Finite Group Databasebeta

← #13000

Subgroups — #13000

Order 256.

256 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 75 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×#470C211yesmaximal, characteristic
2#2911C211yesmaximal, characteristic
3#2141C212yesmaximal, complemented
4C2×#349C211yesmaximal, characteristic
5#1386C212yesmaximal, complemented
6C22×Q16C2211yescharacteristic
7C22×M16C2211yescharacteristic
8C2×C22⋊Q8—21no—
9C2×#117—21no—
10C2×M32C2212yes—
11#470—22no—
12#470C414yescomplemented
13#428—24no—
14#427—21no—
15#427C2212yes—
16C2×#106C2211yescharacteristic
17C2×#103—21no—
18#358—24no—
19#349—22no—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C2×Q16—22no—
23C2×Q16—21no—
24C2×Q16—22no—
25C2×Q16C2×C412yescomplemented
26C2×M16—22no—
27C2×M16—22no—
28C2×M16C2×C412yes—
29C2×M16D412yes—
30C22×C8D411yescharacteristic
31C22×C8C2311yescharacteristic
32C22⋊Q8—24no—
33C22⋊Q8—42no—
34#117—22no—
35#117—22no—
36#117—41no—
37#117—41no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—41no—
41C2×C22⋊C4—41no—
42C2×C22⋊C4—41no—
43C2×C42—41no—
44M32—22no—
45M32—22no—
46C2×C16—22no—
47#107—22no—
48#107D414yes—
49#106—22no—
50#106—21no—
51#106—22no—
52#106C2×C412yescomplemented
53#103—24no—
54C24D811yescharacteristic
55C2×Q8—22no—
56C2×Q8—41no—
57C2×Q8—41no—
58C2×Q8—41no—
59C2×Q8—41no—
60C2×Q8—41no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—42no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4C22⋊C412yes—
69C22×C4D811yescharacteristic
70C22×C4SD1612yes—
71C22×C4C2×D411yescharacteristic
72Q16—24no—
73Q16—42no—
74M16—24no—
75M16—24no—
76C2×C8—21no—
77C2×C8—21no—
78C2×C8C22⋊C412yes—
79C2×C8C22×C411yescharacteristic
80C2×C8C2×D412yes—
81C2×C8C2×D411yescharacteristic
82C2×C8#4112yes—
83C4⋊C4—22no—
84C4⋊C4—24no—
85C4⋊C4—42no—
86C4⋊C4—44no—
87C4⋊C4—41no—
88C22⋊C4—42no—
89C22⋊C4—44no—
90C22⋊C4—42no—
91C42—42no—
92C42—42no—
93C16—42no—
94C23—21no—
95C23—21no—
96C23—42no—
97C23#10212yes—
98C23C2×D811yescharacteristic
99Q8—42no—
100Q8—42no—
101Q8—81no—
102Q8—81no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—44no—
112C2×C4—82no—
113C2×C4—84no—
114C2×C4—81no—
115C2×C4—81no—
116C2×C4#10212yes—
117C2×C4C2×C22⋊C411yescharacteristic
118C2×C4#12011yescharacteristic
119C2×C4#12311yescharacteristic
120C2×C4C2×SD1611yescharacteristic
121C2×C4AGL(1,ℤ/8)11yescharacteristic
122C2×C4#13711yescharacteristic
123C8—22no—
124C8C4×D412yes—
125C8#12112yes—
126C22—22no—
127C22—22no—
128C22—42no—
129C22—42no—
130C22—81no—
131C22C2×#10211yescharacteristic
132C22#44711yescharacteristic
133C22#48011yescharacteristic
134C4—22no—
135C4—82no—
136C4—82no—
137C4—84no—
138C4#38611yescharacteristic
139C4#41711yescharacteristic
140C4#44811yescharacteristic
141C4#48111yescharacteristic
142C2—21no—
143C2—21no—
144C2—42no—
145C2#187411yescharacteristic
146C2#219112yes—
subgroupG/H
center, socleC22#447
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
prime radicalC24D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2141C2semidirect
5. #1386C2semidirect
12. #470C4semidirect
25. C2×Q16C2×C4semidirect
52. #106C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
#29112
#21412
C2×#3492
#13862