Finite Group Databasebeta

← #25648

Subgroups — #25648

Order 256.

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

normal subgroups only

This group has 549 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×#477C211yesmaximal, characteristic, complemented
4#2567C211yesmaximal, characteristic, complemented
5#2042C214yesmaximal, complemented
6#1984C211yesmaximal, characteristic, complemented
7#1982C211yesmaximal, characteristic
8#1862C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10#559—24no—
11#559—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#124—21no—
15C2×#124C2212yes—
16C2×#123—22no—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2212yes—
21#486—24no—
22#477—24no—
23C2×#107C2212yes—
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#102C2214yes—
28C2×#102C2214yes—
29C2×C4×C8C2211yescharacteristic
30#395—22no—
31#391—21no—
32#389C2214yes—
33#384—21no—
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—
48#127—22no—
49#127D414yes—
50#124—28no—
51#124—24no—
52#124—24no—
53#123—216no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C4×D4—28no—
59C4×D4—24no—
60C4×D4—24no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—24no—
77C4⋊C8—22no—
78#103—24no—
79#103—24no—
80#102—28no—
81#102—28no—
82C4×C8—22no—
83#95—21no—
84#95—24no—
85C24—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—42no—
93C2×Q8—82no—
94C2×Q8C2×D412yes—
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—24no—
99C2×D4—24no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—82no—
103C2×D4C2×D414yes—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—28no—
108C22×C4—22no—
109C22×C4—84no—
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2411yescharacteristic
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—24no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—24no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—84no—
129C4⋊C4C2×D412yes—
130C4⋊C4#4114yes—
131C22⋊C4—28no—
132C22⋊C4—28no—
133C22⋊C4—28no—
134C42—24no—
135C42—22no—
136C42—22no—
137C42—42no—
138C42C2×D412yes—
139C42C2×D412yes—
140C23—24no—
141C23—44no—
142C23—42no—
143C23—42no—
144C23—44no—
145C23C22×D411yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—42no—
149Q8—44no—
150Q8—84no—
151D4—28no—
152D4—42no—
153D4—42no—
154D4—48no—
155D4—84no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—88no—
172C2×C4#12012yes—
173C2×C4#12012yes—
174C2×C4#13512yes—
175C2×C4#13712yes—
176C2×C4C22×D411yescharacteristic
177C2×C4C22×D411yescharacteristic
178C2×C4C2×#4112yes—
179C2×C42−1+412yes—
180C8—42no—
181C8—42no—
182C22—44no—
183C22—44no—
184C22—44no—
185C22—84no—
186C22—84no—
187C22C2×#12011yescharacteristic
188C22#54812yes—
189C22C2×#13511yescharacteristic
190C22C2×#13711yescharacteristic
191C22#57612yes—
192C4—22no—
193C4—44no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—82no—
199C4#45214yes—
200C4#53211yescharacteristic
201C4#53611yescharacteristic
202C4#54712yes—
203C2—84no—
204C2#243211yescharacteristic
205C2C2×#45211yescharacteristic
206C2#300811yescharacteristic
207C2#336112yes—
208C2#337512yes—
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×#477C2semidirect
4. #2567C2semidirect
5. #2042C2semidirect
6. #1984C2semidirect
8. #1862C2semidirect
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×#4772
#25672
#20422
#19842
#19822
#18622
C4×#1032