Finite Group Databasebeta

← #9289

Subgroups — #9289

Order 256.

245 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 69 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#3206C211yesmaximal, characteristic
2#2932C211yesmaximal, characteristic
3#1993C211yesmaximal, characteristic, complemented
4#1549C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic, complemented
7#1275C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#492C412yescomplemented
12#487C412yescomplemented
13#461—22no—
14#448—22no—
15#435—22no—
16#435—22no—
17#431C2211yescharacteristic
18#431C2211yescharacteristic
19#423C2211yescharacteristic
20#408—21no—
21C2×#97C2211yescharacteristic
22#382C2211yescharacteristic
23#340—21no—
24#340—21no—
25#332—22no—
26#327—22no—
27C2×#41—21no—
28C22×Q8—21no—
29C2×SD16—24no—
30#131—42no—
31#131—42no—
32C2×M16—22no—
33C2×M16—21no—
34C22×C8—22no—
35C22×C8D411yescharacteristic
36C22×C8D411yescharacteristic
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8C2×C412yes—
41C4⋊Q8C2×C411yescharacteristic
42#127C2×C411yescharacteristic
43#124—21no—
44#121—21no—
45#121D412yes—
46C4×D4—21no—
47C4×D4D412yes—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C42C2311yescharacteristic
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8D412yes—
56#103—24no—
57C22⋊C8—24no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#97—21no—
61#97—21no—
62C4×C8—21no—
63C4×C8C2×C412yes—
64#95—21no—
65#95—21no—
66#41—41no—
67#41—41no—
68#41—41no—
69#41—41no—
70C2×Q8—21no—
71C2×Q8—22no—
72C2×Q8—41no—
73C2×Q8—82no—
74C2×D4—22no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4C22⋊C412yes—
78C22×C4—22no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4#4111yescharacteristic
84SD16—84no—
85M16—44no—
86M16—41no—
87M16—41no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—22no—
91C2×C8—21no—
92C2×C8—22no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—44no—
96C2×C8—41no—
97C2×C8SD1612yescomplemented
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4C22⋊C412yes—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C42C22×C411yescharacteristic
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42#4111yescharacteristic
112C23—22no—
113C23#12111yescharacteristic
114Q8—44no—
115Q8—41no—
116Q8—41no—
117Q8—82no—
118D4—42no—
119D4—42no—
120D4—42no—
121D4—41no—
122D4—41no—
123D4—81no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—82no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4C2×C22⋊C411yescharacteristic
135C2×C4C4×D411yescharacteristic
136C2×C4C4×D411yescharacteristic
137C2×C4#12011yescharacteristic
138C2×C4#12111yescharacteristic
139C2×C4#12311yescharacteristic
140C2×C4C2×SD1611yescharacteristic
141C2×C4#13511yescharacteristic
142C2×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C4#13711yescharacteristic
144C8—22no—
145C8—42no—
146C8—42no—
147C8—41no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#38611yescharacteristic
155C22#46111yescharacteristic
156C22#46311yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—42no—
162C4—82no—
163C4—161no—
164C4C4×SD1611yescharacteristic
165C4#44011yescharacteristic
166C4#45011yescharacteristic
167C4#45211yescharacteristic
168C2—21no—
169C2—21no—
170C2—82no—
171C2#187711yescharacteristic
172C2#190211yescharacteristic
173C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ 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
3. #1993C2semidirect
4. #1549C2semidirect
6. #1286C2semidirect
7. #1275C2semidirect
11. #492C4semidirect
12. #487C4semidirect
97. C2×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
#29322
#19932
#15492
#15172
#12862
#12752