Finite Group Databasebeta

← #25840

Subgroups — #25840

Order 256.

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

normal subgroups only

This group has 503 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×#557C211yesmaximal, characteristic
2C2×#500C211yesmaximal, characteristic
3C2×#475C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic
5#2059C214yesmaximal
6#1988C211yesmaximal, characteristic
7#1984C211yesmaximal, characteristic
8#1863C214yesmaximal
9C4×#103C211yesmaximal, characteristic
10#557—24no—
11#557—24no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—22no—
17C2×#125C2212yes—
18C2×C4×Q8—21no—
19C2×C4×Q8C2212yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22#500—24no—
23#475—24no—
24C2×#107C2214yes—
25C2×#106C2212yes—
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C4×C8C2211yescharacteristic
30#398—22no—
31#391—21no—
32#389C2214yes—
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8D411yescharacteristic
40C22×Q8C2311yescharacteristic
41C22×C8C2312yes—
42C22×C8C2312yes—
43C4⋊Q8—28no—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—44no—
48C4⋊Q8D414yes—
49#125—216no—
50#125—24no—
51#125—24no—
52C4×Q8—28no—
53C4×Q8—24no—
54C4×Q8—24no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×Q8D414yes—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4D412yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C42—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—28no—
78#106—24no—
79C4⋊C8—22no—
80#103—24no—
81#103Q818yescomplemented
82C4×C8—22no—
83#95—21no—
84#95—24no—
85C2×Q8—22no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C2×D412yes—
97C2×Q8C2×Q814yes—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—24no—
112C2×C8C2×Q814yes—
113C4⋊C4—28no—
114C4⋊C4—28no—
115C4⋊C4—28no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—24no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—48no—
126C4⋊C4—42no—
127C4⋊C4C2×D412yes—
128C4⋊C4C2×D414yes—
129C4⋊C4C2×Q814yes—
130C42—28no—
131C42—24no—
132C42—22no—
133C42—22no—
134C42—42no—
135C42C2×D412yes—
136C42C2×D412yes—
137C23C22×D411yescharacteristic
138Q8—22no—
139Q8—24no—
140Q8—44no—
141Q8—48no—
142Q8—84no—
143Q8C4⋊Q814yescomplemented
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#12012yes—
164C2×C4#12012yes—
165C2×C4#13512yes—
166C2×C4AGL(1,ℤ/8)12yes—
167C2×C4C22×D411yescharacteristic
168C2×C4C22×D411yescharacteristic
169C2×C4C22×Q812yes—
170C2×C42−1+412yes—
171C8—24no—
172C8—42no—
173C22C2×#12011yescharacteristic
174C22D4×Q812yes—
175C22C2×#13511yescharacteristic
176C22C2×AGL(1,ℤ/8)11yescharacteristic
177C22#57812yes—
178C4—24no—
179C4—22no—
180C4—44no—
181C4—42no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4#44914yes—
187C4C2×C4⋊Q811yescharacteristic
188C4#53611yescharacteristic
189C4D4×Q812yes—
190C2#243411yescharacteristic
191C2C2×#44911yescharacteristic
192C2#301111yescharacteristic
193C2#338312yes—
194C2#338612yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
81. #103Q8semidirect
143. Q8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#5002
C2×#4752
#25462
#20592
#19882
#19842
#18632
C4×#1032