Finite Group Databasebeta

← #16856181

Subgroups — #16856181

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×#11758519C211yesmaximal, characteristic, complemented
2C2×#11758509C211yesmaximal, characteristic, complemented
3C2×#11758481C211yesmaximal, characteristic, complemented
4#14521623C211yesmaximal, characteristic, complemented
5#14521616C211yesmaximal, characteristic, complemented
6#14521440C212yesmaximal, complemented
7#14521435C212yesmaximal, complemented
8#14521430C212yesmaximal, complemented
9#14521420C212yesmaximal, complemented
10#14521004C211yesmaximal, characteristic, complemented
11C22×SD16×C31C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C22×D4×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C2×#92495C2211yescharacteristic
16C2×SD16×C31—24no—
17C2×SD16×C31C2212yes—
18C2×SD16×C31C2212yes—
19C22×C8×C31C2211yescharacteristic
20#11758530—22no—
21#11758524C2212yescomplemented
22#11758523—22no—
23#11758521C2212yescomplemented
24C2×#92482C2212yescomplemented
25#11758519—24no—
26C2×#92481C2212yescomplemented
27#11758517C2212yes—
28#11758515—22no—
29#11758513C2212yes—
30C2×#92480C2212yes—
31#11758509—24no—
32C2×#92479C2212yes—
33#11758506—22no—
34#11758503C2211yescharacteristic
35C2×#92474C2211yescharacteristic
36C2×Dic124C2211yescharacteristic, complemented
37#11758481—24no—
38C2×D248C2211yescharacteristic, complemented
39C2×#92470C2211yescharacteristic, complemented
40C2×#92470C2211yescharacteristic, complemented
41#11758403C2212yescomplemented
42C8×Dic31C2212yescomplemented
43C24×C31—21no—
44C2×Q8×C31—21no—
45C2×Q8×C31—22no—
46C2×Q8×C31—22no—
47C2×Q8×C31C2312yes—
48C2×D4×C31—22no—
49C2×D4×C31—22no—
50C2×D4×C31—21no—
51C2×D4×C31C2312yes—
52C22×C4×C31—21no—
53C22×C4×C31C2311yescharacteristic
54C2×C22⋊D31—21no—
55C2×C22⋊D31—21no—
56C2×C22⋊D31—24no—
57#92495—22no—
58#92495—22no—
59#92495—22no—
60#92495—22no—
61C2×D124C2311yescharacteristic
62C2×D124C2311yescharacteristic
63C2×C4×D31—21no—
64C2×C4×D31—21no—
65C2×Dic62C2311yescharacteristic
66C2×Dic62C2311yescharacteristic
67SD16×C31—24no—
68SD16×C31—24no—
69C2×C8×C31D414yescomplemented
70C2×C8×C31C2311yescharacteristic
71C2×C8×C31C2311yescharacteristic
72C22⋊Dic31—24no—
73C22⋊Dic31D412yes—
74#92482—24no—
75#92481—24no—
76#92480—24no—
77#92479—24no—
78(C2×D31)⋊C4—24no—
79C4⋊Dic31D412yes—
80Dic31⋊C4—24no—
81C4×Dic31C2312yes—
82#92474D414yescomplemented
83C2×C31⋊C8C2312yes—
84Dic124—22no—
85D248—22no—
86#92470—22no—
87#92470—22no—
88C23×C31—22no—
89C23×C31—42no—
90C23×C31—82no—
91C23×C31C2×D411yescharacteristic
92C22×D31—21no—
93C22×D31—21no—
94Q8×C31—22no—
95Q8×C31—22no—
96Q8×C31—42no—
97D4×C31—22no—
98D4×C31—22no—
99D4×C31—42no—
100C2×C4×C31—22no—
101C2×C4×C31—42no—
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2×D412yes—
104C2×C4×C31C2×D412yes—
105C2×C4×C31C2411yescharacteristic
106C22⋊D31—42no—
107C22⋊D31—42no—
108C22⋊D31—84no—
109C2×Dic31—21no—
110C2×Dic31—21no—
111C2×Dic31C2×D414yes—
112D124—21no—
113D124—21no—
114D124—21no—
115D124—21no—
116C4×D31—42no—
117C4×D31—42no—
118Dic62—21no—
119Dic62—21no—
120Dic62—21no—
121Dic62—21no—
122C8×C31C2×D412yes—
123C8×C31C2×D412yes—
124C31⋊C8C2×D414yes—
125#3154—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31—42no—
128C22×C31—44no—
129C22×C31—84no—
130C22×C31#12712yes—
131C22×C31C22×D411yescharacteristic
132C2×D31—41no—
133C2×D31—41no—
134C2×D31—42no—
135C2×D31—42no—
136C4×C31—42no—
137C4×C31#12712yes—
138C4×C31C22×D411yescharacteristic
139C4×C31C22×D411yescharacteristic
140Dic31—24no—
141Dic31—41no—
142Dic31—41no—
143C2×#137—311no—
144C2×AGL(1,ℤ/8)—311no—
145C2×#135—311no—
146C22×SD16D3111yescharacteristic, complemented
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—82no—
156C2×C31C2×#12711yescharacteristic
157C2×C31#57712yes—
158D31—81no—
159D31—81no—
160C2×#41—311no—
161C2×#41—311no—
162C22×Q8C2×D3111yescharacteristic
163C22×D4C2×D3111yescharacteristic
164#137—624no—
165AGL(1,ℤ/8)—624no—
166#135—624no—
167C2×Q16—312no—
168C2×Q16—311no—
169C2×SD16—24no—
170C2×SD16—312no—
171C2×SD16—312no—
172C2×SD16—311no—
173C2×SD16—311no—
174C2×SD16C2×D3112yes—
175C2×SD16C2×D3112yes—
176C2×D8—312no—
177C2×D8—311no—
178C2×M16—311no—
179C22×C8C2×D3111yescharacteristic
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—622no—
194#41—622no—
195#41—622no—
196#41—622no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—311no—
201C2×Q8—311no—
202C2×Q8C22×D3112yes—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—311no—
207C2×D4—311no—
208C2×D4—624no—
209C2×D4—621no—
210C2×D4—621no—
211C2×D4C22×D3112yes—
212C22×C4—21no—
213C22×C4—621no—
214C22×C4—621no—
215C22×C4C22×D3111yescharacteristic
216Q16—624no—
217Q16—622no—
218SD16—24no—
219SD16—24no—
220SD16—624no—
221SD16—624no—
222SD16—622no—
223SD16—622no—
224D8—624no—
225D8—622no—
226M16—314no—
227C2×C8—312no—
228C2×C8C22⋊D3114yescomplemented
229C2×C8C22×D3111yescharacteristic
230C2×C8C22×D3111yescharacteristic
231C4⋊C4—312no—
232C4⋊C4—624no—
233C22⋊C4—312no—
234C22⋊C4—624no—
235C22⋊C4—624no—
236C42—312no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—621no—
241C23—621no—
242C23D4×D3111yescharacteristic
243Q8—22no—
244Q8—22no—
245Q8—42no—
246Q8—621no—
247Q8—621no—
248Q8—621no—
249Q8—621no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—621no—
254D4—621no—
255D4—621no—
256D4—621no—
257D4—1242no—
258D4—1242no—
259D4—2484no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—314no—
263C2×C4—621no—
264C2×C4—621no—
265C2×C4—1242no—
266C2×C4—1242no—
267C2×C4D4×D3111yescharacteristic
268C2×C4D4×D3112yes—
269C2×C4C2×C22⋊D3112yes—
270C2×C4C23×D3111yescharacteristic
271C8—314no—
272C8C2×C22⋊D3112yes—
273C8C2×C22⋊D3112yes—
274C22—22no—
275C22—42no—
276C22—44no—
277C22—84no—
278C22—1241no—
279C22—1241no—
280C22—1242no—
281C22—1242no—
282C22#1175851712yes—
283C22C2×D4×D3111yescharacteristic
284C4—42no—
285C4—624no—
286C4—1241no—
287C4—1241no—
288C4#1175851712yes—
289C4C2×D4×D3111yescharacteristic
290C4C22×C22⋊D3111yescharacteristic
291C2—22no—
292C2—82no—
293C2—2481no—
294C2—2481no—
295C2C2×#1175851711yescharacteristic
296C2#1452185512yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×SD16×C31C2
socleC22×C31C22×D4
hypercenterC22×SD16D31
hypocenterC31#3154
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
1. C2×#11758519C2semidirect
2. C2×#11758509C2semidirect
3. C2×#11758481C2semidirect
4. #14521623C2semidirect
5. #14521616C2semidirect
6. #14521440C2semidirect
7. #14521435C2semidirect
8. #14521430C2semidirect
9. #14521420C2semidirect
10. #14521004C2semidirect
11. C22×SD16×C31C2semidirect
21. #11758524C22semidirect
23. #11758521C22semidirect
24. C2×#92482C22semidirect
26. C2×#92481C22semidirect
36. C2×Dic124C22semidirect
38. C2×D248C22semidirect
39. C2×#92470C22semidirect
40. C2×#92470C22semidirect
41. #11758403C22semidirect
42. C8×Dic31C22semidirect
69. C2×C8×C31D4semidirect
82. #92474D4semidirect
146. C22×SD16D31semidirect
191. C31#3154semidirect
228. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
C2×#117585092
C2×#117584812
#145216232
#145216162
#145214402
#145214352
#145214302
#145214202
#145210042
C22×SD16×C312
#315431