Finite Group Databasebeta

← #9335

Subgroups — #9335

Order 256.

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

normal subgroups only

This group has 200 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#3234C211yesmaximal, characteristic
2#2980C211yesmaximal, characteristic
3#1553C211yesmaximal, characteristic
4#1534C211yesmaximal, characteristic, complemented
5#1470C211yesmaximal, characteristic, complemented
6#1308C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#125C2211yescharacteristic
10#501—21no—
11#490—21no—
12#481C412yes—
13#480C412yes—
14#478—22no—
15#435—22no—
16#435—22no—
17#432C2211yescharacteristic
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#431C2211yescharacteristic
22#408—22no—
23#332—21no—
24#331—21no—
25#330—22no—
26#324—22no—
27C2×#41—21no—
28C2×M16—22no—
29C22×C8—22no—
30C4⋊Q8D411yescharacteristic
31#127D411yescharacteristic
32#125—21no—
33#125—21no—
34#125—21no—
35#125—21no—
36#124—21no—
37#121—21no—
38#121C2×C412yes—
39C4×D4—21no—
40C4×D4D412yes—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4C2×C411yescharacteristic
44C2×C4⋊C4C2×C411yescharacteristic
45C2×C42C2311yescharacteristic
46#107—22no—
47#106—22no—
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53#102—22no—
54#102—22no—
55C22⋊C8—24no—
56C22⋊C8—21no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8C2×C412yes—
60C22⋊C8D412yes—
61#97—21no—
62#97—21no—
63#97—21no—
64#97D412yes—
65C4×C8—21no—
66#41—81no—
67#41—81no—
68C2×Q8—21no—
69C2×D4—22no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4C22⋊C412yes—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4#4111yescharacteristic
79M16—82no—
80C2×C8—22no—
81C2×C8—22no—
82C2×C8—22no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—82no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4C22⋊C412yes—
96C22⋊C4—22no—
97C22⋊C4—22no—
98C42C2×D411yescharacteristic
99C42C2×D411yescharacteristic
100C42C2×D411yescharacteristic
101C42#4111yescharacteristic
102C23—22no—
103C23C4×D411yescharacteristic
104Q8—81no—
105D4—42no—
106D4—81no—
107D4—81no—
108D4—82no—
109C2×C4—22no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—81no—
119C2×C4C2×C22⋊C411yescharacteristic
120C2×C4C4×D411yescharacteristic
121C2×C4#12011yescharacteristic
122C2×C4#12111yescharacteristic
123C2×C4#12111yescharacteristic
124C2×C4#12311yescharacteristic
125C8—42no—
126C8—42no—
127C8—42no—
128C8—81no—
129C8—81no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—42no—
134C22—42no—
135C22—42no—
136C22#38611yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—42no—
144C4—81no—
145C4—81no—
146C2—41no—
147C2—82no—
148C2#188111yescharacteristic
149C2#190411yescharacteristic
150C2#191511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
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
4. #1534C2semidirect
5. #1470C2semidirect
6. #1308C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#29802
#15532
#15342
#14702
#13082
#12872