Finite Group Databasebeta

← #25908

Subgroups — #25908

Order 256.

660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#489C212yesmaximal, complemented
3#2850C211yesmaximal, characteristic
4#2024C212yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1993C212yesmaximal, complemented
7#1874C212yesmaximal
8#1872C212yesmaximal, complemented
9#1749C211yesmaximal, characteristic, complemented
10C23×Q8—21no—
11C2×C4⋊Q8—28no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2212yes—
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C22×C4⋊C4—21no—
20C22×C4⋊C4C2212yes—
21C22×C42C2211yescharacteristic
22#489—28no—
23C2×#103C2214yes—
24C2×#102C2214yes—
25C2×C22⋊C8C2212yes—
26C2×#97C2212yes—
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#388—21no—
30#388—21no—
31#386—21no—
32#386—21no—
33#382C2211yescharacteristic
34#340C2214yes—
35#340C2214yes—
36#336C2212yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—22no—
40C22×Q8—44no—
41C22×Q8—44no—
42C22×Q8C2311yescharacteristic
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—22no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2314yes—
50C4⋊Q8—28no—
51C4⋊Q8—28no—
52C4⋊Q8—28no—
53C4⋊Q8—24no—
54C4⋊Q8D418yescomplemented
55#124—24no—
56C22⋊Q8—24no—
57#121—24no—
58C2×C4⋊C4—28no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2314yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—24no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#103—28no—
79#102—28no—
80C22⋊C8—24no—
81#97—24no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86C24—21no—
87C24C2×D411yescharacteristic
88C2×Q8—28no—
89C2×Q8—28no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—432no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—42no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—28no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4#4114yes—
125C22×C4C2411yescharacteristic
126C2×C8—24no—
127C2×C8—24no—
128C2×C8—24no—
129C4⋊C4—28no—
130C4⋊C4—28no—
131C4⋊C4—28no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4C2×D418yes—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42C2×D414yes—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23#12412yes—
159C23AGL(1,ℤ/8)12yes—
160C23#13712yes—
161C23C22×D411yescharacteristic
162Q8—416no—
163Q8—416no—
164Q8—44no—
165D4—44no—
166D4—84no—
167C2×C4—28no—
168C2×C4—28no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—28no—
180C2×C4—28no—
181C2×C4—416no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4#12012yes—
194C2×C4#12012yes—
195C2×C4#12412yes—
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4#13712yes—
198C2×C4C22×D412yes—
199C2×C4C2×#4112yes—
200C2×C42−1+412yes—
201C8—44no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#48914yescomplemented
216C22C2×#12011yescharacteristic
217C22C2×#12411yescharacteristic
218C22#53611yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C22C2×AGL(1,ℤ/8)11yescharacteristic
221C22C2×#13711yescharacteristic
222C22C2×#13711yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—48no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4#54814yes—
233C2—24no—
234C2—161no—
235C2#243611yescharacteristic
236C2#300311yescharacteristic
237C2#300411yescharacteristic
238C2C2×#48911yescharacteristic
239C2#314111yescharacteristic
240C2#323811yescharacteristic
241C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×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
1. C22×C4⋊Q8C2semidirect
2. C2×#489C2semidirect
4. #2024C2semidirect
5. #2023C2semidirect
6. #1993C2semidirect
8. #1872C2semidirect
9. #1749C2semidirect
54. C4⋊Q8D4semidirect
215. C22#489semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4892
#28502
#20242
#20232
#19932
#18742
#18722
#17492