Finite Group Databasebeta

← #23934

Subgroups — #23934

Order 256.

636 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#533C211yesmaximal, characteristic
2C2×#451C212yesmaximal
3C2×#391C211yesmaximal, characteristic
4#2024C212yesmaximal, complemented
5#2016C212yesmaximal, complemented
6#1878C214yesmaximal, complemented
7#1872C212yesmaximal
8#1817C211yesmaximal, characteristic
9C23×Q8C2211yescharacteristic
10C22×Q16C2214yes—
11#533—28no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2212yes—
17C2×C4×Q8—21no—
18C2×C4×Q8C2212yes—
19C2×#117—21no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#451—28no—
23C2×#103C2214yes—
24C2×#103C2214yes—
25C2×C22⋊C8C2212yes—
26#391—21no—
27#391—21no—
28#391C2212yes—
29#391C2212yes—
30C2×#95—21no—
31C2×#95—21no—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—24no—
36C22×Q8—21no—
37C22×Q8—21no—
38C22×Q8—24no—
39C22×Q8—42no—
40C22×Q8C2312yes—
41C22×Q8C2312yes—
42C23×C4—21no—
43C23×C4—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C2×Q16—28no—
48C2×Q16—28no—
49C2×Q16—44no—
50C22×C8C2314yes—
51C4⋊Q8—24no—
52C4⋊Q8—24no—
53#125—28no—
54C22⋊Q8—28no—
55C22⋊Q8—24no—
56C22⋊Q8D418yes—
57C4×Q8—28no—
58C4×Q8—24no—
59C4×Q8—24no—
60#117—28no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C42—22no—
72C2×C42—24no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#103—28no—
78#103—44no—
79C22⋊C8—24no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95—22no—
84C24C2×D411yescharacteristic
85C2×Q8—24no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—24no—
90C2×Q8—24no—
91C2×Q8—24no—
92C2×Q8—24no—
93C2×Q8—416no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8C2×D414yes—
99C2×Q8#4114yes—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4Q1614yescomplemented
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D414yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125Q16—416no—
126C2×C8—24no—
127C2×C8—24no—
128C2×C8—24no—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—28no—
132C4⋊C4—24no—
133C4⋊C4—28no—
134C4⋊C4—22no—
135C4⋊C4—24no—
136C4⋊C4—42no—
137C4⋊C4C2×D414yes—
138C22⋊C4—28no—
139C42—28no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23#12012yes—
149C23C2×Q1612yes—
150C23#13712yes—
151C23C22×D411yescharacteristic
152Q8—28no—
153Q8—28no—
154Q8—48no—
155Q8—48no—
156Q8—44no—
157Q8—44no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—28no—
167C2×C4—28no—
168C2×C4—21no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—44no—
175C2×C4—48no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—84no—
186C2×C4#12012yes—
187C2×C4C2×Q1612yes—
188C2×C4C2×Q1612yes—
189C2×C4#13712yes—
190C2×C4C22×D412yes—
191C2×C4C2×#4112yes—
192C2×C4H(5,𝔽2)12yes—
193C8—44no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22#45114yes—
201C22C2×#12011yescharacteristic
202C22#52511yescharacteristic
203C22#53511yescharacteristic
204C22C22×Q1611yescharacteristic
205C22C2×#13711yescharacteristic
206C22C2×#13711yescharacteristic
207C22#57511yescharacteristic
208C4—22no—
209C4—22no—
210C4—22no—
211C4—22no—
212C4—44no—
213C4—44no—
214C4—41no—
215C4—44no—
216C4—82no—
217C4#54614yes—
218C2—24no—
219C2#240411yescharacteristic
220C2C2×#45111yescharacteristic
221C2#300711yescharacteristic
222C2#310411yescharacteristic
223C2#311311yescharacteristic
224C2#320911yescharacteristic
225C2#321511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2024C2semidirect
5. #2016C2semidirect
6. #1878C2semidirect
117. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4512
C2×#3912
#20242
#20162
#18782
#18722
#18172