Finite Group Databasebeta

← #11880

Subgroups — #11880

Order 256.

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

normal subgroups only

This group has 382 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#1800C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1345C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C2×#121—21no—
7C2×#120—21no—
8C2×C4×D4C2211yescharacteristic
9#423—21no—
10C2×#100—21no—
11C2×C22⋊C8—21no—
12#394C2212yes—
13#392C412yes—
14#390—21no—
15#380C411yescharacteristic
16#379—21no—
17#343C2212yescomplemented
18#324C2212yescomplemented
19#323—21no—
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
30C2×M16—22no—
31C22×C8—41no—
32#121—44no—
33#120—42no—
34#120—42no—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C42D411yescharacteristic
46C4⋊C8—41no—
47C4⋊C8—41no—
48#100—42no—
49C22⋊C8—22no—
50C22⋊C8—41no—
51#95C2×C412yes—
52C24—22no—
53C24—42no—
54C24—44no—
55C24—42no—
56C24—42no—
57C24—82no—
58C24C22⋊C411yescharacteristic
59C24C22⋊C411yescharacteristic
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—44no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—82no—
74C2×D4D812yes—
75C2×D4SD1612yes—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22×C411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88M16—42no—
89M16—42no—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—82no—
93C4⋊C4—41no—
94C4⋊C4—42no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—44no—
98C22⋊C4—42no—
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—
125C23#10212yes—
126C23C2×C22⋊C411yescharacteristic
127D4—42no—
128D4—42no—
129D4—44no—
130D4—44no—
131D4—44no—
132D4—44no—
133D4—88no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—81no—
150C2×C4#9911yescharacteristic
151C2×C4#9911yescharacteristic
152C2×C4C2×D811yescharacteristic
153C2×C4C2×SD1611yescharacteristic
154C8—82no—
155C8—82no—
156C22—21no—
157C22—21no—
158C22—22no—
159C22—22no—
160C22—42no—
161C22—42no—
162C22—42no—
163C22—44no—
164C22—44no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—44no—
170C22—81no—
171C22—82no—
172C22—82no—
173C22—816no—
174C22—82no—
175C22—82no—
176C22C2×#9911yescharacteristic
177C22C2×#10211yescharacteristic
178C22#42111yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—82no—
186C4AΣL(1,𝔽16)12yes—
187C2—21no—
188C2—21no—
189C2—42no—
190C2—42no—
191C2—84no—
192C2#151911yescharacteristic
193C2C2×AΣL(1,𝔽16)11yescharacteristic
194C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1800C2semidirect
3. #1459C2semidirect
4. #1345C2semidirect
17. #343C22semidirect
18. #324C22semidirect

Maximal subgroups

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

subgroupindex
#24082
#18002
#14592
#13452