Finite Group Databasebeta

← #9479

Subgroups — #9479

Order 256.

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

normal subgroups only

This group has 419 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#2404C211yesmaximal, characteristic
2#2312C211yesmaximal, characteristic, complemented
3#1789C212yesmaximal, complemented
4#1462C211yesmaximal, characteristic, complemented
5#1284C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C2×#120—21no—
8C22×C22⋊C4C2211yescharacteristic
9C2×#104—24no—
10#407C2212yes—
11#393C412yescomplemented
12#392C412yescomplemented
13#388—24no—
14#386—21no—
15#386—22no—
16#381C2211yescharacteristic
17#380—21no—
18#379—21no—
19C4×C22⋊C4C2212yes—
20#323—22no—
21C25D411yescharacteristic
22C2×#41—22no—
23C2×#41—22no—
24C2×#41—44no—
25C22×Q8C2×C411yescharacteristic
26C22×D4—21no—
27C22×D4—22no—
28C22×D4C2×C411yescharacteristic
29C23×C4—21no—
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C2×M16—22no—
33#120—44no—
34C2×C4⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C412yes—
44C2×C22⋊C4D412yes—
45C2×C42D414yescomplemented
46#104—48no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49#95—22no—
50#95—22no—
51#95—22no—
52#95—21no—
53#95C2×C412yes—
54C24—21no—
55C24—21no—
56C24—21no—
57C24—42no—
58C24—42no—
59C24—81no—
60C24C22⋊C412yes—
61C24C2×D411yescharacteristic
62#41—28no—
63#41—44no—
64#41—44no—
65#41—44no—
66C2×Q8—22no—
67C2×Q8—42no—
68C2×D4—24no—
69C2×D4—22no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—44no—
75C2×D4—44no—
76C2×D4—82no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—24no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—81no—
87C22×C4C22⋊C412yes—
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D412yes—
90C22×C4C2×D411yescharacteristic
91C22×C4#4112yes—
92M16—44no—
93C2×C8—42no—
94C4⋊C4—82no—
95C22⋊C4—24no—
96C22⋊C4—24no—
97C22⋊C4—44no—
98C22⋊C4—44no—
99C22⋊C4—42no—
100C22⋊C4—82no—
101C22⋊C4—82no—
102C22⋊C4—82no—
103C42—44no—
104C23—22no—
105C23—22no—
106C23—22no—
107C23—22no—
108C23—22no—
109C23—22no—
110C23—41no—
111C23—42no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23—41no—
116C23—42no—
117C23—42no—
118C23—41no—
119C23—42no—
120C23—42no—
121C23—81no—
122C23—81no—
123C23—88no—
124C23C2×C22⋊C411yescharacteristic
125C23#12011yescharacteristic
126C23#12311yescharacteristic
127Q8—24no—
128Q8—42no—
129D4—24no—
130D4—42no—
131D4—44no—
132D4—44no—
133D4—88no—
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—48no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—44no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—44no—
151C2×C4—81no—
152C2×C4C4×D412yes—
153C2×C4#12112yes—
154C8—82no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—44no—
163C22—44no—
164C22—44no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—82no—
170C22—82no—
171C22—84no—
172C22—84no—
173C22—81no—
174C22—81no—
175C22#38611yescharacteristic
176C22#42112yes—
177C22B(4,𝔽2)12yes—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—84no—
183C4—82no—
184C2—22no—
185C2—22no—
186C2—41no—
187C2—82no—
188C2—82no—
189C2#188911yescharacteristic
190C2#189711yescharacteristic
191C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2404C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2312C2semidirect
3. #1789C2semidirect
4. #1462C2semidirect
5. #1284C2semidirect
11. #393C4semidirect
12. #392C4semidirect
45. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
#24042
#23122
#17892
#14622
#12842