Finite Group Databasebeta

← #26126790

Subgroups — #26126790

Order 12032.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#20247675C211yesmaximal, characteristic
2C2×#15688892C211yesmaximal, characteristic
3#20241416C212yesmaximal, complemented
4#20241395C212yesmaximal, complemented
5C47×#1463C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×#13093983C411yescharacteristic
8#15689062C411yescharacteristic
9#15689062C2211yescharacteristic
10#15689053C411yescharacteristic
11#15689053C2211yescharacteristic
12#15689004—21no—
13#15689004—21no—
14#15688996—21no—
15#15688986—21no—
16#15688986C411yescharacteristic
17#15688892—21no—
18#15688892C2212yes—
19C22×C4⋊C4×C47C2211yescharacteristic
20C47×#407—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23C2×#10625329—24no—
24C2×#10625329—22no—
25C2×#10625329—41no—
26C2×#10625329—42no—
27C22×C47⋊C8—41no—
28#13093983—41no—
29#13093983—41no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C2×M16×C47—41no—
50C2×C4⋊C4×C47—41no—
51C2×C4⋊C4×C47—41no—
52C2×C4⋊C4×C47C2×C411yescharacteristic
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C4⋊C4×C47D411yescharacteristic
55C2×C4⋊C4×C47C2311yescharacteristic
56C22⋊C8×C47—22no—
57C22⋊C8×C47—41no—
58C24×C47C22⋊C411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80M16×C47—82no—
81C2×C8×C47—42no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C22⋊Dic47—42no—
87C22⋊Dic47—41no—
88C22⋊Dic47—41no—
89C4⋊Dic47—41no—
90C4⋊Dic47—41no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93Dic47⋊C4—41no—
94C4×Dic47—42no—
95#10625329—24no—
96#10625329—44no—
97#10625329—42no—
98C2×C47⋊C8—44no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47#9911yescharacteristic
105C23×C47#9911yescharacteristic
106C23×C47C2×C22⋊C411yescharacteristic
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—81no—
127C2×C4×C47#10412yes—
128C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C47#13711yescharacteristic
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—42no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C8×C47—82no—
142C47⋊C8—44no—
143#11764—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47#35212yes—
154C22×C47C2×#9911yescharacteristic
155C22×C47#41711yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C4×C47—21no—
158C4×C47—22no—
159C4×C47—21no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47—41no—
164Dic47—81no—
165Dic47—81no—
166Dic47—81no—
167Dic47—81no—
168#2484—471no—
169C2×#343—471no—
170#1463D4711yescharacteristic
171#1353—472no—
172#1346—472no—
173C2×C47—21no—
174C2×C47—21no—
175C2×C47—22no—
176C2×C47—41no—
177C2×C47#150411yescharacteristic
178C2×C47C2×#35211yescharacteristic
179C2×C47#213311yescharacteristic
180C22×M16—941no—
181C2×#117—471no—
182C22×C4⋊C4C2×D4711yescharacteristic
183#407—21no—
184#400—471no—
185#400—471no—
186#398—471no—
187#398—471no—
188#387—941no—
189#387—941no—
190#385—941no—
191#381—471no—
192#381—941no—
193#343—472no—
194#343—941no—
195#324—21no—
196#324C2×D4712yes—
197C47#1176411yescharacteristic, complemented, Sylow
198C23×C4C4×D4711yescharacteristic
199C23×C4D18811yescharacteristic
200C23×C4C22⋊D4711yescharacteristic
201C2×M16—41no—
202C2×M16—944no—
203C2×M16—942no—
204C2×M16—1881no—
205C2×M16—1882no—
206C22×C8—1881no—
207#117—1881no—
208#117—1881no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—471no—
212C2×C4⋊C4—471no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4C4×D4711yescharacteristic
216C2×C4⋊C4D18811yescharacteristic
217C2×C4⋊C4C22⋊D4711yescharacteristic
218C2×C4⋊C4C22×D4711yescharacteristic
219C2×C22⋊C4—941no—
220C2×C22⋊C4—941no—
221C2×C42—941no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224#95—471no—
225#95—471no—
226#95—941no—
227#95—941no—
228#95—941no—
229#95—941no—
230#95—941no—
231#95—941no—
232#95—941no—
233C24(C2×D47)⋊C411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—941no—
246C22×C4—941no—
247C22×C4—941no—
248C22×C4—941no—
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4(C2×D47)⋊C411yescharacteristic
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4C2×D18811yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255M16—82no—
256M16—944no—
257M16—1884no—
258M16—1882no—
259C2×C8—42no—
260C2×C8—1884no—
261C2×C8—1882no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—1881no—
267C4⋊C4—1881no—
268C4⋊C4—1882no—
269C4⋊C4—1881no—
270C4⋊C4—1881no—
271C22⋊C4—1882no—
272C22⋊C4—1881no—
273C22⋊C4—1881no—
274C42—1882no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1309393811yescharacteristic
280C23#1309393811yescharacteristic
281C23C2×(C2×D47)⋊C411yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—1881no—
303C2×C4—1881no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4—1882no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4#1309395012yes—
314C2×C4#1309405211yescharacteristic
315C2×C4#1309405711yescharacteristic
316C8—82no—
317C8—1884no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1568891212yes—
328C22C2×#1309393811yescharacteristic
329C22#1568912511yescharacteristic
330C22C2×#1309395011yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3761no—
339C4—3761no—
340C4—3761no—
341C4—3761no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2024189311yescharacteristic
347C2C2×#1568891211yescharacteristic
348C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1463C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1463D47
hypocenterC47#11764
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20241416C2semidirect
4. #20241395C2semidirect
197. C47#11764semidirect

Maximal subgroups

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

subgroupindex
#202476752
C2×#156888922
#202414162
#202413952
C47×#14632
#1176447