Finite Group Databasebeta

← #17222

Subgroups — #17222

Order 256.

527 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 159 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×#400C212yesmaximal
2#1444C212yesmaximal
3#1444C212yesmaximal
4#1442C211yesmaximal, characteristic
5#1442C212yesmaximal
6#1442C212yesmaximal
7#1442C211yesmaximal, characteristic
8#1442C211yesmaximal, characteristic
9#1441C212yesmaximal
10C22×C4⋊C4C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16#400—28no—
17C2×#95C2212yes—
18C2×#95C2212yes—
19C2×#95C2212yes—
20C2×#95C2212yes—
21C2×#95C2212yes—
22C2×#95C2211yescharacteristic
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C23×C4C2312yes—
33C23×C4C2312yes—
34C23×C4C2312yes—
35C23×C4C2312yes—
36C23×C4C2311yescharacteristic
37C23×C4C2312yes—
38C23×C4C2311yescharacteristic
39C23×C4C2312yes—
40C23×C4C2311yescharacteristic
41C2×C4⋊C4—24no—
42C2×C4⋊C4—24no—
43C2×C4⋊C4—24no—
44C2×C4⋊C4—41no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—42no—
56#95—24no—
57#95—24no—
58#95—24no—
59#95—24no—
60#95—24no—
61#95—24no—
62#95—24no—
63#95—22no—
64#95—41no—
65#95—42no—
66#95—42no—
67#95—42no—
68#95—42no—
69#95D414yes—
70#95Q814yes—
71C24C2411yescharacteristic
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
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—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4C2×D412yes—
124C22×C4C2×D414yes—
125C22×C4C2×Q814yes—
126C22×C4C2×Q812yes—
127C22×C4#4112yes—
128C22×C4#4114yes—
129C4⋊C4—44no—
130C4⋊C4—48no—
131C4⋊C4—48no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—48no—
135C23C22×D411yescharacteristic
136C23C22×Q811yescharacteristic
137C23C2×#4111yescharacteristic
138C23C2×#4112yes—
139C23H(5,𝔽2)12yes—
140C23H(5,𝔽2)12yes—
141C23H(5,𝔽2)11yescharacteristic
142C232−1+411yescharacteristic
143C232−1+411yescharacteristic
144C232−1+412yes—
145C232−1+411yescharacteristic
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C22#53512yes—
204C22#53711yescharacteristic
205C22#53912yes—
206C22#53911yescharacteristic
207C22#53912yes—
208C22#54112yes—
209C22#54212yes—
210C22#54412yes—
211C22#54411yescharacteristic
212C22#54711yescharacteristic
213C22#54712yes—
214C22D4×Q811yescharacteristic
215C22#55111yescharacteristic
216C22#55312yes—
217C22#55412yes—
218C22#55711yescharacteristic
219C22H(3,𝔽4)11yescharacteristic
220C22H(3,𝔽4)11yescharacteristic
221C22#56212yes—
222C22#56312yes—
223C22#56311yescharacteristic
224C22#56312yes—
225C22#56411yescharacteristic
226C4—42no—
227C4—44no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—44no—
232C4—82no—
233C4—82no—
234C4—82no—
235C2#271812yes—
236C2#272012yes—
237C2#272211yescharacteristic
238C2#272611yescharacteristic
239C2#276311yescharacteristic
240C2#278211yescharacteristic
241C2#280212yes—
242C2#281911yescharacteristic
243C2#283511yescharacteristic
244C2#283912yes—
245C2#284411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#4002
#14442
#14422
#14412