Finite Group Databasebeta

← #28551

Subgroups — #28551

Order 256.

543 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 139 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×#466C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic, complemented
3#2416C212yesmaximal, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2075C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1934C212yesmaximal, complemented
9#1893C212yesmaximal, complemented
10#1874C212yesmaximal, complemented
11C2×#340C211yesmaximal, characteristic, complemented
12C22×D8C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×#121—22no—
16C2×#121C2212yes—
17C2×#121C2212yes—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20#466—24no—
21C8×D4—24no—
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#396—22no—
31#394—22no—
32#390C2212yes—
33#389C2212yes—
34#386—22no—
35#380C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×D4—22no—
42C22×D4—22no—
43C22×D4D411yescharacteristic
44C22×D4C2312yes—
45C23×C4—22no—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C2×D8—22no—
49C2×D8—42no—
50C22×C8—22no—
51C22×C8—42no—
52C22×C8—42no—
53C22×C8D414yescomplemented
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58#121—24no—
59#121—24no—
60#121—24no—
61#121—42no—
62#121—48no—
63C4×D4—22no—
64C4×D4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4D411yescharacteristic
79C2×C22⋊C4D411yescharacteristic
80C2×C42C2311yescharacteristic
81#107—22no—
82C4⋊C8—22no—
83#102—24no—
84#102—24no—
85#102—42no—
86C22⋊C8—22no—
87C22⋊C8—22no—
88C4×C8—22no—
89#95—22no—
90#95—22no—
91C24—22no—
92C24C2×D411yescharacteristic
93C24C2×D411yescharacteristic
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—82no—
108C2×D4D814yescomplemented
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—82no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4#4112yes—
127C22×C4C2411yescharacteristic
128D8—44no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—44no—
136C2×C8—44no—
137C2×C8C2×D412yes—
138C2×C8C2×D412yes—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4#4114yes—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—44no—
151C22⋊C4—44no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C22⋊C4—82no—
155C42—21no—
156C42—21no—
157C23—22no—
158C23—22no—
159C23—21no—
160C23—22no—
161C23—21no—
162C23—22no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23#12112yes—
171C23C2×D812yes—
172C23C2×D812yes—
173C23#13512yes—
174C23C22×D411yescharacteristic
175D4—28no—
176D4—42no—
177D4—42no—
178D4—42no—
179D4—42no—
180D4—88no—
181D4—88no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—88no—
205C2×C4#12112yes—
206C2×C4C2×D812yes—
207C2×C4C22×D411yescharacteristic
208C2×C4C2×#4112yes—
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C8—22no—
212C8—22no—
213C8—41no—
214C8—41no—
215C22—24no—
216C22—24no—
217C22—22no—
218C22—22no—
219C22—24no—
220C22—22no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22#46614yes—
232C22C2×#12111yescharacteristic
233C22#54612yes—
234C22C22×D811yescharacteristic
235C22C2×#13511yescharacteristic
236C22#57712yes—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—41no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4#53211yescharacteristic
245C4#53411yescharacteristic
246C4#55312yes—
247C2—24no—
248C2—42no—
249C2—162no—
250C2#254311yescharacteristic
251C2C2×#46611yescharacteristic
252C2#306011yescharacteristic
253C2#329412yes—
254C2#330112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#466C2semidirect
2. C2×C8×D4C2semidirect
3. #2416C2semidirect
4. #2086C2semidirect
5. #2075C2semidirect
6. #2054C2semidirect
7. #1999C2semidirect
8. #1934C2semidirect
9. #1893C2semidirect
10. #1874C2semidirect
11. C2×#340C2semidirect
53. C22×C8D4semidirect
108. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×C8×D42
#24162
#20862
#20752
#20542
#19992
#19342
#18932
#18742
C2×#3402