Finite Group Databasebeta

← #16854936

Subgroups — #16854936

Order 3968.

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

normal subgroups only

This group has 994 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×D4×D31C211yesmaximal, characteristic, complemented
2C2×#11758526C211yesmaximal, characteristic, complemented
3#14521556C211yesmaximal, characteristic, complemented
4#14521198C214yesmaximal, complemented
5#14521181C214yesmaximal, complemented
6#14521055C211yesmaximal, characteristic, complemented
7#14521054C211yesmaximal, characteristic, complemented
8#14521012C211yesmaximal, characteristic, complemented
9C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
10C24×D31—21no—
11C2×#41×C31C2211yescharacteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C22×C22⋊D31—21no—
14C2×D4×D31—24no—
15C2×D4×D31—24no—
16C2×D4×D31—22no—
17C2×D4×D31C2214yes—
18C22×D124C2211yescharacteristic
19C22×C4×D31C2211yescharacteristic
20C31×AGL(1,ℤ/8)—24no—
21C2×SD16×C31C2212yescomplemented
22C2×D8×C31C2212yescomplemented
23C2×M16×C31C2211yescharacteristic, complemented
24#11758526—24no—
25#11758523C2212yescomplemented
26C2×#92481C2212yescomplemented
27#11758515C2212yescomplemented
28#11758514—21no—
29#11758512—21no—
30C2×#92479C2212yescomplemented
31C4×C22⋊D31—22no—
32#11758503C2211yescharacteristic, complemented
33C2×#92474C2211yescharacteristic, complemented
34#11758420C2214yescomplemented
35#11758409C2214yescomplemented
36(C2×D31)⋊C8C2214yescomplemented
37C24×C31—21no—
38C23×D31—28no—
39C23×D31—21no—
40C23×D31—24no—
41C23×D31—48no—
42C23×D31D411yescharacteristic
43#41×C31—22no—
44#41×C31—22no—
45C2×Q8×C31C2311yescharacteristic, complemented
46C2×D4×C31—21no—
47C2×D4×C31—21no—
48C2×D4×C31—22no—
49C2×D4×C31D414yescomplemented
50C2×D4×C31C2312yes—
51C2×D4×C31C2311yescharacteristic, complemented
52C22×C4×C31—21no—
53C22×C4×C31C2311yescharacteristic
54C2×C22⋊D31—28no—
55C2×C22⋊D31—22no—
56C2×C22⋊D31—24no—
57C22×Dic31D411yescharacteristic
58D4×D31—216no—
59D4×D31—28no—
60D4×D31—28no—
61C2×D124—21no—
62C2×D124—22no—
63C2×D124D414yescomplemented
64C2×D124C2312yes—
65C2×C4×D31—22no—
66C2×C4×D31—22no—
67C2×C4×D31D412yes—
68C2×C4×D31C2312yes—
69SD16×C31—24no—
70D8×C31—24no—
71M16×C31—22no—
72C2×C8×C31C2312yescomplemented
73C22⋊Dic31—21no—
74C22⋊Dic31—22no—
75#92481—24no—
76#92479—24no—
77(C2×D31)⋊C4—22no—
78C4⋊Dic31C2312yescomplemented
79Dic31⋊C4—22no—
80C4×Dic31—21no—
81#92474—22no—
82C2×C31⋊C8C2312yescomplemented
83C23×C31—24no—
84C23×C31—22no—
85C23×C31—21no—
86C23×C31—44no—
87C23×C31C2×D411yescharacteristic
88C22×D31—22no—
89C22×D31—24no—
90C22×D31—216no—
91C22×D31—24no—
92C22×D31—22no—
93C22×D31—432no—
94C22×D31—48no—
95C22×D31—44no—
96C22×D31—48no—
97C22×D31C2×D412yes—
98Q8×C31—21no—
99Q8×C31—21no—
100D4×C31—24no—
101D4×C31—22no—
102D4×C31—21no—
103D4×C31—21no—
104D4×C31—42no—
105D4×C31C2×D414yes—
106C2×C4×C31—21no—
107C2×C4×C31—42no—
108C2×C4×C31C2×D411yescharacteristic
109C2×C4×C31C2×D412yes—
110C2×C4×C31C2×D412yes—
111C2×C4×C31C2411yescharacteristic
112C22⋊D31—216no—
113C22⋊D31—44no—
114C2×Dic31—22no—
115C2×Dic31—22no—
116C2×Dic31C2×D412yes—
117D124—24no—
118D124—22no—
119D124C2×D414yes—
120C4×D31—24no—
121C4×D31—22no—
122C4×D31—22no—
123C8×C31—22no—
124C31⋊C8—22no—
125#3003—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31—24no—
128C22×C31—48no—
129C22×C31—44no—
130C22×C31—41no—
131C22×C31—42no—
132C22×C31#12012yes—
133C22×C31C22×D411yescharacteristic
134C2×D31—28no—
135C2×D31—22no—
136C2×D31—22no—
137C2×D31—24no—
138C2×D31—432no—
139C2×D31—48no—
140C2×D31—44no—
141C2×D31AGL(1,ℤ/8)14yescomplemented
142C4×C31—41no—
143C4×C31#12012yes—
144C4×C31C22×D411yescharacteristic
145C4×C31C22×D411yescharacteristic
146Dic31—22no—
147Dic31—42no—
148C23×D4—311no—
149C2×AGL(1,ℤ/8)—311no—
150C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
151#525—311no—
152#450—314no—
153#447—314no—
154#418—311no—
155#418—311no—
156#407—311no—
157C2×C31—22no—
158C2×C31—44no—
159C2×C31—81no—
160C2×C31C2×#12011yescharacteristic
161C2×C31C2×AGL(1,ℤ/8)12yes—
162D31—24no—
163D31—44no—
164C25—621no—
165C2×#41C2×D3111yescharacteristic, complemented
166C22×D4—314no—
167C22×D4—311no—
168C22×D4—622no—
169C22×D4—621no—
170C22×D4—624no—
171C22×D4—624no—
172C22×D4C2×D3111yescharacteristic
173C23×C4—311no—
174AGL(1,ℤ/8)—24no—
175AGL(1,ℤ/8)—624no—
176C2×SD16—312no—
177C2×SD16C2×D3112yescomplemented
178C2×D8—312no—
179C2×D8C2×D3112yescomplemented
180C2×M16—311no—
181C2×M16C2×D3111yescharacteristic, complemented
182#123—621no—
183C22⋊Q8—312no—
184#121—312no—
185#120—621no—
186C4×D4—622no—
187#117—311no—
188#102—314no—
189#102—314no—
190C22⋊C8—314no—
191C31#300311yescharacteristic, complemented, Sylow
192C24—21no—
193C24—311no—
194C24—624no—
195C24—628no—
196C24—621no—
197C24—1248no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D3111yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—314no—
205C2×D4—312no—
206C2×D4—624no—
207C2×D4—628no—
208C2×D4—6216no—
209C2×D4—628no—
210C2×D4—628no—
211C2×D4—622no—
212C2×D4—621no—
213C2×D4—622no—
214C2×D4C22⋊D3114yescomplemented
215C2×D4C22×D3112yes—
216C2×D4C22×D3111yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—311no—
219C22×C4—312no—
220C22×C4—312no—
221C22×C4—622no—
222C22×C4—622no—
223C22×C4C22×D3111yescharacteristic
224SD16—24no—
225SD16—624no—
226D8—24no—
227D8—624no—
228M16—22no—
229M16—622no—
230C2×C8—312no—
231C2×C8C22×D3112yescomplemented
232C4⋊C4—312no—
233C4⋊C4—622no—
234C22⋊C4—622no—
235C22⋊C4—622no—
236C22⋊C4—621no—
237C42—621no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—312no—
243C23—624no—
244C23—6216no—
245C23—622no—
246C23—624no—
247C23—622no—
248C23—1244no—
249C23—1248no—
250C23—12432no—
251C23—1248no—
252C23D4×D3111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—314no—
261D4—6216no—
262D4—624no—
263D4—622no—
264D4—1244no—
265D4C2×C22⋊D3114yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—312no—
269C2×C4—622no—
270C2×C4—624no—
271C2×C4—622no—
272C2×C4—622no—
273C2×C4—622no—
274C2×C4D4×D3111yescharacteristic
275C2×C4D4×D3112yes—
276C2×C4C2×C22⋊D3112yes—
277C2×C4C23×D3111yescharacteristic
278C8—22no—
279C8—622no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—314no—
287C22—628no—
288C22—622no—
289C22—622no—
290C22—624no—
291C22—1248no—
292C22—12432no—
293C22—1244no—
294C22#1175851412yes—
295C22C2×D4×D3111yescharacteristic
296C4—41no—
297C4—622no—
298C4—1242no—
299C4#1175851412yes—
300C4C2×D4×D3111yescharacteristic
301C4C22×C22⋊D3111yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—624no—
306C2—1244no—
307C2C2×#1175851411yescharacteristic
308C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3003
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D31C2semidirect
2. C2×#11758526C2semidirect
3. #14521556C2semidirect
4. #14521198C2semidirect
5. #14521181C2semidirect
6. #14521055C2semidirect
7. #14521054C2semidirect
8. #14521012C2semidirect
9. C2×C31×AGL(1,ℤ/8)C2semidirect
11. C2×#41×C31C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×D8×C31C22semidirect
23. C2×M16×C31C22semidirect
25. #11758523C22semidirect
26. C2×#92481C22semidirect
27. #11758515C22semidirect
30. C2×#92479C22semidirect
32. #11758503C22semidirect
33. C2×#92474C22semidirect
34. #11758420C22semidirect
35. #11758409C22semidirect
36. (C2×D31)⋊C8C22semidirect
45. C2×Q8×C31C23semidirect
49. C2×D4×C31D4semidirect
51. C2×D4×C31C23semidirect
63. C2×D124D4semidirect
72. C2×C8×C31C23semidirect
78. C4⋊Dic31C23semidirect
82. C2×C31⋊C8C23semidirect
141. C2×D31AGL(1,ℤ/8)semidirect
150. C2×AGL(1,ℤ/8)D31semidirect
165. C2×#41C2×D31semidirect
177. C2×SD16C2×D31semidirect
179. C2×D8C2×D31semidirect
181. C2×M16C2×D31semidirect
191. C31#3003semidirect
200. C2×Q8C22×D31semidirect
214. C2×D4C22⋊D31semidirect
216. C2×D4C22×D31semidirect
231. C2×C8C22×D31semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117585262
#145215562
#145211982
#145211812
#145210552
#145210542
#145210122
C2×C31×AGL(1,ℤ/8)2
#300331