Finite Group Databasebeta

← #16777716

Subgroups — #16777716

Order 3328.

324 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 38 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#13129768C212yesmaximal
2#13114760C211yesmaximal, characteristic
3C23×Q8×C13C411yescharacteristic
4C2×#88076—22no—
5#11718868C411yescharacteristic
6#11718868C2211yescharacteristic
7#11718284—22no—
8C22×Q8×C13—21no—
9C22×Q8×C13—21no—
10C22×Q8×C13—41no—
11C22×Q8×C13—41no—
12C22×Q8×C13—44no—
13C23×C4×C13C812yes—
14C23×C4×C13C2×C411yescharacteristic
15C2×#6099—21no—
16C2×#6099D412yes—
17#88106—42no—
18#88076—44no—
19C22⋊#1055—41no—
20C22⋊#1055—41no—
21#7341—131nomaximal, Sylow
22C24×C13C2×C811yescharacteristic
23C2×Q8×C13—24no—
24C2×Q8×C13—42no—
25C2×Q8×C13—44no—
26C2×Q8×C13—42no—
27C2×Q8×C13—44no—
28C2×Q8×C13—48no—
29C2×Q8×C13—44no—
30C2×Q8×C13—41no—
31C2×Q8×C13—41no—
32C22×C4×C13—22no—
33C22×C4×C13—21no—
34C22×C4×C13—21no—
35C22×C4×C13—44no—
36C22×C4×C13—42no—
37C22×C4×C13—42no—
38C22×C4×C13C22⋊C411yescharacteristic
39C22×C4×C13M1611yescharacteristic
40#6099—42no—
41#6099—42no—
42C2×#1055—22no—
43C2×#1055—41no—
44#6092—82no—
45#1784—131no—
46#1320—132no—
47C23×C13—21no—
48C23×C13—41no—
49C23×C13—42no—
50C23×C13C22⋊C811yescharacteristic
51Q8×C13—44no—
52Q8×C13—48no—
53Q8×C13—44no—
54C2×C4×C13—22no—
55C2×C4×C13—24no—
56C2×C4×C13—21no—
57C2×C4×C13—42no—
58C2×C4×C13—42no—
59C2×C4×C13—44no—
60C2×C4×C13—42no—
61C2×C4×C13—42no—
62C2×C4×C13—41no—
63C2×C4×C13—44no—
64C2×C4×C13#9911yescharacteristic
65C2×C4×C13#10011yescharacteristic
66#1055—42no—
67#1055—81no—
68C23×Q8#26111yescharacteristic
69C2×#101—262no—
70#407—131no—
71#407—131no—
72#349—262no—
73C22×C13—22no—
74C22×C13—21no—
75C22×C13—41no—
76C22×C13—41no—
77C22×C13—42no—
78C22×C13—42no—
79C22×C13—41no—
80C22×C13#32311yescharacteristic
81C4×C13—21no—
82C4×C13—44no—
83C4×C13—41no—
84C4×C13AΣL(1,𝔽16)12yes—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8—44no—
90C23×C4#105712yes—
91C23×C4C2×#26111yescharacteristic
92C2×M16—132no—
93C2×M16—261no—
94M32—522no—
95#101—524no—
96C22⋊C8—521no—
97C22⋊C8—521no—
98C2×C13—21no—
99C2×C13—42no—
100C2×C13—41no—
101C2×C13#131611yescharacteristic
102C24C2×#105711yescharacteristic
103C2×Q8—24no—
104C2×Q8—42no—
105C2×Q8—44no—
106C2×Q8—42no—
107C2×Q8—44no—
108C2×Q8—48no—
109C2×Q8—44no—
110C2×Q8—41no—
111C2×Q8—41no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4#612211yescharacteristic
119C22×C4C22⋊#26111yescharacteristic
120M16—522no—
121M16—522no—
122C2×C8—262no—
123C2×C8—521no—
124C16—1042no—
125C13#734111yescharacteristic, complemented, Sylow
126C23—21no—
127C23—41no—
128C23—42no—
129C23C22⋊#105711yescharacteristic
130Q8—44no—
131Q8—48no—
132Q8—44no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—21no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—44no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4#8811911yescharacteristic
144C2×C4#8812111yescharacteristic
145C8—522no—
146C8—1041no—
147C22—22no—
148C22—21no—
149C22—41no—
150C22—41no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22#1171830511yescharacteristic
155C4—21no—
156C4—44no—
157C4—41no—
158C4#1171830612yes—
159C2—21no—
160C2—42no—
161C2—41no—
162C2#1312976011yescharacteristic
subgroupG/H
centerC2#13129760
commutator, prime radicalC24×C13C2×C8
FrattiniC23×C4C2×#261
FittingC23×Q8×C13C4
socleC2×C13#1316
cosocle#11718868C22
hypercenterC23×Q8#261
hypocenterC13#7341
LCSG ▷ C24×C13 ▷ C23×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
125. C13#7341semidirect

Maximal subgroups

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

subgroupindex
#131297682
#131147602
#734113