Finite Group Databasebeta

← #25629

Subgroups — #25629

Order 256.

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

normal subgroups only

This group has 589 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×#559C211yesmaximal, characteristic
2C2×#492C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4#2568C211yesmaximal, characteristic, complemented
5#2015C214yesmaximal, complemented
6#1984C211yesmaximal, characteristic, complemented
7#1982C211yesmaximal, characteristic
8#1877C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10C22×Q16C2212yes—
11C22×SD16C2214yes—
12#559—24no—
13#559—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124—21no—
17C2×#124C2212yes—
18C2×#123—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2212yes—
23#492—24no—
24#462—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2214yes—
29C2×C4×C8C2211yescharacteristic
30#400—22no—
31#391—21no—
32#391C2214yes—
33#382—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#336C2214yes—
37C22×Q8D411yescharacteristic
38C22×Q8C2312yes—
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4—21no—
42C22×D4C2312yes—
43C23×C4—22no—
44C2×Q16—24no—
45C2×Q16—44no—
46C2×SD16—28no—
47C2×SD16—28no—
48C2×SD16—44no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#127—22no—
54#127D414yes—
55#124—28no—
56#124—24no—
57#124—24no—
58#123—216no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8D414yes—
63C4×D4—28no—
64C4×D4—24no—
65C4×D4—24no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4D412yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80C4⋊C8—22no—
81#103—24no—
82#103—24no—
83#102—44no—
84C4×C8—22no—
85#95—21no—
86#95—24no—
87#95—24no—
88C24—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—24no—
97C2×Q8—22no—
98C2×Q8—24no—
99C2×Q8C2×D412yes—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—22no—
103C2×D4—24no—
104C2×D4—24no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—82no—
108C2×D4C2×D414yes—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—28no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—84no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120Q16—48no—
121SD16—416no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—24no—
126C2×C8#4114yes—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4C2×D412yes—
136C22⋊C4—28no—
137C22⋊C4—28no—
138C22⋊C4—28no—
139C42—24no—
140C42—22no—
141C42—22no—
142C42—42no—
143C42C2×D412yes—
144C42C2×D412yes—
145C23—24no—
146C23—44no—
147C23—42no—
148C23—42no—
149C23—44no—
150C23C22×D411yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—42no—
154Q8—44no—
155Q8—42no—
156Q8—42no—
157Q8—44no—
158D4—28no—
159D4—42no—
160D4—42no—
161D4—48no—
162D4—84no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—88no—
179C2×C4#12012yes—
180C2×C4#12012yes—
181C2×C4#13512yes—
182C2×C4#13712yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×D411yescharacteristic
185C2×C4C2×#4112yes—
186C2×C4H(5,𝔽2)12yes—
187C8—24no—
188C8—42no—
189C22—44no—
190C22—44no—
191C22—44no—
192C22—84no—
193C22—84no—
194C22C2×#12011yescharacteristic
195C22#55012yes—
196C22C2×#13511yescharacteristic
197C22C2×#13711yescharacteristic
198C22#57712yes—
199C4—22no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—82no—
205C4—82no—
206C4#45214yes—
207C4#53211yescharacteristic
208C4#53511yescharacteristic
209C4#54612yes—
210C2—84no—
211C2#243111yescharacteristic
212C2C2×#45211yescharacteristic
213C2#300911yescharacteristic
214C2#339412yes—
215C2#340012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#462C2semidirect
4. #2568C2semidirect
5. #2015C2semidirect
6. #1984C2semidirect
8. #1877C2semidirect
9. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4922
C2×#4622
#25682
#20152
#19842
#19822
#18772
C4×#1032