Finite Group Databasebeta

← #17632

Subgroups — #17632

Order 256.

437 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 183 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#3502C211yesmaximal, characteristic
2#2983C211yesmaximal, characteristic
3#2356C211yesmaximal, characteristic
4#1518C212yesmaximal
5#1515C214yesmaximal
6#1506C214yesmaximal
7#1503C212yesmaximal
8Q82C414yescomplemented
9#557—24no—
10#557C414yescomplemented
11#544—21no—
12#544—22no—
13#533C2212yes—
14#533C2214yes—
15#519—21no—
16C2×C4×Q8C2211yescharacteristic
17#446—24no—
18#432C2214yes—
19#431C2212yes—
20#391—21no—
21#387—22no—
22#383C2212yes—
23#382C2211yescharacteristic
24C4×C4⋊C4—21no—
25C4×C4⋊C4C2211yescharacteristic
26#376C2212yes—
27#328C2214yes—
28#328C2214yes—
29#328C2218yes—
30C22×Q8D411yescharacteristic
31C4⋊Q8—28no—
32C4⋊Q8—24no—
33C4⋊Q8C2×C412yes—
34C4⋊Q8C2×C414yes—
35C4⋊Q8C2×C412yes—
36C4⋊Q8C2×C414yes—
37#125—28no—
38#125—24no—
39#125—22no—
40#125—24no—
41C22⋊Q8—24no—
42C22⋊Q8—22no—
43C22⋊Q8C2314yes—
44C4×Q8—24no—
45C4×Q8—24no—
46C4×Q8—22no—
47C4×Q8—24no—
48C4×Q8C2×C414yes—
49C4×Q8C2×C418yes—
50C4×Q8C2×C414yes—
51C4×Q8D414yes—
52#117—24no—
53#117—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D412yes—
58C2×C42—22no—
59C2×C42—21no—
60C2×C42C2312yes—
61C2×C42C2311yescharacteristic
62C4⋊C8—22no—
63C4⋊C8—24no—
64C22⋊C8C2314yes—
65#97—24no—
66C4×C8—22no—
67#95—22no—
68#95—22no—
69#95—21no—
70#95—22no—
71#95C2311yescharacteristic
72#95C2312yes—
73#95C2311yescharacteristic
74C2×Q8—22no—
75C2×Q8—24no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8C22⋊C414yes—
79C2×Q8C22×C414yes—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4C2×D412yes—
84C22×C4C2×D412yes—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2411yescharacteristic
88C2×C8—24no—
89C4⋊C4—24no—
90C4⋊C4—22no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—24no—
95C4⋊C4—24no—
96C4⋊C4—28no—
97C4⋊C4—24no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4C22⋊C418yes—
101C4⋊C4C22⋊C414yes—
102C4⋊C4C22×C414yes—
103C22⋊C4—24no—
104C42—28no—
105C42—24no—
106C42—44no—
107C42—42no—
108C42C22×C414yes—
109C42C22×C412yes—
110C42C2×D412yes—
111C42C2×D414yes—
112C23C22×D411yescharacteristic
113Q8—28no—
114Q8—44no—
115Q8—84no—
116Q8—82no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4C2×C22⋊C414yes—
134C2×C4C2×C22⋊C412yes—
135C2×C4C2×C22⋊C414yes—
136C2×C4C2×C22⋊C412yes—
137C2×C4#13712yes—
138C2×C4#13712yes—
139C2×C4C23×C411yescharacteristic
140C2×C4C22×D411yescharacteristic
141C8—44no—
142C22—21no—
143C22—21no—
144C22C22×C22⋊C411yescharacteristic
145C22C2×#13711yescharacteristic
146C22C2×#13711yescharacteristic
147C4—21no—
148C4—21no—
149C4—24no—
150C4—24no—
151C4—22no—
152C4—42no—
153C4—44no—
154C4—44no—
155C4—81no—
156C4—82no—
157C4#41914yes—
158C2—21no—
159C2—21no—
160C2#288411yescharacteristic
161C2C2×#41911yescharacteristic
162C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. Q82C4semidirect
10. #557C4semidirect

Maximal subgroups

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

subgroupindex
#35022
#29832
#23562
#15182
#15152
#15062
#15032