Finite Group Databasebeta

← #10244

Subgroups — #10244

Order 256.

277 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 61 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#3206C211yesmaximal, characteristic
2#3037C211yesmaximal, characteristic
3#2568C211yesmaximal, characteristic, complemented
4#1548C211yesmaximal, characteristic, complemented
5#1502C211yesmaximal, characteristic
6#1479C211yesmaximal, characteristic, complemented
7#1303C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4×Q8C2211yescharacteristic
12#492—21no—
13#487—21no—
14#464—22no—
15#462—21no—
16#461—22no—
17#460—21no—
18#448—22no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#417—21no—
23#400—22no—
24#391—22no—
25#391—21no—
26#382—21no—
27C4×C4⋊C4C2211yescharacteristic
28#333—22no—
29#327—22no—
30Q8⋊C8—22no—
31C2×#41—21no—
32C22×Q8D411yescharacteristic
33C22×Q8D412yes—
34#137—42no—
35#137—42no—
36C2×Q16—22no—
37C2×SD16—22no—
38C2×SD16—24no—
39C2×M16—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8D411yescharacteristic
45#127D411yescharacteristic
46#124—21no—
47#121—21no—
48#121—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×D4—21no—
54C4×D4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C42—21no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D412yes—
66C4⋊C8D412yes—
67#103—22no—
68#103—24no—
69#102—22no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C4×C8—21no—
73C4×C8—21no—
74#95—24no—
75#95—24no—
76#95D412yes—
77#41—41no—
78#41—41no—
79#41—41no—
80#41—41no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—22no—
85C2×Q8—21no—
86C2×Q8—22no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—44no—
90C2×Q8SD1612yescomplemented
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—41no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103Q16—44no—
104SD16—44no—
105SD16—84no—
106M16—41no—
107M16—41no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—41no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—42no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—41no—
125C42—41no—
126C42—41no—
127C42—41no—
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42#4111yescharacteristic
132C23—41no—
133C23#12011yescharacteristic
134Q8—22no—
135Q8—22no—
136Q8—41no—
137Q8—41no—
138Q8—41no—
139Q8—41no—
140Q8—44no—
141Q8—82no—
142D4—41no—
143D4—41no—
144D4—42no—
145D4—81no—
146D4—81no—
147D4—81no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—44no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4#12011yescharacteristic
164C2×C4#12011yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12111yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12711yescharacteristic
169C2×C4C2×SD1611yescharacteristic
170C2×C4#13511yescharacteristic
171C2×C4AGL(1,ℤ/8)11yescharacteristic
172C2×C4#13711yescharacteristic
173C8—42no—
174C8—42no—
175C8—81no—
176C22—21no—
177C22—21no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22#39211yescharacteristic
182C22#44811yescharacteristic
183C22#44911yescharacteristic
184C4—21no—
185C4—21no—
186C4—21no—
187C4—42no—
188C4—82no—
189C4—81no—
190C4—82no—
191C4—81no—
192C4#45011yescharacteristic
193C4#45211yescharacteristic
194C4#45512yes—
195C4#46511yescharacteristic
196C4#46811yescharacteristic
197C2—21no—
198C2—21no—
199C2—161no—
200C2#200011yescharacteristic
201C2#200411yescharacteristic
202C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2568C2semidirect
4. #1548C2semidirect
6. #1479C2semidirect
7. #1303C2semidirect
90. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
#30372
#25682
#15482
#15022
#14792
#13032