Finite Group Databasebeta

← #16854943

Subgroups — #16854943

Order 3968.

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

normal subgroups only

This group has 738 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×Q8×D31C211yesmaximal, characteristic, complemented
2C2×#11758528C211yesmaximal, characteristic, complemented
3#14521556C211yesmaximal, characteristic
4#14521203C214yesmaximal, complemented
5#14521186C214yesmaximal, complemented
6#14521059C211yesmaximal, characteristic, complemented
7#14521058C211yesmaximal, characteristic, complemented
8#14521012C211yesmaximal, characteristic, complemented
9C2×C31×#137C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×Q8×C31C2211yescharacteristic
12C2×Q8×D31—24no—
13C2×Q8×D31—22no—
14C2×Q8×D31—24no—
15C2×Q8×D31C2214yes—
16C22×C4×D31—21no—
17C22×C4×D31C2211yescharacteristic
18C22×Dic62—21no—
19C22×Dic62C2211yescharacteristic
20C31×#137—24no—
21C2×Q16×C31C2212yescomplemented
22C2×SD16×C31C2212yescomplemented
23C2×M16×C31C2211yescharacteristic
24#11758528—24no—
25#11758523C2212yes—
26C2×#92482C2212yescomplemented
27#11758515C2212yes—
28#11758514—21no—
29#11758512—21no—
30C2×#92480C2212yescomplemented
31C4×C22⋊D31—22no—
32#11758503C2211yescharacteristic
33C2×#92474C2211yescharacteristic
34#11758423C2214yescomplemented
35(C2×D31)⋊C8C2214yes—
36#11758404C2214yescomplemented
37C23×D31D411yescharacteristic
38#41×C31—22no—
39#41×C31—22no—
40C2×Q8×C31—22no—
41C2×Q8×C31—21no—
42C2×Q8×C31D414yescomplemented
43C2×Q8×C31C2311yescharacteristic
44C2×Q8×C31C2312yes—
45C2×D4×C31—21no—
46C2×D4×C31C2311yescharacteristic
47C22×C4×C31—21no—
48C22×C4×C31—21no—
49C22×C4×C31C2311yescharacteristic
50C2×C22⋊D31—22no—
51C22×Dic31—21no—
52C22×Dic31D411yescharacteristic
53Q8×D31—216no—
54Q8×D31—28no—
55Q8×D31—28no—
56C2×C4×D31—28no—
57C2×C4×D31—24no—
58C2×C4×D31—22no—
59C2×C4×D31—22no—
60C2×C4×D31D412yes—
61C2×C4×D31C2312yes—
62C2×Dic62—28no—
63C2×Dic62—21no—
64C2×Dic62—22no—
65C2×Dic62—24no—
66C2×Dic62D414yescomplemented
67C2×Dic62C2312yes—
68Q16×C31—24no—
69SD16×C31—24no—
70M16×C31—22no—
71C2×C8×C31C2312yes—
72C22⋊Dic31—21no—
73C22⋊Dic31—22no—
74#92482—24no—
75#92480—24no—
76(C2×D31)⋊C4—22no—
77C4⋊Dic31C2312yes—
78Dic31⋊C4—22no—
79C4×Dic31—21no—
80#92474—22no—
81C2×C31⋊C8C2312yes—
82C23×C31—21no—
83C23×C31C2×D411yescharacteristic
84C22×D31—22no—
85C22×D31—24no—
86C22×D31C2×D412yes—
87Q8×C31—24no—
88Q8×C31—22no—
89Q8×C31—21no—
90Q8×C31—21no—
91Q8×C31C2×D414yes—
92D4×C31—21no—
93D4×C31—21no—
94D4×C31—42no—
95C2×C4×C31—24no—
96C2×C4×C31—22no—
97C2×C4×C31—21no—
98C2×C4×C31—42no—
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—44no—
104C2×Dic31—22no—
105C2×Dic31—24no—
106C2×Dic31—22no—
107C2×Dic31—22no—
108C2×Dic31C2×D412yes—
109C4×D31—216no—
110C4×D31—24no—
111C4×D31—22no—
112C4×D31—22no—
113Dic62—216no—
114Dic62—24no—
115Dic62—22no—
116Dic62C2×D414yes—
117C8×C31—22no—
118C31⋊C8—22no—
119#3004—311nomaximal, Sylow
120C22×C31—22no—
121C22×C31—41no—
122C22×C31—42no—
123C22×C31#12012yes—
124C22×C31C22×D411yescharacteristic
125C2×D31—28no—
126C2×D31—22no—
127C2×D31—22no—
128C2×D31#13714yescomplemented
129C4×C31—24no—
130C4×C31—41no—
131C4×C31#12012yes—
132C4×C31C22×D411yescharacteristic
133C4×C31C22×D411yescharacteristic
134Dic31—22no—
135Dic31—24no—
136Dic31—42no—
137C23×Q8—311no—
138C2×#137—311no—
139C2×#137D3111yescharacteristic, complemented
140#525—311no—
141#451—314no—
142#448—314no—
143#419—311no—
144#419—311no—
145#407—311no—
146C2×C31—22no—
147C2×C31—81no—
148C2×C31C2×#12011yescharacteristic
149C2×C31C2×#13712yes—
150D31—24no—
151C2×#41C2×D3111yescharacteristic
152C22×Q8—314no—
153C22×Q8—311no—
154C22×Q8—621no—
155C22×Q8—622no—
156C22×Q8—624no—
157C22×Q8—624no—
158C22×Q8C2×D3111yescharacteristic
159C23×C4—311no—
160C23×C4—621no—
161#137—24no—
162#137—624no—
163C2×Q16—312no—
164C2×Q16C2×D3112yescomplemented
165C2×SD16—312no—
166C2×SD16C2×D3112yescomplemented
167C2×M16—311no—
168C2×M16C2×D3111yescharacteristic
169#123—621no—
170C22⋊Q8—312no—
171#121—312no—
172#120—621no—
173C4×D4—622no—
174#117—311no—
175#103—314no—
176#103—314no—
177C22⋊C8—314no—
178C31#300411yescharacteristic, complemented, Sylow
179C24—311no—
180#41—22no—
181#41—22no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—314no—
185C2×Q8—312no—
186C2×Q8—624no—
187C2×Q8—628no—
188C2×Q8—628no—
189C2×Q8—6216no—
190C2×Q8—628no—
191C2×Q8—621no—
192C2×Q8—622no—
193C2×Q8C22⋊D3114yescomplemented
194C2×Q8C22×D3111yescharacteristic
195C2×Q8C22×D3112yes—
196C2×D4—21no—
197C2×D4—622no—
198C2×D4C22×D3111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—311no—
202C22×C4—312no—
203C22×C4—312no—
204C22×C4—624no—
205C22×C4—628no—
206C22×C4—621no—
207C22×C4—622no—
208C22×C4—622no—
209C22×C4C22×D3111yescharacteristic
210Q16—24no—
211Q16—624no—
212SD16—24no—
213SD16—624no—
214M16—22no—
215M16—622no—
216C2×C8—312no—
217C2×C8C22×D3112yes—
218C4⋊C4—312no—
219C4⋊C4—622no—
220C22⋊C4—622no—
221C22⋊C4—622no—
222C22⋊C4—621no—
223C42—621no—
224C23—21no—
225C23—312no—
226C23—624no—
227C23—622no—
228C23D4×D3111yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—21no—
232Q8—21no—
233Q8—314no—
234Q8—6216no—
235Q8—624no—
236Q8—622no—
237Q8C2×C22⋊D3114yes—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—1244no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—312no—
247C2×C4—6216no—
248C2×C4—622no—
249C2×C4—624no—
250C2×C4—622no—
251C2×C4—624no—
252C2×C4—622no—
253C2×C4—622no—
254C2×C4—622no—
255C2×C4D4×D3111yescharacteristic
256C2×C4D4×D3112yes—
257C2×C4C2×C22⋊D3112yes—
258C2×C4C23×D3111yescharacteristic
259C8—22no—
260C8—622no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—314no—
265C22—628no—
266C22—622no—
267C22—622no—
268C22#1175851412yes—
269C22C2×D4×D3111yescharacteristic
270C4—24no—
271C4—41no—
272C4—622no—
273C4—624no—
274C4—1242no—
275C4#1175851412yes—
276C4C2×D4×D3111yescharacteristic
277C4C22×C22⋊D3111yescharacteristic
278C2—22no—
279C2—81no—
280C2—624no—
281C2C2×#1175851411yescharacteristic
282C2#137×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#137C2
socleC22×C31C22×D4
prime radical#14521556C2
hypercenterC2×#137D31
hypocenterC31#3004
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D31C2semidirect
2. C2×#11758528C2semidirect
4. #14521203C2semidirect
5. #14521186C2semidirect
6. #14521059C2semidirect
7. #14521058C2semidirect
8. #14521012C2semidirect
9. C2×C31×#137C2semidirect
21. C2×Q16×C31C22semidirect
22. C2×SD16×C31C22semidirect
26. C2×#92482C22semidirect
30. C2×#92480C22semidirect
34. #11758423C22semidirect
36. #11758404C22semidirect
42. C2×Q8×C31D4semidirect
66. C2×Dic62D4semidirect
128. C2×D31#137semidirect
139. C2×#137D31semidirect
164. C2×Q16C2×D31semidirect
166. C2×SD16C2×D31semidirect
178. C31#3004semidirect
193. C2×Q8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D312
C2×#117585282
#145215562
#145212032
#145211862
#145210592
#145210582
#145210122
C2×C31×#1372
#300431