Finite Group Databasebeta

← #21396

Subgroups — #21396

Order 256.

591 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 199 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×#529C211yesmaximal, characteristic
2C2×#432C211yesmaximal, characteristic, complemented
3C4×#117C211yesmaximal, characteristic, complemented
4#1989C212yesmaximal, complemented
5#1983C212yesmaximal, complemented
6#1875C212yesmaximal
7#1873C212yesmaximal, complemented
8#1763C212yesmaximal, complemented
9#1762C212yesmaximal, complemented
10#529—24no—
11#529C418yescomplemented
12C2×#126—21no—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×#123—21no—
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#117C2212yes—
23C22×C42C2211yescharacteristic
24#432—24no—
25C2×C4⋊C8C2212yes—
26C2×#103C2214yes—
27C2×#102C2214yes—
28C2×C22⋊C8C2212yes—
29C2×#97C2212yes—
30C4×C4⋊C4C2212yes—
31C4×C4⋊C4C2212yes—
32C4×C22⋊C4—21no—
33C4×C22⋊C4—21no—
34#376—21no—
35C43—21no—
36#340C2214yes—
37#340C2214yes—
38C22×Q8C2311yescharacteristic
39C22×D4—21no—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2314yes—
46#126—28no—
47#125—22no—
48#125C2×C414yes—
49#124—22no—
50#124C2×C414yes—
51#123—28no—
52C22⋊Q8—22no—
53C22⋊Q8C2×C414yes—
54#121—22no—
55#121C2×C414yes—
56C4×Q8—22no—
57C4×Q8C2×C414yes—
58C4×D4—28no—
59C4×D4—22no—
60C4×D4C2×C414yes—
61#117—28no—
62#117—28no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2314yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C42—21no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42C2×C414yes—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D414yes—
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C4⋊C8—24no—
87#103—28no—
88#102—28no—
89C22⋊C8—24no—
90#97—24no—
91#95—22no—
92#95—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—24no—
97C2×Q8C22×C412yes—
98C2×D4—24no—
99C2×D4—21no—
100C2×D4—24no—
101C2×D4—44no—
102C2×D4C22×C412yes—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4C22⋊C414yes—
113C22×C4C22⋊C414yes—
114C22×C4C22×C412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2411yescharacteristic
122C2×C8—24no—
123C2×C8—24no—
124C2×C8—24no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—24no—
132C4⋊C4C22×C412yes—
133C4⋊C4C22×C412yes—
134C4⋊C4#4118yes—
135C22⋊C4—28no—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C42—28no—
142C42—24no—
143C42—44no—
144C42—44no—
145C42C22⋊C414yes—
146C42C22⋊C414yes—
147C42C22×C412yes—
148C42C22×C412yes—
149C42C2×D412yes—
150C42C2×D412yes—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23C2×C22⋊C412yes—
159C23C22×D411yescharacteristic
160Q8—24no—
161Q8—42no—
162D4—24no—
163D4—48no—
164D4—42no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—28no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—48no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4C2×C22⋊C412yes—
185C2×C4C2×C22⋊C412yes—
186C2×C4C2×C22⋊C412yes—
187C2×C4C2×C22⋊C412yes—
188C2×C4C2×C22⋊C412yes—
189C2×C4C23×C411yescharacteristic
190C2×C4C22×D411yescharacteristic
191C2×C4C2×#4114yes—
192C8—44no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—44no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—84no—
201C22C22×C22⋊C411yescharacteristic
202C22#52512yes—
203C22#57512yes—
204C22#57512yes—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—22no—
210C4—48no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4C4×#4112yes—
215C4#53212yes—
216C2—42no—
217C2—82no—
218C2#229711yescharacteristic
219C2#289611yescharacteristic
220C2#289611yescharacteristic
221C2#311114yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×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
2. C2×#432C2semidirect
3. C4×#117C2semidirect
4. #1989C2semidirect
5. #1983C2semidirect
7. #1873C2semidirect
8. #1763C2semidirect
9. #1762C2semidirect
11. #529C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4322
C4×#1172
#19892
#19832
#18752
#18732
#17632
#17622