Finite Group Databasebeta

← #26112342

Subgroups — #26112342

Order 12032.

762 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246002C211yesmaximal, characteristic, complemented
2#20241956C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic, complemented
4#20241255C212yesmaximal, complemented
5C47×#2617C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688998—24no—
9#15688983—21no—
10#15688983C2212yes—
11#15688857—24no—
12#15688847—22no—
13C2×#121×C47C2211yescharacteristic
14C2×#120×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#397C412yescomplemented
17C47×#394—22no—
18C47×#392C412yescomplemented
19C47×#386—21no—
20C47×#380C2211yescharacteristic
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—42no—
28C2×C22⋊Dic47—42no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C47⋊#95—22no—
32C47⋊#95D414yescomplemented
33C25×C47D411yescharacteristic
34C22×D4×C47—21no—
35C22×D4×C47—22no—
36C22×D4×C47C2×C411yescharacteristic, complemented
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39#121×C47—22no—
40#121×C47—22no—
41#120×C47—42no—
42#120×C47—42no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—22no—
51C2×C22⋊C4×C47C2×C412yes—
52C2×C22⋊C4×C47D412yes—
53#95×C47—22no—
54#95×C47C2×C412yescomplemented
55C24×C47—21no—
56C24×C47—21no—
57C24×C47—21no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—81no—
61C24×C47—81no—
62C24×C47C22⋊C412yes—
63C24×C47D812yes—
64C24×C47SD1612yes—
65C24×C47C2×D411yescharacteristic
66C2×D4×C47—22no—
67C2×D4×C47—44no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C2×D4×C47—82no—
75C2×D4×C47—82no—
76C2×D4×C47—81no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—22no—
80C22×C4×C47—21no—
81C22×C4×C47—81no—
82C22×C4×C47—81no—
83C22×C4×C47C22⋊C412yes—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47#4112yes—
88C22×Dic47—24no—
89C22×Dic47—44no—
90C22×Dic47—82no—
91C22×Dic47—82no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—81no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—24no—
96C22⋊C4×C47—44no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—82no—
101C22⋊Dic47—44no—
102C22⋊Dic47—42no—
103C22⋊Dic47—42no—
104C4⋊Dic47—82no—
105C4⋊Dic47—82no—
106C2×C47⋊C8—42no—
107C2×C47⋊C8—82no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—81no—
126C23×C47—81no—
127C23×C47—82no—
128C23×C47—82no—
129C23×C47—82no—
130C23×C47—82no—
131C23×C47#10214yes—
132C23×C47C2×C22⋊C411yescharacteristic
133C23×C47#12011yescharacteristic
134C23×C47#12311yescharacteristic
135C23×C47C2×D811yescharacteristic
136C23×C47C2×SD1611yescharacteristic
137C23×C47AGL(1,ℤ/8)11yescharacteristic
138C23×C47#13711yescharacteristic
139D4×C47—82no—
140D4×C47—88no—
141D4×C47—81no—
142D4×C47—81no—
143D4×C47—81no—
144D4×C47—81no—
145C2×C4×C47—22no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—42no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—81no—
159C2×C4×C47C4×D412yes—
160C2×C4×C47#12112yes—
161C2×Dic47—44no—
162C2×Dic47—44no—
163C2×Dic47—44no—
164C2×Dic47—88no—
165C47⋊C8—82no—
166#9203—471nomaximal, Sylow
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—24no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47—41no—
177C22×C47—41no—
178C22×C47—44no—
179C22×C47—44no—
180C22×C47—44no—
181C22×C47—42no—
182C22×C47—42no—
183C22×C47—82no—
184C22×C47—82no—
185C22×C47—82no—
186C22×C47—82no—
187C22×C47—82no—
188C22×C47—81no—
189C22×C47—81no—
190C22×C47—81no—
191C22×C47#38611yescharacteristic
192C22×C47C2×#10211yescharacteristic
193C22×C47#41711yescharacteristic
194C22×C47#44711yescharacteristic
195C22×C47#44811yescharacteristic
196C22×C47#48011yescharacteristic
197C22×C47#48111yescharacteristic
198C4×C47—22no—
199C4×C47—22no—
200C4×C47—42no—
201C4×C47—82no—
202C4×C47—81no—
203Dic47—84no—
204#2617D4711yescharacteristic
205#2315—471no—
206#1519—472no—
207C2×#323—471no—
208#1289—472no—
209C2×C47—21no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—21no—
213C2×C47—21no—
214C2×C47—21no—
215C2×C47—44no—
216C2×C47—161no—
217C2×C47#187411yescharacteristic
218C2×C47#191111yescharacteristic
219C2×C47#191311yescharacteristic
220C2×#121C2×D4711yescharacteristic
221C2×#120—21no—
222C22×C22⋊C4—941no—
223C22×C22⋊C4C2×D4711yescharacteristic
224C2×C22⋊C8—472no—
225#397Dic4712yescomplemented
226#394—22no—
227#392Dic4712yescomplemented
228#386—21no—
229#385—944no—
230#380—472no—
231#380—941no—
232#380C2×D4711yescharacteristic
233#327—944no—
234#323—942no—
235C47#920311yescharacteristic, complemented, Sylow
236C25C22⋊D4711yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2×Dic4711yescharacteristic, complemented
240C23×C4—942no—
241C23×C4C22⋊D4711yescharacteristic
242C23×C4C22×D4711yescharacteristic
243C22×C8—942no—
244#121—22no—
245#121—22no—
246#120—42no—
247#120—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—942no—
250C2×C4⋊C4—942no—
251C2×C4⋊C4C2×Dic4711yescharacteristic, complemented
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—942no—
259C2×C22⋊C4—942no—
260C2×C22⋊C4—1882no—
261C2×C22⋊C4—1882no—
262C2×C22⋊C4C2×Dic4712yes—
263C2×C22⋊C4C22⋊D4712yes—
264C22⋊C8—942no—
265C22⋊C8—942no—
266#95—22no—
267#95—474no—
268#95—942no—
269#95C2×Dic4712yescomplemented
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—81no—
276C24—81no—
277C24#1062533412yes—
278C24#1062533512yes—
279C24C22⋊Dic4712yes—
280C24C2×C22⋊D4711yescharacteristic
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—81no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—944no—
299C22×C4—3762no—
300C22×C4—3762no—
301C22×C4—1884no—
302C22×C4C22⋊Dic4712yes—
303C22×C4D4×D4712yes—
304C22×C4#1062535212yes—
305C22×C4C22×Dic4711yescharacteristic
306C22×C4C2×C22⋊D4711yescharacteristic
307C2×C8—3762no—
308C2×C8—1882no—
309C4⋊C4—22no—
310C4⋊C4—81no—
311C4⋊C4—3762no—
312C4⋊C4—3762no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—44no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—82no—
320C22⋊C4—1884no—
321C22⋊C4—1882no—
322C22⋊C4—1882no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—81no—
341C23—81no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23#1309394514yes—
347C23C2×C22⋊Dic4711yescharacteristic
348C23#1309399211yescharacteristic
349C23#1309400511yescharacteristic
350C23C2×#1062533411yescharacteristic
351C23C2×#1062533511yescharacteristic
352C23#1309404911yescharacteristic
353C23#1309405411yescharacteristic
354D4—82no—
355D4—88no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—3768no—
375C2×C4—1884no—
376C2×C4—1884no—
377C2×C4—1884no—
378C2×C4D4×Dic4712yes—
379C2×C4#1309399612yes—
380C8—3762no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#1568900311yescharacteristic
406C22C2×#1309394511yescharacteristic
407C22#1568912411yescharacteristic
408C22#1568925611yescharacteristic
409C22#1568926111yescharacteristic
410C22#1568943411yescharacteristic
411C22#1568943911yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—3764no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—44no—
425C2—161no—
426C2#2024370611yescharacteristic
427C2#2024392111yescharacteristic
428C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#2617C2
socleC22×C47#386
cosocleC23×C4×C47C23
hypercenter#2617D47
hypocenterC47#9203
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. #20246002C2semidirect
2. #20241956C2semidirect
3. C2×#15688847C2semidirect
4. #20241255C2semidirect
16. C47×#397C4semidirect
18. C47×#392C4semidirect
32. C47⋊#95D4semidirect
36. C22×D4×C47C2×C4semidirect
44. C2×C4⋊C4×C47C2×C4semidirect
54. #95×C47C2×C4semidirect
225. #397Dic47semidirect
227. #392Dic47semidirect
235. C47#9203semidirect
239. C22×D4C2×Dic47semidirect
251. C2×C4⋊C4C2×Dic47semidirect
269. #95C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202460022
#202419562
C2×#156888472
#202412552
C47×#26172
#920347