Finite Group Databasebeta

← #16846110

Subgroups — #16846110

Order 3968.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#14521524C211yesmaximal, characteristic, complemented
2#14521054C212yesmaximal, complemented
3C2×#11758421C211yesmaximal, characteristic, complemented
4#14520809C212yesmaximal, complemented
5C2×C31×#127C211yesmaximal, characteristic
6C22×D4×C31—22no—
7C22×D4×C31C2211yescharacteristic
8C22×D4×C31C2211yescharacteristic
9C31×#127—22no—
10C31×#127C414yescomplemented
11C2×C42×C31C2211yescharacteristic
12C2×C22⋊Dic31—22no—
13D4×Dic31—24no—
14#11758503C2212yes—
15C2×#92474C2212yes—
16#11758421—22no—
17#11758420—24no—
18C24×C31—21no—
19C24×C31D412yes—
20C2×D4×C31—28no—
21C2×D4×C31—22no—
22C2×D4×C31—22no—
23C2×D4×C31—21no—
24C2×D4×C31—22no—
25C2×D4×C31—22no—
26C2×D4×C31—48no—
27C2×D4×C31C2×C412yes—
28C2×D4×C31C2×C412yescomplemented
29C2×D4×C31D412yes—
30C22×C4×C31—21no—
31C22×C4×C31C2311yescharacteristic
32C22×Dic31—24no—
33C42×C31—21no—
34C42×C31C2×C412yescomplemented
35C22⋊Dic31—24no—
36C22⋊Dic31—22no—
37C4⋊Dic31D414yescomplemented
38C4×Dic31—22no—
39#92474—22no—
40#92474—22no—
41C2×C31⋊C8—22no—
42C23×C31—22no—
43C23×C31—22no—
44C23×C31—22no—
45C23×C31—44no—
46C23×C31—44no—
47C23×C31—41no—
48C23×C31—41no—
49C23×C31C22⋊C414yes—
50C23×C31C2×D411yescharacteristic
51D4×C31—24no—
52D4×C31—24no—
53D4×C31—24no—
54D4×C31—416no—
55D4×C31—42no—
56C2×C4×C31—24no—
57C2×C4×C31—22no—
58C2×C4×C31C22×C411yescharacteristic
59C2×C4×C31C2×D412yes—
60C2×C4×C31C2×D411yescharacteristic
61C2×C4×C31#4112yes—
62C2×Dic31—24no—
63C2×Dic31—84no—
64C31⋊C8—42no—
65#1887—311nomaximal, Sylow
66C22×C31—22no—
67C22×C31—24no—
68C22×C31—44no—
69C22×C31—44no—
70C22×C31—44no—
71C22×C31—42no—
72C22×C31—82no—
73C22×C31—82no—
74C22×C31—82no—
75C22×C31C2×C22⋊C411yescharacteristic
76C22×C31#12011yescharacteristic
77C22×C31#12311yescharacteristic
78C4×C31—24no—
79C4×C31C4×D412yes—
80C4×C31#12112yes—
81Dic31—44no—
82C2×#127D3111yescharacteristic
83#518—311no—
84#418—312no—
85C2×#100—311no—
86#339—312no—
87C2×C31—21no—
88C2×C31—21no—
89C2×C31—44no—
90C2×C31—82no—
91C2×C31#38611yescharacteristic
92C2×C31#45312yes—
93C22×D4—22no—
94C22×D4C2×D3111yescharacteristic
95C22×D4C2×D3111yescharacteristic
96C2×M16—312no—
97#127—22no—
98#127Dic3114yescomplemented
99C4×D4—624no—
100#117—312no—
101C2×C22⋊C4—622no—
102C2×C42C2×D3111yescharacteristic
103#102—624no—
104#100—622no—
105C31#188711yescharacteristic, complemented, Sylow
106C24—21no—
107C24C22⋊D3112yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—48no—
115C2×D4C2×Dic3112yes—
116C2×D4C2×Dic3112yescomplemented
117C2×D4C22⋊D3112yes—
118C22×C4—21no—
119C22×C4—624no—
120C22×C4C22×D3111yescharacteristic
121M16—622no—
122M16—622no—
123C2×C8—622no—
124C4⋊C4—314no—
125C22⋊C4—624no—
126C22⋊C4—622no—
127C42—21no—
128C42—622no—
129C42C2×Dic3112yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic3114yes—
138C23C2×C22⋊D3111yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—624no—
147C2×C4—2484no—
148C2×C4D4×D3112yes—
149C2×C4#9249712yes—
150C2×C4C22×Dic3111yescharacteristic
151C2×C4C2×C22⋊D3111yescharacteristic
152C8—1242no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22#1175851211yescharacteristic
163C22#1175851411yescharacteristic
164C22C2×C22⋊Dic3111yescharacteristic
165C4—24no—
166C4—1244no—
167C4D4×Dic3112yes—
168C4#1175851612yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#1452092911yescharacteristic
174C2#1452121612yes—
subgroupG/H
centerC22#11758514
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
Fitting, prime radicalC2×C31×#127C2
socleC22×C31#120
cosocleC22×C4×C31C23
hypercenterC2×#127D31
hypocenterC31#1887
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #14521524C2semidirect
2. #14521054C2semidirect
3. C2×#11758421C2semidirect
4. #14520809C2semidirect
10. C31×#127C4semidirect
28. C2×D4×C31C2×C4semidirect
34. C42×C31C2×C4semidirect
37. C4⋊Dic31D4semidirect
98. #127Dic31semidirect
105. C31#1887semidirect
116. C2×D4C2×Dic31semidirect
129. C42C2×Dic31semidirect

Maximal subgroups

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

subgroupindex
#145215242
#145210542
C2×#117584212
#145208092
C2×C31×#1272
#188731