Finite Group Databasebeta

← #27683

Subgroups — #27683

Order 256.

427 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C4×#121C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2093C211yesmaximal, characteristic, complemented
7#1949C212yesmaximal, complemented
8#1875C212yesmaximal
9#1854C212yesmaximal, complemented
10#1804C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×#121C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C2×#117C2212yes—
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C22⋊C8C2212yes—
23#390C2211yescharacteristic
24#389C2211yescharacteristic
25#388—21no—
26#386—21no—
27#384C2211yescharacteristic
28#384C2211yescharacteristic
29#382C2212yes—
30#380—21no—
31#380—21no—
32#380C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51#121—22no—
52#121Q814yes—
53C4×D4—24no—
54C4×D4—44no—
55#117—42no—
56#117—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—24no—
74C2×C42—22no—
75C2×C42—21no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81#106—42no—
82C4⋊C8—24no—
83#102—42no—
84#102—42no—
85C22⋊C8—24no—
86#95—21no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95—21no—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4C2×Q812yes—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×Q812yes—
122C22×C4#4112yes—
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4C2×Q812yes—
143C22⋊C4—24no—
144C22⋊C4—44no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42—44no—
153C42—42no—
154C42#4114yes—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23C22⋊Q812yes—
169C23C22×D411yescharacteristic
170D4—24no—
171D4—48no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—84no—
203C2×C4C22⋊Q812yes—
204C2×C4#12312yes—
205C2×C4#12312yes—
206C2×C4#13512yes—
207C2×C4AGL(1,ℤ/8)12yes—
208C2×C4C22×Q811yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4112yes—
211C2×C4H(5,𝔽2)12yes—
212C8—42no—
213C8—42no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22C2×C22⋊Q811yescharacteristic
229C22C2×#12311yescharacteristic
230C22#53511yescharacteristic
231C22C2×#13511yescharacteristic
232C22C2×AGL(1,ℤ/8)11yescharacteristic
233C22#57511yescharacteristic
234C22#57511yescharacteristic
235C4—22no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4#55312yes—
246C4#55412yes—
247C2—42no—
248C2—82no—
249C2#248711yescharacteristic
250C2#307611yescharacteristic
251C2#307711yescharacteristic
252C2#309111yescharacteristic
253C2#309411yescharacteristic
254C2#320711yescharacteristic
255C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2285C2semidirect
3. #2100C2semidirect
4. #2099C2semidirect
5. #2094C2semidirect
6. #2093C2semidirect
7. #1949C2semidirect
9. #1854C2semidirect
10. #1804C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#22852
#21002
#20992
#20942
#20932
#19492
#18752
#18542
#18042
#17972