Finite Group Databasebeta

← #16847694

Subgroups — #16847694

Order 3968.

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

normal subgroups only

This group has 354 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#14521684C211yesmaximal, characteristic, complemented
2#14521236C212yesmaximal, complemented
3#14520842C211yesmaximal, characteristic, complemented
4#14520837C212yesmaximal, complemented
5C31×#453C211yesmaximal, characteristic, complemented
6C31×H(5,𝔽2)C2211yescharacteristic, complemented
7C2×D4×D31—22no—
8C31×AGL(1,ℤ/8)—21no—
9C31×#127C2211yescharacteristic
10#11758515—22no—
11#11758514—21no—
12#11758514—22no—
13#11758514C2212yes—
14C4×D124—21no—
15C31×#104—21no—
16C31×#100C2211yescharacteristic, complemented
17#11758422—22no—
18#11758422C2212yescomplemented
19C23×D31—21no—
20C23×D31D412yescomplemented
21#41×C31—21no—
22#41×C31—41no—
23C2×D4×C31—22no—
24C2×D4×C31—22no—
25C2×D4×C31—41no—
26C2×D4×C31D411yescharacteristic, complemented
27C2×D4×C31C2311yescharacteristic
28C2×C22⋊D31—22no—
29C2×C22⋊D31—22no—
30C2×C22⋊D31—42no—
31D4×D31—48no—
32C2×D124—21no—
33C2×C4×D31—22no—
34SD16×C31—41no—
35D8×C31—41no—
36M16×C31—21no—
37C42×C31D411yescharacteristic, complemented
38C22⋊Dic31—21no—
39C22⋊Dic31—22no—
40C22⋊Dic31D412yescomplemented
41(C2×D31)⋊C4—22no—
42(C2×D31)⋊C4—42no—
43C4⋊Dic31—21no—
44C23×C31—22no—
45C23×C31—21no—
46C23×C31C2×D412yes—
47C22×D31—22no—
48C22×D31—44no—
49C22×D31—44no—
50C22×D31—42no—
51C22×D31—81no—
52Q8×C31—21no—
53D4×C31—21no—
54D4×C31—41no—
55D4×C31—41no—
56D4×C31—44no—
57D4×C31—42no—
58D4×C31—41no—
59C2×C4×C31—21no—
60C2×C4×C31—22no—
61C2×C4×C31—41no—
62C2×C4×C31C2×D411yescharacteristic
63C22⋊D31—44no—
64C22⋊D31—44no—
65C22⋊D31—84no—
66C2×Dic31—22no—
67C2×Dic31—42no—
68D124—24no—
69C4×D31—44no—
70C8×C31—41no—
71#2196—311nomaximal, Sylow
72C22×C31—22no—
73C22×C31—42no—
74C22×C31—42no—
75C22×C31—41no—
76C22×C31—41no—
77C22×C31—82no—
78C22×C31#12011yescharacteristic
79C2×D31—24no—
80C2×D31—42no—
81C2×D31—42no—
82C2×D31—44no—
83C2×D31—82no—
84C2×D31—84no—
85C4×C31—22no—
86C4×C31—21no—
87C4×C31—41no—
88Dic31—42no—
89Dic31—82no—
90D42—311no—
91B(4,𝔽2)—312no—
92#453D3111yescharacteristic, complemented
93#353—311no—
94AΣL(1,𝔽16)—312no—
95C2×C31—21no—
96C2×C31—42no—
97C2×C31—81no—
98C2×C31—81no—
99C2×C31B(4,𝔽2)11yescharacteristic
100D31—44no—
101H(5,𝔽2)C2×D3111yescharacteristic, complemented
102C22×D4—622no—
103AGL(1,ℤ/8)—21no—
104#127C2×D3111yescharacteristic
105#121—622no—
106#120—312no—
107#120—621no—
108#120—622no—
109C4×D4—621no—
110#104—21no—
111#100C2×D3111yescharacteristic, complemented
112#99—312no—
113#99—622no—
114C31#219611yescharacteristic, complemented, Sylow
115C24—312no—
116C24—621no—
117#41—21no—
118#41—41no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—41no—
122C2×D4—621no—
123C2×D4—622no—
124C2×D4—622no—
125C2×D4—1248no—
126C2×D4—1242no—
127C2×D4C22⋊D3111yescharacteristic, complemented
128C2×D4C22×D3111yescharacteristic
129C22×C4—622no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—621no—
134C22⋊C4—312no—
135C22⋊C4—621no—
136C22⋊C4—622no—
137C22⋊C4—622no—
138C22⋊C4—1242no—
139C42C22⋊D3111yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—622no—
143C23—1242no—
144C23—1244no—
145C23—1244no—
146C23—2481no—
147C23D4×D3112yes—
148Q8—21no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—42no—
154D4—41no—
155D4—624no—
156D4—1244no—
157D4—1244no—
158D4—2484no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—622no—
163C2×C4—1244no—
164C2×C4—1242no—
165C2×C4C2×C22⋊D3111yescharacteristic
166C8—41no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—82no—
173C22—624no—
174C22—1242no—
175C22—1244no—
176C22—1242no—
177C22—2482no—
178C22—2484no—
179C22#1175851411yescharacteristic
180C4—22no—
181C4—21no—
182C4—41no—
183C4—1242no—
184C4—2482no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—81no—
189C2—1244no—
190C2#1452123511yescharacteristic
subgroupG/H
centerC2#14521235
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×#453C2
socleC2×C31B(4,𝔽2)
hypercenter#453D31
hypocenterC31#2196
LCSG ▷ C2×D4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #14521684C2semidirect
2. #14521236C2semidirect
3. #14520842C2semidirect
4. #14520837C2semidirect
5. C31×#453C2semidirect
6. C31×H(5,𝔽2)C22semidirect
16. C31×#100C22semidirect
18. #11758422C22semidirect
20. C23×D31D4semidirect
26. C2×D4×C31D4semidirect
37. C42×C31D4semidirect
40. C22⋊Dic31D4semidirect
92. #453D31semidirect
101. H(5,𝔽2)C2×D31semidirect
111. #100C2×D31semidirect
114. C31#2196semidirect
127. C2×D4C22⋊D31semidirect
139. C42C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145216842
#145212362
#145208422
#145208372
C31×#4532
#219631