Finite Group Databasebeta

← #9512

Subgroups — #9512

Order 256.

279 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 59 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#2921C211yesmaximal, characteristic, complemented
2#2482C211yesmaximal, characteristic
3#1822C211yesmaximal, characteristic, complemented
4#1462C212yesmaximal
5#1306C212yesmaximal, complemented
6C2×#117C2211yescharacteristic
7C22×C22⋊C4C2211yescharacteristic
8#433—24no—
9#423—21no—
10#407C2212yes—
11#407C2212yes—
12C4×M16—21no—
13#399—21no—
14#397—21no—
15#388—21no—
16#388—21no—
17#385C411yescharacteristic
18#385C411yescharacteristic
19#381C2211yescharacteristic
20#379C411yescharacteristic
21#379C411yescharacteristic
22#343—21no—
23#343—21no—
24#323—24no—
25C25C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2311yescharacteristic
28C2×M16—22no—
29C2×M16—22no—
30#117—22no—
31#117—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4C2×C411yescharacteristic
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C4⋊C8—24no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53C22⋊C8—22no—
54C22⋊C8D414yes—
55#97—22no—
56C4×C8—22no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61C24—21no—
62C24—22no—
63C24—42no—
64C24—41no—
65C24—41no—
66C24—81no—
67C24C22×C411yescharacteristic
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—81no—
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4#4112yes—
83M16—42no—
84M16—42no—
85M16—42no—
86M16—42no—
87C2×C8—24no—
88C2×C8—42no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—81no—
92C4⋊C4—81no—
93C22⋊C4—22no—
94C22⋊C4—22no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—42no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—81no—
103C22⋊C4—81no—
104C42—21no—
105C42—21no—
106C42—21no—
107C42—21no—
108C23—21no—
109C23—21no—
110C23—22no—
111C23—24no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—42no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—82no—
122C23—82no—
123C23—84no—
124C23—84no—
125C23C2×C22⋊C411yescharacteristic
126C23C4×D412yes—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—42no—
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—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—82no—
151C2×C4#12011yescharacteristic
152C2×C4#12112yes—
153C2×C4#12311yescharacteristic
154C8—44no—
155C8—82no—
156C22—21no—
157C22—21no—
158C22—22no—
159C22—22no—
160C22—22no—
161C22—42no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—81no—
167C22—81no—
168C22—84no—
169C22—84no—
170C22—81no—
171C22—81no—
172C22#38611yescharacteristic
173C22#41011yescharacteristic
174C22#41011yescharacteristic
175C4—21no—
176C4—22no—
177C4—21no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—81no—
183C4—81no—
184C4—81no—
185C2—21no—
186C2—21no—
187C2—42no—
188C2—82no—
189C2#189111yescharacteristic
190C2#189911yescharacteristic
191C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25C2×C4
LCSG ▷ C24 ▷ 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. #2921C2semidirect
3. #1822C2semidirect
5. #1306C2semidirect

Maximal subgroups

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

subgroupindex
#29212
#24822
#18222
#14622
#13062