Finite Group Databasebeta

← #22223

Subgroups — #22223

Order 256.

421 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 163 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×#489C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#124C211yesmaximal, characteristic
4#1993C211yesmaximal, characteristic, complemented
5#1959C211yesmaximal, characteristic
6#1928C212yesmaximal, complemented
7#1926C212yesmaximal
8#1865C212yesmaximal, complemented
9#1862C212yesmaximal
10#1847C211yesmaximal, characteristic, complemented
11#1745C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#489C418yescomplemented
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#97C2211yescharacteristic
25C2×#97C2211yescharacteristic
26#391—21no—
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#389—21no—
30#389C2212yes—
31#388—21no—
32#388—21no—
33#384C2211yescharacteristic
34#382C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4—21no—
38C4×C22⋊C4—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43#336C2212yes—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8C2×C414yescomplemented
52#124—22no—
53#124C2×C414yes—
54C4×Q8—42no—
55C4×Q8—44no—
56C4×D4—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C42—22no—
72C2×C42—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81#106—42no—
82#103—42no—
83#103C2×C418yescomplemented
84#102—42no—
85#102C2×C418yes—
86#97—22no—
87#97C2×C414yescomplemented
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94C24—21no—
95C2×Q8—24no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8C22×C412yes—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4C22×C412yes—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8C22×C414yes—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4C22×C414yes—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42—22no—
147C42—41no—
148C42—41no—
149C42—42no—
150C42—44no—
151C42C22×C412yes—
152C42#4112yes—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23C22×D411yescharacteristic
161Q8—24no—
162Q8—48no—
163Q8—42no—
164D4—24no—
165D4—42no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—84no—
196C2×C4C4×D412yes—
197C2×C4C4×D412yes—
198C2×C4AGL(1,ℤ/8)12yes—
199C2×C4#13712yes—
200C2×C4C23×C411yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)12yes—
203C2×C42−1+412yes—
204C8—24no—
205C8—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22C2×C4×D411yescharacteristic
214C22#53511yescharacteristic
215C22#53611yescharacteristic
216C22C2×AGL(1,ℤ/8)11yescharacteristic
217C22C2×#13711yescharacteristic
218C22#57511yescharacteristic
219C22#57511yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—44no—
226C4—41no—
227C4—41no—
228C4—82no—
229C4—81no—
230C4—81no—
231C4#52012yes—
232C4#54212yes—
233C2—82no—
234C2#232611yescharacteristic
235C2C4×AGL(1,ℤ/8)11yescharacteristic
236C2C4×#13711yescharacteristic
237C2#321311yescharacteristic
238C2#321611yescharacteristic
239C2#323911yescharacteristic
240C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊Q8C2semidirect
4. #1993C2semidirect
6. #1928C2semidirect
8. #1865C2semidirect
10. #1847C2semidirect
11. #1745C2semidirect
17. #489C4semidirect
51. C4⋊Q8C2×C4semidirect
83. #103C2×C4semidirect
87. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C4×C4⋊Q82
C4×#1242
#19932
#19592
#19282
#19262
#18652
#18622
#18472
#17452