Finite Group Databasebeta

← #24023

Subgroups — #24023

Order 256.

607 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 127 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#3486C211yesmaximal, characteristic
2#3093C212yesmaximal, complemented
3#2310C211yesmaximal, characteristic, complemented
4#2038C214yesmaximal, complemented
5#1897C212yesmaximal, complemented
6#1857C212yesmaximal, complemented
7#1793C212yesmaximal, complemented
8#1784C211yesmaximal, characteristic, complemented
9C2×2−1+4C2211yescharacteristic
10C23×Q8C2211yescharacteristic
11D4×Q8—24no—
12#548—24no—
13#536—22no—
14#536C2212yes—
15#525—22no—
16#525C2212yes—
17#519C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20#485—24no—
21#482—24no—
22#428C2214yes—
23C2×#104C2214yes—
24#417C2214yes—
25C2×#101C2212yes—
26C2×#101C2212yes—
27#407C2212yes—
28#391—22no—
29#391C2212yes—
30#391C2212yes—
31#381—22no—
32#339C2214yes—
332−1+4—24no—
342−1+4—42no—
35C2×#41—22no—
36C2×#41—22no—
37C2×#41C2312yes—
38C22×Q8—22no—
39C22×Q8—24no—
40C22×Q8—24no—
41C22×Q8—42no—
42C22×Q8D412yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2311yescharacteristic
45C23×C4D412yes—
46C23×C4C2311yescharacteristic
47C2×M16C2314yes—
48C4⋊Q8—24no—
49C4⋊Q8—22no—
50#124—24no—
51#124—22no—
52#123—24no—
53#123—22no—
54C22⋊Q8—24no—
55C22⋊Q8—28no—
56C22⋊Q8—24no—
57C22⋊Q8D414yes—
58#121—24no—
59#121D414yes—
60#120—22no—
61C4×Q8—24no—
62C4×D4—28no—
63C4×D4—24no—
64#117—22no—
65#117—24no—
66#117—42no—
67#117C2312yes—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—21no—
73C2×C42C2312yes—
74#107—24no—
75#106—24no—
76#104—44no—
77#103—24no—
78#102—24no—
79#101—24no—
80#101—42no—
81C22⋊C8—24no—
82#95—24no—
83#95—24no—
84C24C2×D411yescharacteristic
85#41—24no—
86#41—24no—
87#41—44no—
88#41—48no—
89#41—44no—
90C2×Q8—22no—
91C2×Q8—24no—
92C2×Q8—24no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—42no—
99C2×Q8—416no—
100C2×Q8—48no—
101C2×Q8C2×D412yes—
102C2×Q8#4114yes—
103C2×D4—22no—
104C2×D4—24no—
105C2×D4—22no—
106C2×D4C2×D412yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D414yes—
119C22×C4C2×D412yes—
120C22×C4C2411yescharacteristic
121M16—24no—
122M16—24no—
123C2×C8—24no—
124C4⋊C4—24no—
125C4⋊C4—28no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—42no—
129C4⋊C4C2×D414yes—
130C22⋊C4—28no—
131C22⋊C4—24no—
132C22⋊C4—22no—
133C22⋊C4—24no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C42—24no—
137C42—22no—
138C42—24no—
139C42—42no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—42no—
144C23#12012yes—
145C23C22×D411yescharacteristic
146Q8—28no—
147Q8—24no—
148Q8—48no—
149Q8—44no—
150Q8—44no—
151D4—24no—
152D4—44no—
153D4—48no—
154D4—42no—
155D4—42no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—48no—
167C2×C4—48no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4#12012yes—
174C2×C4#12314yes—
175C2×C4C22×D412yes—
176C2×C4C2×#4112yes—
177C2×C4H(5,𝔽2)12yes—
178C8—44no—
179C22—22no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—44no—
187C22C2×#12011yescharacteristic
188C22C2×#12311yescharacteristic
189C22#53411yescharacteristic
190C4—22no—
191C4—24no—
192C4—42no—
193C4—42no—
194C4—44no—
195C4—82no—
196C4—82no—
197C4#54614yes—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#240511yescharacteristic
203C2#302012yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3486C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3093C2semidirect
3. #2310C2semidirect
4. #2038C2semidirect
5. #1897C2semidirect
6. #1857C2semidirect
7. #1793C2semidirect
8. #1784C2semidirect

Maximal subgroups

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

subgroupindex
#34862
#30932
#23102
#20382
#18972
#18572
#17932
#17842