Finite Group Databasebeta

← #8304

Subgroups — #8304

Order 256.

619 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#123C211yesmaximal, characteristic
2#1779C212yesmaximal, complemented
3C2×#324C212yesmaximal, complemented
4C2×#323C212yesmaximal, complemented
5C23×D4C411yescharacteristic
6C24×C4C2211yescharacteristic
7C2×#123—24no—
8C2×#123—24no—
9C2×#123—24no—
10C22×C4⋊C4C411yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C22⋊C4C411yescharacteristic
13C22×C22⋊C4C411yescharacteristic
14C22×C22⋊C4C2211yescharacteristic
15C2×C22⋊C8—24no—
16C2×C22⋊C8C2214yes—
17#324—24no—
18#323—24no—
19C25C2×C411yescharacteristic
20C25C2×C411yescharacteristic
21C22×D4—22no—
22C22×D4—22no—
23C22×D4—22no—
24C22×D4—44no—
25C23×C4—22no—
26C23×C4—22no—
27C23×C4—44no—
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2×C411yescharacteristic
32C23×C4D414yes—
33C23×C4C2311yescharacteristic
34C22×C8—24no—
35#123—232no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4D414yes—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4D414yes—
51C22⋊C8—24no—
52C22⋊C8—24no—
53C22⋊C8—48no—
54C24—24no—
55C24—21no—
56C24—21no—
57C24—21no—
58C24—21no—
59C24—24no—
60C24—24no—
61C24—42no—
62C24—42no—
63C24—42no—
64C24C22⋊C412yes—
65C24C22⋊C412yes—
66C24C22×C411yescharacteristic
67C2×D4—28no—
68C2×D4—48no—
69C2×D4—44no—
70C2×D4—48no—
71C2×D4—44no—
72C22×C4—28no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—24no—
82C22×C4—28no—
83C22×C4—42no—
84C22×C4—44no—
85C22×C4—42no—
86C22×C4—48no—
87C22×C4—48no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4C22⋊C412yes—
91C22×C4C22⋊C412yes—
92C22×C4C22⋊C412yes—
93C22×C4C22⋊C412yes—
94C22×C4C2×D412yes—
95C22×C4C2×D412yes—
96C22×C4C2×D412yes—
97C2×C8—48no—
98C2×C8—44no—
99C4⋊C4—28no—
100C4⋊C4—28no—
101C22⋊C4—28no—
102C22⋊C4—28no—
103C22⋊C4—28no—
104C23—22no—
105C23—22no—
106C23—24no—
107C23—24no—
108C23—22no—
109C23—24no—
110C23—22no—
111C23—24no—
112C23—22no—
113C23—24no—
114C23—416no—
115C23—42no—
116C23—42no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—42no—
122C23—42no—
123C23—42no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—88no—
128C23#9914yes—
129C23C2×C22⋊C411yescharacteristic
130C23C2×C22⋊C411yescharacteristic
131C23C2×C22⋊C411yescharacteristic
132C23#13114yes—
133D4—416no—
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—416no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4#12014yes—
155C8—84no—
156C22—28no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—28no—
162C22—24no—
163C22—48no—
164C22—48no—
165C22—44no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—42no—
171C22—84no—
172C22—84no—
173C22—84no—
174C22#37911yescharacteristic
175C22#40812yes—
176C22#40812yes—
177C22#40812yes—
178C22C2×#9912yes—
179C22#41312yes—
180C4—24no—
181C4—42no—
182C4—42no—
183C4—42no—
184C4—42no—
185C2—22no—
186C2—22no—
187C2—44no—
188C2—82no—
189C2#146814yes—
190C2#178011yescharacteristic
191C2#178111yescharacteristic
192C2#178511yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC23×D4C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1779C2semidirect
3. C2×#324C2semidirect
4. C2×#323C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
#17792
C2×#3242
C2×#3232