Finite Group Databasebeta

← #32700

Subgroups — #32700

Order 256.

389 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 89 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#2104C212yesmaximal, complemented
2#2032C212yesmaximal, complemented
3#2029C211yesmaximal, characteristic, complemented
4#2021C211yesmaximal, characteristic
5#1913C212yesmaximal, complemented
6#1898C212yesmaximal, complemented
7#1889C212yesmaximal, complemented
8#1828C212yesmaximal, complemented
9#1819C211yesmaximal, characteristic, complemented
10#543C2211yescharacteristic
11#534C2211yescharacteristic
12C2×#120C2212yes—
13#518C2212yes—
14C22×C22⋊C4C2211yescharacteristic
15C2×#99C2212yes—
16C2×#99C2212yes—
17C2×#99C2212yes—
18C2×#99C2212yes—
19#379C2211yescharacteristic
20#379C2211yescharacteristic
21#379C2212yes—
22#342C2212yes—
23#342C2212yes—
24#342C2211yescharacteristic
25#342C2212yes—
26#342C2212yes—
27#342C2211yescharacteristic
28#342C2212yes—
29#342C2212yes—
30#342C2212yes—
31C25C2311yescharacteristic
32C22×D4—22no—
33C22×D4C2312yes—
34C23×C4—21no—
35#123—22no—
36C22⋊Q8—42no—
37C22⋊Q8—41no—
38#121—41no—
39#120—22no—
40#120—22no—
41#120—22no—
42#120—42no—
43C4×D4—42no—
44C4×D4—42no—
45#117—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—41no—
50C2×C22⋊C4—41no—
51C2×C22⋊C4—41no—
52C2×C22⋊C4—42no—
53C2×C22⋊C4—41no—
54C2×C22⋊C4C2312yes—
55C2×C22⋊C4C2312yes—
56C2×C22⋊C4C2312yes—
57C2×C22⋊C4C2312yes—
58C2×C22⋊C4C2311yescharacteristic
59C2×C22⋊C4C2311yescharacteristic
60C2×C22⋊C4C2312yes—
61#99—42no—
62#99—42no—
63#99—42no—
64#99—42no—
65C24—22no—
66C24—21no—
67C24—21no—
68C24—22no—
69C24—82no—
70C24#4111yescharacteristic
71C24#4111yescharacteristic
72C24#4112yes—
73C24C2411yescharacteristic
74C2×Q8—41no—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—42no—
82C2×D4—42no—
83C2×D4—41no—
84C2×D4—84no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—42no—
95C22×C4—82no—
96C22×C4#4112yes—
97C4⋊C4—41no—
98C4⋊C4—42no—
99C4⋊C4—43no—
100C22⋊C4—22no—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—42no—
117C22⋊C4—82no—
118C22⋊C4—84no—
119C22⋊C4—82no—
120C42—42no—
121C23—21no—
122C23—22no—
123C23—21no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23—44no—
132C23—44no—
133C23—42no—
134C23—41no—
135C23—41no—
136C23—81no—
137C23—81no—
138C23—81no—
139C23—81no—
140C23—82no—
141C23—82no—
142C23—82no—
143C23C2×#4112yes—
144C23C2×#4111yescharacteristic
145C23H(5,𝔽2)11yescharacteristic
146C23H(5,𝔽2)12yes—
147C23H(5,𝔽2)11yescharacteristic
148Q8—161no—
149D4—42no—
150D4—42no—
151D4—42no—
152D4—42no—
153D4—84no—
154D4—84no—
155D4—161no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—162no—
176C22—22no—
177C22—24no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—41no—
188C22—42no—
189C22—41no—
190C22—42no—
191C22—41no—
192C22—82no—
193C22—84no—
194C22—82no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22#53812yes—
199C22#53811yescharacteristic
200C22#55111yescharacteristic
201C22#55112yes—
202C22#56011yescharacteristic
203C4—42no—
204C4—82no—
205C4—82no—
206C4—82no—
207C4—82no—
208C4—81no—
209C4—81no—
210C4—82no—
211C2—21no—
212C2—21no—
213C2—22no—
214C2—22no—
215C2—21no—
216C2—42no—
217C2—81no—
218C2—82no—
219C2#276011yescharacteristic
subgroupG/H
center, socleC2#2760
commutator, Frattini, cosocleC24C24
prime radical#2021C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2104C2semidirect
2. #2032C2semidirect
3. #2029C2semidirect
5. #1913C2semidirect
6. #1898C2semidirect
7. #1889C2semidirect
8. #1828C2semidirect
9. #1819C2semidirect

Maximal subgroups

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

subgroupindex
#21042
#20322
#20292
#20212
#19132
#18982
#18892
#18282
#18192