Finite Group Databasebeta

← #17013

Subgroups — #17013

Order 256.

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

normal subgroups only

This group has 521 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×#400C211yesmaximal, characteristic
2C2×#398C211yesmaximal, characteristic
3#1443C212yesmaximal
4#1443C212yesmaximal
5#1442C212yesmaximal
6#1442C212yesmaximal
7#1441C212yesmaximal
8#1436C212yesmaximal
9C4⋊#95C211yesmaximal, characteristic
10C22×C4⋊C4C2212yes—
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C42C2211yescharacteristic
16#400—24no—
17#398—24no—
18C2×#95C2212yes—
19C2×#95C2211yescharacteristic
20C2×#95C2212yes—
21C2×#95C2212yes—
22C2×#95C2211yescharacteristic
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C23×C4C2312yes—
34C23×C4C2312yes—
35C23×C4C2312yes—
36C23×C4C2312yes—
37C23×C4C2311yescharacteristic
38C23×C4C2311yescharacteristic
39C23×C4C2312yes—
40C23×C4C2312yes—
41C23×C4C2311yescharacteristic
42C2×C4⋊C4—22no—
43C2×C4⋊C4—24no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—41no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4D414yes—
57C2×C42—22no—
58C2×C42—41no—
59C2×C42—41no—
60#95—22no—
61#95—24no—
62#95—22no—
63#95—24no—
64#95—24no—
65#95—24no—
66#95—24no—
67#95—24no—
68#95—24no—
69#95—24no—
70#95—24no—
71#95—42no—
72#95—42no—
73#95—42no—
74#95—42no—
75C24C2411yescharacteristic
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
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×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4#4114yes—
132C22×C4#4114yes—
133C22×C4#4112yes—
134C22×C4#4112yes—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4—48no—
140C42—44no—
141C23C22×D411yescharacteristic
142C23C2×#4111yescharacteristic
143C23C2×#4111yescharacteristic
144C23C2×#4112yes—
145C23C2×#4112yes—
146C23H(5,𝔽2)11yescharacteristic
147C23H(5,𝔽2)11yescharacteristic
148C23H(5,𝔽2)11yescharacteristic
149C232−1+411yescharacteristic
150C232−1+412yes—
151C232−1+412yes—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
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—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4#12414yes—
211C22C2×#12411yescharacteristic
212C22#52912yes—
213C22#53411yescharacteristic
214C22#53711yescharacteristic
215C22#53811yescharacteristic
216C22#53811yescharacteristic
217C22#53911yescharacteristic
218C22#53911yescharacteristic
219C22#53912yes—
220C22#54112yes—
221C22#54112yes—
222C22#54112yes—
223C22#54212yes—
224C22#54712yes—
225C22#54812yes—
226C22#55212yes—
227C22#55212yes—
228C22#55312yes—
229C22#55511yescharacteristic
230C22#55911yescharacteristic
231C22#56311yescharacteristic
232C22#56312yes—
233C22#56311yescharacteristic
234C4—24no—
235C4—42no—
236C4—42no—
237C4—44no—
238C4—44no—
239C4—82no—
240C4—82no—
241C4—82no—
242C4—82no—
243C2#248912yes—
244C2#259311yescharacteristic
245C2#263911yescharacteristic
246C2#265512yes—
247C2#272411yescharacteristic
248C2#272512yes—
249C2#273111yescharacteristic
250C2#276911yescharacteristic
251C2#277012yes—
252C2#279411yescharacteristic
253C2#283311yescharacteristic
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
C2×#3982
#14432
#14422
#14412
#14362
C4⋊#952