Finite Group Databasebeta

← #8300

Subgroups — #8300

Order 256.

579 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 115 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×#117C211yesmaximal, characteristic
2#1817C212yesmaximal, complemented
3C2×#324C212yesmaximal, complemented
4C2×#323C212yesmaximal
5C24×C4C411yescharacteristic
6C2×#117—24no—
7C2×#117—24no—
8C2×#117—24no—
9C22×C4⋊C4C411yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11C22×C22⋊C4C411yescharacteristic
12C22×C22⋊C4C2211yescharacteristic
13C22×C42C411yescharacteristic
14C22×C42C2211yescharacteristic
15C2×C22⋊C8C2214yes—
16#336—24no—
17#324—24no—
18#323—24no—
19C25C2×C411yescharacteristic
20C23×C4—22no—
21C23×C4—22no—
22C23×C4—22no—
23C23×C4—22no—
24C23×C4—22no—
25C23×C4—22no—
26C23×C4C814yescomplemented
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2×C411yescharacteristic
32C23×C4C2311yescharacteristic
33C22×C8—24no—
34#117—232no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4D414yes—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4D414yes—
47C2×C42—22no—
48C2×C42—22no—
49C2×C42—22no—
50C2×C42—22no—
51C2×C42—22no—
52C2×C42D414yes—
53C22⋊C8—24no—
54C22⋊C8—24no—
55C24—21no—
56C24—21no—
57C24—21no—
58C24—21no—
59C24—42no—
60C24—42no—
61C24C22⋊C412yes—
62C24C2×C812yes—
63C24M1612yes—
64C24C22×C411yescharacteristic
65C22×C4—28no—
66C22×C4—28no—
67C22×C4—28no—
68C22×C4—28no—
69C22×C4—28no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—28no—
79C22×C4—28no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4C22⋊C412yes—
91C22×C4C22⋊C412yes—
92C22×C4C22⋊C412yes—
93C22×C4C22⋊C412yes—
94C22×C4C22⋊C412yes—
95C22×C4C2×C812yes—
96C22×C4C2×C812yes—
97C22×C4M1612yes—
98C22×C4M1612yes—
99C22×C4C2×D412yes—
100C22×C4C2×D412yes—
101C22×C4C2×D412yes—
102C2×C8—48no—
103C2×C8—44no—
104C4⋊C4—28no—
105C4⋊C4—28no—
106C22⋊C4—28no—
107C22⋊C4—28no—
108C42—28no—
109C42—28no—
110C23—22no—
111C23—22no—
112C23—22no—
113C23—22no—
114C23—22no—
115C23—22no—
116C23—22no—
117C23—22no—
118C23—42no—
119C23—42no—
120C23—44no—
121C23—44no—
122C23—44no—
123C23—44no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23C22⋊C814yes—
129C23C2×C22⋊C411yescharacteristic
130C23C2×C22⋊C411yescharacteristic
131C23C2×C22⋊C411yescharacteristic
132C23C22×C811yescharacteristic
133C23C2×M1611yescharacteristic
134C23C2×M1611yescharacteristic
135C23C2×M1611yescharacteristic
136C2×C4—232no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4C22⋊C814yes—
165C2×C4C22⋊C814yes—
166C2×C4#12014yes—
167C8—84no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—24no—
177C22—44no—
178C22—44no—
179C22—44no—
180C22—44no—
181C22—44no—
182C22—44no—
183C22—42no—
184C22—42no—
185C22#37911yescharacteristic
186C22C2×C22⋊C811yescharacteristic
187C22C2×C22⋊C811yescharacteristic
188C22C2×C22⋊C811yescharacteristic
189C22#40711yescharacteristic
190C22#40711yescharacteristic
191C22#40711yescharacteristic
192C22#41012yes—
193C22#41312yes—
194C4—24no—
195C4—24no—
196C4—24no—
197C4—42no—
198C4—42no—
199C4—42no—
200C2—21no—
201C2—21no—
202C2—21no—
203C2—21no—
204C2—44no—
205C2#145812yes—
206C2#146412yes—
207C2#177911yescharacteristic
208C2#178211yescharacteristic
209C2#178511yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1817C2semidirect
3. C2×#324C2semidirect
26. C23×C4C8semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
#18172
C2×#3242
C2×#3232