Finite Group Databasebeta

← #29148

Subgroups — #29148

Order 256.

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

normal subgroups only

This group has 415 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
1#2868C211yesmaximal, characteristic, complemented
2C4×#124C211yesmaximal, characteristic
3#2024C212yesmaximal, complemented
4#2023C212yesmaximal
5#1959C212yesmaximal
6#1836C212yesmaximal, 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
16C2×C4⋊C8C2212yes—
17C2×#103C2214yes—
18C2×#102C2214yes—
19C2×#97C2212yes—
20#391C2211yescharacteristic
21#390C2211yescharacteristic
22#388—21no—
23#388—21no—
24#384—21no—
25#384—21no—
26#384C2212yes—
27#384C2212yes—
28#382C2211yescharacteristic
29C4×C22⋊C4—21no—
30C4×C22⋊C4—21no—
31C43C2211yescharacteristic
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8—22no—
36C22×Q8C2311yescharacteristic
37C22×D4C2311yescharacteristic
38C23×C4—21no—
39C22×C8C2314yes—
40C4⋊Q8—44no—
41C4⋊Q8—44no—
42C4⋊Q8—44no—
43#124—22no—
44#124Q814yescomplemented
45C4×Q8—24no—
46C4×D4—24no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2314yes—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69C4⋊C8—24no—
70#103—44no—
71#102—44no—
72#97—24no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77C24—21no—
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—44no—
82C2×Q8—44no—
83C2×Q8—44no—
84C2×Q8C2×Q812yes—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4—22no—
88C2×D4C2×Q812yes—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—24no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—24no—
104C2×C8—24no—
105C2×C8—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—44no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—44no—
114C4⋊C4—42no—
115C4⋊C4—44no—
116C22⋊C4—24no—
117C22⋊C4—24no—
118C22⋊C4—42no—
119C22⋊C4—42no—
120C22⋊C4—42no—
121C42—22no—
122C42—22no—
123C42—22no—
124C42—22no—
125C42—22no—
126C42—42no—
127C42C2×Q812yes—
128C42#4112yes—
129C42#4114yes—
130C23—22no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23C22×D411yescharacteristic
138Q8—24no—
139Q8—42no—
140Q8—88no—
141D4—24no—
142D4—42no—
143C2×C4—21no—
144C2×C4—21no—
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—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—44no—
161C2×C4—44no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—84no—
166C2×C4C22⋊Q812yes—
167C2×C4C22⋊Q812yes—
168C2×C4#12412yes—
169C2×C4#12412yes—
170C2×C4AGL(1,ℤ/8)12yes—
171C2×C4AGL(1,ℤ/8)12yes—
172C2×C4#13712yes—
173C2×C4#13712yes—
174C2×C4C22×Q811yescharacteristic
175C2×C4C2×#4112yes—
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)12yes—
178C8—44no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22C2×C22⋊Q811yescharacteristic
187C22C2×#12411yescharacteristic
188C22#53511yescharacteristic
189C22C2×AGL(1,ℤ/8)11yescharacteristic
190C22C2×AGL(1,ℤ/8)11yescharacteristic
191C22C2×#13711yescharacteristic
192C22C2×#13711yescharacteristic
193C4—22no—
194C4—22no—
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—81no—
200C4—81no—
201C4—84no—
202C4#48914yes—
203C4#55412yes—
204C4#55512yes—
205C2—82no—
206C2#256311yescharacteristic
207C2#307911yescharacteristic
208C2#307911yescharacteristic
209C2C2×#48911yescharacteristic
210C2#314111yescharacteristic
211C2#321711yescharacteristic
212C2#321811yescharacteristic
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
1. #2868C2semidirect
3. #2024C2semidirect
6. #1836C2semidirect
7. #1803C2semidirect
9. #1767C2semidirect
44. #124Q8semidirect

Maximal subgroups

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

subgroupindex
#28682
C4×#1242
#20242
#20232
#19592
#18362
#18032
#18022
#17672