Finite Group Databasebeta

← #25569

Subgroups — #25569

Order 256.

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

normal subgroups only

This group has 653 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
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#486C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic, complemented
4#2541C211yesmaximal, characteristic
5#2000C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1874C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10C22×SD16C2212yes—
11#554—24no—
12#554—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#127—21no—
15C2×#127C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—22no—
18C2×C22⋊Q8C2212yes—
19C2×#121C2214yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4C2212yes—
24#486—24no—
25#460—24no—
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#102C2214yes—
29C2×#102C2212yes—
30C2×C22⋊C8C2214yes—
31C2×C4×C8C2211yescharacteristic
32#392—22no—
33#390—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8C2311yescharacteristic
39C22×D4—24no—
40C22×D4—21no—
41C22×D4—21no—
42C22×D4D411yescharacteristic
43C22×D4C2312yes—
44C23×C4—22no—
45C23×C4C2312yes—
46C2×SD16—24no—
47C2×SD16—44no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8D414yescomplemented
52#127—22no—
53#127—22no—
54#127—44no—
55#125—28no—
56C22⋊Q8—216no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59#121—28no—
60#121—44no—
61C4×Q8—22no—
62C4×Q8D414yescomplemented
63C4×D4—28no—
64C4×D4—22no—
65C4×D4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4D412yes—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C2311yescharacteristic
82C4⋊C8—22no—
83#103—24no—
84#102—24no—
85#102—28no—
86C22⋊C8—44no—
87C4×C8—22no—
88#95—21no—
89C24—22no—
90C24C2×D412yes—
91C2×Q8—24no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—82no—
96C2×Q8C2×D412yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—48no—
104C2×D4—48no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—82no—
110C2×D4—84no—
111C2×D4SD1614yescomplemented
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—28no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—84no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126SD16—48no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—24no—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4C2×D412yes—
141C4⋊C4C2×D414yes—
142C4⋊C4#4114yes—
143C22⋊C4—28no—
144C22⋊C4—24no—
145C22⋊C4—48no—
146C42—24no—
147C42—42no—
148C42C2×D412yes—
149C42C2×D412yes—
150C23—22no—
151C23—24no—
152C23—22no—
153C23—24no—
154C23—42no—
155C23—44no—
156C23—42no—
157C23—42no—
158C23—44no—
159C23C2×SD1614yes—
160C23C22×D411yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—42no—
164Q8—84no—
165D4—28no—
166D4—42no—
167D4—42no—
168D4—44no—
169D4—84no—
170D4—816no—
171D4—84no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—88no—
188C2×C4#12012yes—
189C2×C4#12012yes—
190C2×C4C2×SD1612yes—
191C2×C4#13712yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D411yescharacteristic
194C2×C4C2×#4112yes—
195C2×C4H(5,𝔽2)12yes—
196C8—42no—
197C8—42no—
198C22—24no—
199C22—28no—
200C22—44no—
201C22—42no—
202C22—42no—
203C22—84no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22—84no—
208C22C2×#12011yescharacteristic
209C22#54612yes—
210C22C22×SD1611yescharacteristic
211C22C2×#13711yescharacteristic
212C22#57612yes—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—44no—
218C4—44no—
219C4#44814yes—
220C4#53211yescharacteristic
221C4#53511yescharacteristic
222C4#55012yes—
223C2—44no—
224C2—162no—
225C2#243111yescharacteristic
226C2C2×#44811yescharacteristic
227C2#300811yescharacteristic
228C2#329512yes—
229C2#333312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#486C2semidirect
3. C2×#460C2semidirect
5. #2000C2semidirect
6. #1987C2semidirect
7. #1954C2semidirect
8. #1874C2semidirect
9. C4×#103C2semidirect
51. C4⋊Q8D4semidirect
62. C4×Q8D4semidirect
111. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4862
C2×#4602
#25412
#20002
#19872
#19542
#18742
C4×#1032