Finite Group Databasebeta

← #26213671

Subgroups — #26213671

Order 12032.

2130 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256591C211yesmaximal, characteristic
2C2×#41×Dic47C211yesmaximal, characteristic
3#20243099C216yesmaximal, complemented
4C2×#15688884C216yesmaximal, complemented
5C2×C47×#568C211yesmaximal, characteristic
6#15689606C418yescomplemented
7#15689601—21no—
8#15689598—23no—
9#41×Dic47—24no—
10#41×Dic47—24no—
11#41×Dic47C418yescomplemented
12C2×Q8×Dic47—21no—
13C2×D4×Dic47—23no—
14C2×#13093983C2216yes—
15C2×#13093983C2213yes—
16C22×C4×Dic47C2213yes—
17#15688884C2214yes—
18#15688884C22112yes—
19C22×#41×C47C2211yescharacteristic
20C47×#568—24no—
21C47×#568C418yescomplemented
22C2×#131×C47—24no—
23C22×M16×C47C2216yes—
24C23×Dic47—23no—
25Q8×Dic47—28no—
26Q8×Dic47—28no—
27D4×Dic47—224no—
28D4×Dic47—224no—
29#13093983—26no—
30#13093983—26no—
31#13093983—26no—
32#13093983C2×C4112yescomplemented
33#13093983C2×C4112yescomplemented
34#13093983C2314yes—
35C2×C4⋊Dic47—23no—
36C2×C4⋊Dic47C2×C4112yescomplemented
37C2×C22⋊Dic47—23no—
38C2×C22⋊Dic47—26no—
39C2×C4×Dic47—23no—
40C2×C4×Dic47—26no—
41C2×C4×Dic47—26no—
42C2×C4×Dic47C2×C4112yescomplemented
43C2×C4×Dic47C2311yescharacteristic
44C2×C4×Dic47C2313yes—
45C2×#41×C47C2×C414yes—
46C2×#41×C47C2×C414yes—
47C2×#41×C47C2×C414yes—
48C2×#41×C47D414yes—
49C2×#41×C47D414yes—
50C2×#41×C47Q814yes—
51C22×Q8×C47D411yescharacteristic
52C22×D4×C47D413yes—
53C23×C4×C47C2313yes—
54#131×C47—232no—
55C2×M16×C47—26no—
56C2×M16×C47—212no—
57C2×M16×C47C2×C4124yescomplemented
58C2×M16×C47C2314yes—
59C22×C8×C47—26no—
60C24×C47C2×D413yes—
61#41×C47C42116yes—
62#41×C47C22⋊C4116yes—
63#41×C47C4⋊C4116yes—
64#41×C47C4⋊C4116yes—
65C2×Q8×C47C22⋊C414yes—
66C2×Q8×C47C22⋊C414yes—
67C2×Q8×C47C4⋊C414yes—
68C2×D4×C47C22⋊C4112yes—
69C2×D4×C47C22⋊C4112yes—
70C2×D4×C47C4⋊C4112yes—
71C22×C4×C47C22×C416yes—
72C22×C4×C47C22×C416yes—
73C22×C4×C47C22×C416yes—
74C22×C4×C47C2×D413yes—
75C22×C4×C47C2×D416yes—
76C22×C4×C47C2×D416yes—
77C22×C4×C47C2×Q816yes—
78C22×C4×C47C2411yescharacteristic
79C22×Dic47—21no—
80C22×Dic47—212no—
81C22×Dic47—23no—
82C22×Dic47—212no—
83C22×Dic47—412no—
84M16×C47—224no—
85M16×C47—24no—
86M16×C47C22×C418yes—
87C2×C8×C47—224no—
88C2×C8×C47—24no—
89C22⋊Dic47—212no—
90C22⋊Dic47—212no—
91C22⋊Dic47—24no—
92C4⋊Dic47—212no—
93C4⋊Dic47—212no—
94C4⋊Dic47C22×C418yes—
95C4×Dic47—212no—
96C4×Dic47—212no—
97C4×Dic47—21no—
98C4×Dic47—24no—
99C4×Dic47—23no—
100C4×Dic47C22×C412yes—
101C4×Dic47C22×C416yes—
102C23×C47—46no—
103C23×C47C2×C22⋊C416yes—
104C23×C47C2×C22⋊C416yes—
105C23×C47C2×C4⋊C416yes—
106C23×C47C22×D411yescharacteristic
107Q8×C47#95116yes—
108D4×C47#95148yes—
109C2×C4×C47C2×C42112yes—
110C2×C4×C47C2×C22⋊C416yes—
111C2×C4×C47C2×C22⋊C416yes—
112C2×C4×C47C2×C22⋊C4112yes—
113C2×C4×C47C2×C4⋊C4112yes—
114C2×C4×C47C2×C4⋊C4112yes—
115C2×C4×C47C2×C4⋊C416yes—
116C2×C4×C47C23×C411yescharacteristic
117C2×C4×C47C23×C411yescharacteristic
118C2×C4×C47C23×C411yescharacteristic
119C2×C4×C47C22×D411yescharacteristic
120C2×C4×C47C22×D411yescharacteristic
121C2×C4×C47C22×Q811yescharacteristic
122C2×Dic47—24no—
123C2×Dic47—22no—
124C2×Dic47—28no—
125C2×Dic47—26no—
126C2×Dic47—424no—
127C2×Dic47—46no—
128C2×Dic47—812no—
129C8×C47—28no—
130#20343—471nomaximal, Sylow
131C22×C47—22no—
132C22×C47—212no—
133C22×C47—46no—
134C22×C47—46no—
135C22×C47C2×#95112yes—
136C22×C47C22×C22⋊C411yescharacteristic
137C22×C47C22×C22⋊C411yescharacteristic
138C22×C47C22×C4⋊C411yescharacteristic
139C4×C47C2×#95112yes—
140C4×C47C22×C4211yescharacteristic
141C4×C47C22×C22⋊C411yescharacteristic
142C4×C47C22×C4⋊C411yescharacteristic
143C4×C47C22×C4⋊C411yescharacteristic
144Dic47—24no—
145Dic47—46no—
146Dic47—48no—
147C2×#568D4711yescharacteristic
148#3428—471no—
149C2×C4×#41—471no—
150#1733—476no—
151C2×#339—476no—
152C2×C47—21no—
153C2×C47—21no—
154C2×C47—212no—
155C2×C47C22×#9511yescharacteristic
156C2×C47#290112yes—
157C22×#41C2×D4711yescharacteristic
158#568—24no—
159#568Dic4718yescomplemented
160C2×#131—24no—
161C22×M16C2×D4716yes—
162#520—478no—
163#519—941no—
164#518—943no—
165C4×#41—478no—
166C4×#41—944no—
167C4×#41—944no—
168C2×C4×Q8—941no—
169C2×C4×D4—943no—
170C2×#117—473no—
171C2×#117—476no—
172C22×C42—473no—
173#339—474no—
174#339—4712no—
175C47#2034311yescharacteristic, complemented, Sylow
176C2×#41Dic9414yes—
177C2×#41C4×D4714yes—
178C2×#41C4×D4714yes—
179C2×#41C2×Dic4714yes—
180C2×#41C22⋊D4714yes—
181C2×#41C22⋊D4714yes—
182C22×Q8D18811yescharacteristic
183C22×D4D18813yes—
184C23×C4—943no—
185C23×C4C22×D4713yes—
186#131—232no—
187C2×M16—26no—
188C2×M16—212no—
189C2×M16C2×Dic47124yescomplemented
190C2×M16C22×D4714yes—
191C22×C8—26no—
192C4×Q8—948no—
193C4×Q8—948no—
194C4×D4—9424no—
195C4×D4—9424no—
196#117—4712no—
197#117—4712no—
198#117—474no—
199#117—946no—
200#117—946no—
201#117—946no—
202C2×C4⋊C4—4712no—
203C2×C4⋊C4—943no—
204C2×C22⋊C4—943no—
205C2×C22⋊C4—946no—
206C2×C42—4712no—
207C2×C42—473no—
208C2×C42—471no—
209C2×C42—943no—
210C2×C42—946no—
211C2×C42—946no—
212C24C2×D18813yes—
213#41C4×Dic47116yes—
214#41Dic47⋊C4116yes—
215#41Dic47⋊C4116yes—
216#41C22⋊Dic47116yes—
217C2×Q8C4⋊Dic4714yes—
218C2×Q8(C2×D47)⋊C414yes—
219C2×Q8(C2×D47)⋊C414yes—
220C2×D4C4⋊Dic47112yes—
221C2×D4(C2×D47)⋊C4112yes—
222C2×D4(C2×D47)⋊C4112yes—
223C22×C4—941no—
224C22×C4—9412no—
225C22×C4—9412no—
226C22×C4—943no—
227C22×C4—18812no—
228C22×C4C2×Dic9416yes—
229C22×C4C2×C4×D4716yes—
230C22×C4C2×C4×D4716yes—
231C22×C4C2×D18813yes—
232C22×C4C22×Dic4716yes—
233C22×C4C2×C22⋊D4716yes—
234C22×C4C2×C22⋊D4716yes—
235C22×C4C23×D4711yescharacteristic
236M16—224no—
237M16—24no—
238M16C22×Dic4718yes—
239C2×C8—224no—
240C2×C8—24no—
241C4⋊C4—478no—
242C4⋊C4—9412no—
243C4⋊C4—9412no—
244C22⋊C4—9412no—
245C22⋊C4—9412no—
246C22⋊C4—944no—
247C42—472no—
248C42—476no—
249C42—9412no—
250C42—9412no—
251C42—941no—
252C42—943no—
253C42—944no—
254C23—46no—
255C23C2×(C2×D47)⋊C416yes—
256C23C2×(C2×D47)⋊C416yes—
257C23C2×C4⋊Dic4716yes—
258C23C22×D18811yescharacteristic
259Q8C47⋊#95116yes—
260D4C47⋊#95148yes—
261C2×C4—944no—
262C2×C4—942no—
263C2×C4—946no—
264C2×C4—948no—
265C2×C4—37612no—
266C2×C4—18824no—
267C2×C4—1886no—
268C2×C4C2×C4×Dic47112yes—
269C2×C4C2×(C2×D47)⋊C416yes—
270C2×C4C2×(C2×D47)⋊C416yes—
271C2×C4C2×C22⋊Dic47112yes—
272C2×C4C2×Dic47⋊C4112yes—
273C2×C4C2×Dic47⋊C4112yes—
274C2×C4C2×C4⋊Dic4716yes—
275C2×C4C22×C4×D4711yescharacteristic
276C2×C4C22×C4×D4711yescharacteristic
277C2×C4C23×Dic4711yescharacteristic
278C2×C4C22×C22⋊D4711yescharacteristic
279C2×C4C22×C22⋊D4711yescharacteristic
280C2×C4C22×Dic9411yescharacteristic
281C8—28no—
282C22—22no—
283C22—212no—
284C22—46no—
285C22—46no—
286C22C2×C47⋊#95112yes—
287C22C22×(C2×D47)⋊C411yescharacteristic
288C22C22×(C2×D47)⋊C411yescharacteristic
289C22C22×C4⋊Dic4711yescharacteristic
290C4—944no—
291C4—1886no—
292C4—1888no—
293C4C2×C47⋊#95112yes—
294C4C22×C4×Dic4711yescharacteristic
295C4C22×C22⋊Dic4711yescharacteristic
296C4C22×Dic47⋊C411yescharacteristic
297C4C22×Dic47⋊C411yescharacteristic
298C2—21no—
299C2—21no—
300C2—212no—
301C2C22×C47⋊#9511yescharacteristic
302C2#2025193312yes—
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC2×C47×#568C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#568D47
hypocenterC47#20343
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
3. #20243099C2semidirect
4. C2×#15688884C2semidirect
6. #15689606C4semidirect
11. #41×Dic47C4semidirect
21. C47×#568C4semidirect
32. #13093983C2×C4semidirect
33. #13093983C2×C4semidirect
36. C2×C4⋊Dic47C2×C4semidirect
42. C2×C4×Dic47C2×C4semidirect
57. C2×M16×C47C2×C4semidirect
159. #568Dic47semidirect
175. C47#20343semidirect
189. C2×M16C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202565912
C2×#41×Dic472
#202430992
C2×#156888842
C2×C47×#5682
#2034347