Finite Group Databasebeta

← #27880

Subgroups — #27880

Order 256.

355 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 119 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#2851C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2077C212yesmaximal
4#2075C212yesmaximal
5#1992C211yesmaximal, characteristic, complemented
6#1992C211yesmaximal, characteristic, complemented
7#1921C212yesmaximal, complemented
8#1919C212yesmaximal, complemented
9#1847C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic
12C2×#125C2211yescharacteristic
13C2×#117—21no—
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C22⋊C8C2212yes—
18C2×#97C2211yescharacteristic
19C2×#97C2211yescharacteristic
20#389C2211yescharacteristic
21#389C2211yescharacteristic
22#387—21no—
23#387—21no—
24#385—21no—
25#385—21no—
26#384C2211yescharacteristic
27#384C2211yescharacteristic
28#382—21no—
29#382C2211yescharacteristic
30#380—21no—
31#380C2211yescharacteristic
32#380C2212yes—
33#380C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48#125—22no—
49#125—22no—
50#117—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—22no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—24no—
76#106—24no—
77C22⋊C8—24no—
78#97—22no—
79#97—22no—
80#95—22no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88C24C2×D411yescharacteristic
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8#4114yes—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—24no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—44no—
137C42—22no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—44no—
142C42#4112yes—
143C42#4112yes—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23AGL(1,ℤ/8)12yes—
151C23C22×D411yescharacteristic
152C2×C4—28no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4#12312yes—
187C2×C4#12312yes—
188C2×C4#13712yes—
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4112yes—
191C2×C4C2×#4111yescharacteristic
192C2×C42−1+412yes—
193C8—24no—
194C8—42no—
195C22—22no—
196C22—22no—
197C22—22no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22C2×#12311yescharacteristic
203C22#53211yescharacteristic
204C22#53611yescharacteristic
205C22C2×AGL(1,ℤ/8)11yescharacteristic
206C22C2×#13711yescharacteristic
207C22#57511yescharacteristic
208C22#57511yescharacteristic
209C4—22no—
210C4—22no—
211C4—22no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—81no—
216C4—81no—
217C4—82no—
218C4—81no—
219C4—82no—
220C4#55212yes—
221C4#55612yes—
222C2—42no—
223C2#249911yescharacteristic
224C2#309211yescharacteristic
225C2#309311yescharacteristic
226C2#317211yescharacteristic
227C2#318011yescharacteristic
228C2#323411yescharacteristic
229C2#323711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
5. #1992C2semidirect
6. #1992C2semidirect
7. #1921C2semidirect
8. #1919C2semidirect
9. #1847C2semidirect
10. #1836C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#22852
#20772
#20752
#19922
#19212
#19192
#18472
#18362
#17492