Finite Group Databasebeta

← #25652

Subgroups — #25652

Order 256.

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

normal subgroups only

This group has 559 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×#559C211yesmaximal, characteristic
2C2×#486C211yesmaximal, characteristic
3C2×#475C211yesmaximal, characteristic, complemented
4#2546C211yesmaximal, characteristic, complemented
5#2042C214yesmaximal, complemented
6#1984C211yesmaximal, characteristic, complemented
7#1982C211yesmaximal, characteristic
8#1864C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10#559—24no—
11#559—24no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#127C2211yescharacteristic
15C2×#124—21no—
16C2×#124C2212yes—
17C2×#123—22no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2212yes—
22#486—24no—
23#475—24no—
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2214yes—
29C2×#102C2214yes—
30C2×C4×C8C2211yescharacteristic
31#398—22no—
32#391—21no—
33#389C2214yes—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8D411yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4—21no—
42C22×D4C2312yes—
43C23×C4—22no—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—44no—
49#127—22no—
50#127D414yes—
51#124—28no—
52#124—24no—
53#124—24no—
54#123—216no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×Q8D414yes—
59C4×D4—28no—
60C4×D4—24no—
61C4×D4—24no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D412yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#106—24no—
78C4⋊C8—22no—
79#103—24no—
80#103—24no—
81#102—28no—
82#102—28no—
83C4×C8—22no—
84#95—21no—
85#95—24no—
86C24—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C2×D412yes—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—82no—
105C2×D4C2×D414yes—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—28no—
110C22×C4—22no—
111C22×C4—84no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—24no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—48no—
130C4⋊C4—42no—
131C4⋊C4C2×D412yes—
132C4⋊C4#4114yes—
133C22⋊C4—28no—
134C22⋊C4—28no—
135C22⋊C4—28no—
136C42—24no—
137C42—22no—
138C42—22no—
139C42—42no—
140C42C2×D412yes—
141C42C2×D412yes—
142C23—24no—
143C23—44no—
144C23—42no—
145C23—42no—
146C23—44no—
147C23C22×D411yescharacteristic
148Q8—22no—
149Q8—22no—
150Q8—42no—
151Q8—44no—
152Q8—84no—
153D4—28no—
154D4—42no—
155D4—42no—
156D4—48no—
157D4—84no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—88no—
175C2×C4#12012yes—
176C2×C4#12012yes—
177C2×C4#13512yes—
178C2×C4AGL(1,ℤ/8)12yes—
179C2×C4C22×D411yescharacteristic
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4112yes—
182C2×C42−1+412yes—
183C8—42no—
184C8—42no—
185C22—44no—
186C22—44no—
187C22—44no—
188C22—84no—
189C22—84no—
190C22C2×#12011yescharacteristic
191C22#54812yes—
192C22C2×#13511yescharacteristic
193C22C2×AGL(1,ℤ/8)11yescharacteristic
194C22#57712yes—
195C4—22no—
196C4—44no—
197C4—42no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4#44914yes—
203C4#53211yescharacteristic
204C4#53611yescharacteristic
205C4#54712yes—
206C2—84no—
207C2#243211yescharacteristic
208C2C2×#44911yescharacteristic
209C2#300911yescharacteristic
210C2#336512yes—
211C2#337312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#475C2semidirect
4. #2546C2semidirect
5. #2042C2semidirect
6. #1984C2semidirect
8. #1864C2semidirect
9. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4862
C2×#4752
#25462
#20422
#19842
#19822
#18642
C4×#1032