Finite Group Databasebeta

← #23554

Subgroups — #23554

Order 256.

601 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 195 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×#428C212yesmaximal
3C2×#389C211yesmaximal, characteristic
4#2059C214yesmaximal, complemented
5#2058C212yesmaximal, complemented
6#2057C212yesmaximal, complemented
7#1736C212yesmaximal
8#1728C211yesmaximal, characteristic
9C22×M16C2212yes—
10#533—28no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2212yes—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C4×Q8—21no—
17C2×C4×Q8—21no—
18C2×#117C2212yes—
19C22×C4⋊C4—21no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#428—28no—
24C2×#107C2214yes—
25C2×#106C2214yes—
26#389—21no—
27#389—21no—
28#389C2212yes—
29#389C2212yes—
30C2×#95—21no—
31#340C2214yes—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—21no—
36C22×Q8—21no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C2×M16—24no—
43C2×M16—24no—
44C2×M16Q818yescomplemented
45C22×C8C2314yes—
46C4⋊Q8—24no—
47C4⋊Q8—24no—
48#125—24no—
49#125—24no—
50C22⋊Q8—28no—
51C22⋊Q8—28no—
52C4×Q8—28no—
53C4×Q8—28no—
54#117—24no—
55#117D418yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—44no—
67C2×C4⋊C4Q814yescomplemented
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—22no—
73C2×C42—22no—
74C2×C42—24no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—28no—
80#106—28no—
81#95—22no—
82#95—22no—
83C24C2×D411yescharacteristic
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—42no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—24no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D414yes—
112C22×C4C2×Q814yes—
113C22×C4C2×Q812yes—
114C22×C4#4112yes—
115C22×C4C2411yescharacteristic
116M16—216no—
117C2×C8—24no—
118C2×C8—24no—
119C2×C8C2×Q818yes—
120C4⋊C4—28no—
121C4⋊C4—28no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—24no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—44no—
132C4⋊C4C2×D414yes—
133C4⋊C4C2×D414yes—
134C4⋊C4C2×Q814yes—
135C22⋊C4—28no—
136C42—22no—
137C42—22no—
138C42—28no—
139C42—22no—
140C42—22no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23#12012yes—
146C23C22⋊Q812yes—
147C23C4⋊Q812yes—
148C23C22×D411yescharacteristic
149Q8—48no—
150Q8—48no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—48no—
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—42no—
176C2×C4—84no—
177C2×C4#12012yes—
178C2×C4C22⋊Q818yes—
179C2×C4C22⋊Q812yes—
180C2×C4C4⋊Q818yes—
181C2×C4C4⋊Q812yes—
182C2×C4AGL(1,ℤ/8)12yes—
183C2×C4#13712yes—
184C2×C4C22×D412yes—
185C2×C4C22×Q812yes—
186C2×C4C22×Q811yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C8—28no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22#39514yes—
196C22C2×#12011yescharacteristic
197C22C2×C22⋊Q811yescharacteristic
198C22C2×C4⋊Q811yescharacteristic
199C22C2×AGL(1,ℤ/8)11yescharacteristic
200C22C2×#13711yescharacteristic
201C22#57511yescharacteristic
202C22#57511yescharacteristic
203C4—22no—
204C4—22no—
205C4—44no—
206C4—41no—
207C4—44no—
208C4—41no—
209C4—44no—
210C4—82no—
211C4#39514yes—
212C4C2×C22⋊Q812yes—
213C4C2×C4⋊Q812yes—
214C2—24no—
215C2C2×#39511yescharacteristic
216C2#300611yescharacteristic
217C2#300711yescharacteristic
218C2#307711yescharacteristic
219C2#307811yescharacteristic
220C2#316511yescharacteristic
221C2#316611yescharacteristic
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. #2059C2semidirect
5. #2058C2semidirect
6. #2057C2semidirect
44. C2×M16Q8semidirect
67. C2×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4282
C2×#3892
#20592
#20582
#20572
#17362
#17282