Finite Group Databasebeta

← #9980

Subgroups — #9980

Order 256.

411 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 49 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#3006C212yesmaximal, complemented
2#2815C211yesmaximal, characteristic, complemented
3#1782C211yesmaximal, characteristic, complemented
4#1510C212yesmaximal, complemented
5#1306C211yesmaximal, characteristic, complemented
6C22×#41C2211yescharacteristic
7C2×AGL(1,ℤ/8)—22no—
8C2×#121C2212yes—
9#449—24no—
10#448—22no—
11#447—22no—
12#417—22no—
13C2×#99—22no—
14#407C2212yes—
15#397—22no—
16#392—22no—
17#392—22no—
18#381C2211yescharacteristic
19#381C2211yescharacteristic
20#327—24no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41—44no—
24C22×Q8D411yescharacteristic, complemented
25C22×D4—21no—
26C22×D4—22no—
27C22×D4—22no—
28C22×D4D412yes—
29C22×D4D411yescharacteristic, complemented
30C23×C4—21no—
31C23×C4C2311yescharacteristic
32AGL(1,ℤ/8)—48no—
33C2×SD16—24no—
34C2×D8—24no—
35C2×M16—22no—
36#121—22no—
37#121—22no—
38#121—22no—
39#121—22no—
40C2×C4⋊C4D412yes—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—24no—
44C2×C22⋊C4D412yes—
45#103—24no—
46#102—24no—
47#99—44no—
48C22⋊C8—22no—
49C22⋊C8D414yescomplemented
50#95—22no—
51#95—22no—
52#95—22no—
53C24—21no—
54C24—22no—
55C24C2×D411yescharacteristic
56#41—28no—
57#41—44no—
58#41—44no—
59#41—44no—
60C2×Q8—22no—
61C2×Q8—42no—
62C2×D4—24no—
63C2×D4—22no—
64C2×D4—22no—
65C2×D4—22no—
66C2×D4—44no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—84no—
74C2×D4—84no—
75C2×D4—82no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—41no—
84C22×C4C2×D412yes—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D412yes—
87C22×C4#4111yescharacteristic
88SD16—48no—
89D8—48no—
90M16—44no—
91C2×C8—24no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—84no—
99C23—22no—
100C23—22no—
101C23—42no—
102C23—41no—
103C23—42no—
104C23—42no—
105C23—44no—
106C23—41no—
107C23—42no—
108C23—44no—
109C23—82no—
110C23#12011yescharacteristic
111C23#12111yescharacteristic
112C23#12711yescharacteristic
113Q8—24no—
114Q8—42no—
115D4—24no—
116D4—42no—
117D4—44no—
118D4—44no—
119D4—44no—
120D4—82no—
121D4—84no—
122D4—84no—
123D4—88no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—48no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—44no—
133C2×C4—44no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—81no—
137C2×C4—82no—
138C2×C4#12012yes—
139C2×C4#12112yes—
140C8—44no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—21no—
145C22—21no—
146C22—42no—
147C22—82no—
148C22—84no—
149C22—84no—
150C22—82no—
151C22—82no—
152C22—82no—
153C22—82no—
154C22—82no—
155C22—82no—
156C22#39211yescharacteristic
157C22B(4,𝔽2)12yes—
158C4—22no—
159C4—22no—
160C4—42no—
161C4—82no—
162C4—82no—
163C4—161no—
164C2—22no—
165C2—41no—
166C2—41no—
167C2—82no—
168C2—162no—
169C2#200812yes—
170C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3006C2semidirect
2. #2815C2semidirect
3. #1782C2semidirect
4. #1510C2semidirect
5. #1306C2semidirect
24. C22×Q8D4semidirect
29. C22×D4D4semidirect
49. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#30062
#28152
#17822
#15102
#13062