Finite Group Databasebeta

← #7596

Subgroups — #7596

Order 256.

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

normal subgroups only

This group has 203 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#2295C211yesmaximal, characteristic
2#1839C211yesmaximal, characteristic
3#1831C211yesmaximal, characteristic
4#1272C212yesmaximal
5#1271C212yesmaximal
6C2×C4×Q8C411yescharacteristic
7C2×C4⋊C8—21no—
8#391C411yescharacteristic
9#391C2211yescharacteristic
10#389C411yescharacteristic
11#389C2211yescharacteristic
12#387—21no—
13#387—21no—
14C4×C4⋊C4—21no—
15C4×C4⋊C4C411yescharacteristic
16#376—21no—
17C43C2211yescharacteristic
18#336—21no—
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×C8—22no—
25C22×C8—22no—
26C4×Q8—22no—
27C4×Q8C814yescomplemented
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C42—21no—
35C2×C42—21no—
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C2×C42D411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42C2311yescharacteristic
43C4⋊C8—42no—
44#95—21no—
45#95—21no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51C2×Q8—21no—
52C2×Q8—41no—
53C2×Q8—41no—
54C2×Q8C2×C812yes—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22×C411yescharacteristic
64C22×C4C2×D411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C2×C8—42no—
67C2×C8—42no—
68C2×C8—42no—
69C2×C8—42no—
70C2×C8—42no—
71C2×C8—42no—
72C4⋊C4—24no—
73C4⋊C4—21no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—42no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4C2×C812yes—
80C42—24no—
81C42—21no—
82C42—21no—
83C42—22no—
84C42—41no—
85C42—41no—
86C42—41no—
87C42—41no—
88C42—42no—
89C42C2×C812yes—
90C42M1612yes—
91C23C2×C22⋊C411yescharacteristic
92Q8—42no—
93Q8—42no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4C22⋊C812yes—
123C2×C4C22⋊C812yes—
124C2×C4C22×C811yescharacteristic
125C2×C4C2×M1611yescharacteristic
126C2×C4AGL(1,ℤ/8)11yescharacteristic
127C2×C4#13711yescharacteristic
128C2×C4#13712yes—
129C8—82no—
130C8—82no—
131C22C2×C22⋊C811yescharacteristic
132C22#41011yescharacteristic
133C22#41311yescharacteristic
134C22#41711yescharacteristic
135C22#41911yescharacteristic
136C22#42111yescharacteristic
137C22#42111yescharacteristic
138C4—21no—
139C4—21no—
140C4—22no—
141C4—22no—
142C4—21no—
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—41no—
149C4—81no—
150C4—81no—
151C2#145811yescharacteristic
152C2#147711yescharacteristic
153C2#147911yescharacteristic
154C2#151211yescharacteristic
155C2#151511yescharacteristic
156C2#153511yescharacteristic
157C2#155911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
27. C4×Q8C8semidirect

Maximal subgroups

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

subgroupindex
#22952
#18392
#18312
#12722
#12712