Finite Group Databasebeta

← #26127049

Subgroups — #26127049

Order 12032.

434 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 91 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#20243376C211yesmaximal, characteristic
2#20241729C211yesmaximal, characteristic
3#20241412C212yesmaximal, complemented
4#20241388C212yesmaximal, complemented
5C47×#2395C211yesmaximal, characteristic
6#15689180—21no—
7C2×#13093939—21no—
8#15688983—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×C4×D4×C47C2211yescharacteristic
13C2×#117×C47C411yescharacteristic, complemented
14C47×#400C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#398C411yescharacteristic, complemented
17C47×#397—21no—
18C47×#397C2211yescharacteristic
19C47×#394—21no—
20C4×C4⋊C4×C47—21no—
21C4×C22⋊C4×C47—21no—
22C4×C22⋊C4×C47C411yescharacteristic, complemented
23C2×C4⋊Dic47—41no—
24C2×C22⋊Dic47—22no—
25C4⋊(C47⋊C8)—41no—
26#13093939—42no—
27C22⋊(C47⋊C8)—22no—
28#13093935—41no—
29C47⋊#95—41no—
30C22×D4×C47D411yescharacteristic
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C4×D4×C47—41no—
34C4×D4×C47—41no—
35#117×C47—41no—
36#117×C47—41no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47C2311yescharacteristic
48C2×C42×C47—21no—
49C2×C42×C47—21no—
50C2×C42×C47D411yescharacteristic
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×D4×C47—21no—
60C2×D4×C47—21no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×Dic47—42no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—42no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—42no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C42×C47—21no—
92C42×C47—21no—
93C42×C47—42no—
94C42×C47—41no—
95C42×C47—41no—
96C22⋊Dic47—42no—
97C22⋊Dic47—42no—
98C4⋊Dic47—82no—
99C2×C47⋊C8—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47C2×C22⋊C411yescharacteristic
110C23×C47AGL(1,ℤ/8)11yescharacteristic
111C23×C47#13711yescharacteristic
112D4×C47—42no—
113D4×C47—42no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—81no—
140C2×C4×C47#9911yescharacteristic
141C2×C4×C47#9911yescharacteristic
142C2×Dic47—82no—
143C2×Dic47—82no—
144C2×Dic47—82no—
145C47⋊C8—82no—
146#11822—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—81no—
159C22×C47—81no—
160C22×C47—81no—
161C22×C47C2×#9911yescharacteristic
162C22×C47#41711yescharacteristic
163C22×C47#42111yescharacteristic
164C4×C47—21no—
165C4×C47—42no—
166C4×C47—41no—
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—41no—
170C4×C47—81no—
171C4×C47—81no—
172C4×C47—81no—
173C4×C47#35212yes—
174Dic47—162no—
175#2395D4711yescharacteristic
176#1799—471no—
177#1466—471no—
178#1352—472no—
179#1344—472no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—41no—
183C2×C47—41no—
184C2×C47—81no—
185C2×C47#151311yescharacteristic
186C2×C47C2×#35211yescharacteristic
187C2×C47#212611yescharacteristic
188C2×C4×D4C2×D4711yescharacteristic
189C2×#117Dic4711yescharacteristic, complemented
190#432—941no—
191C2×#99—941no—
192#400Dic4711yescharacteristic, complemented
193#399C2×D4711yescharacteristic
194#398Dic4711yescharacteristic, complemented
195#397—21no—
196#397C2×D4711yescharacteristic
197#394—21no—
198#380—941no—
199C4×C4⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4Dic4711yescharacteristic, complemented
202#342—472no—
203#323—472no—
204#323—941no—
205C47#1182211yescharacteristic, complemented, Sylow
206C22×D4C22⋊D4711yescharacteristic
207C23×C4C2×Dic4711yescharacteristic
208C23×C4C22⋊D4711yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1881no—
216C2×C4⋊C4C2×Dic4711yescharacteristic
217C2×C4⋊C4C2×Dic4711yescharacteristic
218C2×C4⋊C4C2×Dic4711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—942no—
224C2×C22⋊C4C22⋊D4711yescharacteristic
225C2×C22⋊C4C22×D4711yescharacteristic
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C22⋊D4711yescharacteristic
229C4⋊C8—1881no—
230#99—1882no—
231C22⋊C8—942no—
232#97—1881no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—1881no—
238#95C2×Dic4711yescharacteristic
239#95C2×Dic4711yescharacteristic
240C24C22⋊Dic4711yescharacteristic
241C24C2×C22⋊D4711yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—1882no—
257C22×C4C22⋊Dic4711yescharacteristic
258C22×C4C22⋊Dic4711yescharacteristic
259C22×C4C22⋊Dic4711yescharacteristic
260C22×C4C22×Dic4711yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262C2×C8—1882no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—3762no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—1882no—
277C22⋊C4—1882no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—41no—
282C42—41no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊Dic4711yescharacteristic
293C23#1309404911yescharacteristic
294C23#1309405411yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—3762no—
324C2×C4—3762no—
325C2×C4—3762no—
326C2×C4#1309393911yescharacteristic
327C2×C4#1309393911yescharacteristic
328C8—3762no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#1309393911yescharacteristic
344C22#1568912411yescharacteristic
345C22#1568914011yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—7522no—
356C4#1568891312yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2#2024193111yescharacteristic
363C2C2×#1568891311yescharacteristic
364C2#2024505311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2395C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2395D47
hypocenterC47#11822
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2395

Complements

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

NH ≅ G/Ntype
3. #20241412C2semidirect
4. #20241388C2semidirect
13. C2×#117×C47C4semidirect
14. C47×#400C4semidirect
16. C47×#398C4semidirect
22. C4×C22⋊C4×C47C4semidirect
189. C2×#117Dic47semidirect
192. #400Dic47semidirect
194. #398Dic47semidirect
201. C4×C22⋊C4Dic47semidirect
205. C47#11822semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417292
#202414122
#202413882
C47×#23952
#1182247