Finite Group Databasebeta

← #25819

Subgroups — #25819

Order 256.

539 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 159 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×#125C211yesmaximal, characteristic
2C2×#501C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic
4#2059C214yesmaximal, complemented
5#1992C212yesmaximal, complemented
6#1853C212yesmaximal
7#1852C212yesmaximal
8#1749C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#125—28no—
11C2×#125—21no—
12C2×#125—21no—
13C2×#125C2212yes—
14C2×#125C2212yes—
15C2×C22⋊Q8—21no—
16C22×C4⋊C4—21no—
17C22×C4⋊C4—21no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#501—28no—
22C2×#107C2214yes—
23C2×#106C2214yes—
24C2×C22⋊C8C2212yes—
25C2×#97C2212yes—
26#391—21no—
27#389—21no—
28#389C2212yes—
29#384C2211yescharacteristic
30#380—21no—
31#380C2212yes—
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8—21no—
36C23×C4—21no—
37C23×C4—21no—
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8C2314yes—
42C4⋊Q8—24no—
43#125—28no—
44#125—28no—
45#125—24no—
46#125—24no—
47#125—24no—
48#125D418yes—
49C22⋊Q8—44no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—28no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C42—24no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—28no—
75#106—28no—
76C22⋊C8Q818yescomplemented
77#97—24no—
78#95—22no—
79#95—22no—
80C24C2×D411yescharacteristic
81C2×Q8—42no—
82C2×Q8—42no—
83C2×Q8—42no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—24no—
87C22×C4—24no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×Q814yes—
108C22×C4C2411yescharacteristic
109C2×C8—24no—
110C2×C8—24no—
111C2×C8C2×Q818yes—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—28no—
115C4⋊C4—28no—
116C4⋊C4—28no—
117C4⋊C4—28no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4—42no—
126C4⋊C4C2×D414yes—
127C4⋊C4C2×D414yes—
128C22⋊C4—44no—
129C42—22no—
130C42—22no—
131C42—22no—
132C42C2×D414yes—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23C4⋊Q812yes—
138C23AGL(1,ℤ/8)12yes—
139C23#13712yes—
140C23C22×D411yescharacteristic
141Q8—84no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—28no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—48no—
160C2×C4—48no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4#12012yes—
172C2×C4#12012yes—
173C2×C4C4⋊Q812yes—
174C2×C4C22×D412yes—
175C2×C4C22×Q812yes—
176C2×C42−1+412yes—
177C8—28no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22#50114yescomplemented
185C22C2×#12011yescharacteristic
186C22C2×C4⋊Q811yescharacteristic
187C22#53611yescharacteristic
188C22C2×AGL(1,ℤ/8)11yescharacteristic
189C22C2×#13711yescharacteristic
190C22#57511yescharacteristic
191C22#57511yescharacteristic
192C4—22no—
193C4—22no—
194C4—22no—
195C4—41no—
196C4—44no—
197C4—41no—
198C4—44no—
199C4—82no—
200C4—82no—
201C4D4×Q814yes—
202C2—24no—
203C2#243411yescharacteristic
204C2#300611yescharacteristic
205C2#300711yescharacteristic
206C2C2×#50111yescharacteristic
207C2#316611yescharacteristic
208C2#323411yescharacteristic
209C2#323611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#501C2semidirect
4. #2059C2semidirect
5. #1992C2semidirect
76. C22⋊C8Q8semidirect
184. C22#501semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#5012
#28572
#20592
#19922
#18532
#18522
#17492