Finite Group Databasebeta

← #24533

Subgroups — #24533

Order 256.

504 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 135 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×#448C211yesmaximal, characteristic
2C22×#103C211yesmaximal, characteristic, complemented
3#2504C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5#2074C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic
8#1880C212yesmaximal, complemented
9#1850C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1805C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19#448—24no—
20C2×C4⋊C8C2212yes—
21C2×#103—24no—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#397—22no—
29#391—22no—
30#391C2212yes—
31#390C2212yes—
32#386—21no—
33#386—21no—
34#382C2212yes—
35#380—21no—
36#380C2211yescharacteristic
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C22×Q8D414yes—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×SD16—22no—
56C2×SD16—22no—
57C2×SD16—24no—
58C2×SD16—42no—
59C22×C8—22no—
60C22×C8—42no—
61C22×C8C2312yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64#121—22no—
65#121—22no—
66C4×Q8—24no—
67C4×Q8—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4D411yescharacteristic
80C2×C22⋊C4D411yescharacteristic
81C2×C42—22no—
82C2×C42C2312yes—
83C4⋊C8—42no—
84#103—24no—
85#103—24no—
86#103—22no—
87#103—22no—
88#102—42no—
89C22⋊C8—22no—
90#95—22no—
91#95—22no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95D412yes—
96C24—21no—
97C24C2×D411yescharacteristic
98C2×Q8—24no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—28no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—48no—
108C2×Q8C2×D412yes—
109C2×Q8C2×D412yes—
110C2×Q8#4114yes—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—82no—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D412yes—
133C22×C4#4112yes—
134C22×C4C2411yescharacteristic
135SD16—44no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—44no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—44no—
152C4⋊C4—42no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C42—22no—
160C42—22no—
161C42—44no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—42no—
166C23—41no—
167C23—41no—
168C23—42no—
169C23#12112yes—
170C23#13512yes—
171C23#13712yes—
172C23C22×D411yescharacteristic
173Q8—22no—
174Q8—24no—
175Q8—22no—
176Q8—22no—
177Q8—48no—
178Q8—42no—
179Q8#12314yes—
180D4—41no—
181D4—41no—
182D4—42no—
183D4—84no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—48no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4#12112yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C2×#4112yes—
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)12yes—
214C8—41no—
215C8—42no—
216C8—41no—
217C22—22no—
218C22—22no—
219C22—24no—
220C22—24no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22—82no—
226C22#46414yes—
227C22C2×#12111yescharacteristic
228C22#54612yes—
229C22C2×#13511yescharacteristic
230C22C2×#13711yescharacteristic
231C22#57712yes—
232C4—22no—
233C4—24no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—81no—
238C4—82no—
239C4—82no—
240C4C2×#12311yescharacteristic
241C4#53411yescharacteristic
242C4#55112yes—
243C2—24no—
244C2—161no—
245C2#241211yescharacteristic
246C2C2×#46411yescharacteristic
247C2#304311yescharacteristic
248C2#330112yes—
249C2#332012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#103C2semidirect
4. Q8×C22⋊C4C2semidirect
5. #2074C2semidirect
6. #2055C2semidirect
8. #1880C2semidirect
9. #1850C2semidirect
10. #1824C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
C22×#1032
#25042
Q8×C22⋊C42
#20742
#20552
#18932
#18802
#18502
#18242
#18052