Finite Group Databasebeta

← #16854969

Subgroups — #16854969

Order 3968.

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

normal subgroups only

This group has 754 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
2C2×#11758519C212yesmaximal, complemented
3#14521209C214yesmaximal, complemented
4#14521204C212yesmaximal, complemented
5#14521199C212yesmaximal, complemented
6#14521048C212yesmaximal, complemented
7#14521013C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
9C22×Q8×C31C2211yescharacteristic, complemented
10C23×C4×C31C2211yescharacteristic
11C22×C22⋊D31—21no—
12C2×#92495—21no—
13C2×#92495—21no—
14C2×#92495—28no—
15C2×#92495C2212yes—
16C2×#92495C2212yes—
17C22×D124C2211yescharacteristic
18C22×C4×D31—21no—
19C22×Dic62C2211yescharacteristic
20C31×C22⋊Q8—21no—
21C31×C22⋊Q8—21no—
22C31×C22⋊Q8C2212yes—
23C31×C22⋊Q8C2212yescomplemented
24C2×C4⋊C4×C31—21no—
25C2×C4⋊C4×C31C2211yescharacteristic, complemented
26C2×C22⋊C4×C31—21no—
27C2×#92482C2214yescomplemented
28#11758519—28no—
29C2×#92481C2214yescomplemented
30C2×#92474C2212yescomplemented
31C22⋊(C31⋊C8)C2212yes—
32C22⋊(C31⋊C8)C2212yescomplemented
33#11758398C2214yescomplemented
34#11758397C2214yescomplemented
35C24×C31D411yescharacteristic
36C23×D31—21no—
37C2×Q8×C31—21no—
38C2×Q8×C31—24no—
39C2×Q8×C31C2312yes—
40C22×C4×C31—24no—
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31D411yescharacteristic
44C22×C4×C31D411yescharacteristic
45C22×C4×C31D411yescharacteristic
46C22×C4×C31C2311yescharacteristic
47C22×C4×C31C2311yescharacteristic
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—28no—
50C2×C22⋊D31—22no—
51C2×C22⋊D31—22no—
52C22×Dic31—21no—
53#92495—28no—
54#92495—28no—
55#92495—24no—
56#92495—24no—
57#92495—24no—
58#92495D418yes—
59C2×D124—21no—
60C2×D124—24no—
61C2×D124C2312yes—
62C2×C4×D31—28no—
63C2×C4×D31—22no—
64C2×C4×D31—22no—
65C2×Dic62—21no—
66C2×Dic62—24no—
67C2×Dic62C2312yes—
68C4⋊C4×C31—22no—
69C4⋊C4×C31—22no—
70C4⋊C4×C31—21no—
71C4⋊C4×C31C2312yes—
72C22⋊C4×C31—22no—
73C22⋊C4×C31—22no—
74#92482—28no—
75#92481—28no—
76#92474—24no—
77C2×C31⋊C8C2314yes—
78C23×C31—22no—
79C23×C31—22no—
80C23×C31—41no—
81C23×C31C2×D411yescharacteristic
82C23×C31C2×D411yescharacteristic
83C23×C31C2×D411yescharacteristic
84C22×D31—24no—
85C22×D31—22no—
86C22×D31—42no—
87C22×D31—42no—
88Q8×C31—24no—
89Q8×C31—42no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31—42no—
98C2×C4×C31—42no—
99C2×C4×C31C2×D414yes—
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2411yescharacteristic
104C22⋊D31—28no—
105C22⋊D31—28no—
106C2×Dic31—24no—
107C2×Dic31—22no—
108D124—24no—
109D124—22no—
110D124C2×D414yes—
111C4×D31—28no—
112C4×D31—28no—
113Dic62—24no—
114Dic62—22no—
115Dic62C2×D414yes—
116C31⋊C8—24no—
117#3007—311nomaximal, Sylow
118C22×C31—21no—
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—44no—
126C22×C31#12012yes—
127C22×C31#13712yescomplemented
128C22×C31C22×D411yescharacteristic
129C2×D31—24no—
130C2×D31—48no—
131C2×D31—42no—
132C2×D31—42no—
133C4×C31—21no—
134C4×C31—21no—
135C4×C31—42no—
136C4×C31—42no—
137C4×C31#12012yes—
138C4×C31C22×D412yes—
139Dic31—24no—
140C22×#41—311no—
141C2×#137—312no—
142C2×C22⋊Q8D3111yescharacteristic, complemented
143#452—314no—
144#451—312no—
145#450—312no—
146#417—312no—
147#407—311no—
148C2×C31—22no—
149C2×C31—22no—
150C2×C31—41no—
151C2×C31C2×#12011yescharacteristic
152C2×C31C2×#13711yescharacteristic
153C2×C31#57511yescharacteristic
154D31—44no—
155C2×#41—312no—
156C2×#41—312no—
157C2×#41—628no—
158C2×#41—621no—
159C2×#41—621no—
160C22×Q8—311no—
161C22×Q8C2×D3111yescharacteristic, complemented
162C22×D4—311no—
163C22×D4—621no—
164C23×C4—621no—
165C23×C4C2×D3111yescharacteristic
166#137—628no—
167C2×Q16—314no—
168C2×SD16—314no—
169C2×M16—312no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8C2×D3112yes—
173C22⋊Q8C2×D3112yescomplemented
174C2×C4⋊C4—21no—
175C2×C4⋊C4C2×D3111yescharacteristic, complemented
176C2×C22⋊C4—21no—
177#103—314no—
178#102—314no—
179C22⋊C8—312no—
180C22⋊C8—312no—
181C31#300711yescharacteristic, complemented, Sylow
182C24—621no—
183C24C22⋊D3111yescharacteristic
184#41—318no—
185#41—624no—
186#41—624no—
187#41—628no—
188#41—628no—
189#41—624no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—312no—
193C2×Q8—624no—
194C2×Q8—621no—
195C2×Q8C22×D3112yes—
196C2×D4—312no—
197C2×D4—624no—
198C2×D4—628no—
199C2×D4—622no—
200C2×D4—622no—
201C2×D4—621no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—628no—
206C22×C4—621no—
207C22×C4—622no—
208C22×C4—622no—
209C22×C4C22⋊D3111yescharacteristic
210C22×C4C22⋊D3111yescharacteristic
211C22×C4C22⋊D3111yescharacteristic
212C22×C4C22×D3111yescharacteristic
213C22×C4C22×D3111yescharacteristic
214C22×C4C22×D3111yescharacteristic
215Q16—628no—
216SD16—628no—
217M16—624no—
218C2×C8—314no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D3112yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—624no—
229C23—622no—
230C23—1242no—
231C23—1242no—
232C23C2×C22⋊D3111yescharacteristic
233C23C2×C22⋊D3111yescharacteristic
234C23C2×C22⋊D3111yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—314no—
238Q8—624no—
239Q8—622no—
240D4—314no—
241D4—624no—
242D4—628no—
243D4—628no—
244D4—622no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—628no—
255C2×C4—624no—
256C2×C4—622no—
257C2×C4—628no—
258C2×C4D4×D3114yes—
259C2×C4C2×C22⋊D3111yescharacteristic
260C2×C4C2×C22⋊D3111yescharacteristic
261C2×C4C2×C22⋊D3111yescharacteristic
262C2×C4C23×D3111yescharacteristic
263C8—624no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—624no—
273C22—1248no—
274C22—1242no—
275C22—1242no—
276C22#1175851412yes—
277C22#1175851912yescomplemented
278C22C22×C22⋊D3111yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—624no—
284C4#1175851412yes—
285C4C2×D4×D3112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—1244no—
290C2C2×#1175851411yescharacteristic
291C2C2×#1175851911yescharacteristic
292C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×C22⋊Q8C2
socleC22×C31C22×D4
prime radicalC22×#92495C2
hypercenterC2×C22⋊Q8D31
hypocenterC31#3007
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. #14521209C2semidirect
4. #14521204C2semidirect
5. #14521199C2semidirect
6. #14521048C2semidirect
7. #14521013C2semidirect
8. C2×C31×C22⋊Q8C2semidirect
9. C22×Q8×C31C22semidirect
23. C31×C22⋊Q8C22semidirect
25. C2×C4⋊C4×C31C22semidirect
27. C2×#92482C22semidirect
29. C2×#92481C22semidirect
30. C2×#92474C22semidirect
32. C22⋊(C31⋊C8)C22semidirect
33. #11758398C22semidirect
34. #11758397C22semidirect
127. C22×C31#137semidirect
142. C2×C22⋊Q8D31semidirect
161. C22×Q8C2×D31semidirect
173. C22⋊Q8C2×D31semidirect
175. C2×C4⋊C4C2×D31semidirect
181. C31#3007semidirect
277. C22#11758519semidirect

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117585192
#145212092
#145212042
#145211992
#145210482
#145210132
C2×C31×C22⋊Q82
#300731