Finite Group Databasebeta

← #25839

Subgroups — #25839

Order 256.

531 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 159 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×#500C211yesmaximal, characteristic, complemented
3C2×#476C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#2059C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1862C214yesmaximal
9C4×#102C211yesmaximal, characteristic
10#554—24no—
11#554—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2212yes—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8—22no—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2212yes—
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4—21no—
21#500—24no—
22#476—24no—
23C2×#107C2214yes—
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×C4×C8C2211yescharacteristic
29#395—22no—
30#391—21no—
31#389C2214yes—
32#384—21no—
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8D411yescharacteristic
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4—22no—
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8D414yes—
49#125—24no—
50#125—24no—
51C22⋊Q8—28no—
52C22⋊Q8—216no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—22no—
58C4×D4D414yes—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D412yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—28no—
77#106—24no—
78C4⋊C8—22no—
79#102—24no—
80#102Q818yescomplemented
81C4×C8—22no—
82#95—21no—
83#95—24no—
84C24—21no—
85C2×Q8—22no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—42no—
90C2×Q8—82no—
91C2×Q8—82no—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4C2×D412yes—
95C2×D4C2×Q814yes—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—28no—
100C22×C4—24no—
101C22×C4—22no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—84no—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—24no—
114C2×C8C2×Q814yes—
115C4⋊C4—28no—
116C4⋊C4—28no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—84no—
127C4⋊C4C2×D412yes—
128C4⋊C4C2×D414yes—
129C4⋊C4C2×Q814yes—
130C22⋊C4—28no—
131C22⋊C4—24no—
132C42—22no—
133C42—22no—
134C42—42no—
135C42C2×D412yes—
136C42C2×D412yes—
137C23—22no—
138C23—24no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23C22×D411yescharacteristic
143Q8—44no—
144Q8—84no—
145Q8—84no—
146D4—22no—
147D4—24no—
148D4C4⋊Q814yes—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—48no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—88no—
167C2×C4#12012yes—
168C2×C4#12012yes—
169C2×C4#13512yes—
170C2×C4#13712yes—
171C2×C4C22×D411yescharacteristic
172C2×C4C22×D411yescharacteristic
173C2×C4C22×Q812yes—
174C2×C42−1+412yes—
175C8—24no—
176C8—42no—
177C22—24no—
178C22—44no—
179C22—44no—
180C22—42no—
181C22—42no—
182C22C2×#12011yescharacteristic
183C22D4×Q812yes—
184C22C2×#13511yescharacteristic
185C22C2×#13711yescharacteristic
186C22#57612yes—
187C4—22no—
188C4—44no—
189C4—42no—
190C4—42no—
191C4—42no—
192C4—44no—
193C4—82no—
194C4#45214yes—
195C4C2×C4⋊Q811yescharacteristic
196C4#53611yescharacteristic
197C4D4×Q812yes—
198C2—44no—
199C2#243411yescharacteristic
200C2C2×#45211yescharacteristic
201C2#300811yescharacteristic
202C2#338012yes—
203C2#338512yes—
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×#500C2semidirect
4. #2567C2semidirect
5. #2059C2semidirect
6. #1987C2semidirect
7. #1984C2semidirect
80. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5002
C2×#4762
#25672
#20592
#19872
#19842
#18622
C4×#1022