Finite Group Databasebeta

← #16856083

Subgroups — #16856083

Order 3968.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#14521602C211yesmaximal, characteristic, complemented
2C2×#11758517C211yesmaximal, characteristic
3#14521412C214yesmaximal, complemented
4#14521394C214yesmaximal, complemented
5#14521055C211yesmaximal, characteristic, complemented
6#14521054C211yesmaximal, characteristic, complemented
7#14521053C211yesmaximal, characteristic, complemented
8M16×Dic31C211yesmaximal, characteristic, complemented
9C31×#418C211yesmaximal, characteristic, complemented
10C22×D4×C31C2211yescharacteristic
11C22×C22⋊D31—22no—
12C22×D124C2211yescharacteristic
13C2×M16×C31C2211yescharacteristic
14C31×#117C2211yescharacteristic
15#11758517—22no—
16#11758517C2212yes—
17#11758517C2212yes—
18#11758503C2211yescharacteristic
19C2×C4×Dic31C2211yescharacteristic
20C2×#92474C2211yescharacteristic
21#11758467C2212yescomplemented
22#11758465C2212yescomplemented
23#11758455—22no—
24C4×Dic62—22no—
25C31×#102C2214yescomplemented
26#11758420C2214yescomplemented
27#11758409C2214yescomplemented
28#11758403C2212yescomplemented
29C8×Dic31C2212yescomplemented
30#11758397C2214yescomplemented
31C24×C31—21no—
32C23×D31—21no—
33C2×D4×C31—21no—
34C2×D4×C31—22no—
35C2×D4×C31—22no—
36C2×D4×C31C2312yes—
37C22×C4×C31C2311yescharacteristic
38C2×C22⋊D31—24no—
39C2×C22⋊D31—24no—
40C2×C22⋊D31—48no—
41C22×Dic31D412yes—
42C2×D124—22no—
43C2×D124—22no—
44C2×D124—21no—
45C2×D124C2312yes—
46C2×Dic62—22no—
47M16×C31—22no—
48C2×C8×C31C2312yes—
49C4⋊C4×C31C2312yes—
50C22⋊C4×C31—21no—
51C42×C31—21no—
52C22⋊Dic31—21no—
53C4⋊Dic31—22no—
54C4⋊Dic31C2312yes—
55Dic31⋊C4—22no—
56Dic31⋊C4—22no—
57C4×Dic31—21no—
58C4×Dic31D412yes—
59C4×Dic31C2311yescharacteristic
60C4×Dic31C2311yescharacteristic
61#92474—22no—
62C2×C31⋊C8C2312yes—
63C23×C31—22no—
64C23×C31—44no—
65C23×C31—42no—
66C23×C31C2×D411yescharacteristic
67C22×D31—22no—
68C22×D31—44no—
69C22×D31—42no—
70D4×C31—22no—
71D4×C31—22no—
72D4×C31—42no—
73C2×C4×C31—22no—
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31#4112yes—
76C2×C4×C31#4112yes—
77C2×C4×C31C2411yescharacteristic
78C22⋊D31—48no—
79C22⋊D31—48no—
80C2×Dic31—24no—
81C2×Dic31—22no—
82C2×Dic31C2×D412yes—
83C2×Dic31C2×D412yes—
84D124—22no—
85D124—22no—
86D124—42no—
87Dic62—44no—
88C8×C31—22no—
89C31⋊C8—22no—
90#3138—311nomaximal, Sylow
91C22×C31—22no—
92C22×C31—42no—
93C22×C31—44no—
94C22×C31—84no—
95C22×C31#12412yes—
96C22×C31C22×D411yescharacteristic
97C2×D31—42no—
98C2×D31—44no—
99C2×D31—84no—
100C4×C31—42no—
101C4×C31#12412yes—
102C4×C31C2×#4111yescharacteristic
103C4×C31C2×#4111yescharacteristic
104Dic31—22no—
105Dic31—42no—
106Dic31AGL(1,ℤ/8)14yes—
107#533—311no—
108C2×#127—311no—
109#490—314no—
110#486—314no—
111#418—311no—
112#418—311no—
113#418—311no—
114#418D3111yescharacteristic, complemented
115C4×M16—311no—
116C2×C31—22no—
117C2×C31—82no—
118C2×C31C2×#12411yescharacteristic
119C2×C31C2×AGL(1,ℤ/8)12yes—
120D31—82no—
121C22×D4—311no—
122C22×D4—622no—
123C22×D4C2×D3111yescharacteristic
124C2×M16—311no—
125C2×M16C2×D3111yescharacteristic
126C4⋊Q8—312no—
127#127—312no—
128#127—312no—
129#127—622no—
130#125—312no—
131C22⋊Q8—622no—
132C4×Q8—622no—
133#117—311no—
134#117C2×D3111yescharacteristic
135C2×C42—311no—
136#102—314no—
137#102—314no—
138#102—314no—
139#102C2×D3114yescomplemented
140#97—312no—
141C4×C8—312no—
142C31#313811yescharacteristic, complemented, Sylow
143C24—21no—
144C24—621no—
145C2×Q8—622no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—312no—
150C2×D4—624no—
151C2×D4—624no—
152C2×D4—622no—
153C2×D4—621no—
154C2×D4—622no—
155C2×D4—1248no—
156C2×D4C22×D3112yes—
157C22×C4—312no—
158C22×C4C22×D3111yescharacteristic
159M16—22no—
160M16—622no—
161C2×C8—312no—
162C2×C8C22×D3112yes—
163C4⋊C4—312no—
164C4⋊C4—622no—
165C4⋊C4—622no—
166C4⋊C4—622no—
167C4⋊C4C22×D3112yes—
168C22⋊C4—21no—
169C22⋊C4—621no—
170C42—21no—
171C42—312no—
172C42—311no—
173C42—311no—
174C42—621no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—622no—
179C23—1244no—
180C23—1242no—
181C23D4×D3111yescharacteristic
182Q8—1244no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—622no—
187D4—622no—
188D4—1248no—
189D4—1248no—
190D4—1242no—
191C2×C4—22no—
192C2×C4—312no—
193C2×C4—312no—
194C2×C4—624no—
195C2×C4—622no—
196C2×C4#9249512yes—
197C2×C4D4×D3111yescharacteristic
198C2×C4#9249712yes—
199C2×C4C23×D3111yescharacteristic
200C8—22no—
201C8—622no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—1242no—
207C22—1244no—
208C22—2484no—
209C22#1175846212yes—
210C22C2×D4×D3111yescharacteristic
211C4—42no—
212C4—314no—
213C4—622no—
214C4—1242no—
215C4#1175846212yes—
216C4C2×#9249511yescharacteristic
217C4C2×#9249711yescharacteristic
218C2—22no—
219C2—82no—
220C2—2482no—
221C2C2×#1175846211yescharacteristic
222C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#418C2
socleC22×C31C22×D4
prime radicalC2×#11758517C2
hypercenter#418D31
hypocenterC31#3138
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #14521602C2semidirect
3. #14521412C2semidirect
4. #14521394C2semidirect
5. #14521055C2semidirect
6. #14521054C2semidirect
7. #14521053C2semidirect
8. M16×Dic31C2semidirect
9. C31×#418C2semidirect
21. #11758467C22semidirect
22. #11758465C22semidirect
25. C31×#102C22semidirect
26. #11758420C22semidirect
27. #11758409C22semidirect
28. #11758403C22semidirect
29. C8×Dic31C22semidirect
30. #11758397C22semidirect
114. #418D31semidirect
139. #102C2×D31semidirect
142. C31#3138semidirect

Maximal subgroups

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

subgroupindex
#145216022
C2×#117585172
#145214122
#145213942
#145210552
#145210542
#145210532
M16×Dic312
C31×#4182
#313831