Finite Group Databasebeta

← #17627

Subgroups — #17627

Order 256.

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

normal subgroups only

This group has 487 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#3501C211yesmaximal, characteristic
2#2975C211yesmaximal, characteristic
3#2357C211yesmaximal, characteristic
4#1518C212yesmaximal, complemented
5#1511C214yesmaximal, complemented
6#1508C214yesmaximal, complemented
7#1503C212yesmaximal, complemented
8#554—24no—
9#554C414yescomplemented
10D4×Q8C414yescomplemented
11#544—21no—
12#543—22no—
13#533C2212yes—
14C2×C22⋊Q8C2214yes—
15#518—21no—
16C2×C4×D4C2211yescharacteristic
17#436—24no—
18#431C2212yes—
19#407C2214yes—
20#390—21no—
21#388—22no—
22#382C2211yescharacteristic
23#381C2212yes—
24C4×C4⋊C4—21no—
25C4×C4⋊C4C2211yescharacteristic
26C4×C22⋊C4C2212yes—
27#328C2214yes—
28#328C2214yes—
29#328C2218yes—
30C22×Q8C2×C414yes—
31C22×D4D411yescharacteristic
32C23×C4C2312yes—
33C2×M16—24no—
34C4⋊Q8—24no—
35C4⋊Q8C2×C412yes—
36C4⋊Q8C2×C412yes—
37#125—24no—
38#125—22no—
39C22⋊Q8—24no—
40C22⋊Q8—24no—
41C22⋊Q8—22no—
42C22⋊Q8—28no—
43C22⋊Q8—28no—
44C22⋊Q8C2×C418yes—
45C22⋊Q8C2314yes—
46C4×Q8—22no—
47C4×Q8C2×C414yes—
48C4×D4—24no—
49C4×D4—24no—
50C4×D4C2×C414yes—
51C4×D4D414yes—
52#117—22no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2×C414yes—
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—24no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4D412yes—
62C2×C42—21no—
63C2×C42—22no—
64C2×C42C2311yescharacteristic
65C4⋊C8—22no—
66C22⋊C8—24no—
67C22⋊C8C2314yes—
68C4×C8—22no—
69#95—22no—
70#95—21no—
71#95C2311yescharacteristic
72#95C2312yes—
73#95C2311yescharacteristic
74C24C2×D412yes—
75C2×Q8—22no—
76C2×Q8—24no—
77C2×Q8—84no—
78C2×Q8C22×C414yes—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4C22⋊C414yes—
82C22×C4—21no—
83C22×C4—24no—
84C22×C4—24no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—82no—
88C22×C4C22×C414yes—
89C22×C4C2×D412yes—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D414yes—
92C22×C4C2×D411yescharacteristic
93C22×C4C2411yescharacteristic
94M16—84no—
95C2×C8—24no—
96C4⋊C4—28no—
97C4⋊C4—24no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—24no—
102C4⋊C4—24no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4C22⋊C414yes—
106C4⋊C4C22×C414yes—
107C22⋊C4—28no—
108C22⋊C4—24no—
109C22⋊C4—44no—
110C22⋊C4—82no—
111C22⋊C4C22⋊C418yes—
112C42—24no—
113C42—42no—
114C42—82no—
115C42C22×C412yes—
116C42C2×D412yes—
117C23—22no—
118C23—22no—
119C23—42no—
120C23—42no—
121C23C2×C22⋊C414yes—
122C23C22×D411yescharacteristic
123Q8—44no—
124Q8—82no—
125Q8—84no—
126D4—28no—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—44no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—84no—
144C2×C4—84no—
145C2×C4C2×C22⋊C412yes—
146C2×C4C2×C22⋊C414yes—
147C2×C4C2×C22⋊C412yes—
148C2×C4#13712yes—
149C2×C4#13712yes—
150C2×C4C23×C411yescharacteristic
151C2×C4C22×D411yescharacteristic
152C8—44no—
153C22—21no—
154C22—21no—
155C22—24no—
156C22—24no—
157C22—24no—
158C22—82no—
159C22—82no—
160C22C22×C22⋊C411yescharacteristic
161C22C2×#13711yescharacteristic
162C22C2×#13711yescharacteristic
163C4—21no—
164C4—21no—
165C4—24no—
166C4—22no—
167C4—42no—
168C4—44no—
169C4—44no—
170C4—81no—
171C4—82no—
172C4#41914yes—
173C2—21no—
174C2—21no—
175C2—44no—
176C2#288311yescharacteristic
177C2C2×#41911yescharacteristic
178C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1518C2semidirect
5. #1511C2semidirect
6. #1508C2semidirect
7. #1503C2semidirect
9. #554C4semidirect
10. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#35012
#29752
#23572
#15182
#15112
#15082
#15032