Finite Group Databasebeta

← #8350

Subgroups — #8350

Order 256.

557 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 89 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#3496C211yesmaximal, characteristic
2#1893C212yesmaximal, complemented
3#1869C211yesmaximal, characteristic, complemented
4#1799C211yesmaximal, characteristic, complemented
5#1519C212yesmaximal, complemented
6C2×H(5,𝔽2)—21no—
7C23×C8C2211yescharacteristic, complemented
8#547—22no—
9#546—24no—
10D42—22no—
11#537C412yescomplemented
12#534—21no—
13#534C412yescomplemented
14#532—21no—
15C2×#121—21no—
16C2×#121C2212yes—
17C2×C4×D4C2211yescharacteristic
18C2×#102—22no—
19C2×#99—21no—
20C2×C22⋊C8—21no—
21C2×C22⋊C8C2211yescharacteristic, complemented
22#380C2212yescomplemented
23#342—21no—
24#340—22no—
25#336—21no—
26#327—24no—
27H(5,𝔽2)—44no—
28H(5,𝔽2)—42no—
29H(5,𝔽2)—42no—
30C2×#41—24no—
31C2×#41—22no—
32C22×D4—24no—
33C22×D4—22no—
34C22×D4—22no—
35C22×D4—21no—
36C22×D4D411yescharacteristic
37C22×D4D412yes—
38C23×C4D411yescharacteristic
39C23×C4C2311yescharacteristic
40C22×C8—21no—
41C22×C8—22no—
42C22×C8—21no—
43C22×C8—42no—
44C4⋊Q8—21no—
45#127—21no—
46#124—22no—
47#123—24no—
48#123—24no—
49C22⋊Q8—24no—
50#121—22no—
51#121—22no—
52#121—24no—
53#121—22no—
54#121—22no—
55#121—22no—
56#121—22no—
57#121C2×C414yes—
58#121D414yes—
59#120—24no—
60#120—24no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4C2×C411yescharacteristic
68C2×C4⋊C4D412yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4C2×C411yescharacteristic
73C2×C22⋊C4D411yescharacteristic
74C2×C42D411yescharacteristic
75#102—44no—
76#99—42no—
77C22⋊C8—22no—
78C22⋊C8—42no—
79#95—22no—
80C24—22no—
81C24—22no—
82C24C22⋊C411yescharacteristic
83C24C2×D411yescharacteristic
84#41—44no—
85#41—48no—
86#41—48no—
87#41—44no—
88C2×Q8—22no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—48no—
101C2×D4—48no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—81no—
109C2×D4—81no—
110C2×D4C22⋊C414yes—
111C2×D4D812yes—
112C2×D4SD1612yes—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4C22⋊C411yescharacteristic
124C22×C4C22⋊C411yescharacteristic
125C22×C4C22⋊C411yescharacteristic
126C22×C4C22×C411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D412yes—
130C2×C8—22no—
131C2×C8—24no—
132C2×C8—42no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4C22⋊C414yes—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—22no—
150C22⋊C4—24no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—21no—
154C42—21no—
155C23—21no—
156C23—21no—
157C23—24no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—44no—
163C23—42no—
164C23—42no—
165C23—82no—
166C23—82no—
167C23—82no—
168C23—82no—
169C23#10212yescomplemented
170C23C2×C22⋊C411yescharacteristic
171C23#12012yes—
172C23C2×D811yescharacteristic
173C23C2×SD1611yescharacteristic
174C23AGL(1,ℤ/8)12yes—
175Q8—44no—
176D4—24no—
177D4—24no—
178D4—44no—
179D4—44no—
180D4—48no—
181D4—44no—
182D4—44no—
183D4—44no—
184D4—44no—
185D4—42no—
186D4—42no—
187D4—82no—
188D4—82no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#10212yescomplemented
213C2×C4C2×C22⋊C412yes—
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C8—42no—
217C8—81no—
218C22—21no—
219C22—22no—
220C22—21no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—81no—
233C22—162no—
234C22—162no—
235C22—164no—
236C22#37911yescharacteristic
237C22C2×#10211yescharacteristic
238C22#41811yescharacteristic
239C22#44712yes—
240C22#45012yes—
241C4—22no—
242C4—21no—
243C4—22no—
244C4—41no—
245C4—44no—
246C4—41no—
247C4—162no—
248C4B(4,𝔽2)11yescharacteristic
249C4#45811yescharacteristic
250C2—22no—
251C2—21no—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—84no—
256C2#178611yescharacteristic
257C2#179011yescharacteristic
258C2#179411yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1893C2semidirect
3. #1869C2semidirect
4. #1799C2semidirect
5. #1519C2semidirect
7. C23×C8C22semidirect
11. #537C4semidirect
13. #534C4semidirect
21. C2×C22⋊C8C22semidirect
22. #380C22semidirect
169. C23#102semidirect
212. C2×C4#102semidirect

Maximal subgroups

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

subgroupindex
#34962
#18932
#18692
#17992
#15192