Finite Group Databasebeta

← #27680

Subgroups — #27680

Order 256.

443 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#384C211yesmaximal, characteristic
3#2099C212yesmaximal, complemented
4#2096C212yesmaximal, complemented
5#1948C212yesmaximal, complemented
6#1872C212yesmaximal
7#1853C212yesmaximal
8#1803C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic
10C2×C22⋊Q8C2211yescharacteristic
11C2×C4×Q8C2211yescharacteristic
12C22×C4⋊C4—21no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4C2212yes—
15C22×C42C2211yescharacteristic
16C2×#106C2214yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2214yes—
19C2×C22⋊C8C2212yes—
20#391C2211yescharacteristic
21#389C2211yescharacteristic
22#387—21no—
23#385—21no—
24#384—21no—
25#384—21no—
26#384C2212yes—
27#384C2212yes—
28#380C2211yescharacteristic
29C4×C4⋊C4—21no—
30C4×C4⋊C4C2211yescharacteristic
31C4×C22⋊C4—21no—
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8C2311yescharacteristic
36C23×C4—22no—
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4C2311yescharacteristic
40C22×C8C2314yes—
41C22⋊Q8—22no—
42C22⋊Q8Q814yes—
43C4×Q8—24no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—44no—
55C2×C4⋊C4C2314yes—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—24no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#106—44no—
69C4⋊C8—24no—
70#103—44no—
71C22⋊C8—24no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77C24C2×D411yescharacteristic
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8C2×Q812yes—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4—44no—
97C22×C4SD1614yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×Q812yes—
104C22×C4#4112yes—
105C22×C4C2411yescharacteristic
106C2×C8—24no—
107C2×C8—24no—
108C2×C8—24no—
109C4⋊C4—24no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—44no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4C2×Q812yes—
123C22⋊C4—24no—
124C22⋊C4—42no—
125C42—22no—
126C42—22no—
127C42—22no—
128C42—42no—
129C42—42no—
130C42#4114yes—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23C22⋊Q812yes—
136C23C2×SD1612yes—
137C23#13712yes—
138C23C22×D411yescharacteristic
139Q8—24no—
140Q8—42no—
141C2×C4—28no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—28no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—44no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—88no—
169C2×C4C22⋊Q812yes—
170C2×C4#12312yes—
171C2×C4#12312yes—
172C2×C4C2×SD1612yes—
173C2×C4C2×SD1612yes—
174C2×C4AGL(1,ℤ/8)12yes—
175C2×C4#13712yes—
176C2×C4C22×Q811yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4112yes—
179C2×C4H(5,𝔽2)12yes—
180C8—44no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22#47514yes—
188C22C2×C22⋊Q811yescharacteristic
189C22C2×#12311yescharacteristic
190C22#53511yescharacteristic
191C22C22×SD1611yescharacteristic
192C22C2×AGL(1,ℤ/8)11yescharacteristic
193C22C2×#13711yescharacteristic
194C22C2×#13711yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—22no—
199C4—42no—
200C4—42no—
201C4—81no—
202C4—84no—
203C4—81no—
204C4#48314yes—
205C4#55312yes—
206C4#55412yes—
207C2—24no—
208C2#248711yescharacteristic
209C2C2×#47511yescharacteristic
210C2#307911yescharacteristic
211C2C2×#48311yescharacteristic
212C2#309211yescharacteristic
213C2#320811yescharacteristic
214C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2099C2semidirect
4. #2096C2semidirect
5. #1948C2semidirect
8. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
C2×#3842
#20992
#20962
#19482
#18722
#18532
#18032
#17972