Finite Group Databasebeta

← #27889

Subgroups — #27889

Order 256.

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

normal subgroups only

This group has 399 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#2857C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2035C214yesmaximal
4#1956C211yesmaximal, characteristic, complemented
5#1956C211yesmaximal, characteristic, complemented
6#1927C214yesmaximal, complemented
7#1844C211yesmaximal, characteristic, complemented
8#1834C211yesmaximal, characteristic, complemented
9C4×C22⋊C8C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C22⋊Q8C2212yes—
12C2×#117—21no—
13C22×C42C2211yescharacteristic
14C2×#106C2214yes—
15C2×#103C2214yes—
16C2×#103C2214yes—
17C2×C22⋊C8C2212yes—
18C2×C4×C8C2211yescharacteristic
19C2×C4×C8C2211yescharacteristic
20#391C2212yes—
21#389—21no—
22#389—21no—
23#384C2211yescharacteristic
24#384C2211yescharacteristic
25#384C2211yescharacteristic
26#382—21no—
27#380—21no—
28#380—21no—
29#380—21no—
30#380C2212yes—
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2212yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8C2312yes—
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C22×C8C2312yes—
40C22×C8C2312yes—
41C4⋊Q8—22no—
42C4⋊Q8—22no—
43C22⋊Q8—44no—
44#117—44no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C42—22no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#106—44no—
67#103—28no—
68#103—44no—
69C22⋊C8—24no—
70C4×C8—22no—
71C4×C8—22no—
72#95—22no—
73#95—22no—
74#95—22no—
75#95—22no—
76C24C2×D411yescharacteristic
77C2×Q8—24no—
78C2×Q8—22no—
79C2×Q8—22no—
80C2×Q8—24no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4SD1614yes—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8#4114yes—
107C4⋊C4—24no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—44no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C22⋊C4—44no—
120C22⋊C4—44no—
121C42—22no—
122C42—22no—
123C42—21no—
124C42—21no—
125C42—44no—
126C42#4112yes—
127C42#4112yes—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23C2×SD1612yes—
135C23C22×D411yescharacteristic
136Q8—44no—
137Q8—44no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—28no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—44no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4#12312yes—
163C2×C4#12312yes—
164C2×C4C2×SD1612yes—
165C2×C4C2×SD1612yes—
166C2×C4#13512yes—
167C2×C4#13712yes—
168C2×C4#13712yes—
169C2×C4C2×#4111yescharacteristic
170C2×C4C2×#4112yes—
171C2×C4C2×#4111yescharacteristic
172C2×C4H(5,𝔽2)12yes—
173C8—24no—
174C8—42no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22C2×#12311yescharacteristic
183C22#53211yescharacteristic
184C22#53511yescharacteristic
185C22C22×SD1611yescharacteristic
186C22C2×#13511yescharacteristic
187C22C2×#13711yescharacteristic
188C22C2×#13711yescharacteristic
189C4—22no—
190C4—41no—
191C4—41no—
192C4—82no—
193C4—82no—
194C4—82no—
195C4—82no—
196C4#48314yes—
197C4#48514yes—
198C4#55312yes—
199C4#55512yes—
200C2—42no—
201C2#250011yescharacteristic
202C2C2×#48311yescharacteristic
203C2C2×#48511yescharacteristic
204C2#316711yescharacteristic
205C2#318511yescharacteristic
206C2#320611yescharacteristic
207C2#321011yescharacteristic
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
4. #1956C2semidirect
5. #1956C2semidirect
6. #1927C2semidirect
7. #1844C2semidirect
8. #1834C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22852
#20352
#19562
#19272
#18442
#18342
C4×C22⋊C82