Finite Group Databasebeta

← #9475

Subgroups — #9475

Order 256.

479 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 65 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#2273C211yesmaximal, characteristic, complemented
3#1789C212yesmaximal, complemented
4#1462C211yesmaximal, characteristic, complemented
5#1284C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C2×#120—21no—
8C22×C22⋊C4—21no—
9C22×C22⋊C4C2211yescharacteristic
10C2×#104—24no—
11#407C2212yes—
12#393C412yescomplemented
13#392C412yescomplemented
14#386—21no—
15#386—22no—
16#381C2211yescharacteristic
17C4×C22⋊C4—21no—
18C4×C22⋊C4—24no—
19C4×C22⋊C4C2212yes—
20#323—22no—
21C25D411yescharacteristic
22C2×#41—22no—
23C2×#41—22no—
24C2×#41—44no—
25C22×Q8C2×C411yescharacteristic, complemented
26C22×D4—21no—
27C22×D4—22no—
28C22×D4C2×C411yescharacteristic, complemented
29C23×C4—22no—
30C23×C4—21no—
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C2×M16—22no—
34#120—44no—
35C2×C22⋊C4—28no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C412yes—
44C2×C22⋊C4D412yes—
45C2×C42—22no—
46C2×C42D414yescomplemented
47#104—48no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#95—22no—
51#95—22no—
52#95—22no—
53#95—21no—
54#95C2×C412yescomplemented
55C24—21no—
56C24—24no—
57C24—21no—
58C24—21no—
59C24—42no—
60C24—81no—
61C24C22⋊C412yes—
62C24C2×D411yescharacteristic
63#41—28no—
64#41—44no—
65#41—44no—
66#41—44no—
67C2×Q8—22no—
68C2×Q8—42no—
69C2×D4—24no—
70C2×D4—22no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—44no—
76C2×D4—44no—
77C2×D4—82no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—24no—
81C22×C4—28no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—81no—
89C22×C4—82no—
90C22×C4—82no—
91C22×C4C22⋊C412yes—
92C22×C4C22×C411yescharacteristic
93C22×C4C2×D412yes—
94C22×C4C2×D411yescharacteristic
95C22×C4#4112yes—
96M16—44no—
97C2×C8—42no—
98C22⋊C4—28no—
99C22⋊C4—28no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C22⋊C4—44no—
103C22⋊C4—42no—
104C22⋊C4—82no—
105C42—44no—
106C42—82no—
107C23—22no—
108C23—22no—
109C23—24no—
110C23—22no—
111C23—22no—
112C23—22no—
113C23—22no—
114C23—22no—
115C23—41no—
116C23—42no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—41no—
121C23—42no—
122C23—41no—
123C23—42no—
124C23—81no—
125C23—81no—
126C23—88no—
127C23#10414yescomplemented
128C23C2×C22⋊C411yescharacteristic
129C23#12011yescharacteristic
130C23#12311yescharacteristic
131Q8—24no—
132Q8—42no—
133D4—24no—
134D4—42no—
135D4—44no—
136D4—44no—
137D4—88no—
138C2×C4—24no—
139C2×C4—28no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—48no—
146C2×C4—44no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—44no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—81no—
156C2×C4—88no—
157C2×C4C4×D412yes—
158C2×C4#12112yes—
159C8—82no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—24no—
166C22—21no—
167C22—21no—
168C22—44no—
169C22—44no—
170C22—44no—
171C22—44no—
172C22—44no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—82no—
177C22—82no—
178C22—84no—
179C22—81no—
180C22—81no—
181C22#38611yescharacteristic
182C22C2×#10412yes—
183C22B(4,𝔽2)12yes—
184C4—22no—
185C4—22no—
186C4—48no—
187C4—42no—
188C4—42no—
189C4—82no—
190C2—22no—
191C2—22no—
192C2—41no—
193C2—44no—
194C2—82no—
195C2#188911yescharacteristic
196C2#189611yescharacteristic
197C2#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. #2273C2semidirect
3. #1789C2semidirect
4. #1462C2semidirect
5. #1284C2semidirect
12. #393C4semidirect
13. #392C4semidirect
25. C22×Q8C2×C4semidirect
28. C22×D4C2×C4semidirect
46. C2×C42D4semidirect
54. #95C2×C4semidirect
127. C23#104semidirect

Maximal subgroups

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

subgroupindex
#24042
#22732
#17892
#14622
#12842