Finite Group Databasebeta

← #8440

Subgroups — #8440

Order 256.

442 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 79 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#3481C211yesmaximal, characteristic
2#1962C211yesmaximal, characteristic, complemented
3C4×#104C212yesmaximal, complemented
4#1525C211yesmaximal, characteristic, complemented
5#1489C212yesmaximal, complemented
6#559C411yescharacteristic, complemented
7#554C412yescomplemented
8#553—22no—
9#551—22no—
10#546—22no—
11D42C411yescharacteristic, complemented
12#532C2211yescharacteristic
13#529—22no—
14#525—22no—
15C4×#41C2212yes—
16C2×C4×D4—22no—
17#423C2211yescharacteristic, complemented
18C2×#104—22no—
19C2×#104—22no—
20C4×M16C2212yescomplemented
21C43C2211yescharacteristic, complemented
22#339—22no—
23#329—22no—
24Q8⋊C8—22no—
25D4⋊C8—22no—
26C2×#41D412yes—
27C2×#41D412yes—
28C22×D4—22no—
29C23×C4—22no—
30C2×M16—22no—
31C2×M16—21no—
32C4⋊Q8C2×C411yescharacteristic
33#127C2×C411yescharacteristic
34#126—24no—
35#125—22no—
36#124—22no—
37#124D412yes—
38#123—22no—
39#123—24no—
40#123—22no—
41C22⋊Q8—24no—
42C22⋊Q8—22no—
43#121—24no—
44#121—22no—
45#121—22no—
46#120—22no—
47#120—22no—
48C4×Q8C2×C412yes—
49C4×Q8D412yes—
50C4×D4—24no—
51C4×D4—24no—
52C4×D4—22no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4C2×C412yes—
58C4×D4D412yes—
59#117—22no—
60#117—22no—
61#117D412yes—
62C2×C4⋊C4—22no—
63C2×C22⋊C4—24no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42—21no—
67C2×C42C2311yescharacteristic
68C4⋊C8—22no—
69#104—28no—
70#104—44no—
71#97—22no—
72C4×C8—22no—
73C24—22no—
74#41—24no—
75#41—42no—
76#41—42no—
77#41—42no—
78C2×Q8C22⋊C412yes—
79C2×Q8C22⋊C412yes—
80C2×D4—24no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—44no—
85C2×D4—44no—
86C2×D4C22⋊C412yes—
87C2×D4C22⋊C412yes—
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—41no—
95C22×C4—82no—
96C22×C4—82no—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99M16—24no—
100M16—42no—
101M16—42no—
102C2×C8—42no—
103C2×C8—41no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4C22⋊C412yes—
110C4⋊C4C22⋊C412yes—
111C22⋊C4—24no—
112C22⋊C4—24no—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—24no—
119C42—22no—
120C42—22no—
121C42—22no—
122C42—24no—
123C42—22no—
124C42—42no—
125C42—41no—
126C42C22×C411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D412yes—
129C23—22no—
130C23—22no—
131C23—48no—
132C23—82no—
133C23#12011yescharacteristic
134Q8—22no—
135Q8—22no—
136Q8—42no—
137D4—22no—
138D4—42no—
139D4—44no—
140D4—48no—
141D4—42no—
142D4—42no—
143D4—82no—
144D4#10414yescomplemented
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—44no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—82no—
163C2×C4—84no—
164C2×C4C2×C22⋊C411yescharacteristic
165C2×C4C2×C22⋊C412yes—
166C2×C4#12012yes—
167C2×C4#12011yescharacteristic
168C8—44no—
169C8—81no—
170C22—21no—
171C22—22no—
172C22—24no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—44no—
177C22—88no—
178C22#37911yescharacteristic
179C4—21no—
180C4—22no—
181C4—44no—
182C4—42no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4C2×#10412yes—
187C4#42112yes—
188C4#45311yescharacteristic
189C4#45511yescharacteristic
190C2—41no—
191C2—44no—
192C2—82no—
193C2#178912yes—
194C2#179211yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3481C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1962C2semidirect
3. C4×#104C2semidirect
4. #1525C2semidirect
5. #1489C2semidirect
6. #559C4semidirect
7. #554C4semidirect
11. D42C4semidirect
17. #423C22semidirect
20. C4×M16C22semidirect
21. C43C22semidirect
144. D4#104semidirect

Maximal subgroups

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

subgroupindex
#34812
#19622
C4×#1042
#15252
#14892