Finite Group Databasebeta

← #29637

Subgroups — #29637

Order 256.

679 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 679 conjugacy classes of subgroups — too many to draw. Only its 119 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#3449C211yesmaximal, characteristic
2#2484C211yesmaximal, characteristic, complemented
3#2406C212yesmaximal, complemented
4#2044C214yesmaximal, complemented
5#2022C214yesmaximal, complemented
6#1820C211yesmaximal, characteristic, complemented
7#1782C212yesmaximal, complemented
8C22×#41C2212yes—
9#537—21no—
10#537—21no—
11#536—22no—
12#535—22no—
13#534C2214yes—
14#532—22no—
15#525—24no—
16C2×#123C2214yes—
17C2×#117C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C2×#99C2218yes—
20#400—21no—
21#398—21no—
22#395—22no—
23#394—22no—
24#387—22no—
25#386C2214yes—
26#385—24no—
27#385C2212yes—
28#381C2212yes—
29#381C2212yes—
30#381C2211yescharacteristic
31#342C2214yes—
32C2×#41—22no—
33C2×#41—22no—
34C2×#41—22no—
35C2×#41—22no—
36C2×#41—48no—
37C22×Q8D412yes—
38C22×D4D412yes—
39C22×D4C2314yes—
40C23×C4D414yes—
41C23×C4C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C4⋊Q8—22no—
45#127—22no—
46#124—24no—
47#123—28no—
48#123—48no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51#121—24no—
52#121—24no—
53#121—24no—
54#121—24no—
55#120—28no—
56C4×D4—24no—
57C4×D4—24no—
58#117—22no—
59#117—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4D411yescharacteristic
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4C2314yes—
71C2×C22⋊C4C2312yes—
72C2×C22⋊C4C2312yes—
73C2×C42D412yes—
74#99—48no—
75#95—22no—
76#95—21no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—21no—
81#95—24no—
82#95—22no—
83C24C2×D414yes—
84C24C2411yescharacteristic
85#41—48no—
86#41—48no—
87#41—48no—
88#41—48no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—44no—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—28no—
95C2×D4—28no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—44no—
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4C2×D414yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—44no—
126C4⋊C4—48no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C22⋊C4—28no—
131C22⋊C4—28no—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—28no—
135C22⋊C4—48no—
136C22⋊C4—48no—
137C42—22no—
138C42—22no—
139C23—24no—
140C23—24no—
141C23—24no—
142C23—41no—
143C23—41no—
144C23—44no—
145C23—44no—
146C23C22×D412yes—
147C23C22×D411yescharacteristic
148C23H(5,𝔽2)14yes—
149Q8—44no—
150Q8—44no—
151D4—44no—
152D4—44no—
153D4—48no—
154D4—48no—
155D4—84no—
156D4—84no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—48no—
169C2×C4—48no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—41no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4#12011yescharacteristic
185C2×C4#12011yescharacteristic
186C2×C4#12012yes—
187C22—21no—
188C22—22no—
189C22—21no—
190C22—22no—
191C22—44no—
192C22—48no—
193C22—44no—
194C22—44no—
195C22—44no—
196C22—84no—
197C22C2×#12011yescharacteristic
198C22#53414yes—
199C22#53512yes—
200C4—21no—
201C4—22no—
202C4—21no—
203C4—44no—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—84no—
208C4—82no—
209C4—81no—
210C2—24no—
211C2—22no—
212C2—84no—
213C2#261311yescharacteristic
214C2#302511yescharacteristic
215C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#3449C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2484C2semidirect
3. #2406C2semidirect
4. #2044C2semidirect
5. #2022C2semidirect
6. #1820C2semidirect
7. #1782C2semidirect

Maximal subgroups

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

subgroupindex
#34492
#24842
#24062
#20442
#20222
#18202
#17822