Finite Group Databasebeta

← #16856185

Subgroups — #16856185

Order 3968.

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

normal subgroups only

This group has 564 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×#11758486C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3C22×#92480C211yesmaximal, characteristic, complemented
4#14521620C211yesmaximal, characteristic, complemented
5#14521616C211yesmaximal, characteristic, complemented
6#14521440C212yesmaximal, complemented
7#14521437C212yesmaximal, complemented
8#14521432C212yesmaximal, complemented
9#14521420C212yesmaximal, complemented
10#14521007C211yesmaximal, characteristic, complemented
11C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×D4×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C22×Dic62C2211yescharacteristic
16C31×AGL(1,ℤ/8)—24no—
17C2×SD16×C31C2212yes—
18C2×D8×C31C2212yes—
19C2×M16×C31C2211yescharacteristic
20#11758530—22no—
21#11758527—24no—
22#11758521C2212yescomplemented
23C2×#92482C2211yescharacteristic, complemented
24C2×#92481C2211yescharacteristic, complemented
25#11758517C2212yes—
26#11758515—22no—
27#11758513C2212yes—
28#11758513C2212yescomplemented
29#11758512—22no—
30C2×#92480—24no—
31C2×#92480C2212yes—
32C2×#92480C2212yes—
33C2×#92480C2211yescharacteristic, complemented
34C2×#92479C2211yescharacteristic, complemented
35#11758503C2211yescharacteristic
36#11758501—22no—
37C22×C31⋊C8C2211yescharacteristic
38#11758486—24no—
39C2×Dic124C2212yescomplemented
40C2×#92470C2212yescomplemented
41#11758403C2212yescomplemented
42C8×Dic31C2212yescomplemented
43C24×C31—21no—
44#41×C31—22no—
45#41×C31—22no—
46C2×Q8×C31C2311yescharacteristic
47C2×D4×C31—21no—
48C2×D4×C31—22no—
49C2×D4×C31—22no—
50C2×D4×C31—21no—
51C2×D4×C31C2311yescharacteristic
52C2×D4×C31C2312yes—
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55C2×C22⋊D31—21no—
56C2×C22⋊D31—24no—
57C22×Dic31—21no—
58#92495—22no—
59#92495—22no—
60C2×D124C2311yescharacteristic
61C2×C4×D31—21no—
62C2×Dic62—22no—
63C2×Dic62—22no—
64C2×Dic62—21no—
65C2×Dic62C2312yes—
66C2×Dic62C2311yescharacteristic
67SD16×C31—24no—
68D8×C31—24no—
69M16×C31D414yescomplemented
70C2×C8×C31C2312yes—
71C22⋊Dic31—24no—
72C22⋊Dic31—24no—
73C22⋊Dic31D412yes—
74#92482—22no—
75#92481—22no—
76#92480—24no—
77#92480—24no—
78#92480—22no—
79#92479—22no—
80C4⋊Dic31D412yes—
81Dic31⋊C4—24no—
82C4×Dic31C2312yes—
83C2×C31⋊C8D414yescomplemented
84C2×C31⋊C8C2311yescharacteristic
85C2×C31⋊C8C2311yescharacteristic
86Dic124—24no—
87#92470—24no—
88C23×C31—21no—
89C23×C31—22no—
90C23×C31—42no—
91C23×C31—82no—
92C23×C31C2×D411yescharacteristic
93C22×D31—21no—
94Q8×C31—21no—
95Q8×C31—21no—
96D4×C31—21no—
97D4×C31—22no—
98D4×C31—21no—
99D4×C31—22no—
100D4×C31—42no—
101D4×C31—42no—
102C2×C4×C31—21no—
103C2×C4×C31—42no—
104C2×C4×C31C2×D411yescharacteristic
105C2×C4×C31C2×D412yes—
106C2×C4×C31C2×D412yes—
107C2×C4×C31C2411yescharacteristic
108C22⋊D31—42no—
109C22⋊D31—84no—
110C2×Dic31—22no—
111C2×Dic31—21no—
112C2×Dic31—42no—
113C2×Dic31C2×D414yes—
114D124—21no—
115D124—21no—
116C4×D31—42no—
117Dic62—22no—
118Dic62—21no—
119Dic62—22no—
120Dic62—21no—
121Dic62—42no—
122C8×C31C2×D414yes—
123C31⋊C8C2×D412yes—
124C31⋊C8C2×D412yes—
125#3154—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31—41no—
128C22×C31—42no—
129C22×C31—44no—
130C22×C31—42no—
131C22×C31—84no—
132C22×C31#12712yes—
133C22×C31C22×D411yescharacteristic
134C2×D31—41no—
135C2×D31—42no—
136C4×C31—41no—
137C4×C31#12712yes—
138C4×C31C22×D411yescharacteristic
139C4×C31C22×D411yescharacteristic
140Dic31—24no—
141Dic31—42no—
142Dic31—41no—
143C2×#137—311no—
144C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
145C2×#135—311no—
146C22×SD16—311no—
147#536—311no—
148#535—311no—
149#497—312no—
150#496—312no—
151#495—312no—
152#492—312no—
153#405—311no—
154C2×C31—22no—
155C2×C31—81no—
156C2×C31—82no—
157C2×C31C2×#12711yescharacteristic
158C2×C31#57712yes—
159D31—81no—
160C2×#41—311no—
161C2×#41C2×D3111yescharacteristic
162C22×Q8—311no—
163C22×D4C2×D3111yescharacteristic
164#137—624no—
165AGL(1,ℤ/8)—24no—
166#135—624no—
167C2×Q16—311no—
168C2×Q16—312no—
169C2×SD16—311no—
170C2×SD16—312no—
171C2×SD16—312no—
172C2×SD16—311no—
173C2×SD16—312no—
174C2×SD16—624no—
175C2×SD16C2×D3112yes—
176C2×D8—311no—
177C2×D8C2×D3112yes—
178C2×M16C2×D3111yescharacteristic
179C22×C8—311no—
180C4⋊Q8—312no—
181#127—312no—
182#124—312no—
183#124—312no—
184#123—622no—
185C22⋊Q8—622no—
186#121—622no—
187#120—622no—
188#117—311no—
189#97—312no—
190C4×C8—312no—
191C31#315411yescharacteristic, complemented, Sylow
192C24—21no—
193#41—22no—
194#41—22no—
195#41—622no—
196#41—622no—
197C2×Q8—311no—
198C2×Q8—312no—
199C2×Q8—621no—
200C2×Q8—622no—
201C2×Q8—622no—
202C2×Q8C22×D3111yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—311no—
208C2×D4—624no—
209C2×D4—621no—
210C2×D4C22×D3111yescharacteristic
211C2×D4C22×D3112yes—
212C22×C4—21no—
213C22×C4—621no—
214C22×C4—621no—
215C22×C4C22×D3111yescharacteristic
216Q16—622no—
217Q16—624no—
218SD16—24no—
219SD16—624no—
220SD16—624no—
221SD16—622no—
222SD16—622no—
223SD16—624no—
224D8—24no—
225D8—622no—
226M16C22⋊D3114yescomplemented
227C2×C8—314no—
228C2×C8—311no—
229C2×C8—311no—
230C2×C8C22×D3112yes—
231C4⋊C4—312no—
232C4⋊C4—624no—
233C22⋊C4—312no—
234C22⋊C4—624no—
235C22⋊C4—624no—
236C42—312no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—621no—
242C23D4×D3111yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—622no—
246Q8—621no—
247Q8—622no—
248Q8—621no—
249Q8—1242no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—621no—
257D4—621no—
258D4—1242no—
259D4—2484no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—314no—
263C2×C4—621no—
264C2×C4—622no—
265C2×C4—1242no—
266C2×C4—1242no—
267C2×C4D4×D3111yescharacteristic
268C2×C4D4×D3112yes—
269C2×C4C2×C22⋊D3112yes—
270C2×C4C23×D3111yescharacteristic
271C8—312no—
272C8—312no—
273C8C2×C22⋊D3114yes—
274C22—22no—
275C22—44no—
276C22—42no—
277C22—41no—
278C22—42no—
279C22—84no—
280C22—1242no—
281C22—1241no—
282C22#1175851712yes—
283C22C2×D4×D3111yescharacteristic
284C4—41no—
285C4—624no—
286C4—1241no—
287C4—1242no—
288C4#1175851712yes—
289C4C2×D4×D3111yescharacteristic
290C4C22×C22⋊D3111yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—2481no—
295C2C2×#1175851711yescharacteristic
296C2#1452185912yes—
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#3154
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. C2×#11758486C2semidirect
2. C2×#11758527C2semidirect
3. C22×#92480C2semidirect
4. #14521620C2semidirect
5. #14521616C2semidirect
6. #14521440C2semidirect
7. #14521437C2semidirect
8. #14521432C2semidirect
9. #14521420C2semidirect
10. #14521007C2semidirect
11. C2×C31×AGL(1,ℤ/8)C2semidirect
22. #11758521C22semidirect
23. C2×#92482C22semidirect
24. C2×#92481C22semidirect
28. #11758513C22semidirect
33. C2×#92480C22semidirect
34. C2×#92479C22semidirect
39. C2×Dic124C22semidirect
40. C2×#92470C22semidirect
41. #11758403C22semidirect
42. C8×Dic31C22semidirect
69. M16×C31D4semidirect
83. C2×C31⋊C8D4semidirect
144. C2×AGL(1,ℤ/8)D31semidirect
191. C31#3154semidirect
226. M16C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584862
C2×#117585272
C22×#924802
#145216202
#145216162
#145214402
#145214372
#145214322
#145214202
#145210072
C2×C31×AGL(1,ℤ/8)2
#315431