Finite Group Databasebeta

← #16847734

Subgroups — #16847734

Order 3968.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#14521799C211yesmaximal, characteristic, complemented
2#14521234C212yesmaximal, complemented
3#14521221C211yesmaximal, characteristic
4#14520847C212yesmaximal, complemented
5C31×#355C211yesmaximal, characteristic, complemented
6#11758564C2211yescharacteristic
7C31×#124C2211yescharacteristic
8#11758521C2212yes—
9#11758512—21no—
10#11758509—21no—
11#11758486—22no—
12#11758456—22no—
13#11758455—22no—
14#11758447—21no—
15C31×#101C2212yes—
16#11758424C2211yescharacteristic
17#11758413—22no—
18#11758393—21no—
19C2×Q8×C31C2311yescharacteristic
20C2×D4×C31D411yescharacteristic
21C22×Dic31—21no—
22#92499—41no—
23Q8×D31—41no—
24#92495—21no—
25#92495—22no—
26C2×Dic62D412yes—
27M16×C31—22no—
28C22⋊C4×C31—22no—
29C42×C31D411yescharacteristic
30C22⋊Dic31—22no—
31C22⋊Dic31—21no—
32#92480—41no—
33#92479—41no—
34C4⋊Dic31—21no—
35C4⋊Dic31—21no—
36Dic31⋊C4—22no—
37Dic31⋊C4—22no—
38Dic31⋊C4—22no—
39C4×Dic31D412yes—
40#92474—21no—
41Dic124—42no—
42#92470—42no—
43C23×C31—21no—
44Q8×C31—41no—
45D4×C31—41no—
46C2×C4×C31—21no—
47C2×C4×C31C2×D411yescharacteristic
48C2×C4×C31C2×D412yes—
49C22⋊D31—41no—
50C2×Dic31—22no—
51C2×Dic31—22no—
52C2×Dic31—81no—
53D124—22no—
54D124—21no—
55C4×D31—41no—
56C4×D31—42no—
57Dic62—22no—
58Dic62—21no—
59Dic62—42no—
60C8×C31—42no—
61C31⋊C8—41no—
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—
70Dic31—42no—
71Dic31—41no—
72Dic31—42no—
73#563—311no—
74#456—312no—
75#454—311no—
76#356—312no—
77#355D3111yescharacteristic, complemented
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⋊Q8—312no—
86#126—622no—
87#125—621no—
88#124C2×D3111yescharacteristic
89#123—621no—
90C22⋊Q8—622no—
91#104—622no—
92#104—621no—
93#101—311no—
94#101C2×D3112yes—
95C31#220311yescharacteristic, complemented, Sylow
96#41—621no—
97#41—622no—
98#41—1241no—
99C2×Q8—312no—
100C2×Q8—1241no—
101C2×Q8C22×D3111yescharacteristic
102C2×D4D12411yescharacteristic
103C22×C4—621no—
104Q16—1242no—
105SD16—1242no—
106SD16—1241no—
107D8—1241no—
108M16—22no—
109M16—621no—
110C4⋊C4—622no—
111C4⋊C4—621no—
112C4⋊C4—621no—
113C4⋊C4—622no—
114C4⋊C4—622no—
115C22⋊C4—22no—
116C22⋊C4—622no—
117C22⋊C4—621no—
118C42—312no—
119C42D12411yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—621no—
123Q8—622no—
124Q8—1242no—
125D4—41no—
126D4—621no—
127D4—622no—
128D4—1241no—
129C2×C4—21no—
130C2×C4—622no—
131C2×C4—622no—
132C2×C4—1242no—
133C2×C4—1241no—
134C2×C4—2481no—
135C2×C4C2×D12411yescharacteristic
136C2×C4D4×D3112yes—
137C8—42no—
138C8—1241no—
139C22—42no—
140C22—41no—
141C22—1241no—
142C22#1175845911yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—1242no—
147C4—1242no—
148C4—1241no—
149C2—21no—
150C2—81no—
151C2—2481no—
152C2#1452123711yescharacteristic
subgroupG/H
centerC2#14521237
commutator, cosocleC2×Q8×C31C23
FrattiniC2×Q8C22×D31
FittingC31×#355C2
socleC2×C31B(4,𝔽2)
prime radical#14521221C2
hypercenter#355D31
hypocenterC31#2203
LCSG ▷ C2×Q8×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #14521799C2semidirect
2. #14521234C2semidirect
4. #14520847C2semidirect
5. C31×#355C2semidirect
77. #355D31semidirect
95. C31#2203semidirect

Maximal subgroups

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

subgroupindex
#145217992
#145212342
#145212212
#145208472
C31×#3552
#220331