Finite Group Databasebeta

← #28726

Subgroups — #28726

Order 256.

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

normal subgroups only

This group has 435 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×#489C212yesmaximal
2#2868C211yesmaximal, characteristic, complemented
3C4×#124C211yesmaximal, characteristic
4#2072C212yesmaximal, complemented
5#2071C212yesmaximal
6#1993C212yesmaximal, complemented
7#1803C212yesmaximal, complemented
8#1802C212yesmaximal
9#1767C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8—21no—
12C2×C4⋊Q8C2212yes—
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#489—28no—
17C2×C4⋊C8C2212yes—
18C2×#103C2214yes—
19C2×#102C2214yes—
20C2×#97C2212yes—
21#391C2211yescharacteristic
22#390C2211yescharacteristic
23#388—21no—
24#388—21no—
25#384—21no—
26#384—21no—
27#384C2212yes—
28#384C2212yes—
29#382C2211yescharacteristic
30C4×C22⋊C4—21no—
31C4×C22⋊C4—21no—
32C43C2211yescharacteristic
33#340C2214yes—
34#340C2214yes—
35#336C2212yes—
36C22×Q8—22no—
37C22×Q8C2311yescharacteristic
38C22×D4C2311yescharacteristic
39C23×C4—21no—
40C22×C8C2314yes—
41C4⋊Q8—24no—
42C4⋊Q8—24no—
43C4⋊Q8—44no—
44C4⋊Q8—44no—
45#124—22no—
46#124D414yes—
47C4×Q8—24no—
48C4×D4—24no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2314yes—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C42—21no—
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71C4⋊C8—24no—
72#103—28no—
73#102—28no—
74#97—24no—
75#95—21no—
76#95—21no—
77#95—21no—
78#95—21no—
79C24—21no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—44no—
84C2×Q8—44no—
85C2×Q8—44no—
86C2×Q8C2×D412yes—
87C2×D4—22no—
88C2×D4—21no—
89C2×D4—22no—
90C2×D4C2×D412yes—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—24no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—24no—
106C2×C8—24no—
107C2×C8—24no—
108C4⋊C4—24no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—24no—
112C4⋊C4—44no—
113C4⋊C4—44no—
114C4⋊C4—44no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4—44no—
118C22⋊C4—24no—
119C22⋊C4—24no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C42—22no—
124C42—22no—
125C42—22no—
126C42—22no—
127C42—22no—
128C42—42no—
129C42C2×D412yes—
130C42#4112yes—
131C42#4114yes—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C22×D411yescharacteristic
140Q8—24no—
141Q8—42no—
142Q8—88no—
143D4—24no—
144D4—42no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—84no—
168C2×C4#12112yes—
169C2×C4#12112yes—
170C2×C4#12412yes—
171C2×C4#12412yes—
172C2×C4AGL(1,ℤ/8)12yes—
173C2×C4AGL(1,ℤ/8)12yes—
174C2×C4#13712yes—
175C2×C4#13712yes—
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4112yes—
178C2×C4C2×#4111yescharacteristic
179C2×C42−1+412yes—
180C8—44no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22C2×#12111yescharacteristic
189C22C2×#12411yescharacteristic
190C22#53611yescharacteristic
191C22C2×AGL(1,ℤ/8)11yescharacteristic
192C22C2×AGL(1,ℤ/8)11yescharacteristic
193C22C2×#13711yescharacteristic
194C22C2×#13711yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—22no—
199C4—22no—
200C4—42no—
201C4—81no—
202C4—81no—
203C4—84no—
204C4#48914yescomplemented
205C4#54712yes—
206C4#55612yes—
207C2—82no—
208C2#254711yescharacteristic
209C2#303811yescharacteristic
210C2#303911yescharacteristic
211C2C2×#48911yescharacteristic
212C2#314111yescharacteristic
213C2#323811yescharacteristic
214C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2868C2semidirect
4. #2072C2semidirect
6. #1993C2semidirect
7. #1803C2semidirect
9. #1767C2semidirect
204. C4#489semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
#28682
C4×#1242
#20722
#20712
#19932
#18032
#18022
#17672