Finite Group Databasebeta

← #11772781

Subgroups — #11772781

Order 1152.

318 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C32×#456C211yesmaximal, characteristic
2#10589414C211yesmaximal, characteristic, complemented
3#10589412C211yesmaximal, characteristic, complemented
4#67773—34nomaximal
5C32×2−1+4C411yescharacteristic
6C32×#137—21no—
7C32×C4⋊Q8C2211yescharacteristic
8C32×#104—21no—
9C32×#101C411yescharacteristic
10#63663—21no—
11C4⋊(C3⋊(C3⋊C8))—21no—
12C3×#456D314yes—
13#4548—34no—
14#4546—34no—
15C32×#41—21no—
16C32×#41—41no—
17C32×#41—41no—
18C2×C32×Q8—41no—
19C2×C32×Q8C2×C411yescharacteristic
20C2×C32×Q8D411yescharacteristic
21C32×Q16—41no—
22C32×SD16—41no—
23C32×M16—21no—
24C32×C4⋊C4—41no—
25C32×C42D411yescharacteristic
26#3748—41no—
27C2×C3⋊(C3⋊C8)—41no—
28#1405—91noSylow
29C3×2−1+4Dic314yes—
30C3×#137—24no—
31C3×C4⋊Q8C2×D314yes—
32C3×#104—24no—
33C3×#101Dic314yes—
34#836—64no—
35C4⋊(C3⋊C8)—64no—
36C32×Q8—21no—
37C32×Q8—41no—
38C32×Q8—41no—
39C32×Q8—41no—
40C32×D4—21no—
41C32×D4—41no—
42C32×D4—41no—
43C2×C32×C4—21no—
44C2×C32×C4—21no—
45C2×C32×C4—21no—
46C2×C32×C4—41no—
47C2×C32×C4—42no—
48C2×C32×C4C22⋊C411yescharacteristic
49C32×C8—41no—
50C3⋊(C3⋊C8)—81no—
51C3⋊(C3⋊C8)—81no—
52#456C3⋊D311yescharacteristic
53#356—91no—
54#333—91no—
55C3×#41—24no—
56C3×#41—44no—
57C3×#41—44no—
58C2×C3×Q8—44no—
59C2×C3×Q8C2×Dic314yes—
60C2×C3×Q8#6714yes—
61C3×Q16—44no—
62C3×SD16—44no—
63C3×M16—24no—
64C3×C4⋊C4—44no—
65C3×C42#6714yes—
66#208—124no—
67C2×C3⋊C8—124no—
68C22×C32—41no—
69C22×C32#9911yescharacteristic
70C32×C4—21no—
71C32×C4—21no—
72C32×C4—21no—
73C32×C4—41no—
74C32×C4—41no—
75C32×C4—41no—
762−1+4C3⋊Dic311yescharacteristic
77#137—21no—
78C4⋊Q8C2×C3⋊D311yescharacteristic
79C4⋊C8—181no—
80#104—21no—
81#101—181no—
82#101C3⋊Dic311yescharacteristic
83C3×Q8—24no—
84C3×Q8—44no—
85C3×Q8—44no—
86C3×Q8—44no—
87C3×D4—24no—
88C3×D4—44no—
89C3×D4—44no—
90C2×C3×C4—24no—
91C2×C3×C4—24no—
92C2×C3×C4—24no—
93C2×C3×C4—44no—
94C2×C3×C4—48no—
95C2×C3×C4#21714yes—
96C3×C8—44no—
97C3⋊C8—244no—
98C3⋊C8—244no—
99C2×C32—21no—
100C2×C32—81no—
101C2×C32AΣL(1,𝔽16)11yescharacteristic
102#41—21no—
103#41—41no—
104#41—41no—
105C2×Q8—41no—
106C2×Q8C2×C3⋊Dic311yescharacteristic
107C2×Q8#63811yescharacteristic
108Q16—41no—
109SD16—41no—
110M16—21no—
111M16—361no—
112C2×C8—361no—
113C4⋊C4—41no—
114C42#63811yescharacteristic
115C22×C3—44no—
116C22×C3#83414yes—
117C3×C4—24no—
118C3×C4—24no—
119C3×C4—24no—
120C3×C4—44no—
121C3×C4—44no—
122C3×C4—44no—
123C32#140511yescharacteristic, complemented, Sylow
124Q8—21no—
125Q8—41no—
126Q8—41no—
127Q8—41no—
128D4—21no—
129D4—41no—
130D4—41no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4#375711yescharacteristic
137C8—41no—
138C8—721no—
139C8—721no—
140C2×C3—24no—
141C2×C3—84no—
142C2×C3#453614yes—
143C22—41no—
144C22#6366111yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—41no—
149C4—41no—
150C4—41no—
151C3#6777314yescomplemented
152C2—21no—
153C2—81no—
154C2#1058940211yescharacteristic
subgroupG/H
centerC2#10589402
commutatorC2×C32×Q8C2×C4
FrattiniC4⋊Q8C2×C3⋊D3
FittingC32×#456C2
socleC2×C32AΣL(1,𝔽16)
prime radicalC32×2−1+4C4
cosocleC32×C4⋊Q8C22
hypercenter#456C3⋊D3
hypocenterC32#1405
LCSG ▷ C2×C32×Q8 ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #10589414C2semidirect
3. #10589412C2semidirect
123. C32#1405semidirect
151. C3#67773semidirect

Maximal subgroups

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

subgroupindex
C32×#4562
#105894142
#105894122
#677733