Finite Group Databasebeta

← #28552

Subgroups — #28552

Order 256.

501 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 127 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#2414C212yesmaximal, complemented
4#2087C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1934C212yesmaximal, complemented
9#1893C212yesmaximal, complemented
10#1875C212yesmaximal, complemented
11#1735C211yesmaximal, characteristic, complemented
12C22×D8C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×#121C2212yes—
16C2×#121C2212yes—
17C2×C4×D4C2211yescharacteristic
18C2×#117C2212yes—
19#466—24no—
20C8×D4—24no—
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#394—22no—
30#392—22no—
31#390—22no—
32#390C2212yes—
33#389C2212yes—
34#388—22no—
35#380C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×D4—22no—
42C22×D4—22no—
43C22×D4D411yescharacteristic
44C22×D4C2312yes—
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C2×D8—22no—
48C2×D8—42no—
49C22×C8—22no—
50C22×C8—42no—
51C22×C8—42no—
52C22×C8D414yescomplemented
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57#121—24no—
58#121—24no—
59#121—24no—
60#121—42no—
61C4×D4—22no—
62C4×D4—22no—
63#117—24no—
64#117—42no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D411yescharacteristic
77C2×C22⋊C4D411yescharacteristic
78C2×C42—22no—
79C2×C42C2311yescharacteristic
80#107—22no—
81C4⋊C8—22no—
82#102—24no—
83#102—24no—
84#102—42no—
85C22⋊C8—22no—
86C22⋊C8—22no—
87C4×C8—22no—
88#95—22no—
89#95—22no—
90C24—22no—
91C24C2×D411yescharacteristic
92C24C2×D411yescharacteristic
93C2×D4—22no—
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—82no—
106C2×D4—82no—
107C2×D4—82no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4#4112yes—
123C22×C4C2411yescharacteristic
124D8—44no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—82no—
142C4⋊C4#4114yes—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C22⋊C4—82no—
151C42—21no—
152C42—21no—
153C42—44no—
154C23—21no—
155C23—22no—
156C23—22no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—22no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23#12112yes—
170C23#13512yes—
171C23C22×D411yescharacteristic
172D4—28no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—88no—
178D4—88no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4#12112yes—
202C2×C4#13512yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4112yes—
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C8—22no—
208C8—22no—
209C8—41no—
210C8—41no—
211C22—24no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—41no—
220C22—41no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22C2×#12111yescharacteristic
229C22#54612yes—
230C22C2×#13511yescharacteristic
231C22C2×#13511yescharacteristic
232C22#57612yes—
233C4—22no—
234C4—22no—
235C4—44no—
236C4—41no—
237C4—41no—
238C4—82no—
239C4—82no—
240C4#53211yescharacteristic
241C4#53411yescharacteristic
242C4#55312yes—
243C2—42no—
244C2—42no—
245C2—162no—
246C2#254311yescharacteristic
247C2#305011yescharacteristic
248C2#305811yescharacteristic
249C2#329712yes—
250C2#330512yes—
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. #2414C2semidirect
4. #2087C2semidirect
5. #2076C2semidirect
6. #2054C2semidirect
7. #1999C2semidirect
8. #1934C2semidirect
9. #1893C2semidirect
10. #1875C2semidirect
11. #1735C2semidirect
52. C22×C8D4semidirect

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
#24142
#20872
#20762
#20542
#19992
#19342
#18932
#18752
#17352