Finite Group Databasebeta

← #29943

Subgroups — #29943

Order 256.

348 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 117 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#2099C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2077C212yesmaximal
8#2076C212yesmaximal
9#1989C212yesmaximal, complemented
10#1949C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic
12C2×#125C2211yescharacteristic
13C2×#117—21no—
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2212yes—
18C2×C22⋊C8C2212yes—
19#389C2211yescharacteristic
20#389C2211yescharacteristic
21#387—21no—
22#387—21no—
23#385—21no—
24#385—21no—
25#384C2211yescharacteristic
26#384C2211yescharacteristic
27#382—21no—
28#382C2211yescharacteristic
29#380—21no—
30#380C2211yescharacteristic
31#380C2212yes—
32#380C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48#125—24no—
49#117—44no—
50C2×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C42—24no—
68C2×C42—22no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#106—42no—
75C4⋊C8—24no—
76C22⋊C8—42no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85C24C2×D411yescharacteristic
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C22⋊C4—42no—
134C42—22no—
135C42—22no—
136C42—22no—
137C42—44no—
138C42#4114yes—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23#13712yes—
146C23C22×D411yescharacteristic
147C2×C4—28no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4#12312yes—
181C2×C4#12312yes—
182C2×C4#13512yes—
183C2×C4AGL(1,ℤ/8)12yes—
184C2×C4C2×#4112yes—
185C2×C4H(5,𝔽2)11yescharacteristic
186C2×C42−1+411yescharacteristic
187C2×C42−1+412yes—
188C8—42no—
189C8—42no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22C2×#12311yescharacteristic
198C22#53611yescharacteristic
199C22#53711yescharacteristic
200C22C2×#13511yescharacteristic
201C22C2×AGL(1,ℤ/8)11yescharacteristic
202C22C2×#13711yescharacteristic
203C22#57511yescharacteristic
204C4—22no—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—41no—
209C4—81no—
210C4—81no—
211C4—82no—
212C4—81no—
213C4—82no—
214C4—81no—
215C4#54112yes—
216C4#54212yes—
217C2—42no—
218C2#262511yescharacteristic
219C2#309111yescharacteristic
220C2#309211yescharacteristic
221C2#322811yescharacteristic
222C2#323611yescharacteristic
223C2#324811yescharacteristic
224C2#325111yescharacteristic
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. #2099C2semidirect
4. #2099C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect
9. #1989C2semidirect
10. #1949C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#22852
#20992
#20842
#20832
#20772
#20762
#19892
#19492
#17972