Finite Group Databasebeta

← #16856173

Subgroups — #16856173

Order 3968.

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

normal subgroups only

This group has 572 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×#11758528C211yesmaximal, characteristic, complemented
3C2×#11758509C211yesmaximal, characteristic, complemented
4#14521595C211yesmaximal, characteristic, complemented
5C2×#11758513C211yesmaximal, characteristic, complemented
6#14521440C212yesmaximal, complemented
7#14521437C212yesmaximal, complemented
8#14521432C212yesmaximal, complemented
9#14521430C212yesmaximal, complemented
10M16×Dic31C211yesmaximal, 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
20C2×C22⋊Dic31—22no—
21#11758528—24no—
22#11758524C2211yescharacteristic, complemented
23#11758521C2211yescharacteristic, complemented
24C2×#92482C2212yescomplemented
25#11758517C2211yescharacteristic, complemented
26#11758516—22no—
27#11758513—22no—
28#11758513C2212yes—
29#11758513C2212yes—
30#11758513C2211yescharacteristic, complemented
31C2×#92480C2212yes—
32C2×#92480C2212yescomplemented
33#11758509—24no—
34C2×#92479C2212yes—
35C4×C22⋊D31—22no—
36C2×C4×Dic31C2211yescharacteristic
37C2×#92474C2211yescharacteristic
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×C31C2312yes—
52C2×D4×C31C2311yescharacteristic
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55C2×C22⋊D31—21no—
56C2×C22⋊D31—22no—
57C22×Dic31—21no—
58C22×Dic31D412yes—
59#92495—22no—
60#92495—22no—
61C2×D124C2311yescharacteristic
62C2×C4×D31—21no—
63C2×Dic62—22no—
64C2×Dic62—22no—
65C2×Dic62—21no—
66C2×Dic62C2312yes—
67C2×Dic62C2311yescharacteristic
68SD16×C31—24no—
69D8×C31—24no—
70M16×C31D414yescomplemented
71C2×C8×C31C2312yes—
72C22⋊Dic31—22no—
73C22⋊Dic31—24no—
74C22⋊Dic31—24no—
75#92482—24no—
76#92480—24no—
77#92480—24no—
78#92479—24no—
79(C2×D31)⋊C4—22no—
80Dic31⋊C4—22no—
81C4×Dic31D412yes—
82C4×Dic31C2311yescharacteristic
83C4×Dic31C2311yescharacteristic
84#92474D414yescomplemented
85C2×C31⋊C8C2312yes—
86Dic124—24no—
87#92470—24no—
88C23×C31—22no—
89C23×C31—21no—
90C23×C31—44no—
91C23×C31—42no—
92C23×C31C2×D411yescharacteristic
93C22×D31—21no—
94Q8×C31—21no—
95Q8×C31—21no—
96D4×C31—22no—
97D4×C31—21no—
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—44no—
109C22⋊D31—42no—
110C2×Dic31—21no—
111C2×Dic31—22no—
112C2×Dic31—42no—
113C2×Dic31—42no—
114C2×Dic31C2×D412yes—
115C2×Dic31C2×D412yes—
116D124—21no—
117D124—21no—
118C4×D31—42no—
119Dic62—22no—
120Dic62—21no—
121Dic62—21no—
122Dic62—22no—
123Dic62—42no—
124C8×C31C2×D414yes—
125C31⋊C8C2×D414yes—
126#3153—311nomaximal, Sylow
127C22×C31—22no—
128C22×C31—41no—
129C22×C31—42no—
130C22×C31—42no—
131C22×C31—44no—
132C22×C31—84no—
133C22×C31#12712yes—
134C22×C31C22×D411yescharacteristic
135C2×D31—41no—
136C2×D31—42no—
137C4×C31—41no—
138C4×C31#12712yes—
139C4×C31C22×D411yescharacteristic
140C4×C31C22×D411yescharacteristic
141Dic31—22no—
142Dic31—22no—
143Dic31—42no—
144Dic31—41no—
145C2×#137—311no—
146C2×#137—311no—
147C2×AGL(1,ℤ/8)—311no—
148C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
149#532—311no—
150C2×#124—311no—
151#497—312no—
152#496—312no—
153#495—312no—
154#495—312no—
155C4×M16—311no—
156C2×C31—22no—
157C2×C31—81no—
158C2×C31—82no—
159C2×C31C2×#12711yescharacteristic
160C2×C31#57512yes—
161D31—81no—
162C2×#41—311no—
163C2×#41C2×D3111yescharacteristic
164C22×Q8—311no—
165C22×D4C2×D3111yescharacteristic
166#137—624no—
167#137—624no—
168AGL(1,ℤ/8)—24no—
169AGL(1,ℤ/8)—624no—
170C2×Q16—312no—
171C2×Q16—312no—
172C2×SD16—312no—
173C2×SD16—312no—
174C2×SD16—312no—
175C2×SD16C2×D3112yes—
176C2×D8—312no—
177C2×D8C2×D3112yes—
178C2×M16—311no—
179C2×M16C2×D3111yescharacteristic
180C4⋊Q8—311no—
181#127—311no—
182#124—311no—
183#124—312no—
184#124—312no—
185#124—311no—
186#124—622no—
187#121—622no—
188C4×D4—622no—
189C2×C22⋊C4—622no—
190C2×C42—311no—
191#97—312no—
192C4×C8—312no—
193C31#315311yescharacteristic, complemented, Sylow
194C24—21no—
195#41—22no—
196#41—22no—
197#41—622no—
198#41—622no—
199C2×Q8—312no—
200C2×Q8—311no—
201C2×Q8—621no—
202C2×Q8—622no—
203C2×Q8—622no—
204C2×Q8C22×D3111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—311no—
210C2×D4—622no—
211C2×D4—621no—
212C2×D4C22×D3112yes—
213C2×D4C22×D3111yescharacteristic
214C22×C4—21no—
215C22×C4—312no—
216C22×C4—621no—
217C22×C4—621no—
218C22×C4C22×D3111yescharacteristic
219Q16—624no—
220Q16—624no—
221SD16—24no—
222SD16—624no—
223SD16—624no—
224SD16—624no—
225D8—24no—
226D8—624no—
227M16—314no—
228M16C22⋊D3114yescomplemented
229C2×C8—312no—
230C2×C8C22×D3112yes—
231C4⋊C4—622no—
232C22⋊C4—622no—
233C22⋊C4—624no—
234C22⋊C4—624no—
235C22⋊C4—622no—
236C42—312no—
237C42—311no—
238C42—311no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—621no—
244C23D4×D3111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—621no—
248Q8—622no—
249Q8—622no—
250Q8—621no—
251Q8—1242no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—621no—
259D4—621no—
260D4—1244no—
261D4—1242no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—312no—
265C2×C4—312no—
266C2×C4—622no—
267C2×C4—621no—
268C2×C4—1242no—
269C2×C4—1242no—
270C2×C4—1242no—
271C2×C4D4×D3111yescharacteristic
272C2×C4D4×D3112yes—
273C2×C4C2×C22⋊D3112yes—
274C2×C4C23×D3111yescharacteristic
275C8—314no—
276C8C2×C22⋊D3114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—1241no—
284C22—1242no—
285C22#1175851712yes—
286C22C2×D4×D3111yescharacteristic
287C4—41no—
288C4—622no—
289C4—622no—
290C4—1241no—
291C4—1242no—
292C4#1175851712yes—
293C4C2×D4×D3111yescharacteristic
294C4C22×C22⋊D3111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—2481no—
299C2C2×#1175851711yescharacteristic
300C2#1452184512yes—
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#3153
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×#11758528C2semidirect
3. C2×#11758509C2semidirect
4. #14521595C2semidirect
5. C2×#11758513C2semidirect
6. #14521440C2semidirect
7. #14521437C2semidirect
8. #14521432C2semidirect
9. #14521430C2semidirect
10. M16×Dic31C2semidirect
11. C2×C31×AGL(1,ℤ/8)C2semidirect
22. #11758524C22semidirect
23. #11758521C22semidirect
24. C2×#92482C22semidirect
25. #11758517C22semidirect
30. #11758513C22semidirect
32. C2×#92480C22semidirect
39. C2×Dic124C22semidirect
40. C2×#92470C22semidirect
41. #11758403C22semidirect
42. C8×Dic31C22semidirect
70. M16×C31D4semidirect
84. #92474D4semidirect
148. C2×AGL(1,ℤ/8)D31semidirect
193. C31#3153semidirect
228. 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×#117585282
C2×#117585092
#145215952
C2×#117585132
#145214402
#145214372
#145214322
#145214302
M16×Dic312
C2×C31×AGL(1,ℤ/8)2
#315331