Finite Group Databasebeta

← #29655

Subgroups — #29655

Order 256.

453 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 119 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#3458C211yesmaximal, characteristic
2#3095C212yesmaximal
3#2499C211yesmaximal, characteristic
4#2061C212yesmaximal, complemented
5#2060C212yesmaximal, complemented
6#2028C212yesmaximal, complemented
7#2028C212yesmaximal, complemented
8#1826C212yesmaximal
9#1822C211yesmaximal, characteristic
10#544—22no—
11#544—22no—
12#544C2212yes—
13#544C2212yes—
14#543—22no—
15#533—24no—
16#533—22no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#117C2212yes—
19C2×#117C2211yescharacteristic
20#485—24no—
21#484—22no—
22#483—22no—
23#428C2214yes—
24#419C2214yes—
25#407C2212yes—
26#397—22no—
27#391C2211yescharacteristic
28#391C2211yescharacteristic
29#387—22no—
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32C4×C22⋊C4C2211yescharacteristic
33#343C2212yes—
34#343C2212yes—
35#339C2214yes—
36#339C2214yes—
37C22×Q8C2311yescharacteristic
38C23×C4C2311yescharacteristic
39C2×M16C2314yes—
40C4⋊Q8—22no—
41C4⋊Q8—22no—
42C4⋊Q8—24no—
43C4⋊Q8—24no—
44#125—24no—
45#125—24no—
46#125—24no—
47C22⋊Q8—28no—
48C22⋊Q8—28no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C4×Q8—24no—
54C4×Q8—24no—
55#117—22no—
56#117—22no—
57#117—22no—
58#117—22no—
59#117D414yes—
60#117C2314yes—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4D412yes—
72C2×C42—22no—
73C2×C42D412yes—
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—24no—
78#103—24no—
79#103—24no—
80C22⋊C8—22no—
81C22⋊C8—22no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95—22no—
86C24C2×D411yescharacteristic
87C2×Q8—24no—
88C2×Q8—24no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2411yescharacteristic
106M16—24no—
107M16—24no—
108C2×C8—24no—
109C4⋊C4—28no—
110C4⋊C4—28no—
111C4⋊C4—24no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4C2×D414yes—
122C4⋊C4C2×D414yes—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C42—24no—
129C42—22no—
130C42—22no—
131C42—24no—
132C42—22no—
133C42—22no—
134C23—21no—
135C23—21no—
136C23—41no—
137C23—41no—
138C23#12012yes—
139C23C22×D411yescharacteristic
140Q8—44no—
141Q8—84no—
142Q8—84no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4#12011yescharacteristic
167C2×C4#12011yescharacteristic
168C2×C4C22×D412yes—
169C2×C42−1+412yes—
170C2×C42−1+412yes—
171C8—44no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—81no—
178C22—81no—
179C22C2×#12011yescharacteristic
180C22#53611yescharacteristic
181C22#53611yescharacteristic
182C4—21no—
183C4—21no—
184C4—42no—
185C4—42no—
186C4—44no—
187C4—42no—
188C4—44no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4#53614yes—
193C2—21no—
194C2—21no—
195C2—42no—
196C2#261411yescharacteristic
197C2#301811yescharacteristic
198C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2061C2semidirect
5. #2060C2semidirect
6. #2028C2semidirect
7. #2028C2semidirect

Maximal subgroups

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

subgroupindex
#34582
#30952
#24992
#20612
#20602
#20282
#18262
#18222