Finite Group Databasebeta

← #16847707

Subgroups — #16847707

Order 3968.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#14521238C212yesmaximal, complemented
2#14521216C211yesmaximal, characteristic
3#14520842C211yesmaximal, characteristic, complemented
4#14520839C212yesmaximal, complemented
5C31×#559C211yesmaximal, characteristic, complemented
6#11758561C2211yescharacteristic
7C31×#127C2211yescharacteristic
8C31×#124—22no—
9C31×#123—21no—
10C31×#123C2212yes—
11C4×Q8×C31—21no—
12C4×D4×C31—22no—
13#11758509—21no—
14#11758506—22no—
15#11758422C2212yes—
16#11758421C2211yescharacteristic
17#11758394—22no—
18#11758393—21no—
19C2×Q8×C31—21no—
20C2×D4×C31—22no—
21C2×D4×C31D411yescharacteristic
22C2×D4×C31C2311yescharacteristic
23C22×C4×C31—21no—
24C22×C4×C31D412yes—
25C2×C22⋊D31—22no—
26#92497—41no—
27D4×D31—41no—
28#92495—21no—
29C4⋊C4×C31—21no—
30C4⋊C4×C31—22no—
31C22⋊C4×C31—22no—
32C22⋊C4×C31—21no—
33C22⋊C4×C31—22no—
34C22⋊C4×C31D412yes—
35C42×C31—22no—
36C42×C31D411yescharacteristic
37C22⋊Dic31—22no—
38#92480—41no—
39#92479—41no—
40(C2×D31)⋊C4—42no—
41Dic31⋊C4—42no—
42#92474—21no—
43C23×C31—21no—
44C23×C31C2×D412yes—
45C22×D31—22no—
46Q8×C31—42no—
47D4×C31—41no—
48D4×C31—41no—
49D4×C31—44no—
50C2×C4×C31—21no—
51C2×C4×C31—22no—
52C2×C4×C31—22no—
53C2×C4×C31—42no—
54C2×C4×C31—81no—
55C2×C4×C31C2×D411yescharacteristic
56C22⋊D31—42no—
57C22⋊D31—41no—
58C2×Dic31—22no—
59C2×Dic31—42no—
60D124—21no—
61C4×D31—41no—
62Dic62—21no—
63C31⋊C8—41no—
64#2198—311nomaximal, Sylow
65C22×C31—22no—
66C22×C31—42no—
67C22×C31—42no—
68C22×C31—41no—
69C22×C31#12011yescharacteristic
70C2×D31—41no—
71C2×D31—82no—
72C4×C31—22no—
73C4×C31—21no—
74C4×C31—42no—
75C4×C31—42no—
76Dic31—41no—
77Dic31—82no—
78#559D3111yescharacteristic, complemented
79#458—312no—
80#453—311no—
81#353—311no—
82#352—312no—
83C2×C31—21no—
84C2×C31—42no—
85C2×C31—81no—
86C2×C31B(4,𝔽2)11yescharacteristic
87D31—81no—
88H(5,𝔽2)—311no—
89AGL(1,ℤ/8)—621no—
90#127C2×D3111yescharacteristic
91#124—22no—
92#123—21no—
93#123—622no—
94#123C2×D3112yes—
95C4×Q8—21no—
96C4×D4—22no—
97#104—621no—
98#100—311no—
99#99—312no—
100#99—622no—
101C31#219811yescharacteristic, complemented, Sylow
102#41—621no—
103#41—1241no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—622no—
107C2×D4—1241no—
108C2×D4C22⋊D3111yescharacteristic
109C2×D4C22×D3111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D3112yes—
112SD16—1241no—
113D8—1241no—
114M16—621no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—1242no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—622no—
122C22⋊C4—1242no—
123C22⋊C4C22⋊D3112yes—
124C42—22no—
125C42C22⋊D3111yescharacteristic
126C23—21no—
127C23—622no—
128C23C2×C22⋊D3112yes—
129Q8—42no—
130Q8—621no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—621no—
135D4—1242no—
136D4—1241no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—622no—
143C2×C4—1241no—
144C2×C4—1242no—
145C2×C4C2×C22⋊D3111yescharacteristic
146C8—1241no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—1241no—
152C22—2482no—
153C22#1175853011yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—1241no—
159C4—2482no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—2481no—
164C2#1452123611yescharacteristic
subgroupG/H
centerC2#14521236
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×#559C2
socleC2×C31B(4,𝔽2)
prime radical#14521216C2
hypercenter#559D31
hypocenterC31#2198
LCSG ▷ C2×D4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #14521238C2semidirect
3. #14520842C2semidirect
4. #14520839C2semidirect
5. C31×#559C2semidirect
78. #559D31semidirect
101. C31#2198semidirect

Maximal subgroups

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

subgroupindex
#145212382
#145212162
#145208422
#145208392
C31×#5592
#219831