Finite Group Databasebeta

← #9049

Subgroups — #9049

Order 256.

272 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 69 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#3100C211yesmaximal, characteristic
2#2932C211yesmaximal, characteristic, complemented
3#1955C211yesmaximal, characteristic, complemented
4#1540C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic, complemented
6#1301C211yesmaximal, characteristic, complemented
7#1281C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#480C412yes—
12#474—21no—
13#459—21no—
14#447C412yes—
15C4×D8—22no—
16#435—22no—
17#435—22no—
18#431C2211yescharacteristic
19#431C2211yescharacteristic
20#423C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22#408—21no—
23C2×C22⋊C8—21no—
24C2×C22⋊C8—21no—
25C2×#97C2211yescharacteristic
26#332—21no—
27#331—21no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×D4C2×C411yescharacteristic
31C23×C4—21no—
32C2×D8—22no—
33#131—42no—
34#131—42no—
35C2×M16—22no—
36C2×M16—21no—
37C22×C8—22no—
38C22×C8—21no—
39C22×C8D411yescharacteristic
40C22×C8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127D411yescharacteristic
43#124—21no—
44#121—21no—
45#121C2×C412yes—
46C4×D4—21no—
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4D412yes—
52C2×C4⋊C4C2×C411yescharacteristic
53C2×C22⋊C4—21no—
54C2×C42C2311yescharacteristic
55#107—22no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8D412yes—
62#102—22no—
63#102—22no—
64C22⋊C8—24no—
65C22⋊C8—21no—
66C22⋊C8—42no—
67C22⋊C8—42no—
68C22⋊C8C2×C412yes—
69#97—21no—
70#97—21no—
71C4×C8—21no—
72C4×C8—21no—
73C24—21no—
74#41—41no—
75#41—41no—
76#41—41no—
77#41—41no—
78C2×Q8—21no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—42no—
85C2×D4C22⋊C412yes—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C22×C411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4#4111yescharacteristic
97D8—44no—
98M16—44no—
99M16—41no—
100M16—41no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—44no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8D812yescomplemented
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4C22⋊C412yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C42#4111yescharacteristic
126C23—22no—
127C23—22no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23C4×D411yescharacteristic
135Q8—41no—
136Q8—41no—
137D4—22no—
138D4—22no—
139D4—42no—
140D4—41no—
141D4—41no—
142D4—42no—
143D4—42no—
144D4—41no—
145D4—41no—
146D4—81no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4C2×C22⋊C411yescharacteristic
160C2×C4C4×D411yescharacteristic
161C2×C4#12011yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12311yescharacteristic
165C2×C4C2×D811yescharacteristic
166C2×C4#13511yescharacteristic
167C2×C4AGL(1,ℤ/8)11yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C8—22no—
170C8—42no—
171C8—42no—
172C8—41no—
173C8—81no—
174C8—81no—
175C22—21no—
176C22—21no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22#38611yescharacteristic
188C22C4×D811yescharacteristic
189C22#44211yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—42no—
195C4—81no—
196C4#44711yescharacteristic
197C4#44911yescharacteristic
198C4#45911yescharacteristic
199C4#46311yescharacteristic
200C2—21no—
201C2—21no—
202C2—82no—
203C2—82no—
204C2#187911yescharacteristic
205C2#188111yescharacteristic
206C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3100C2
LCSG ▷ C22×C4 ▷ 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. #2932C2semidirect
3. #1955C2semidirect
4. #1540C2semidirect
5. #1486C2semidirect
6. #1301C2semidirect
7. #1281C2semidirect
113. C2×C8D8semidirect

Maximal subgroups

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

subgroupindex
#31002
#29322
#19552
#15402
#14862
#13012
#12812