Finite Group Databasebeta

← #16854984

Subgroups — #16854984

Order 3968.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11758561C211yesmaximal, characteristic
2C2×#11758519C211yesmaximal, characteristic, complemented
3#14521623C211yesmaximal, characteristic, complemented
4#14521208C214yesmaximal, complemented
5#14521198C214yesmaximal, complemented
6#14521054C211yesmaximal, characteristic, complemented
7#14521046C211yesmaximal, characteristic, complemented
8#14521015C211yesmaximal, characteristic, complemented
9C22×SD16×C31C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic, complemented
11C22×D4×C31C2211yescharacteristic
12C22×C22⋊D31—22no—
13#11758561—24no—
14#11758561—24no—
15C2×#92497—21no—
16C2×#92497C2212yes—
17C2×D4×D31—21no—
18C2×D4×D31C2212yes—
19C2×#92495C2211yescharacteristic
20C2×#92495C2211yescharacteristic
21C2×SD16×C31—24no—
22C2×SD16×C31C2214yescomplemented
23C22×C8×C31C2211yescharacteristic, complemented
24#11758524—21no—
25#11758523C2214yescomplemented
26#11758521—21no—
27C2×#92482C2212yescomplemented
28#11758519—24no—
29C2×#92481C2212yescomplemented
30#11758506—22no—
31#11758503C2211yescharacteristic, complemented
32C2×#92474C2211yescharacteristic, complemented
33#11758420C2214yescomplemented
34#11758409C2212yescomplemented
35(C2×D31)⋊C8C2214yescomplemented
36#11758404C2212yescomplemented
37C24×C31—21no—
38C23×D31—22no—
39C2×Q8×C31—21no—
40C2×Q8×C31—22no—
41C2×Q8×C31—22no—
42C2×Q8×C31C2312yes—
43C2×D4×C31—21no—
44C2×D4×C31—22no—
45C2×D4×C31D414yes—
46C2×D4×C31C2312yes—
47C22×C4×C31—21no—
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—216no—
50C2×C22⋊D31—21no—
51C2×C22⋊D31—22no—
52C2×C22⋊D31—22no—
53C2×C22⋊D31D412yes—
54C22×Dic31—22no—
55#92497—28no—
56#92497—24no—
57#92497—24no—
58D4×D31—28no—
59D4×D31—24no—
60D4×D31—24no—
61#92495—22no—
62#92495—22no—
63#92495—22no—
64#92495D414yes—
65C2×D124D411yescharacteristic
66C2×D124C2311yescharacteristic
67C2×C4×D31—21no—
68C2×C4×D31C2312yes—
69C2×Dic62D411yescharacteristic
70C2×Dic62C2311yescharacteristic
71SD16×C31—28no—
72C2×C8×C31—22no—
73C2×C8×C31C2312yes—
74C22⋊Dic31—21no—
75#92482—24no—
76#92481—24no—
77(C2×D31)⋊C4—24no—
78C4⋊Dic31C2312yes—
79Dic31⋊C4—24no—
80C4×Dic31—21no—
81#92474—22no—
82C2×C31⋊C8C2312yes—
83C23×C31—22no—
84C23×C31—24no—
85C23×C31—82no—
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88C22×D31—28no—
89C22×D31—84no—
90C22×D31C2×D412yes—
91Q8×C31—22no—
92Q8×C31—22no—
93Q8×C31—42no—
94D4×C31—24no—
95D4×C31—22no—
96D4×C31C2×D414yes—
97C2×C4×C31—22no—
98C2×C4×C31—42no—
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—28no—
104C22⋊D31—28no—
105C22⋊D31—24no—
106C22⋊D31—24no—
107C2×Dic31—28no—
108C2×Dic31—21no—
109C2×Dic31—22no—
110C2×Dic31C2×D412yes—
111D124—22no—
112D124—21no—
113D124C2×D412yes—
114C4×D31—24no—
115C4×D31—22no—
116C4×D31—22no—
117Dic62—22no—
118Dic62—21no—
119Dic62C2×D412yes—
120C8×C31—22no—
121C31⋊C8—22no—
122#3009—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—24no—
125C22×C31—44no—
126C22×C31—84no—
127C22×C31#12012yes—
128C22×C31C22×D411yescharacteristic
129C2×D31—22no—
130C2×D31—22no—
131C2×D31—24no—
132C2×D31—42no—
133C2×D31—88no—
134C4×C31—42no—
135C4×C31#12012yes—
136C4×C31C22×D411yescharacteristic
137C4×C31C22×D411yescharacteristic
138Dic31—22no—
139Dic31—22no—
140Dic31—42no—
141C2×H(5,𝔽2)—311no—
142C2×#137—311no—
143C22×SD16D3111yescharacteristic, complemented
144#536—311no—
145#452—314no—
146#450—314no—
147#418—311no—
148#416—311no—
149#408—311no—
150C2×C31—22no—
151C2×C31—44no—
152C2×C31C2×#12011yescharacteristic
153C2×C31#57712yes—
154D31—42no—
155D31—42no—
156H(5,𝔽2)—624no—
157H(5,𝔽2)—624no—
158C2×#41—311no—
159C2×#41—312no—
160C2×#41—311no—
161C2×#41—621no—
162C22×Q8C2×D3111yescharacteristic, complemented
163C22×D4—312no—
164C22×D4—621no—
165C22×D4—622no—
166C22×D4C2×D3111yescharacteristic
167#137—624no—
168C2×Q16—312no—
169C2×SD16—24no—
170C2×SD16—312no—
171C2×SD16C2×D3114yescomplemented
172C2×M16—311no—
173C22×C8C2×D3111yescharacteristic, complemented
174C4⋊Q8—621no—
175#124—621no—
176#123—622no—
177C22⋊Q8—314no—
178#117—311no—
179#103—312no—
180#102—314no—
181#102—312no—
182C22⋊C8—314no—
183C31#300911yescharacteristic, complemented, Sylow
184C24—21no—
185C24—622no—
186#41—314no—
187#41—624no—
188#41—628no—
189#41—624no—
190#41—622no—
191#41—622no—
192#41—622no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—311no—
197C2×Q8—311no—
198C2×Q8C22×D3112yes—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—312no—
202C2×D4—311no—
203C2×D4—311no—
204C2×D4—622no—
205C2×D4—624no—
206C2×D4—628no—
207C2×D4—622no—
208C2×D4—6216no—
209C2×D4—621no—
210C2×D4—624no—
211C2×D4C22⋊D3114yes—
212C2×D4C22×D3112yes—
213C22×C4—21no—
214C22×C4—312no—
215C22×C4—622no—
216C22×C4—621no—
217C22×C4C22×D3111yescharacteristic
218Q16—624no—
219SD16—28no—
220SD16—624no—
221M16—622no—
222C2×C8—22no—
223C2×C8—312no—
224C2×C8C22×D3112yes—
225C4⋊C4—312no—
226C4⋊C4—624no—
227C22⋊C4—624no—
228C22⋊C4—621no—
229C42—621no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—312no—
234C23—628no—
235C23—621no—
236C23—2484no—
237C23D4×D3111yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—42no—
241Q8—312no—
242Q8—622no—
243Q8—621no—
244D4—24no—
245D4—22no—
246D4—312no—
247D4—628no—
248D4—628no—
249D4—624no—
250D4—624no—
251D4—622no—
252D4—621no—
253D4C2×C22⋊D3114yes—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—312no—
257C2×C4—628no—
258C2×C4—621no—
259C2×C4—624no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4—622no—
263C2×C4D4×D3111yescharacteristic
264C2×C4D4×D3112yes—
265C2×C4C2×C22⋊D3112yes—
266C2×C4C23×D3111yescharacteristic
267C8—22no—
268C8—622no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—622no—
274C22—622no—
275C22—624no—
276C22—1242no—
277C22—2488no—
278C22#1175851412yes—
279C22C2×D4×D3111yescharacteristic
280C4—42no—
281C4—622no—
282C4—622no—
283C4—1242no—
284C4#1175851412yes—
285C4C2×D4×D3111yescharacteristic
286C4C22×C22⋊D3111yescharacteristic
287C2—22no—
288C2—44no—
289C2—1242no—
290C2—1242no—
291C2C2×#1175851411yescharacteristic
292C2#1452185512yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×SD16×C31C2
socleC22×C31C22×D4
prime radicalC2×#11758561C2
hypercenterC22×SD16D31
hypocenterC31#3009
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. #14521623C2semidirect
4. #14521208C2semidirect
5. #14521198C2semidirect
6. #14521054C2semidirect
7. #14521046C2semidirect
8. #14521015C2semidirect
9. C22×SD16×C31C2semidirect
10. C22×Q8×C31C22semidirect
22. C2×SD16×C31C22semidirect
23. C22×C8×C31C22semidirect
25. #11758523C22semidirect
27. C2×#92482C22semidirect
29. C2×#92481C22semidirect
31. #11758503C22semidirect
32. C2×#92474C22semidirect
33. #11758420C22semidirect
34. #11758409C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758404C22semidirect
143. C22×SD16D31semidirect
162. C22×Q8C2×D31semidirect
171. C2×SD16C2×D31semidirect
173. C22×C8C2×D31semidirect
183. C31#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117585612
C2×#117585192
#145216232
#145212082
#145211982
#145210542
#145210462
#145210152
C22×SD16×C312
#300931