Finite Group Databasebeta

← #8441

Subgroups — #8441

Order 256.

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

normal subgroups only

This group has 432 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#3483C211yesmaximal, characteristic
2#1962C211yesmaximal, characteristic, complemented
3C4×#104C212yesmaximal, complemented
4#1525C211yesmaximal, characteristic, complemented
5#1489C212yesmaximal, complemented
6Q82C411yescharacteristic, complemented
7#555—22no—
8#555—22no—
9#555C411yescharacteristic, complemented
10#550—22no—
11#550C412yescomplemented
12#532—22no—
13#532—22no—
14#532C2211yescharacteristic
15C4×#41—22no—
16C4×#41C2212yes—
17#423C2211yescharacteristic
18C2×#104—22no—
19C2×#104—22no—
20C4×M16C2212yes—
21C43C2211yescharacteristic
22#339—22no—
23#329—22no—
24Q8⋊C8—22no—
25D4⋊C8—22no—
26C2×#41—22no—
27C2×#41—22no—
28C2×#41D412yes—
29C2×#41D412yes—
30C2×M16—22no—
31C2×M16—21no—
32C4⋊Q8—22no—
33C4⋊Q8—22no—
34C4⋊Q8C2×C411yescharacteristic
35#127—22no—
36#127—22no—
37#127C2×C411yescharacteristic
38#124—22no—
39#124—24no—
40#124—22no—
41#124D412yes—
42#121—24no—
43#121—24no—
44#121—22no—
45#121—24no—
46#121—22no—
47#121—22no—
48C4×Q8—22no—
49C4×Q8C2×C412yes—
50C4×Q8D412yes—
51C4×D4—24no—
52C4×D4—24no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4C2×C412yes—
58C4×D4D412yes—
59#117—24no—
60#117—22no—
61#117—22no—
62#117D412yes—
63C2×C42—22no—
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—
73#41—24no—
74#41—42no—
75#41—42no—
76#41—42no—
77#41—44no—
78#41—44no—
79#41—82no—
80#41—82no—
81C2×Q8—22no—
82C2×Q8C22⋊C412yes—
83C2×Q8C22⋊C412yes—
84C2×D4—24no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—24no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4C22⋊C412yes—
91C2×D4C22⋊C412yes—
92C22×C4—24no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—41no—
98C22×C4C2×D412yes—
99C22×C4C2×D411yescharacteristic
100M16—24no—
101M16—42no—
102M16—42no—
103C2×C8—42no—
104C2×C8—41no—
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—24no—
116C22⋊C4—22no—
117C42—22no—
118C42—22no—
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#12011yescharacteristic
132Q8—22no—
133Q8—42no—
134Q8—82no—
135Q8#10414yescomplemented
136D4—22no—
137D4—22no—
138D4—44no—
139D4—42no—
140D4—48no—
141D4—42no—
142D4—42no—
143D4—82no—
144D4—82no—
145D4—84no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—44no—
155C2×C4—48no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4C2×C22⋊C411yescharacteristic
165C2×C4C2×C22⋊C412yes—
166C2×C4#12012yes—
167C2×C4#12011yescharacteristic
168C8—44no—
169C8—81no—
170C22—21no—
171C22—22no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—44no—
176C22—42no—
177C22#37911yescharacteristic
178C4—21no—
179C4—22no—
180C4—24no—
181C4—44no—
182C4—42no—
183C4—42no—
184C4—42no—
185C4C2×#10412yes—
186C4#42112yes—
187C4#45311yescharacteristic
188C4#45511yescharacteristic
189C2—41no—
190C2—82no—
191C2—82no—
192C2#178912yes—
193C2#179211yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3483C2
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. Q82C4semidirect
9. #555C4semidirect
11. #550C4semidirect
135. Q8#104semidirect

Maximal subgroups

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

subgroupindex
#34832
#19622
C4×#1042
#15252
#14892