Finite Group Databasebeta

← #16854947

Subgroups — #16854947

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×#11758486C211yesmaximal, characteristic, complemented
3C2×#11758519C211yesmaximal, characteristic, complemented
4#14521561C211yesmaximal, characteristic
5#14521206C214yesmaximal, complemented
6#14521189C214yesmaximal, complemented
7#14521057C211yesmaximal, characteristic, complemented
8#14521012C211yesmaximal, characteristic, complemented
9C31×#419C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C2×Q8×D31—24no—
12C2×Q8×D31—22no—
13C2×Q8×D31—24no—
14C2×Q8×D31C2214yes—
15C2×#92495C2211yescharacteristic
16C22×C4×D31—21no—
17C22×C4×D31C2211yescharacteristic
18C22×Dic62—21no—
19C22×Dic62C2211yescharacteristic
20C2×M16×C31C2211yescharacteristic
21C31×#117C2211yescharacteristic
22C2×#92482C2212yescomplemented
23#11758519—24no—
24C2×#92481C2212yescomplemented
25C2×#92474C2211yescharacteristic
26#11758486—24no—
27C2×Dic124C2212yescomplemented
28C2×#92470C2212yescomplemented
29#11758474C2212yes—
30#11758472C2212yes—
31#11758463—21no—
32#11758459—21no—
33C4×D124—22no—
34C31×#103C2214yescomplemented
35(C2×D31)⋊C8C2214yes—
36#11758398C2214yescomplemented
37C23×D31D411yescharacteristic
38C2×Q8×C31—22no—
39C2×Q8×C31—21no—
40C2×Q8×C31D414yescomplemented
41C2×Q8×C31C2312yes—
42C22×C4×C31—21no—
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—21no—
45C22×Dic31—21no—
46C22×Dic31D411yescharacteristic
47Q8×D31—216no—
48Q8×D31—28no—
49Q8×D31—28no—
50#92495—22no—
51#92495—22no—
52C2×D124—22no—
53C2×D124C2311yescharacteristic
54C2×C4×D31—28no—
55C2×C4×D31—24no—
56C2×C4×D31—22no—
57C2×C4×D31—22no—
58C2×C4×D31—21no—
59C2×C4×D31D412yes—
60C2×C4×D31C2312yes—
61C2×Dic62—28no—
62C2×Dic62—21no—
63C2×Dic62—22no—
64C2×Dic62—24no—
65C2×Dic62D414yescomplemented
66C2×Dic62C2311yescharacteristic
67C2×Dic62C2312yes—
68M16×C31—22no—
69C2×C8×C31C2312yes—
70C4⋊C4×C31C2312yes—
71C22⋊C4×C31—21no—
72C42×C31—21no—
73#92482—24no—
74#92481—24no—
75(C2×D31)⋊C4—22no—
76(C2×D31)⋊C4—22no—
77C4⋊Dic31—22no—
78#92474—22no—
79C2×C31⋊C8C2312yes—
80Dic124—24no—
81#92470—24no—
82C23×C31C2×D411yescharacteristic
83C22×D31—22no—
84C22×D31—24no—
85C22×D31—21no—
86C22×D31C2×D412yes—
87Q8×C31—24no—
88Q8×C31—22no—
89Q8×C31C2×D414yes—
90C2×C4×C31—24no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31C2×D412yes—
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2×D412yes—
96C2×C4×C31C2411yescharacteristic
97C22⋊D31—42no—
98C2×Dic31—22no—
99C2×Dic31—24no—
100C2×Dic31—22no—
101C2×Dic31—21no—
102C2×Dic31C2×D412yes—
103D124—21no—
104D124—21no—
105D124—44no—
106C4×D31—216no—
107C4×D31—24no—
108C4×D31—22no—
109C4×D31—22no—
110C4×D31—42no—
111Dic62—216no—
112Dic62—24no—
113Dic62—22no—
114Dic62—21no—
115Dic62—21no—
116Dic62C2×D414yes—
117C8×C31—22no—
118C31⋊C8—22no—
119#3004—311nomaximal, Sylow
120C22×C31—22no—
121C22×C31#12012yes—
122C22×C31C22×D411yescharacteristic
123C2×D31—28no—
124C2×D31—22no—
125C2×D31—22no—
126C2×D31—41no—
127C2×D31—42no—
128C2×D31#13714yescomplemented
129C4×C31—24no—
130C4×C31—42no—
131C4×C31#12012yes—
132C4×C31C22×D411yescharacteristic
133C4×C31C22×D411yescharacteristic
134Dic31—22no—
135Dic31—24no—
136Dic31—41no—
137C23×Q8—311no—
138C2×#137—311no—
139C2×#137—311no—
140#525—311no—
141#451—314no—
142#448—314no—
143#419—311no—
144#419D3111yescharacteristic, complemented
145#407—311no—
146C2×C31—22no—
147C2×C31C2×#12011yescharacteristic
148C2×C31C2×#13712yes—
149D31—24no—
150D31—81no—
151C2×#41—311no—
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—624no—
162#137—624no—
163C2×Q16—312no—
164C2×Q16—312no—
165C2×SD16—312no—
166C2×SD16—312no—
167C2×M16—311no—
168C2×M16C2×D3111yescharacteristic
169#123—621no—
170C22⋊Q8—312no—
171#121—312no—
172#120—621no—
173C4×D4—622no—
174#117C2×D3111yescharacteristic
175#103—314no—
176#103C2×D3114yescomplemented
177C22⋊C8—314no—
178C31#300411yescharacteristic, complemented, Sylow
179C24—311no—
180#41—622no—
181#41—622no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—314no—
185C2×Q8—312no—
186C2×Q8—311no—
187C2×Q8—624no—
188C2×Q8—628no—
189C2×Q8—628no—
190C2×Q8—6216no—
191C2×Q8—628no—
192C2×Q8—622no—
193C2×Q8—621no—
194C2×Q8D12414yescomplemented
195C2×Q8C22×D3112yes—
196C2×D4—311no—
197C2×D4—622no—
198C2×D4—621no—
199C22×C4—21no—
200C22×C4—311no—
201C22×C4—312no—
202C22×C4—312no—
203C22×C4—624no—
204C22×C4—628no—
205C22×C4—621no—
206C22×C4—622no—
207C22×C4—622no—
208C22×C4—621no—
209C22×C4C22×D3111yescharacteristic
210Q16—624no—
211Q16—624no—
212SD16—624no—
213SD16—624no—
214M16—22no—
215M16—622no—
216C2×C8—312no—
217C2×C8C22×D3112yes—
218C4⋊C4—622no—
219C4⋊C4C22×D3112yes—
220C22⋊C4—21no—
221C22⋊C4—622no—
222C22⋊C4—622no—
223C42—21no—
224C23—312no—
225C23—624no—
226C23—622no—
227C23—621no—
228C23D4×D3111yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—314no—
232Q8—6216no—
233Q8—624no—
234Q8—622no—
235Q8—621no—
236Q8—621no—
237Q8C2×D12414yes—
238D4—621no—
239D4—621no—
240D4—1244no—
241D4—1242no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—312no—
246C2×C4—6216no—
247C2×C4—622no—
248C2×C4—624no—
249C2×C4—622no—
250C2×C4—624no—
251C2×C4—622no—
252C2×C4—622no—
253C2×C4—621no—
254C2×C4—1242no—
255C2×C4C2×D12412yes—
256C2×C4D4×D3111yescharacteristic
257C2×C4D4×D3112yes—
258C2×C4C23×D3111yescharacteristic
259C8—22no—
260C8—622no—
261C22—22no—
262C22—314no—
263C22—628no—
264C22—622no—
265C22—622no—
266C22—1241no—
267C22—1242no—
268C22#1175845912yes—
269C22C2×D4×D3111yescharacteristic
270C4—24no—
271C4—42no—
272C4—622no—
273C4—624no—
274C4—1241no—
275C4#1175845912yes—
276C4C22×D12411yescharacteristic
277C4C2×D4×D3111yescharacteristic
278C2—22no—
279C2—624no—
280C2—2481no—
281C2C2×#1175845911yescharacteristic
282C2#137×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#419C2
socleC22×C31C22×D4
prime radical#14521561C2
hypercenter#419D31
hypocenterC31#3004
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D31C2semidirect
2. C2×#11758486C2semidirect
3. C2×#11758519C2semidirect
5. #14521206C2semidirect
6. #14521189C2semidirect
7. #14521057C2semidirect
8. #14521012C2semidirect
9. C31×#419C2semidirect
22. C2×#92482C22semidirect
24. C2×#92481C22semidirect
27. C2×Dic124C22semidirect
28. C2×#92470C22semidirect
34. C31×#103C22semidirect
36. #11758398C22semidirect
40. C2×Q8×C31D4semidirect
65. C2×Dic62D4semidirect
128. C2×D31#137semidirect
144. #419D31semidirect
176. #103C2×D31semidirect
178. C31#3004semidirect
194. C2×Q8D124semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D312
C2×#117584862
C2×#117585192
#145215612
#145212062
#145211892
#145210572
#145210122
C31×#4192
#300431