Finite Group Databasebeta

← #27616

Subgroups — #27616

Order 256.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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×#389C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2085C212yesmaximal, complemented
5#1921C212yesmaximal, complemented
6#1920C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic
8#1852C211yesmaximal, characteristic
9#1823C211yesmaximal, characteristic
10#1823C211yesmaximal, characteristic
11#1817C211yesmaximal, characteristic
12C22×C4⋊C4—21no—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C22⋊C8C2211yescharacteristic
19C2×C22⋊C8C2211yescharacteristic
20#389—21no—
21#389—21no—
22#389C2212yes—
23#389C2212yes—
24#383—21no—
25#382C2211yescharacteristic
26#382C2211yescharacteristic
27#381—21no—
28#380C2211yescharacteristic
29#380C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32C4×C22⋊C4—21no—
33#376—21no—
34C2×#95—21no—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—44no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—24no—
75#106—24no—
76C22⋊C8—22no—
77C22⋊C8—22no—
78#95—22no—
79#95—22no—
80#95—21no—
81#95—21no—
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—24no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4#4112yes—
115C22×C4#4112yes—
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—42no—
134C4⋊C4#4114yes—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C42—22no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—42no—
144C42—42no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23#12312yes—
150C23#13512yes—
151C23AGL(1,ℤ/8)12yes—
152C23C22×D411yescharacteristic
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—44no—
170C2×C4—48no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—84no—
190C2×C4#12312yes—
191C2×C4#13712yes—
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4112yes—
195C2×C42−1+412yes—
196C8—42no—
197C8—42no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22#48214yes—
205C22C2×#12311yescharacteristic
206C22#52511yescharacteristic
207C22#53611yescharacteristic
208C22C2×#13511yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22C2×#13711yescharacteristic
211C22#57511yescharacteristic
212C4—22no—
213C4—22no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—44no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4#55212yes—
224C4#55212yes—
225C2—24no—
226C2#248411yescharacteristic
227C2C2×#48211yescharacteristic
228C2#309511yescharacteristic
229C2#310811yescharacteristic
230C2#311411yescharacteristic
231C2#323211yescharacteristic
232C2#323811yescharacteristic
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
3. #2101C2semidirect
4. #2085C2semidirect
5. #1921C2semidirect
6. #1920C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
#22832
#21012
#20852
#19212
#19202
#18532
#18522
#18232
#18172