Finite Group Databasebeta

← #18723

Subgroups — #18723

Order 256.

466 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#3548C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic, complemented
4#3141C211yesmaximal, characteristic, complemented
5#1643C214yesmaximal, complemented
6#1642C214yesmaximal, complemented
7#1543C211yesmaximal, characteristic, complemented
8#1485C211yesmaximal, characteristic, complemented
9#1484C211yesmaximal, characteristic, complemented
10#562—22no—
11#554—21no—
12#554C2212yes—
13#550—21no—
14#550C2212yes—
15#547—21no—
16#547—21no—
17#540—22no—
18#537—22no—
19#537—22no—
20#533C2211yescharacteristic
21#532C2211yescharacteristic
22#532C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#498—21no—
25#492—21no—
26#489—22no—
27#487—21no—
28#487—21no—
29#487C2212yes—
30#486C2212yes—
31#483—22no—
32#475C2214yes—
33#461C2214yes—
34#448—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#417—22no—
38C4×M16C2211yescharacteristic
39#333C2212yes—
40#331C2212yes—
41Q8⋊C8C2214yes—
42D4⋊C8C2214yes—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—21no—
46C2×#41D412yes—
47C22×Q8—21no—
48C2×SD16—24no—
49C2×M16—21no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D411yescharacteristic
54C4⋊Q8D411yescharacteristic
55C4⋊Q8D411yescharacteristic
56C4⋊Q8C2312yes—
57C4⋊Q8C2311yescharacteristic
58#127—22no—
59#127D411yescharacteristic
60#127C2311yescharacteristic
61#125—22no—
62#125—21no—
63#124—21no—
64#124D412yes—
65#123—24no—
66#123—24no—
67C22⋊Q8—22no—
68C22⋊Q8—24no—
69C22⋊Q8—22no—
70C22⋊Q8—24no—
71C22⋊Q8—21no—
72C22⋊Q8D412yes—
73#121—24no—
74#121—24no—
75#121—22no—
76#121—24no—
77#121—24no—
78#121—21no—
79#121—21no—
80#121—21no—
81#121—22no—
82#121D412yes—
83C4×Q8—21no—
84C4×Q8C2312yes—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4C2312yes—
91#117—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C42C2311yescharacteristic
97#106—24no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—24no—
101#103—24no—
102#103—24no—
103#102—24no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106#97—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110#41—44no—
111#41—82no—
112#41—84no—
113#41—84no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—41no—
118C2×Q8—82no—
119C2×Q8C2×D412yes—
120C2×Q8C2×D412yes—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4C2×D412yes—
130C2×D4C2×D412yes—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139SD16—84no—
140M16—42no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C42—22no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2411yescharacteristic
172C23—22no—
173C23—22no—
174C23#12011yescharacteristic
175Q8—24no—
176Q8—42no—
177Q8—42no—
178Q8—44no—
179Q8—81no—
180Q8—82no—
181D4—24no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—42no—
186D4—81no—
187D4—82no—
188D4—82no—
189D4—84no—
190D4—84no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—84no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#12011yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C8—42no—
211C8—42no—
212C8—42no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22C2×#12011yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4#45512yes—
232C4#57712yes—
233C4#57712yes—
234C2—41no—
235C2—82no—
236C2—82no—
237C2#300911yescharacteristic
238C2#301011yescharacteristic
239C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3229C2semidirect
3. #3206C2semidirect
4. #3141C2semidirect
5. #1643C2semidirect
6. #1642C2semidirect
7. #1543C2semidirect
8. #1485C2semidirect
9. #1484C2semidirect

Maximal subgroups

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

subgroupindex
#35482
#32292
#32062
#31412
#16432
#16422
#15432
#14852
#14842