Finite Group Databasebeta

← #9197

Subgroups — #9197

Order 256.

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

normal subgroups only

This group has 312 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#2905C211yesmaximal, characteristic, complemented
2#2619C211yesmaximal, characteristic
3#1545C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic, complemented
5#1286C212yesmaximal, complemented
6C22×M16—21no—
7C2×#127C2211yescharacteristic
8C2×#121—21no—
9C2×#117C2211yescharacteristic
10#431C2212yes—
11#424—24no—
12#423—21no—
13#423C2212yes—
14#394—21no—
15#394C412yes—
16#392—21no—
17#392C412yes—
18#390—21no—
19#382C2211yescharacteristic
20#331—24no—
21#324—21no—
22#323—21no—
23C22×D4—22no—
24C22×D4—21no—
25C22×D4—21no—
26C22×D4C2×C412yes—
27C23×C4D411yescharacteristic
28C2×M16—22no—
29C2×M16—22no—
30C2×M16—42no—
31C2×M16—42no—
32C22×C8—22no—
33#127—22no—
34#127—22no—
35#121—42no—
36#121—42no—
37#117—22no—
38#117—22no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4D411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42C2311yescharacteristic
49C4⋊C8—24no—
50C4⋊C8—22no—
51C4⋊C8—22no—
52C4⋊C8D414yescomplemented
53C22⋊C8—22no—
54C4×C8—22no—
55#95—21no—
56#95C2×C412yes—
57C24—22no—
58C24C22⋊C411yescharacteristic
59C2×D4—24no—
60C2×D4—44no—
61C2×D4—41no—
62C2×D4—42no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—41no—
68C2×D4—82no—
69C2×D4—82no—
70C2×D4—82no—
71C2×D4—82no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4D812yes—
81C22×C4SD1612yes—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—44no—
86M16—42no—
87M16—42no—
88C2×C8—24no—
89C2×C8—44no—
90C2×C8—42no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C22⋊C4—44no—
98C22⋊C4—82no—
99C22⋊C4—82no—
100C42—21no—
101C42—21no—
102C42C2×D412yes—
103C42#4112yes—
104C23—21no—
105C23—21no—
106C23—41no—
107C23—41no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—42no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23#10212yes—
116C23C2×C22⋊C411yescharacteristic
117D4—84no—
118D4—84no—
119D4—88no—
120D4—81no—
121D4—81no—
122D4—81no—
123D4—81no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4#10212yes—
140C2×C4C4×D412yes—
141C2×C4#12011yescharacteristic
142C2×C4#12112yes—
143C2×C4#12311yescharacteristic
144C2×C4C2×D811yescharacteristic
145C2×C4C2×SD1611yescharacteristic
146C2×C4AGL(1,ℤ/8)11yescharacteristic
147C2×C4#13711yescharacteristic
148C8—44no—
149C8—82no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—81no—
157C22—82no—
158C22—82no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22—82no—
163C22—82no—
164C22#38611yescharacteristic
165C22C2×#10211yescharacteristic
166C22#41711yescharacteristic
167C4—22no—
168C4—22no—
169C4—22no—
170C4—41no—
171C4—41no—
172C4—82no—
173C4#44711yescharacteristic
174C4#44811yescharacteristic
175C4#48011yescharacteristic
176C4#48111yescharacteristic
177C2—21no—
178C2—21no—
179C2—42no—
180C2—162no—
181C2#187411yescharacteristic
182C2#191011yescharacteristic
183C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#2619C2
LCSG ▷ C22×C4 ▷ 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
1. #2905C2semidirect
3. #1545C2semidirect
4. #1458C2semidirect
5. #1286C2semidirect
52. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#29052
#26192
#15452
#14582
#12862