Finite Group Databasebeta

← #26369784

Subgroups — #26369784

Order 12032.

498 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252320C211yesmaximal, characteristic
2#20245295C212yesmaximal, complemented
3#20245295C212yesmaximal, complemented
4#20245293C212yesmaximal, complemented
5#20245290C212yesmaximal, complemented
6#20245148C211yesmaximal, characteristic
7C2×Dic47⋊C16C211yesmaximal, characteristic
8#20244985C211yesmaximal, characteristic
9#20244984C211yesmaximal, characteristic
10M32×Dic47C211yesmaximal, characteristic
11C47×#2151C211yesmaximal, characteristic
12#15689676C411yescharacteristic
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16C2211yescharacteristic
15#15689251C412yes—
16#15689251C412yes—
17C8×Dic94C2212yes—
18C8×Dic94C2212yes—
19#15689151C411yescharacteristic
20#15689150C411yescharacteristic
21#15689146C411yescharacteristic
22#15689078C2211yescharacteristic
23C2×C8×Dic47C2211yescharacteristic
24Dic47⋊C16C2211yescharacteristic
25Dic47⋊C16C2212yes—
26Dic47⋊C16C2211yescharacteristic
27Dic47⋊C16C2211yescharacteristic
28Dic47⋊C16C2211yescharacteristic
29Dic47⋊C16C2212yes—
30#15688899C2212yes—
31#15688898C2212yes—
32#15688898C2212yes—
33C4×C47⋊C16C2212yes—
34C16×Dic47C2212yes—
35C16×Dic47C2212yes—
36C2×M32×C47C2211yescharacteristic
37C22×C16×C47C2211yescharacteristic
38C47×#405C2211yescharacteristic
39C47×C4⋊C16C2212yes—
40C47×C4⋊C16C2212yes—
41C2×#10625329C2×C411yescharacteristic
42C22×C47⋊C8C2×C411yescharacteristic
43#13094023C811yescharacteristic
44#13094023C811yescharacteristic
45#13094015C811yescharacteristic
46#13094015C811yescharacteristic
47#13094001C812yes—
48#13094001C812yes—
49C4×Dic94C2×C412yes—
50C4×Dic94C2×C412yes—
51#13093983C2×C411yescharacteristic
52C2×C4×Dic47C2×C411yescharacteristic
53#13093964Q814yescomplemented
54C2×C47⋊C16—22no—
55C2×C47⋊C16C2311yescharacteristic
56C2×C47⋊C16C2311yescharacteristic
57C2×C47⋊C16C2312yes—
58C4⋊(C47⋊C8)C2×C412yes—
59C4⋊(C47⋊C8)C2×C412yes—
60Dic47⋊C8C2×C412yes—
61Dic47⋊C8C2×C412yes—
62Dic47⋊C8C2×C412yes—
63Dic47⋊C8C2×C412yes—
64#13093934C2×C412yes—
65C8×Dic47C2×C411yescharacteristic
66C8×Dic47C2×C411yescharacteristic
67C8×Dic47C2311yescharacteristic
68C8×Dic47C2311yescharacteristic
69C8×Dic47C2312yes—
70C2×M16×C47C2×C411yescharacteristic
71C22×C8×C47C2311yescharacteristic
72#117×C47C2×C411yescharacteristic
73M32×C47—22no—
74C2×C16×C47—22no—
75C2×C16×C47C2311yescharacteristic
76C2×C16×C47C2311yescharacteristic
77C2×C16×C47C2312yes—
78#97×C47C2×C412yes—
79C4×C8×C47C2312yes—
80C22×C4×C47C22×C411yescharacteristic
81C22×Dic47C2×C811yescharacteristic
82C22×Dic47C2×C811yescharacteristic
83C2×Dic94C2×C812yes—
84C2×Dic94C2×C812yes—
85M16×C47—22no—
86C2×C8×C47C22×C412yes—
87C2×C8×C47C22×C411yescharacteristic
88C2×C8×C47C2×Q812yes—
89C2×C8×C47#4112yes—
90C2×C8×C47C2411yescharacteristic
91C4⋊C4×C47C2×C811yescharacteristic
92C4⋊C4×C47C2×C811yescharacteristic
93C22⋊C4×C47C2×C812yes—
94C42×C47C22×C412yes—
95C22⋊Dic47C2×C812yes—
96C4⋊Dic47C2×C811yescharacteristic
97C4⋊Dic47C2×C811yescharacteristic
98C4⋊Dic47C2×C812yes—
99C4⋊Dic47C2×C812yes—
100Dic47⋊C4C2×C812yes—
101Dic47⋊C4C2×C812yes—
102Dic47⋊C4C2×C812yes—
103Dic47⋊C4C2×C812yes—
104C4×Dic47C2×C811yescharacteristic
105C4×Dic47C2×C811yescharacteristic
106C4×Dic47C22×C411yescharacteristic
107C4×Dic47C22×C411yescharacteristic
108C4×Dic47C22×C412yes—
109#10625329—22no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8C22×C411yescharacteristic
112C2×C47⋊C8C22×C411yescharacteristic
113C2×C47⋊C8C22×C412yes—
114C16×C47—21no—
115C16×C47—21no—
116C16×C47—22no—
117C47⋊C16—21no—
118C47⋊C16—21no—
119C47⋊C16C2×Q814yes—
120C23×C47C22×C811yescharacteristic
121C2×C4×C47C4×Q812yes—
122C2×C4×C47C22×C812yes—
123C2×C4×C47C22×C812yes—
124C2×C4×C47C22×C811yescharacteristic
125C2×C4×C47#13112yes—
126C2×C4×C47C23×C411yescharacteristic
127C2×Dic47—22no—
128C2×Dic47—41no—
129C2×Dic47C22×C811yescharacteristic
130C2×Dic47C22×C811yescharacteristic
131C2×Dic47C22×C811yescharacteristic
132C2×Dic47C22×C812yes—
133C2×Dic47C22×C812yes—
134C2×Dic47C22×C811yescharacteristic
135Dic94—24no—
136Dic94—42no—
137C8×C47—22no—
138C8×C47C4×Q812yes—
139C8×C47C22×Q811yescharacteristic
140C8×C47C2×#4111yescharacteristic
141C47⋊C8—21no—
142C47⋊C8—21no—
143C47⋊C8—22no—
144#33476—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47C8×Q812yes—
148C22×C47C23×C811yescharacteristic
149C4×C47—22no—
150C4×C47—22no—
151C4×C47C8×Q812yes—
152C4×C47C2×C4×Q811yescharacteristic
153C4×C47C2×#13111yescharacteristic
154Dic47—22no—
155Dic47—21no—
156Dic47—21no—
157Dic47—22no—
158Dic47—21no—
159Dic47#50412yes—
160#2960—471no—
161#2183—472no—
162#2183—472no—
163#2183—472no—
164Q8×C16—472no—
165#2151—471no—
166#2151D4711yescharacteristic
167C2×C4⋊C16—471no—
168#2108—471no—
169#2108—471no—
170C4×M32—471no—
171C2×C47—22no—
172C2×C47C2×C8×Q811yescharacteristic
173C2×C47C2×#50411yescharacteristic
174C2×C47#340711yescharacteristic
175#533—471no—
176C2×M32—471no—
177C2×M32C2×D4711yescharacteristic
178C22×C16—471no—
179C22×C16C2×D4711yescharacteristic
180#446—472no—
181#446—472no—
182C8×Q8—472no—
183C8×Q8—472no—
184#424—471no—
185#424—471no—
186#423—471no—
187#405—471no—
188#405C2×D4711yescharacteristic
189C2×C4×C8—471no—
190C4⋊C16—471no—
191C4⋊C16—471no—
192C4⋊C16—471no—
193C4⋊C16—471no—
194C4⋊C16—472no—
195C4⋊C16—472no—
196C4⋊C16C2×D4712yes—
197C4⋊C16C2×D4712yes—
198#346—472no—
199#346—472no—
200#346—472no—
201C4×C16—472no—
202C4×C16—472no—
203C4×C16—472no—
204C47#3347611yescharacteristic, complemented, Sylow
205C2×M16—471no—
206C2×M16C4×D4711yescharacteristic
207C22×C8—471no—
208C22×C8C22×D4711yescharacteristic
209C4⋊Q8—471no—
210C4⋊Q8—471no—
211#125—471no—
212#125—471no—
213C22⋊Q8—472no—
214C22⋊Q8—472no—
215C4×Q8—472no—
216C4×Q8—472no—
217#117—471no—
218#117C4×D4711yescharacteristic
219C2×C42—471no—
220M32—22no—
221M32—474no—
222C2×C16—22no—
223C2×C16—472no—
224C2×C16—471no—
225C2×C16—471no—
226C2×C16—942no—
227C2×C16C22×D4711yescharacteristic
228C2×C16C22×D4712yes—
229C2×C16C22×D4711yescharacteristic
230C4⋊C8—472no—
231C4⋊C8—472no—
232C4⋊C8—472no—
233C4⋊C8—472no—
234C4⋊C8—472no—
235C4⋊C8—472no—
236#97—472no—
237#97C4×D4712yes—
238C4×C8—471no—
239C4×C8—471no—
240C4×C8—471no—
241C4×C8—472no—
242C4×C8—471no—
243C4×C8C22×D4712yes—
244C2×Q8—472no—
245C2×Q8—472no—
246C22×C4—471no—
247C22×C4—471no—
248C22×C4C2×C4×D4711yescharacteristic
249M16—22no—
250M16—942no—
251C2×C8—471no—
252C2×C8—471no—
253C2×C8—472no—
254C2×C8—942no—
255C2×C8C2×C4×D4712yes—
256C2×C8C2×C4×D4711yescharacteristic
257C2×C8#1062535212yes—
258C2×C8Q8×D4712yes—
259C2×C8C23×D4711yescharacteristic
260C4⋊C4—472no—
261C4⋊C4—472no—
262C4⋊C4—472no—
263C4⋊C4—472no—
264C4⋊C4—471no—
265C4⋊C4—471no—
266C4⋊C4—472no—
267C4⋊C4—472no—
268C4⋊C4C8×D4711yescharacteristic
269C4⋊C4C8×D4711yescharacteristic
270C22⋊C4—472no—
271C22⋊C4C8×D4712yes—
272C42—471no—
273C42—471no—
274C42—471no—
275C42—471no—
276C42—472no—
277C42C2×C4×D4712yes—
278C16—21no—
279C16—21no—
280C16—22no—
281C16—474no—
282C16—941no—
283C16—941no—
284C23C2×C8×D4711yescharacteristic
285Q8—944no—
286Q8—1882no—
287C2×C4—471no—
288C2×C4—471no—
289C2×C4—471no—
290C2×C4—471no—
291C2×C4—472no—
292C2×C4—472no—
293C2×C4—942no—
294C2×C4—1881no—
295C2×C4#1309398912yes—
296C2×C4C2×C8×D4712yes—
297C2×C4C2×C8×D4712yes—
298C2×C4C2×C8×D4711yescharacteristic
299C2×C4#1309403212yes—
300C2×C4C22×C4×D4711yescharacteristic
301C8—22no—
302C8—941no—
303C8—941no—
304C8—942no—
305C8#1309398912yes—
306C8C2×Q8×D4711yescharacteristic
307C8C2×#1062535211yescharacteristic
308C22—21no—
309C22—21no—
310C22#1568924712yes—
311C22C22×C8×D4711yescharacteristic
312C4—22no—
313C4—22no—
314C4—472no—
315C4—941no—
316C4—942no—
317C4—941no—
318C4—941no—
319C4—942no—
320C4#1568924712yes—
321C4C2×#1309398911yescharacteristic
322C4C2×#1309403211yescharacteristic
323C2—22no—
324C2C2×#1568924711yescharacteristic
325C2#504×D4711yescharacteristic
326C2#2025648911yescharacteristic
subgroupG/H
center, FrattiniC2×C8C23×D47
commutator, socleC22×C47C23×C8
FittingC47×#2151C2
prime radicalC23×C47C22×C8
cosocleC2×C8×C47C24
hypercenter#2151D47
hypocenterC47#33476
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C8 ◁ #2151

Complements

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

NH ≅ G/Ntype
2. #20245295C2semidirect
3. #20245295C2semidirect
4. #20245293C2semidirect
5. #20245290C2semidirect
53. #13093964Q8semidirect
204. C47#33476semidirect

Maximal subgroups

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

subgroupindex
#202523202
#202452952
#202452932
#202452902
#202451482
C2×Dic47⋊C162
#202449852
#202449842
M32×Dic472
C47×#21512
#3347647