Finite Group Databasebeta

← #25568

Subgroups — #25568

Order 256.

595 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 139 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×#555C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4#2542C211yesmaximal, characteristic, complemented
5#2000C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1873C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C22×SD16C2212yes—
11#555—24no—
12#555—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124—22no—
16C2×C22⋊Q8C2214yes—
17C2×#121—21no—
18C2×#121C2212yes—
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#117C2212yes—
23#487—24no—
24#461—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2214yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2214yes—
30C2×C4×C8C2211yescharacteristic
31#399—22no—
32#390—21no—
33#384—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8C2312yes—
39C22×D4—22no—
40C22×D4D411yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4—21no—
43C23×C4C2312yes—
44C2×SD16—24no—
45C2×SD16—44no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8D414yes—
52#124—216no—
53C22⋊Q8—28no—
54C22⋊Q8—44no—
55#121—28no—
56#121—24no—
57#121—24no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4D414yes—
62#117—28no—
63#117—24no—
64#117—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4D412yes—
78C2×C42—22no—
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81C4⋊C8—22no—
82#103—24no—
83#103—28no—
84#102—24no—
85C22⋊C8—44no—
86C4×C8—22no—
87#95—21no—
88C24—21no—
89C24C2×D412yes—
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—24no—
94C2×Q8—24no—
95C2×Q8—82no—
96C2×D4—28no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—84no—
104C2×D4C2×D412yes—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120SD16—48no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—24no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—48no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4C2×D412yes—
135C4⋊C4C2×D414yes—
136C4⋊C4#4114yes—
137C22⋊C4—28no—
138C22⋊C4—28no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—48no—
142C42—28no—
143C42—42no—
144C42C2×D412yes—
145C42C2×D412yes—
146C23—22no—
147C23—22no—
148C23—24no—
149C23—22no—
150C23—22no—
151C23—24no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23C22×D411yescharacteristic
157Q8—42no—
158Q8—42no—
159Q8—44no—
160Q8—84no—
161D4—22no—
162D4—22no—
163D4—42no—
164D4—44no—
165D4—88no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—48no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4#12012yes—
184C2×C4#12012yes—
185C2×C4#13512yes—
186C2×C4#13712yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C22×D411yescharacteristic
189C2×C4C2×#4112yes—
190C2×C4H(5,𝔽2)12yes—
191C8—42no—
192C8—42no—
193C22—44no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—44no—
201C22—82no—
202C22—82no—
203C22C2×#12011yescharacteristic
204C22#54612yes—
205C22C2×#13511yescharacteristic
206C22C2×#13711yescharacteristic
207C22#57712yes—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—44no—
212C4—44no—
213C4—82no—
214C4#45214yes—
215C4#53211yescharacteristic
216C4#53511yescharacteristic
217C4#55012yes—
218C2—82no—
219C2—82no—
220C2#243111yescharacteristic
221C2C2×#45211yescharacteristic
222C2#300911yescharacteristic
223C2#330312yes—
224C2#332912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#555C2
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
2. C2×#487C2semidirect
3. C2×#461C2semidirect
4. #2542C2semidirect
5. #2000C2semidirect
6. #1987C2semidirect
7. #1954C2semidirect
8. #1873C2semidirect
9. C4×#102C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4872
C2×#4612
#25422
#20002
#19872
#19542
#18732
C4×#1022