Finite Group Databasebeta

← #16854965

Subgroups — #16854965

Order 3968.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#92495C211yesmaximal, characteristic, complemented
2C2×#11758485C212yesmaximal, complemented
3C2×#11758507C211yesmaximal, characteristic, complemented
4#14521196C214yesmaximal, complemented
5#14521190C212yesmaximal, complemented
6#14521185C212yesmaximal, complemented
7#14521051C212yesmaximal, complemented
8C31×#407C211yesmaximal, characteristic, complemented
9C23×C4×C31C2211yescharacteristic
10C22×C22⋊D31—21no—
11C22×C22⋊D31—21no—
12C2×#92495—21no—
13C2×#92495—21no—
14C2×#92495—28no—
15C2×#92495C2212yes—
16C2×#92495C2212yescomplemented
17C22×D124C2211yescharacteristic, complemented
18C22×D124C2211yescharacteristic, complemented
19C22×C4×D31—21no—
20C22×Dic62C2211yescharacteristic, complemented
21C2×M16×C31C2212yescomplemented
22#11758507—21no—
23#11758507—21no—
24#11758507C2212yes—
25#11758507C2212yescomplemented
26C2×(C2×D31)⋊C4—21no—
27C2×C4⋊Dic31C2211yescharacteristic, complemented
28#11758485—28no—
29C2×D248C2214yescomplemented
30C2×#92470C2214yescomplemented
31C31×C22⋊C8C2212yescomplemented
32C31×C22⋊C8C2212yescomplemented
33#11758409C2214yescomplemented
34#11758404C2214yescomplemented
35C24×C31D411yescharacteristic
36C23×D31—21no—
37C23×D31—21no—
38C22×C4×C31—24no—
39C22×C4×C31D411yescharacteristic
40C22×C4×C31D411yescharacteristic
41C22×C4×C31D411yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2311yescharacteristic
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—28no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31—22no—
49C2×C22⋊D31—22no—
50C2×C22⋊D31—44no—
51C22×Dic31—21no—
52C22×Dic31—21no—
53#92495—28no—
54#92495—28no—
55#92495—24no—
56#92495—24no—
57#92495—24no—
58#92495D418yescomplemented
59C2×D124—21no—
60C2×D124—24no—
61C2×D124—24no—
62C2×D124—21no—
63C2×D124C2312yes—
64C2×D124C2312yes—
65C2×C4×D31—28no—
66C2×C4×D31—22no—
67C2×C4×D31—22no—
68C2×Dic62—21no—
69C2×Dic62—24no—
70C2×Dic62C2312yes—
71M16×C31—24no—
72C2×C8×C31C2314yes—
73(C2×D31)⋊C4—22no—
74(C2×D31)⋊C4—22no—
75C4⋊Dic31—21no—
76C4⋊Dic31C2312yes—
77D248—28no—
78#92470—28no—
79C23×C31—22no—
80C23×C31—22no—
81C23×C31—41no—
82C23×C31C2×D411yescharacteristic
83C23×C31C2×D411yescharacteristic
84C23×C31C2×D411yescharacteristic
85C22×D31—24no—
86C22×D31—22no—
87C22×D31—22no—
88C22×D31—42no—
89C22×D31—42no—
90C22×D31—42no—
91C22×D31—42no—
92C22×D31—42no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D414yes—
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—28no—
104C22⋊D31—28no—
105C22⋊D31—44no—
106C22⋊D31—44no—
107C2×Dic31—24no—
108C2×Dic31—22no—
109C2×Dic31—22no—
110C2×Dic31—42no—
111D124—24no—
112D124—22no—
113D124—24no—
114D124—42no—
115D124C2×D414yescomplemented
116C4×D31—28no—
117C4×D31—28no—
118Dic62—24no—
119Dic62—22no—
120Dic62C2×D414yescomplemented
121C8×C31—24no—
122#3006—311nomaximal, Sylow
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—21no—
126C22×C31—21no—
127C22×C31—21no—
128C22×C31—21no—
129C22×C31—21no—
130C22×C31—44no—
131C22×C31#12012yes—
132C22×C31AGL(1,ℤ/8)12yescomplemented
133C22×C31C22×D411yescharacteristic
134C2×D31—24no—
135C2×D31—48no—
136C2×D31—42no—
137C2×D31—42no—
138C2×D31—42no—
139C2×D31—42no—
140C2×D31—42no—
141C2×D31—84no—
142C4×C31—21no—
143C4×C31—21no—
144C4×C31#12012yes—
145C4×C31C22×D412yes—
146Dic31—24no—
147Dic31—42no—
148C22×#41—311no—
149C2×AGL(1,ℤ/8)—312no—
150C2×#121—311no—
151#449—314no—
152#448—312no—
153#447—312no—
154#417—312no—
155#407D3111yescharacteristic, complemented
156C2×C31—22no—
157C2×C31—22no—
158C2×C31—41no—
159C2×C31C2×#12011yescharacteristic
160C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
161C2×C31#57511yescharacteristic
162D31—44no—
163D31—82no—
164C2×#41—312no—
165C2×#41—312no—
166C2×#41—628no—
167C2×#41—621no—
168C2×#41—621no—
169C22×Q8—311no—
170C22×D4—311no—
171C22×D4—311no—
172C22×D4—621no—
173C22×D4—621no—
174C23×C4—621no—
175C23×C4C2×D3111yescharacteristic
176AGL(1,ℤ/8)—628no—
177C2×SD16—314no—
178C2×D8—314no—
179C2×M16C2×D3112yescomplemented
180#121—312no—
181#121—312no—
182#121—621no—
183#121—621no—
184C2×C4⋊C4—311no—
185C2×C22⋊C4—621no—
186#103—314no—
187#102—314no—
188C22⋊C8C2×D3112yescomplemented
189C22⋊C8C2×D3112yescomplemented
190C31#300611yescharacteristic, complemented, Sylow
191C24—621no—
192C24—621no—
193C24D12411yescharacteristic
194#41—318no—
195#41—624no—
196#41—624no—
197#41—628no—
198#41—628no—
199#41—624no—
200C2×Q8—312no—
201C2×Q8—624no—
202C2×Q8—621no—
203C2×D4—312no—
204C2×D4—312no—
205C2×D4—624no—
206C2×D4—628no—
207C2×D4—622no—
208C2×D4—622no—
209C2×D4—622no—
210C2×D4—622no—
211C2×D4—621no—
212C2×D4—621no—
213C2×D4—624no—
214C2×D4—1244no—
215C22×C4—24no—
216C22×C4—628no—
217C22×C4—621no—
218C22×C4—622no—
219C22×C4—622no—
220C22×C4—621no—
221C22×C4D12411yescharacteristic
222C22×C4D12411yescharacteristic
223C22×C4D12411yescharacteristic
224C22×C4C22×D3111yescharacteristic
225C22×C4C22×D3111yescharacteristic
226C22×C4C22×D3111yescharacteristic
227SD16—628no—
228D8—628no—
229M16—24no—
230C2×C8C22×D3114yes—
231C4⋊C4—312no—
232C4⋊C4—621no—
233C22⋊C4—622no—
234C22⋊C4—622no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—624no—
239C23—622no—
240C23—622no—
241C23—1242no—
242C23—1242no—
243C23—1242no—
244C23—1242no—
245C23—1242no—
246C23C2×D12411yescharacteristic
247C23C2×D12411yescharacteristic
248C23C2×D12411yescharacteristic
249Q8—314no—
250Q8—624no—
251Q8—622no—
252D4—314no—
253D4—624no—
254D4—628no—
255D4—628no—
256D4—622no—
257D4—624no—
258D4—1244no—
259D4—1244no—
260D4—1242no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—628no—
267C2×C4—624no—
268C2×C4—622no—
269C2×C4—628no—
270C2×C4—622no—
271C2×C4—1242no—
272C2×C4C2×D12411yescharacteristic
273C2×C4C2×D12411yescharacteristic
274C2×C4C2×D12411yescharacteristic
275C2×C4D4×D3114yes—
276C2×C4C23×D3111yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—624no—
287C22—1248no—
288C22—1242no—
289C22—1242no—
290C22—1242no—
291C22—1242no—
292C22—1242no—
293C22—2484no—
294C22#1175845912yes—
295C22#1175848512yescomplemented
296C22C22×D12411yescharacteristic
297C4—21no—
298C4—21no—
299C4—624no—
300C4—1242no—
301C4#1175845912yes—
302C4C2×D4×D3112yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—1244no—
307C2—2482no—
308C2C2×#1175845911yescharacteristic
309C2C2×#1175848511yescharacteristic
310C2#1452184911yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#407C2
socleC22×C31C22×D4
hypercenter#407D31
hypocenterC31#3006
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C2×#11758485C2semidirect
3. C2×#11758507C2semidirect
4. #14521196C2semidirect
5. #14521190C2semidirect
6. #14521185C2semidirect
7. #14521051C2semidirect
8. C31×#407C2semidirect
16. C2×#92495C22semidirect
17. C22×D124C22semidirect
18. C22×D124C22semidirect
20. C22×Dic62C22semidirect
21. C2×M16×C31C22semidirect
25. #11758507C22semidirect
27. C2×C4⋊Dic31C22semidirect
29. C2×D248C22semidirect
30. C2×#92470C22semidirect
31. C31×C22⋊C8C22semidirect
32. C31×C22⋊C8C22semidirect
33. #11758409C22semidirect
34. #11758404C22semidirect
58. #92495D4semidirect
115. D124C2×D4semidirect
120. Dic62C2×D4semidirect
132. C22×C31AGL(1,ℤ/8)semidirect
155. #407D31semidirect
179. C2×M16C2×D31semidirect
188. C22⋊C8C2×D31semidirect
189. C22⋊C8C2×D31semidirect
190. C31#3006semidirect
295. C22#11758485semidirect

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117584852
C2×#117585072
#145211962
#145211902
#145211852
#145210512
C31×#4072
#300631