Finite Group Databasebeta

← #12923

Subgroups — #12923

Order 256.

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

normal subgroups only

This group has 212 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
1#2911C211yesmaximal, characteristic
2#2159C212yesmaximal, complemented
3C2×#349C211yesmaximal, characteristic
4#1809C211yesmaximal, characteristic
5#1386C212yesmaximal, complemented
6C22×M16C2211yescharacteristic
7C2×#117—21no—
8C2×M32C2212yes—
9#428—22no—
10#428—22no—
11#427—21no—
12#427C2212yes—
13C2×#107C2211yescharacteristic
14C2×#106C2211yescharacteristic
15#380—21no—
16#367—22no—
17#366—22no—
18#349C414yescomplemented
19#340—21no—
20C23×C4D411yescharacteristic
21C2×M16—22no—
22C2×M16—22no—
23C2×M16C2×C412yes—
24C2×M16Q812yes—
25C22×C8D411yescharacteristic
26C22×C8C2311yescharacteristic
27#117—22no—
28#117—22no—
29#117—41no—
30#117—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C22⋊C4—41no—
36C2×C22⋊C4—41no—
37C2×C42—41no—
38M32—22no—
39M32C2×C414yescomplemented
40C2×C16—22no—
41#107—21no—
42#107—21no—
43#107—22no—
44#107D412yes—
45#107D412yes—
46#106—22no—
47#106—22no—
48#106—22no—
49#95—41no—
50C24D811yescharacteristic
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4C4⋊C412yes—
60C22×C4D811yescharacteristic
61C22×C4Q1612yes—
62C22×C4C2×D411yescharacteristic
63M16—24no—
64M16—24no—
65C2×C8—21no—
66C2×C8—21no—
67C2×C8C4⋊C412yes—
68C2×C8C22×C411yescharacteristic
69C2×C8C2×D412yes—
70C2×C8C2×Q811yescharacteristic
71C2×C8#4112yes—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—82no—
83C42—42no—
84C42—42no—
85C16—24no—
86C23—21no—
87C23—21no—
88C23—41no—
89C23—41no—
90C23#10712yes—
91C23C2×D811yescharacteristic
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—82no—
101C2×C4—82no—
102C2×C4—81no—
103C2×C4—81no—
104C2×C4—81no—
105C2×C4—81no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4#10712yes—
109C2×C4C2×C4⋊C411yescharacteristic
110C2×C4#12011yescharacteristic
111C2×C4#12311yescharacteristic
112C2×C4C2×Q1611yescharacteristic
113C2×C4AGL(1,ℤ/8)11yescharacteristic
114C2×C4#13711yescharacteristic
115C8—22no—
116C8C4×D412yes—
117C8C22⋊Q812yes—
118C22—21no—
119C22—21no—
120C22—22no—
121C22—42no—
122C22—41no—
123C22—41no—
124C22—81no—
125C22C2×#10711yescharacteristic
126C22#44711yescharacteristic
127C22#48011yescharacteristic
128C4—22no—
129C4—82no—
130C4—82no—
131C4—161no—
132C4—161no—
133C4#38511yescharacteristic
134C4#42811yescharacteristic
135C4#45111yescharacteristic
136C4#48411yescharacteristic
137C2—21no—
138C2—21no—
139C2—42no—
140C2#185211yescharacteristic
141C2#219011yescharacteristic
142C2#219111yescharacteristic
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
2. #2159C2semidirect
5. #1386C2semidirect
18. #349C4semidirect
39. M32C2×C4semidirect

Maximal subgroups

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

subgroupindex
#29112
#21592
C2×#3492
#18092
#13862