Finite Group Databasebeta

← #11785503

Subgroups — #11785503

Order 1152.

342 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×#64289C211yesmaximal, characteristic
2#10593832C212yesmaximal
3#10593543C212yesmaximal, complemented
4#10590091C212yesmaximal, complemented
5C2×#3832C2211yescharacteristic
6#64289—22no—
7#64289—22no—
8#64286C2214yes—
9C2×#3788C412yescomplemented
10C2×#3788C2211yescharacteristic
11C2×#3787C412yescomplemented
12C2×#3787C2211yescharacteristic
13#63774—24no—
14#63746—22no—
15#63745—22no—
16C2×C32×Q8Q811yescharacteristic
17#3832—22no—
18#3832—22no—
19C2×C4⋊(C3⋊D3)D412yes—
20C2×C4⋊(C3⋊D3)Q811yescharacteristic
21C2×C4×C3⋊D3C2×C412yes—
22C2×C4×C3⋊D3C2311yescharacteristic
23C22⋊#157—24no—
24C2×#622C2×C412yes—
25C2×#622C2×C411yescharacteristic
26C2×#622C2×C411yescharacteristic
27C4⋊#157—24no—
28C4×#157—24no—
29#3788—22no—
30#3788—24no—
31#3787—22no—
32#3787—22no—
33#3787D412yes—
34#3787D412yes—
35#1812—91nomaximal, Sylow
36C2×#239—121no—
37C22×C3⋊D3C4⋊C412yes—
38C22×C3⋊D3C2×D411yescharacteristic
39C2×#157—44no—
40C32×Q8—22no—
41C2×C32×C4C4⋊C412yes—
42C2×C32×C4C2×Q811yescharacteristic
43C2×C3⋊Dic3C22×C411yescharacteristic
44C4⋊(C3⋊D3)—24no—
45C4⋊(C3⋊D3)—22no—
46C4×C3⋊D3—22no—
47C4×C3⋊D3—22no—
48C4×C3⋊D3C2×D412yes—
49C4×C3⋊D3#4112yes—
50#622—22no—
51#622C22⋊C411yescharacteristic
52#622C22⋊C411yescharacteristic
53#622C22⋊C412yes—
54C2×#131—91no—
55#427—92no—
56#410—92no—
57H(3,ℤ/4)—92no—
58C2×C3×Q8—41no—
59#239—124no—
60#239—124no—
61C2×D12—122no—
62C2×D12—121no—
63C2×C4×D3—122no—
64C2×C4×D3—121no—
65C22×C32C2×C4⋊C411yescharacteristic
66C2×C3⋊D3—22no—
67C2×C3⋊D3—41no—
68C2×C3⋊D3—42no—
69C2×C3⋊D3#12112yes—
70#157—84no—
71C32×C4—22no—
72C32×C4C22⋊Q812yes—
73C3⋊Dic3C2×C22⋊C411yescharacteristic
74C3⋊Dic3#11711yescharacteristic
75C2×#41—91no—
76#131—182no—
77#131—182no—
78C2×M16—92no—
79C2×M16—91no—
80C22×C8—91no—
81C22×C8—92no—
82#117—94no—
83#107—182no—
84#106—182no—
85#99—184no—
86C22×D3—122no—
87C22×D3—121no—
88C3×Q8—44no—
89C2×C3×C4—41no—
90C2×C3×C4—42no—
91C2×Dic3—121no—
92D12—128no—
93D12—124no—
94C4×D3—122no—
95C4×D3—122no—
96C4×D3—124no—
97C4×D3—124no—
98C2×C32—21no—
99C2×C32#38011yescharacteristic
100C3⋊D3—22no—
101C3⋊D3—42no—
102#41—182no—
103#41—182no—
104C2×Q8PSU(3,𝔽2)11yescharacteristic
105C2×D4—91no—
106C2×D4—92no—
107C22×C4—91no—
108C22×C4—92no—
109M16—184no—
110M16—182no—
111C2×C8—92no—
112C2×C8—92no—
113C2×C8—91no—
114C2×C8—91no—
115C2×C8—92no—
116C2×C8—182no—
117C2×C8—182no—
118C4⋊C4—184no—
119C22⋊C4—184no—
120C42—184no—
121C22×C3—41no—
122C2×D3—122no—
123C2×D3—124no—
124C2×D3—242no—
125C2×D3—244no—
126C3×C4—44no—
127C3×C4—42no—
128Dic3—121no—
129Dic3—121no—
130C32#181211yescharacteristic, complemented, Sylow
131C23—91no—
132C23—92no—
133Q8—22no—
134D4—184no—
135D4—182no—
136C2×C4—91no—
137C2×C4—92no—
138C2×C4—92no—
139C2×C4—182no—
140C2×C4—182no—
141C2×C4—364no—
142C2×C4#377712yes—
143C2×C4C2×PSU(3,𝔽2)11yescharacteristic
144C8—91no—
145C8—91no—
146C8—92no—
147C8—182no—
148C2×C3—42no—
149C2×C3—41no—
150D3—242no—
151D3—244no—
152C22—92no—
153C22—182no—
154C22—361no—
155C22—362no—
156C22C2×#377711yescharacteristic
157C4—22no—
158C4—91no—
159C4—91no—
160C4—724no—
161C4C22⋊PSU(3,𝔽2)12yes—
162C3—41no—
163C2—21no—
164C2—182no—
165C2—362no—
166C2#1059356811yescharacteristic
subgroupG/H
centerC2#10593568
commutatorC2×C3⋊Dic3C22×C4
FrattiniC2×C4C2×PSU(3,𝔽2)
FittingC2×C32×Q8Q8
socleC2×C32#380
prime radicalC2×#3832C22
cosocleC2×C4×C3⋊D3C23
hypercenterC2×Q8PSU(3,𝔽2)
hypocenterC32#1812
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #10593543C2semidirect
4. #10590091C2semidirect
9. C2×#3788C4semidirect
11. C2×#3787C4semidirect
130. C32#1812semidirect

Maximal subgroups

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

subgroupindex
C2×#642892
#105938322
#105935432
#105900912
#18129