Finite Group Databasebeta

← #32669

Subgroups — #32669

Order 256.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#2568C211yesmaximal, characteristic, complemented
2#2548C211yesmaximal, characteristic
3#2055C212yesmaximal, complemented
4#2014C212yesmaximal
5#2013C212yesmaximal, complemented
6#1984C211yesmaximal, characteristic, complemented
7#1968C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1926C212yesmaximal, complemented
10C22×SD16C2212yes—
11C22×SD16C2212yes—
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#124—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C4×C8C2211yescharacteristic
23#400—21no—
24#400—21no—
25#394—21no—
26#394—21no—
27#391—21no—
28#391C2212yes—
29#391C2212yes—
30#390—21no—
31#390C2212yes—
32#390C2212yes—
33#382—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36#340C2212yes—
37#336C2212yes—
38#336C2212yes—
39C22×Q8D411yescharacteristic
40C22×Q8C2312yes—
41C22×D4—21no—
42C22×D4—21no—
43C22×D4C2312yes—
44C2×SD16—24no—
45C2×SD16—42no—
46C2×SD16—42no—
47C2×SD16—42no—
48C2×SD16—42no—
49C2×SD16—42no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—41no—
53C4⋊Q8—41no—
54#127—41no—
55#127—41no—
56#124—44no—
57C4×Q8—41no—
58C4×Q8—41no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#106—42no—
76C4⋊C8—41no—
77#103—24no—
78#103—42no—
79#102—24no—
80#102—42no—
81C4×C8—41no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88C24—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—41no—
96C2×Q8—41no—
97C2×Q8SD1614yescomplemented
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—82no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119SD16—48no—
120SD16—48no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8#4114yes—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—42no—
147C42—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23C22×D411yescharacteristic
156Q8—44no—
157Q8—42no—
158Q8—42no—
159Q8—42no—
160Q8—42no—
161D4—42no—
162D4—42no—
163D4—42no—
164D4—42no—
165D4—84no—
166D4—84no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4C2×SD1612yes—
189C2×C4C2×SD1612yes—
190C2×C4C2×SD1612yes—
191C2×C4C2×#4112yes—
192C2×C4H(5,𝔽2)11yescharacteristic
193C2×C4H(5,𝔽2)12yes—
194C2×C4H(5,𝔽2)11yescharacteristic
195C8—24no—
196C8—42no—
197C22—82no—
198C22—82no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22#53411yescharacteristic
205C22#55012yes—
206C22C22×SD1611yescharacteristic
207C22#57511yescharacteristic
208C22#57712yes—
209C4—22no—
210C4—22no—
211C4—22no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4#53812yes—
219C4#55511yescharacteristic
220C4H(3,𝔽4)11yescharacteristic
221C2—162no—
222C2#273311yescharacteristic
223C2#318711yescharacteristic
224C2#320011yescharacteristic
225C2#339212yes—
226C2#340112yes—
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
1. #2568C2semidirect
3. #2055C2semidirect
5. #2013C2semidirect
6. #1984C2semidirect
7. #1968C2semidirect
8. #1927C2semidirect
9. #1926C2semidirect
97. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#25682
#25482
#20552
#20142
#20132
#19842
#19682
#19272
#19262