22×C4—942no—
347C22×C4—1881no—
348C22×C4—1881no—
349C22×C4—1882no—
350C22×C4—1882no—
351C22×C4—1882no—
352C22×C4—1882no—
353C22×C4—1882no—
354C22×C4—1882no—
355C22×C4D4×D4711yescharacteristic
356C22×C4D4×D4711yescharacteristic
357C22×C4C23×D4711yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—474no—
361C2×C8—942no—
362C2×C8—942no—
363C2×C8C2×C4×D4714yes—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—942no—
371C4⋊C4—942no—
372C4⋊C4—944no—
373C4⋊C4—1882no—
374C4⋊C4—1882no—
375C4⋊C4—1882no—
376C4⋊C4—1882no—
377C22⋊C4—1884no—
378C42—474no—
379C42—474no—
380C42—472no—
381C42—472no—
382C42—1884no—
383C42C2×C4×D4712yes—
384C42Q8×D4712yes—
385C23—472no—
386C23—472no—
387C23—472no—
388C23—941no—
389C23—941no—
390C23—941no—
391C23—941no—
392C23C2×D4×D4711yescharacteristic
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—472no—
407C2×C4—472no—
408C2×C4—472no—
409C2×C4—948no—
410C2×C4—941no—
411C2×C4—941no—
412C2×C4—941no—
413C2×C4—941no—
414C2×C4—941no—
415C2×C4—941no—
416C2×C4—944no—
417C2×C4—944no—
418C2×C4—3764no—
419C2×C4—3764no—
420C2×C4—1884no—
421C2×C4—1882no—
422C2×C4—1881no—
423C2×C4—1881no—
424C2×C4—1881no—
425C2×C4—1882no—
426C2×C4—1881no—
427C2×C4—1881no—
428C2×C4—1881no—
429C2×C4—1882no—
430C2×C4C4⋊C4×D4712yes—
431C2×C4C4⋊C4×D4712yes—
432C2×C4C22×C4×D4711yescharacteristic
433C2×C4C2×Q8×D4711yescharacteristic
434C2×C4#1309406912yes—
435C2×C4#1309407212yes—
436C8—24no—
437C8—944no—
438C22—474no—
439C22—942no—
440C22—942no—
441C22—942no—
442C22—942no—
443C22—942no—
444C22—942no—
445C22C2×C4⋊C4×D4711yescharacteristic
446C22#1568968611yescharacteristic
447C22#1568969811yescharacteristic
448C22AGL(1,ℤ/8)×D4711yescharacteristic
449C22#137×D4711yescharacteristic
450C22#1568991911yescharacteristic
451C22#1568992211yescharacteristic
452C4—22no—
453C4—22no—
454C4—81no—
455C4—82no—
456C4—81no—
457C4—942no—
458C4—942no—
459C4—942no—
460C4—3762no—
461C4—3761no—
462C4—3761no—
463C4#1568960412yes—
464C4#1568975012yes—
465C2—944no—
466C2#2024600911yescharacteristic
467C2#428×D4711yescharacteristic
468C2#2025197511yescharacteristic
469C2#2025437011yescharacteristic
470C2#2025438611yescharacteristic
471C2#2025457911yescharacteristic
472C2#2025459511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1847C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1847D47
hypocenterC47#21986
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #20244157C2semidirect
5. #20244153C2semidirect
8. #20243554C2semidirect
9. #20243522C2semidirect
10. #20243521C2semidirect
11. C47×#1847C2semidirect
16. #15689178C4semidirect
69. Dic47⋊C8C2×C4semidirect
71. #13093935C2×C4semidirect
89. #97×C47C2×C4semidirect
237. #1847D47semidirect
280. C47#21986semidirect
318. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891782
#202457932
#382×D472
#202441572
#202441532
#202435792
#202435742
#202435542
#202435222
#202435212
C47×#18472
#2198647