Finite Group Databasebeta

← #16847732

Subgroups — #16847732

Order 3968.

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

normal subgroups only

This group has 216 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#14521231C212yesmaximal, complemented
2#14521221C211yesmaximal, characteristic
3#14520848C212yesmaximal, complemented
4#14520846C211yesmaximal, characteristic, complemented
5C31×#563C211yesmaximal, characteristic, complemented
6#11758564C2211yescharacteristic
7C31×C4⋊Q8C2212yes—
8C31×#126—22no—
9C31×#125—21no—
10C31×#124C2211yescharacteristic
11C31×#123—21no—
12C31×C22⋊Q8—22no—
13#11758519—22no—
14#11758509—21no—
15#11758424C2212yes—
16#11758424C2211yescharacteristic
17#11758393—21no—
18#11758393—22no—
19C2×Q8×C31D412yes—
20C2×Q8×C31C2311yescharacteristic
21C2×D4×C31D411yescharacteristic
22C22×C4×C31—21no—
23#92499—41no—
24Q8×D31—41no—
25#92495—21no—
26#92495—22no—
27C4⋊C4×C31—21no—
28C4⋊C4×C31—22no—
29C4⋊C4×C31—21no—
30C4⋊C4×C31—22no—
31C4⋊C4×C31—22no—
32C22⋊C4×C31—22no—
33C22⋊C4×C31—21no—
34C22⋊C4×C31—22no—
35C42×C31D411yescharacteristic
36C42×C31D412yes—
37#92482—42no—
38#92481—42no—
39#92480—41no—
40#92479—41no—
41#92474—21no—
42#92474—22no—
43C23×C31—21no—
44Q8×C31—41no—
45Q8×C31—42no—
46D4×C31—41no—
47C2×C4×C31—21no—
48C2×C4×C31—22no—
49C2×C4×C31—22no—
50C2×C4×C31—81no—
51C2×C4×C31C2×D412yes—
52C2×C4×C31C2×D411yescharacteristic
53C22⋊D31—41no—
54D124—21no—
55D124—22no—
56C4×D31—41no—
57C4×D31—42no—
58Dic62—21no—
59Dic62—22no—
60C31⋊C8—41no—
61C31⋊C8—42no—
62#2203—311nomaximal, Sylow
63C22×C31—42no—
64C22×C31—41no—
65C22×C31#12011yescharacteristic
66C2×D31—41no—
67C4×C31—21no—
68C4×C31—22no—
69C4×C31—41no—
70C4×C31—42no—
71C4×C31—42no—
72Dic31—41no—
73#563D3111yescharacteristic, complemented
74#456—312no—
75#454—311no—
76#356—312no—
77#355—311no—
78C2×C31—21no—
79C2×C31—81no—
80C2×C31B(4,𝔽2)11yescharacteristic
81D31—81no—
822−1+4—311no—
83#137—622no—
84AGL(1,ℤ/8)—621no—
85C4⋊Q8C2×D3112yes—
86#126—22no—
87#125—21no—
88#124C2×D3111yescharacteristic
89#123—21no—
90C22⋊Q8—22no—
91#104—621no—
92#104—622no—
93#101—312no—
94#101—311no—
95C31#220311yescharacteristic, complemented, Sylow
96#41—621no—
97#41—622no—
98#41—1241no—
99C2×Q8—1241no—
100C2×Q8C22⋊D3112yes—
101C2×Q8C22×D3111yescharacteristic
102C2×D4C22⋊D3111yescharacteristic
103C22×C4—21no—
104Q16—1242no—
105SD16—1241no—
106SD16—1242no—
107D8—1241no—
108M16—621no—
109M16—622no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D3111yescharacteristic
119C42C22⋊D3112yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—621no—
124Q8—622no—
125D4—41no—
126D4—621no—
127D4—622no—
128D4—1241no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—1241no—
134C2×C4—1242no—
135C2×C4C2×C22⋊D3112yes—
136C2×C4C2×C22⋊D3111yescharacteristic
137C8—1241no—
138C8—1242no—
139C22—42no—
140C22—41no—
141C22—1241no—
142C22#1175853011yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—1241no—
149C2—21no—
150C2—81no—
151C2—2481no—
152C2#1452123611yescharacteristic
subgroupG/H
centerC2#14521236
commutator, cosocleC2×Q8×C31C23
FrattiniC2×Q8C22×D31
FittingC31×#563C2
socleC2×C31B(4,𝔽2)
prime radical#14521221C2
hypercenter#563D31
hypocenterC31#2203
LCSG ▷ C2×Q8×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #14521231C2semidirect
3. #14520848C2semidirect
4. #14520846C2semidirect
5. C31×#563C2semidirect
73. #563D31semidirect
95. C31#2203semidirect

Maximal subgroups

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

subgroupindex
#145212312
#145212212
#145208482
#145208462
C31×#5632
#220331