Finite Group Databasebeta

← #26126789

Subgroups — #26126789

Order 12032.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#15688892C211yesmaximal, characteristic
2#20241718C211yesmaximal, characteristic
3#20241415C212yesmaximal, complemented
4#20241394C212yesmaximal, complemented
5C47×#2484C211yesmaximal, characteristic
6C22×#10625329—21no—
7#15689087—21no—
8#15688892—21no—
9#15688892C2212yes—
10#15688849—21no—
11#15688849C2212yes—
12C2×#117×C47C411yescharacteristic
13C22×C4⋊C4×C47C2211yescharacteristic
14C47×#400C411yescharacteristic
15C47×#400C2211yescharacteristic
16C47×#398C411yescharacteristic
17C47×#398C2211yescharacteristic
18C47×#387—21no—
19C47×#387—21no—
20C47×#385—21no—
21C47×#381—21no—
22C47×#381C411yescharacteristic
23C2×#10625329—24no—
24C2×#10625329—22no—
25C2×#10625329—41no—
26C2×#10625329—42no—
27C2×#10625329—41no—
28C22×C47⋊C8—41no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—41no—
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34#117×C47—41no—
35#117×C47—41no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—41no—
39C2×C4⋊C4×C47—41no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47C2311yescharacteristic
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C42×C47—21no—
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47C2×C411yescharacteristic
57#95×C47C2×C411yescharacteristic
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—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—42no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C22⋊C4×C47—42no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C42×C47—42no—
93#10625329—24no—
94#10625329—44no—
95#10625329—42no—
96#10625329—82no—
97C2×C47⋊C8—44no—
98C2×C47⋊C8—42no—
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—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
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—81no—
138C2×C4×C47#10412yes—
139C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C47#13711yescharacteristic
141C47⋊C8—44no—
142C47⋊C8—82no—
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—
164C4×C47—81no—
165C4×C47—81no—
166C4×C47—81no—
167C4×C47—81no—
168#2484D4711yescharacteristic
169C2×#343—471no—
170#1463—471no—
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×#117Dic4711yescharacteristic
182C22×C4⋊C4C2×D4711yescharacteristic
183#407—941no—
184#400Dic4711yescharacteristic
185#400C2×D4711yescharacteristic
186#398Dic4711yescharacteristic
187#398C2×D4711yescharacteristic
188#387—21no—
189#387—21no—
190#385—21no—
191#381—21no—
192#381Dic4711yescharacteristic
193#343—472no—
194#343—941no—
195#324—472no—
196#324—941no—
197C47#1176411yescharacteristic, complemented, Sylow
198C23×C4C2×Dic4711yescharacteristic
199C23×C4C22⋊D4711yescharacteristic
200C23×C4C22⋊D4711yescharacteristic
201C2×M16—944no—
202C2×M16—942no—
203C2×M16—1881no—
204C2×M16—1881no—
205C2×M16—1882no—
206C22×C8—1881no—
207#117—41no—
208#117—41no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×Dic4711yescharacteristic
214C2×C4⋊C4C2×Dic4711yescharacteristic
215C2×C4⋊C4C2×Dic4711yescharacteristic
216C2×C4⋊C4C22⋊D4711yescharacteristic
217C2×C4⋊C4C22⋊D4711yescharacteristic
218C2×C4⋊C4C22×D4711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C22⋊C8—942no—
223C22⋊C8—1881no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C2×Dic4711yescharacteristic
232#95C2×Dic4711yescharacteristic
233C24C22⋊Dic4711yescharacteristic
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—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C22⋊Dic4711yescharacteristic
250C22×C4C22⋊Dic4711yescharacteristic
251C22×C4C22⋊Dic4711yescharacteristic
252C22×C4C22×Dic4711yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255M16—944no—
256M16—3762no—
257M16—1884no—
258M16—1882no—
259C2×C8—1882no—
260C2×C8—1884no—
261C2×C8—1882no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—42no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1309393911yescharacteristic
280C23#1309393911yescharacteristic
281C23C2×C22⋊Dic4711yescharacteristic
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—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4#1309395112yes—
314C2×C4#1309404911yescharacteristic
315C2×C4#1309405411yescharacteristic
316C8—3762no—
317C8—1884no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1568891312yes—
328C22C2×#1309393911yescharacteristic
329C22#1568912411yescharacteristic
330C22C2×#1309395111yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2024189211yescharacteristic
347C2C2×#1568891311yescharacteristic
348C2#2024507711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2484C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2484D47
hypocenterC47#11764
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
3. #20241415C2semidirect
4. #20241394C2semidirect
197. C47#11764semidirect

Maximal subgroups

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

subgroupindex
C2×#156888922
#202417182
#202414152
#202413942
C47×#24842
#1176447