Finite Group Databasebeta

← #11873

Subgroups — #11873

Order 256.

388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 59 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#2408C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic, complemented
3#1460C211yesmaximal, characteristic, complemented
4#1343C214yesmaximal, complemented
5C23×D4C411yescharacteristic, complemented
6C2×#121—21no—
7C2×#120—21no—
8C2×C4×D4C2211yescharacteristic
9#431—21no—
10C2×#99—21no—
11#394C412yescomplemented
12#392C2212yes—
13#390—21no—
14#380—21no—
15#380C411yescharacteristic, complemented
16#379—21no—
17#342C2212yescomplemented
18#323—21no—
19#323C2212yescomplemented
20C25—21no—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4—41no—
24C22×D4—41no—
25C22×D4—44no—
26C22×D4C2×C412yes—
27C22×D4D411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30#121—44no—
31#120—42no—
32#120—42no—
33C4×D4—41no—
34C4×D4—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—41no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4D411yescharacteristic
44C2×C22⋊C4C2311yescharacteristic
45C2×C42D411yescharacteristic
46C4⋊C8—41no—
47#99—42no—
48C22⋊C8—22no—
49C4×C8—41no—
50#95—41no—
51#95C2×C412yes—
52C24—22no—
53C24—42no—
54C24—44no—
55C24—42no—
56C24—42no—
57C24—82no—
58C24C22⋊C411yescharacteristic
59C24C2×D411yescharacteristic
60C2×D4—24no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—42no—
64C2×D4—42no—
65C2×D4—48no—
66C2×D4—44no—
67C2×D4—44no—
68C2×D4—44no—
69C2×D4—44no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—82no—
75C2×D4D812yes—
76C2×D4SD1612yes—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C2×C8—42no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—82no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—44no—
97C22⋊C4—42no—
98C22⋊C4—44no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—82no—
102C42—21no—
103C42—21no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—24no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—42no—
114C23—42no—
115C23—42no—
116C23—48no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—84no—
121C23—88no—
122C23—88no—
123C23—84no—
124C23#10212yes—
125C23C2×C22⋊C411yescharacteristic
126C23C2×D811yescharacteristic
127C23C2×SD1611yescharacteristic
128D4—42no—
129D4—42no—
130D4—44no—
131D4—44no—
132D4—44no—
133D4—44no—
134D4—84no—
135D4—84no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4#9911yescharacteristic
156C2×C4#9911yescharacteristic
157C2×C4#10212yes—
158C8—82no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—22no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—42no—
167C22—42no—
168C22—44no—
169C22—44no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—44no—
175C22—81no—
176C22—82no—
177C22—82no—
178C22—816no—
179C22—82no—
180C22—82no—
181C22C2×#9911yescharacteristic
182C22C2×#10211yescharacteristic
183C22#42111yescharacteristic
184C4—21no—
185C4—22no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—82no—
190C4—162no—
191C4AΣL(1,𝔽16)12yes—
192C2—21no—
193C2—21no—
194C2—41no—
195C2—42no—
196C2—41no—
197C2—84no—
198C2#151911yescharacteristic
199C2C2×AΣL(1,𝔽16)11yescharacteristic
200C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1799C2semidirect
3. #1460C2semidirect
4. #1343C2semidirect
5. C23×D4C4semidirect
11. #394C4semidirect
15. #380C4semidirect
17. #342C22semidirect
19. #323C22semidirect

Maximal subgroups

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

subgroupindex
#24082
#17992
#14602
#13432