Finite Group Databasebeta

← #27651

Subgroups — #27651

Order 256.

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

normal subgroups only

This group has 461 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×#481C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic
3C4×#121C211yesmaximal, characteristic
4C2×#382C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2073C212yesmaximal, complemented
7#1949C212yesmaximal, complemented
8#1824C211yesmaximal, characteristic, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1804C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×#121C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#481—24no—
19#480—24no—
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27#390C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#386—21no—
31#382—21no—
32#382—21no—
33#382C2212yes—
34#382C2212yes—
35#380C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38C2×#95—21no—
39C2×#95—21no—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2312yes—
55#121—22no—
56#121D414yes—
57C4×D4—24no—
58C4×D4—44no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C42—24no—
77C2×C42—21no—
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—24no—
84C4⋊C8—24no—
85#102—24no—
86#102—24no—
87C22⋊C8—22no—
88C22⋊C8—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—21no—
97C24C2×D411yescharacteristic
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4C2×D412yes—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4#4112yes—
134C22×C4C2411yescharacteristic
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4C2×D412yes—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—42no—
161C42#4114yes—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23#12112yes—
174C23#13512yes—
175C23AGL(1,ℤ/8)12yes—
176C23C22×D411yescharacteristic
177D4—24no—
178D4—48no—
179D4—42no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4#12112yes—
215C2×C4#12312yes—
216C2×C4#12312yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4112yes—
219C2×C4C2×#4111yescharacteristic
220C2×C42−1+412yes—
221C8—42no—
222C8—42no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22#46314yes—
237C22C2×#12111yescharacteristic
238C22C2×#12311yescharacteristic
239C22#53611yescharacteristic
240C22C2×#13511yescharacteristic
241C22C2×AGL(1,ℤ/8)11yescharacteristic
242C22#57511yescharacteristic
243C22#57511yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—41no—
249C4—42no—
250C4—81no—
251C4—81no—
252C4—82no—
253C4—82no—
254C4#54712yes—
255C4#55212yes—
256C2—24no—
257C2—82no—
258C2#248611yescharacteristic
259C2C2×#46311yescharacteristic
260C2#304011yescharacteristic
261C2#309111yescharacteristic
262C2#309411yescharacteristic
263C2#323211yescharacteristic
264C2#323411yescharacteristic
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
4. C2×#382C2semidirect
5. #2085C2semidirect
6. #2073C2semidirect
7. #1949C2semidirect
8. #1824C2semidirect
9. #1823C2semidirect
10. #1804C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4802
C4×#1212
C2×#3822
#20852
#20732
#19492
#18242
#18232
#18042
#17972