Finite Group Databasebeta

← #25934

Subgroups — #25934

Order 256.

539 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 143 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×#554C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic, complemented
5#2026C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1877C214yesmaximal
9C4×#102C211yesmaximal, characteristic
10C22×SD16C2212yes—
11#554—24no—
12#554—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—22no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2212yes—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4—21no—
22#487—24no—
23#461—24no—
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2214yes—
26C2×#103C2212yes—
27C2×#102C2212yes—
28C2×C4×C8C2211yescharacteristic
29#400—22no—
30#391—21no—
31#391C2214yes—
32#382—21no—
33C4×C4⋊C4C2211yescharacteristic
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8D411yescharacteristic
39C22×Q8C2312yes—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4—22no—
43C2×SD16—24no—
44C2×SD16—28no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8D414yes—
51#125—24no—
52#125—24no—
53C22⋊Q8—28no—
54C22⋊Q8—216no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4D414yes—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77C4⋊C8—22no—
78#103—24no—
79#103—28no—
80#102—24no—
81C4×C8—22no—
82#95—21no—
83#95—24no—
84#95—24no—
85C24—21no—
86C2×Q8—22no—
87C2×Q8—24no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—24no—
93C2×Q8—24no—
94C2×Q8—82no—
95C2×D4—21no—
96C2×D4—22no—
97C2×D4C2×D412yes—
98C2×D4#4114yes—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—28no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—84no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113SD16—48no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—24no—
118C2×C8—22no—
119C4⋊C4—28no—
120C4⋊C4—28no—
121C4⋊C4—22no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4C2×D414yes—
131C4⋊C4C2×D412yes—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C42—22no—
135C42—22no—
136C42—42no—
137C42C2×D412yes—
138C42C2×D412yes—
139C23—22no—
140C23—24no—
141C23—41no—
142C23—41no—
143C23—42no—
144C23C22×D411yescharacteristic
145Q8—44no—
146Q8—42no—
147Q8—42no—
148Q8—44no—
149Q8—84no—
150D4—22no—
151D4—24no—
152D4#12414yes—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—44no—
164C2×C4—48no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—88no—
171C2×C4#12012yes—
172C2×C4#12012yes—
173C2×C4#13512yes—
174C2×C4#13712yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4112yes—
178C2×C42−1+412yes—
179C8—42no—
180C8—42no—
181C22—24no—
182C22—44no—
183C22—44no—
184C22—42no—
185C22—42no—
186C22C2×#12011yescharacteristic
187C22#54812yes—
188C22C2×#13511yescharacteristic
189C22C2×#13711yescharacteristic
190C22#57712yes—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—44no—
196C4—82no—
197C4—82no—
198C4#45214yes—
199C4C2×#12411yescharacteristic
200C4#53611yescharacteristic
201C4#54812yes—
202C2—44no—
203C2#243611yescharacteristic
204C2C2×#45211yescharacteristic
205C2#300911yescharacteristic
206C2#336012yes—
207C2#337212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. C2×#487C2semidirect
4. #2568C2semidirect
5. #2026C2semidirect
6. #1987C2semidirect
7. #1984C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4872
C2×#4612
#25682
#20262
#19872
#19842
#18772
C4×#1022