Finite Group Databasebeta

← #9492

Subgroups — #9492

Order 256.

300 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 63 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#2873C211yesmaximal, characteristic, complemented
2#2598C211yesmaximal, characteristic
3#1490C212yesmaximal, complemented
4#1464C211yesmaximal, characteristic, complemented
5#1288C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4×D4C2211yescharacteristic
8C2×#117C2211yescharacteristic
9#432C2212yes—
10#405—24no—
11C4×M16—21no—
12C4×M16C2212yes—
13#400C411yescharacteristic
14#399C411yescharacteristic
15#394C411yescharacteristic
16#392C411yescharacteristic
17#388—21no—
18#386—21no—
19#386—21no—
20#385—21no—
21#383C2211yescharacteristic
22D4⋊C8—24no—
23#324—21no—
24#323—21no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C2×M16—24no—
30C2×M16—24no—
31C2×M16—22no—
32C2×M16—22no—
33C22×C8—22no—
34C4×D4—22no—
35C4×D4—22no—
36#117—22no—
37#117—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51C4⋊C8—22no—
52C22⋊C8—22no—
53#97—22no—
54#97—22no—
55#97—22no—
56C4×C8—22no—
57C4×C8D414yescomplemented
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95C2×C411yescharacteristic
63#95C2×C411yescharacteristic
64C24—21no—
65C24C22⋊C411yescharacteristic
66C2×D4—22no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—81no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—42no—
79C22×C4—81no—
80C22×C4—81no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87M16—28no—
88M16—24no—
89M16—24no—
90C2×C8—28no—
91C2×C8—24no—
92C2×C8—42no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—81no—
97C22⋊C4—22no—
98C22⋊C4—22no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—81no—
104C22⋊C4—81no—
105C42—21no—
106C42—21no—
107C42C2×D412yes—
108C42#4112yes—
109C23—21no—
110C23—21no—
111C23—22no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23#10412yes—
121C23C2×C22⋊C411yescharacteristic
122D4—24no—
123D4—42no—
124D4—84no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—24no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—84no—
148C2×C4#10412yes—
149C2×C4C4×D412yes—
150C2×C4#12011yescharacteristic
151C2×C4#12112yes—
152C2×C4#12311yescharacteristic
153C8—28no—
154C8—82no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22—42no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22#38611yescharacteristic
170C22C2×#10411yescharacteristic
171C22C2×#10411yescharacteristic
172C4—22no—
173C4—21no—
174C4—22no—
175C4—21no—
176C4—22no—
177C4—41no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—81no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#189111yescharacteristic
187C2#189611yescharacteristic
188C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#392C4
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. #2873C2semidirect
3. #1490C2semidirect
4. #1464C2semidirect
5. #1288C2semidirect
57. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#28732
#25982
#14902
#14642
#12882