Finite Group Databasebeta

← #9334

Subgroups — #9334

Order 256.

199 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 57 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#3215C211yesmaximal, characteristic
2#2987C211yesmaximal, characteristic, complemented
3#1554C211yesmaximal, characteristic, complemented
4#1533C211yesmaximal, characteristic
5#1469C211yesmaximal, characteristic
6#1308C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#497—21no—
11#489—21no—
12#464—22no—
13#451C412yes—
14#450C412yes—
15#446—22no—
16#446—22no—
17#433—22no—
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#333—21no—
24#332—21no—
25#329—22no—
26#323—22no—
27C22×Q8C2×C411yescharacteristic
28C22×D4C2×C411yescharacteristic
29C2×Q16—22no—
30C2×SD16—22no—
31C4⋊Q8D411yescharacteristic
32C4⋊Q8D411yescharacteristic
33#125—21no—
34#124—21no—
35#124—21no—
36#124—21no—
37#124—21no—
38C22⋊Q8—21no—
39C22⋊Q8C2×C412yes—
40C4×Q8—21no—
41C4×Q8D412yes—
42#117—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C42C2311yescharacteristic
46C4⋊C8—24no—
47C4⋊C8—21no—
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52#103—22no—
53#102—22no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57C22⋊C8C2×C412yes—
58C22⋊C8D412yes—
59#97—22no—
60#97—21no—
61#97—21no—
62#97—21no—
63#97D412yes—
64C4×C8—22no—
65C4×C8—21no—
66C24—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8—81no—
71C2×Q8C22⋊C412yes—
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—81no—
75C22×C4—21no—
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4#4111yescharacteristic
79Q16—82no—
80SD16—82no—
81C2×C8—22no—
82C2×C8—22no—
83C2×C8—22no—
84C2×C8—41no—
85C2×C8—41no—
86C4⋊C4—22no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—22no—
91C4⋊C4C22⋊C412yes—
92C22⋊C4—22no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C42—22no—
98C42C2×D411yescharacteristic
99C42C2×D411yescharacteristic
100C42C2×D411yescharacteristic
101C42#4111yescharacteristic
102C23—42no—
103C23—41no—
104C23—81no—
105C23—81no—
106C23C4×D411yescharacteristic
107Q8—42no—
108Q8—42no—
109Q8—81no—
110Q8—81no—
111D4—42no—
112D4—81no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—81no—
120C2×C4C2×C22⋊C411yescharacteristic
121C2×C4C4×D411yescharacteristic
122C2×C4#12011yescharacteristic
123C2×C4#12111yescharacteristic
124C2×C4#12111yescharacteristic
125C2×C4#12311yescharacteristic
126C8—42no—
127C8—42no—
128C8—42no—
129C8—81no—
130C8—81no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—162no—
138C22#38611yescharacteristic
139C4—21no—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—42no—
146C4—42no—
147C4—81no—
148C2—41no—
149C2—161no—
150C2#188111yescharacteristic
151C2#190511yescharacteristic
152C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2987C2semidirect
3. #1554C2semidirect
6. #1308C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32152
#29872
#15542
#15332
#14692
#13082
#12872