Finite Group Databasebeta

← #23989

Subgroups — #23989

Order 256.

605 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 123 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#3491C211yesmaximal, characteristic
2#2614C212yesmaximal, complemented
3#2310C211yesmaximal, characteristic, complemented
4#2030C212yesmaximal, complemented
5#2022C212yesmaximal
6#1899C212yesmaximal, complemented
7#1890C214yesmaximal, complemented
8#1820C211yesmaximal, characteristic, complemented
9C23×Q8C2211yescharacteristic
10#552—24no—
11#551—24no—
12#543C2211yescharacteristic
13#539—22no—
14#534C2211yescharacteristic
15#529—22no—
16C2×#123C2214yes—
17C2×#123C2212yes—
18#518C2212yes—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#99C2214yes—
22#395—24no—
23#393—24no—
24#391—22no—
25#391—22no—
26#381C2212yes—
27#381C2212yes—
28#381C2214yes—
29#381C2212yes—
30#342C2212yes—
31#342C2212yes—
32#342C2214yes—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8—24no—
36C22×Q8—24no—
37C22×Q8—42no—
38C22×D4C2312yes—
39C23×C4C2312yes—
40C23×C4C2311yescharacteristic
41#126—24no—
42#126—24no—
43#125—22no—
44#125—24no—
45#124—22no—
46#123—24no—
47#123—22no—
48#123—22no—
49#123—24no—
50#123—22no—
51#123—28no—
52#123—44no—
53C22⋊Q8—24no—
54C22⋊Q8—22no—
55C22⋊Q8—24no—
56#121—22no—
57#120—22no—
58C4×Q8—22no—
59C4×D4—22no—
60C4×D4—24no—
61C4×D4—24no—
62#117—24no—
63#117—24no—
64#117—22no—
65#117—42no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4D412yes—
69C2×C4⋊C4D414yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4D414yes—
73C2×C22⋊C4C2312yes—
74C2×C22⋊C4C2314yes—
75C2×C22⋊C4C2311yescharacteristic
76C2×C22⋊C4C2311yescharacteristic
77C2×C22⋊C4C2312yes—
78C2×C42—21no—
79C2×C42D412yes—
80#99—44no—
81#95—24no—
82#95—24no—
83#95—28no—
84#95—24no—
85C24C2×D412yes—
86C24C2411yescharacteristic
87C2×Q8—24no—
88C2×Q8—24no—
89C2×Q8—22no—
90C2×Q8—416no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4C2×D412yes—
110C22×C4C2×D414yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D414yes—
116C22×C4#4114yes—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—48no—
125C4⋊C4—42no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—22no—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—48no—
138C42—24no—
139C42—22no—
140C42—22no—
141C42—42no—
142C23—22no—
143C23—24no—
144C23—42no—
145C23—44no—
146C23C22×D411yescharacteristic
147C23C22×D412yes—
148C23C2×#4112yes—
149C23H(5,𝔽2)12yes—
150Q8—48no—
151Q8—48no—
152Q8—82no—
153D4—42no—
154D4—42no—
155D4—44no—
156D4—82no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—28no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—28no—
168C2×C4—24no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—48no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—84no—
180C2×C4#12012yes—
181C2×C4#12012yes—
182C22—22no—
183C22—24no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—44no—
188C22—84no—
189C22C2×#12011yescharacteristic
190C22#52511yescharacteristic
191C22#53511yescharacteristic
192C22#54614yes—
193C4—22no—
194C4—22no—
195C4—44no—
196C4—42no—
197C4—42no—
198C4—44no—
199C4—42no—
200C4—82no—
201C4—84no—
202C2—21no—
203C2—24no—
204C2—21no—
205C2—82no—
206C2#240411yescharacteristic
207C2#302812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#534C22
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. #2614C2semidirect
3. #2310C2semidirect
4. #2030C2semidirect
6. #1899C2semidirect
7. #1890C2semidirect
8. #1820C2semidirect

Maximal subgroups

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

subgroupindex
#34912
#26142
#23102
#20302
#20222
#18992
#18902
#18202