Finite Group Databasebeta

← #22085

Subgroups — #22085

Order 256.

442 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 169 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×#484C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#389C211yesmaximal, characteristic
4#1943C212yesmaximal, complemented
5#1935C212yesmaximal, complemented
6#1918C212yesmaximal, complemented
7#1872C211yesmaximal, characteristic
8#1865C212yesmaximal, complemented
9#1853C211yesmaximal, characteristic
10#1824C211yesmaximal, characteristic
11#1749C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C22×C4⋊C4—21no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#484C418yescomplemented
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×#97C2212yes—
26#391C2211yescharacteristic
27#389—21no—
28#389—21no—
29#389C2212yes—
30#389C2212yes—
31#389C2211yescharacteristic
32#387—21no—
33#385—21no—
34#380C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52C22⋊Q8—22no—
53C22⋊Q8C2×C414yes—
54C4×Q8—42no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4C2×C414yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107C2×C418yescomplemented
81#106—42no—
82#103—42no—
83#103C2×C418yescomplemented
84C22⋊C8—22no—
85C22⋊C8C2×C414yes—
86#97—42no—
87#95—22no—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8C22×C412yes—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C22×C412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4#4112yes—
126C22×C4C2411yescharacteristic
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C22×C414yes—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4C22×C414yes—
147C4⋊C4C22×C412yes—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C42—22no—
151C42—22no—
152C42—41no—
153C42—41no—
154C42—42no—
155C42—42no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23C4×D412yes—
161C23#13712yes—
162C23#13712yes—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—42no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—84no—
203C2×C4C4×D412yes—
204C2×C4AGL(1,ℤ/8)12yes—
205C2×C4C23×C411yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)12yes—
208C2×C42−1+412yes—
209C8—24no—
210C8—42no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22#44114yes—
218C22C2×C4×D411yescharacteristic
219C22#53411yescharacteristic
220C22#53611yescharacteristic
221C22C2×AGL(1,ℤ/8)11yescharacteristic
222C22C2×#13711yescharacteristic
223C22C2×#13711yescharacteristic
224C22#57511yescharacteristic
225C4—22no—
226C4—22no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—42no—
232C4—44no—
233C4—82no—
234C4—81no—
235C4—81no—
236C4#52012yes—
237C4#53912yes—
238C2—24no—
239C2#232211yescharacteristic
240C2C2×#44111yescharacteristic
241C2C4×AGL(1,ℤ/8)11yescharacteristic
242C2#319311yescharacteristic
243C2#319611yescharacteristic
244C2#323611yescharacteristic
245C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
4. #1943C2semidirect
5. #1935C2semidirect
6. #1918C2semidirect
8. #1865C2semidirect
18. #484C4semidirect
80. #107C2×C4semidirect
83. #103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C4×C22⋊Q82
C2×#3892
#19432
#19352
#19182
#18722
#18652
#18532
#18242
#17492