Finite Group Databasebeta

← #26127021

Subgroups — #26127021

Order 12032.

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

normal subgroups only

This group has 388 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#20243471C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241418C212yesmaximal, complemented
4#20241415C212yesmaximal, complemented
5C47×#2396C211yesmaximal, characteristic
6#15689180—21no—
7#15689087—21no—
8#15688892—21no—
9#15688892C2212yes—
10#15688849C2212yes—
11#15688847—21no—
12C2×C4×Q8×C47C411yescharacteristic
13C2×#117×C47C2211yescharacteristic
14C47×#400C2211yescharacteristic
15C47×#399C411yescharacteristic
16C47×#398—21no—
17C47×#396—21no—
18C47×#395C2211yescharacteristic
19C47×#393C411yescharacteristic
20C4×C4⋊C4×C47—21no—
21C4×C4⋊C4×C47C411yescharacteristic
22C4×C22⋊C4×C47—21no—
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—41no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—41no—
27C22⋊(C47⋊C8)—41no—
28#13093935—41no—
29C22×Q8×C47C2×C411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C4×Q8×C47—41no—
32C4×Q8×C47—41no—
33#117×C47—41no—
34#117×C47—41no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C42×C47—21no—
47C2×C42×C47—21no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47D411yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47C2×C411yescharacteristic
55#95×C47C2×C411yescharacteristic
56C24×C47C22⋊C411yescharacteristic
57C2×Q8×C47—42no—
58C2×Q8×C47—41no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—41no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—42no—
79C4⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—42no—
83C22⋊C4×C47—42no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—42no—
89C42×C47—41no—
90C42×C47—41no—
91#10625329—42no—
92#10625329—42no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47C2×C22⋊C411yescharacteristic
101Q8×C47—42no—
102Q8×C47—42no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
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#9911yescharacteristic
130C2×C4×C47#9911yescharacteristic
131C2×C4×C47#13712yes—
132C47⋊C8—82no—
133C47⋊C8—82no—
134#11816—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47C2×#9911yescharacteristic
143C22×C47#41911yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—42no—
149C4×C47—41no—
150C4×C47—42no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—81no—
155C4×C47—81no—
156#2396D4711yescharacteristic
157#1822—471no—
158#1464—471no—
159#1354—472no—
160#1353—472no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—81no—
164C2×C47#151111yescharacteristic
165C2×C47#212011yescharacteristic
166C2×C47#213211yescharacteristic
167C2×C4×Q8Dic4711yescharacteristic
168C2×#117C2×D4711yescharacteristic
169#432—941no—
170#407—941no—
171#400C2×D4711yescharacteristic
172#399Dic4711yescharacteristic
173#398—21no—
174#396—21no—
175#395C2×D4711yescharacteristic
176#393Dic4711yescharacteristic
177C4×C4⋊C4—21no—
178C4×C4⋊C4Dic4711yescharacteristic
179C4×C22⋊C4—21no—
180#343—472no—
181#343—941no—
182#324—472no—
183#323—941no—
184C47#1181611yescharacteristic, complemented, Sylow
185C22×Q8C2×Dic4711yescharacteristic
186C23×C4C22⋊D4711yescharacteristic
187C2×M16—942no—
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×Dic4711yescharacteristic
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C4⋊C4C22×D4711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4C2×Dic4711yescharacteristic
202C2×C22⋊C4C22⋊D4711yescharacteristic
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C2×Dic4711yescharacteristic
206C2×C42C22⋊D4711yescharacteristic
207C4⋊C8—1881no—
208C22⋊C8—942no—
209C22⋊C8—1881no—
210C22⋊C8—1881no—
211#97—1881no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95C2×Dic4711yescharacteristic
217#95C2×Dic4711yescharacteristic
218C24C22⋊Dic4711yescharacteristic
219C2×Q8—42no—
220C2×Q8—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22⋊Dic4711yescharacteristic
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22×Dic4711yescharacteristic
233C22×C4C2×C22⋊D4711yescharacteristic
234C22×C4C2×C22⋊D4711yescharacteristic
235M16—1882no—
236M16—1882no—
237C2×C8—1882no—
238C2×C8—1882no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—42no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊Dic4711yescharacteristic
263Q8—42no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4#1309393911yescharacteristic
292C2×C4#1309393911yescharacteristic
293C2×C4#1309405512yes—
294C8—3762no—
295C8—3762no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1309393911yescharacteristic
304C22#1568913211yescharacteristic
305C22#1568914011yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2024192111yescharacteristic
321C2#2024502911yescharacteristic
322C2#2024507311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2396C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2396D47
hypocenterC47#11816
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
3. #20241418C2semidirect
4. #20241415C2semidirect
184. C47#11816semidirect

Maximal subgroups

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

subgroupindex
#202434712
#202417222
#202414182
#202414152
C47×#23962
#1181647