Finite Group Databasebeta

← #24072

Subgroups — #24072

Order 256.

501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 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×#541C211yesmaximal, characteristic
2C2×#485C212yesmaximal
3#2316C211yesmaximal, characteristic
4#2059C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#1872C212yesmaximal
7#1865C212yesmaximal, complemented
8#1863C212yesmaximal, complemented
9#1818C211yesmaximal, characteristic
10#541—28no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×#126—22no—
13C2×#125—22no—
14C2×#125C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C4×Q8C2212yes—
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#485—28no—
21C2×#107C2212yes—
22C2×#106C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×C22⋊C8C2212yes—
28#389—22no—
29#389C2212yes—
30#389C2212yes—
31#380—22no—
32#340C2212yes—
33#340C2212yes—
34#336C2212yes—
35#336C2211yescharacteristic
36#336C2211yescharacteristic
37C22×Q8C2312yes—
38C23×C4—21no—
39C23×C4C2311yescharacteristic
40C22×C8C2312yes—
41C22×C8C2312yes—
42C4⋊Q8—24no—
43#126—216no—
44#125—216no—
45#125—24no—
46C22⋊Q8—24no—
47C22⋊Q8D418yes—
48C4×Q8—24no—
49C4×Q8—24no—
50#117—24no—
51#117—24no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4D412yes—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—24no—
72#106—24no—
73#103—24no—
74#103—24no—
75#103—24no—
76#103—24no—
77C22⋊C8—24no—
78#95—22no—
79#95—22no—
80C24C2×D411yescharacteristic
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8C2×D414yes—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—82no—
96C22×C4C2×D412yes—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D414yes—
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C4⋊C4—24no—
108C4⋊C4—28no—
109C4⋊C4—28no—
110C4⋊C4—28no—
111C4⋊C4—28no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—82no—
118C4⋊C4—82no—
119C4⋊C4C2×D414yes—
120C4⋊C4#4114yes—
121C22⋊C4—28no—
122C22⋊C4—24no—
123C22⋊C4—42no—
124C42—28no—
125C42—22no—
126C42—22no—
127C42—42no—
128C23—21no—
129C23—21no—
130C23—22no—
131C23—22no—
132C23#12012yes—
133C23C22×D411yescharacteristic
134Q8—24no—
135Q8—24no—
136Q8—42no—
137Q8—42no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4#12012yes—
162C2×C4C22×D412yes—
163C2×C4C2×#4112yes—
164C2×C42−1+411yescharacteristic
165C2×C42−1+411yescharacteristic
166C8—42no—
167C8—42no—
168C22—22no—
169C22—24no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22C2×#12011yescharacteristic
174C22#52511yescharacteristic
175C22#53611yescharacteristic
176C22#57712yes—
177C22#57812yes—
178C4—22no—
179C4—44no—
180C4—44no—
181C4—42no—
182C4—42no—
183C4—44no—
184C4—82no—
185C4#54712yes—
186C4#54712yes—
187C2—42no—
188C2#240611yescharacteristic
189C2#301312yes—
190C2#312311yescharacteristic
191C2#312711yescharacteristic
192C2#324212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2059C2semidirect
5. #2043C2semidirect
7. #1865C2semidirect
8. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4852
#23162
#20592
#20432
#18722
#18652
#18632
#18182