Finite Group Databasebeta

← #24064

Subgroups — #24064

Order 256.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#533C211yesmaximal, characteristic
2C2×#484C212yesmaximal
3C2×#389C211yesmaximal, characteristic
4#2057C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#1872C212yesmaximal
7#1863C214yesmaximal, complemented
8#1817C211yesmaximal, characteristic
9#533—28no—
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#125—21no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2212yes—
15C2×C4×Q8—21no—
16C2×C4×Q8C2212yes—
17C2×#117—21no—
18C22×C4⋊C4—21no—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#484—28no—
23C2×#107C2214yes—
24C2×#103C2214yes—
25C2×#103C2214yes—
26C2×C22⋊C8C2212yes—
27#389—21no—
28#389—21no—
29#389C2212yes—
30#389C2212yes—
31C2×#95—21no—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—21no—
36C22×Q8C2312yes—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2314yes—
43C4⋊Q8—24no—
44C4⋊Q8—24no—
45#125—28no—
46C22⋊Q8—28no—
47C22⋊Q8—24no—
48C22⋊Q8D418yes—
49C4×Q8—28no—
50C4×Q8—24no—
51C4×Q8—24no—
52#117—28no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C42—22no—
70C2×C42—24no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—28no—
76#103—28no—
77#103—28no—
78C22⋊C8—24no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C2×Q8—24no—
83C2×Q8—22no—
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8C2×D414yes—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—41no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4Q1614yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D414yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2411yescharacteristic
114C2×C8—24no—
115C2×C8—24no—
116C2×C8—24no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—28no—
121C4⋊C4—24no—
122C4⋊C4—28no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—24no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4C2×D414yes—
130C4⋊C4#4114yes—
131C22⋊C4—28no—
132C42—28no—
133C42—22no—
134C42—22no—
135C42—22no—
136C42—22no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23#12012yes—
142C23C2×Q1612yes—
143C23#13712yes—
144C23C22×D411yescharacteristic
145Q8—28no—
146Q8—48no—
147Q8—44no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—28no—
157C2×C4—28no—
158C2×C4—21no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—44no—
165C2×C4—48no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—84no—
176C2×C4#12012yes—
177C2×C4C2×Q1612yes—
178C2×C4C2×Q1612yes—
179C2×C4AGL(1,ℤ/8)12yes—
180C2×C4C22×D412yes—
181C2×C4C2×#4112yes—
182C2×C42−1+412yes—
183C8—44no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22#45114yes—
191C22C2×#12011yescharacteristic
192C22#52511yescharacteristic
193C22#53611yescharacteristic
194C22C22×Q1611yescharacteristic
195C22C2×AGL(1,ℤ/8)11yescharacteristic
196C22C2×#13711yescharacteristic
197C22#57511yescharacteristic
198C4—22no—
199C4—22no—
200C4—22no—
201C4—22no—
202C4—44no—
203C4—44no—
204C4—41no—
205C4—44no—
206C4—82no—
207C4#54714yes—
208C2—24no—
209C2#240611yescharacteristic
210C2C2×#45111yescharacteristic
211C2#300611yescharacteristic
212C2#310411yescharacteristic
213C2#311011yescharacteristic
214C2#323011yescharacteristic
215C2#323611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2057C2semidirect
5. #2043C2semidirect
7. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4842
C2×#3892
#20572
#20432
#18722
#18632
#18172