Finite Group Databasebeta

← #16846258

Subgroups — #16846258

Order 3968.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#14521613C211yesmaximal, characteristic
2#14521033C211yesmaximal, characteristic, complemented
3C2×#11758411C212yesmaximal, complemented
4#14520821C212yesmaximal, complemented
5C31×#568C211yesmaximal, characteristic, complemented
6C2×#41×C31C2211yescharacteristic
7C22×D124C2211yescharacteristic
8C31×#131—24no—
9C2×M16×C31—22no—
10C2×M16×C31C2212yes—
11#11758524—21no—
12#11758517—21no—
13#11758514C414yes—
14#11758507—22no—
15#11758503C2211yescharacteristic
16C2×#92474C2212yes—
17#11758424—21no—
18#11758421—21no—
19#11758411—24no—
20C23×D31C2×C412yes—
21#41×C31—22no—
22#41×C31—22no—
23C2×Q8×C31D411yescharacteristic
24C2×D4×C31C2×C412yes—
25C2×D4×C31D411yescharacteristic
26C22×C4×C31D412yes—
27C22×C4×C31C2311yescharacteristic
28C2×C22⋊D31—24no—
29C2×D124—22no—
30C2×D124—21no—
31C2×D124—21no—
32C2×D124—22no—
33M16×C31—24no—
34M16×C31—22no—
35M16×C31D414yescomplemented
36C2×C8×C31—24no—
37C2×C8×C31—22no—
38C22⋊Dic31C2×C412yes—
39(C2×D31)⋊C4—24no—
40C4⋊Dic31—21no—
41C4×Dic31—21no—
42#92474—22no—
43#92474—22no—
44C2×C31⋊C8—22no—
45C23×C31C22⋊C412yes—
46C23×C31C22×C411yescharacteristic
47C22×D31—22no—
48C22×D31—24no—
49C22×D31—82no—
50Q8×C31—22no—
51D4×C31—24no—
52D4×C31—22no—
53C2×C4×C31—22no—
54C2×C4×C31—22no—
55C2×C4×C31C22⋊C412yes—
56C2×C4×C31C2×D411yescharacteristic
57C2×C4×C31C2×D412yes—
58C2×C4×C31C2×D411yescharacteristic
59C2×C4×C31#4112yes—
60C22⋊D31—84no—
61C2×Dic31—22no—
62D124—41no—
63D124—42no—
64D124—41no—
65C8×C31—24no—
66C31⋊C8—42no—
67#1914—311nomaximal, Sylow
68C22×C31—22no—
69C22×C31—41no—
70C22×C31—42no—
71C22×C31C2×C22⋊C411yescharacteristic
72C22×C31C4×D412yes—
73C2×D31—44no—
74C2×D31—42no—
75C2×D31—84no—
76C4×C31—22no—
77C4×C31#12011yescharacteristic
78C4×C31#12112yes—
79C4×C31#12311yescharacteristic
80Dic31—42no—
81#568D3111yescharacteristic, complemented
82#535—311no—
83#413—311no—
84C2×#100—312no—
85#344—312no—
86C2×C31—21no—
87C2×C31—22no—
88C2×C31—42no—
89C2×C31#38611yescharacteristic
90D31—82no—
91C2×#41C2×D3111yescharacteristic
92C22×D4—311no—
93#131—24no—
94C2×M16—22no—
95C2×M16—312no—
96C2×M16C2×D3112yes—
97#127—621no—
98#124—621no—
99#121—622no—
100#120—314no—
101#117—311no—
102#101—621no—
103#100—621no—
104#100—624no—
105C31#191411yescharacteristic, complemented, Sylow
106C24—312no—
107#41—22no—
108#41—22no—
109C2×Q8D12411yescharacteristic
110C2×D4—624no—
111C2×D4—621no—
112C2×D4—622no—
113C2×D4—622no—
114C2×D4—621no—
115C2×D4C4×D3112yes—
116C2×D4D12411yescharacteristic
117C22×C4C22⋊D3112yes—
118C22×C4C22×D3111yescharacteristic
119M16—24no—
120M16—22no—
121M16—622no—
122M16—622no—
123M16C22⋊D3114yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—622no—
127C4⋊C4—621no—
128C22⋊C4—312no—
129C22⋊C4—624no—
130C42—621no—
131C23—622no—
132C23—624no—
133C23—2482no—
134C23(C2×D31)⋊C412yes—
135C23C2×C4×D3111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—1241no—
140D4—1242no—
141D4—1241no—
142D4—2484no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—622no—
146C2×C4(C2×D31)⋊C412yes—
147C2×C4C2×D12411yescharacteristic
148C2×C4#9249512yes—
149C2×C4C2×C22⋊D3111yescharacteristic
150C2×C4C2×C22⋊D3112yes—
151C8—24no—
152C8—1242no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—1244no—
157C22—1242no—
158C22—2484no—
159C22C2×(C2×D31)⋊C411yescharacteristic
160C22C4×C22⋊D3112yes—
161C4—22no—
162C4—1242no—
163C4#1175850611yescharacteristic
164C4#1175850712yes—
165C4#1175853011yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—2482no—
170C2#1452092811yescharacteristic
subgroupG/H
centerC2#14520928
commutatorC23×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#568C2
socleC2×C31#386
prime radical#14521613C2
cosocleC22×C4×C31C23
hypercenter#568D31
hypocenterC31#1914
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #14521033C2semidirect
3. C2×#11758411C2semidirect
4. #14520821C2semidirect
5. C31×#568C2semidirect
35. M16×C31D4semidirect
81. #568D31semidirect
105. C31#1914semidirect
123. M16C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145216132
#145210332
C2×#117584112
#145208212
C31×#5682
#191431