Finite Group Databasebeta

← #11772786

Subgroups — #11772786

Order 1152.

248 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 20 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#10592945C211yesmaximal, characteristic
2#10590220C211yesmaximal, characteristic, complemented
3#10590218C211yesmaximal, characteristic, complemented
4C32×2−1+4C411yescharacteristic
5#64160—21no—
6#64153C2211yescharacteristic
7#63780—21no—
8#63779—21no—
9#63664—21no—
10#63663C411yescharacteristic
11#5246—62no—
12C32×#41—21no—
13C32×#41—41no—
14C32×#41—41no—
15C2×C32×Q8—41no—
16C2×C32×Q8C2×C411yescharacteristic
17C2×#634D411yescharacteristic
18#3792—41no—
19C2×#622—41no—
20#3756—41no—
21#3754—41no—
22#3750—41no—
23C4×C3⋊Dic3D411yescharacteristic
24#3748—21no—
25#1405—91nomaximal, Sylow
26C3×2−1+4—22no—
27#951—64no—
28#944—62no—
29#837—64no—
30#836—62no—
31C32×Q8—21no—
32C32×Q8—41no—
33C32×Q8—41no—
34C32×D4—21no—
35C32×D4—41no—
36C32×D4—41no—
37C2×C32×C4—21no—
38C2×C32×C4—41no—
39C2×C32×C4—42no—
40C2×C32×C4C22⋊C411yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3⋊Dic3—21no—
43#634—41no—
44#622—81no—
45#622—81no—
46C3⋊(C3⋊C8)—41no—
47#456—91no—
48#356—91no—
49#333—91no—
50C3×#41—24no—
51C3×#41—44no—
52C3×#41—44no—
53C2×C3×Q8—22no—
54C2×C3×Q8—44no—
55C2×Dic6—62no—
56#216—124no—
57#214—124no—
58Dic3⋊C4—124no—
59C4×Dic3—62no—
60#208—64no—
61C22×C32—41no—
62C22×C32#9911yescharacteristic
63C32×C4—21no—
64C32×C4—41no—
65C32×C4—41no—
66C3⋊Dic3—21no—
67C3⋊Dic3—21no—
68C3⋊Dic3—41no—
692−1+4#15711yescharacteristic
70#137—181no—
71C4⋊Q8—91no—
72C4⋊C8—181no—
73#104—181no—
74#101—91no—
75#101—181no—
76C3×Q8—24no—
77C3×Q8—44no—
78C3×Q8—44no—
79C3×D4—24no—
80C3×D4—44no—
81C3×D4—44no—
82C2×C3×C4—24no—
83C2×C3×C4—22no—
84C2×C3×C4—44no—
85C2×C3×C4—48no—
86C2×Dic3—122no—
87C2×Dic3—122no—
88Dic6—124no—
89C3⋊C8—124no—
90C2×C32—21no—
91C2×C32—81no—
92C2×C32AΣL(1,𝔽16)11yescharacteristic
93#41—21no—
94#41—41no—
95#41—41no—
96C2×Q8—41no—
97C2×Q8—91no—
98C2×Q8C2×#15711yescharacteristic
99Q16—361no—
100SD16—361no—
101M16—181no—
102M16—361no—
103C2×C8—361no—
104C4⋊C4—361no—
105C42—91no—
106C22×C3—22no—
107C22×C3—44no—
108C3×C4—24no—
109C3×C4—44no—
110C3×C4—44no—
111Dic3—122no—
112Dic3—122no—
113Dic3—242no—
114C32#140511yescharacteristic, complemented, Sylow
115Q8—21no—
116Q8—41no—
117Q8—41no—
118Q8—361no—
119D4—21no—
120D4—41no—
121D4—41no—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—181no—
126C2×C4—181no—
127C2×C4C22⋊#15711yescharacteristic
128C8—361no—
129C8—721no—
130C8—721no—
131C2×C3—22no—
132C2×C3—42no—
133C2×C3—84no—
134C22—41no—
135C22#6378611yescharacteristic
136C4—21no—
137C4—41no—
138C4—41no—
139C4—181no—
140C4—181no—
141C4—361no—
142C3—22no—
143C2—21no—
144C2—81no—
145C2#1059022311yescharacteristic
subgroupG/H
centerC2#10590223
commutatorC2×C32×Q8C2×C4
FrattiniC2×Q8C2×#157
Fitting, prime radicalC32×2−1+4C4
socleC2×C32AΣL(1,𝔽16)
cosocle#64153C22
hypercenter2−1+4#157
hypocenterC32#1405
LCSG ▷ C2×C32×Q8 ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
2. #10590220C2semidirect
3. #10590218C2semidirect
114. C32#1405semidirect

Maximal subgroups

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

subgroupindex
#105929452
#105902202
#105902182
#14059