Finite Group Databasebeta

← #27663

Subgroups — #27663

Order 256.

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

normal subgroups only

This group has 430 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×#435C211yesmaximal, characteristic
2C22×#106C211yesmaximal, characteristic, complemented
3#2487C211yesmaximal, characteristic
4#2486C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2077C212yesmaximal, complemented
7#1943C212yesmaximal, complemented
8#1824C211yesmaximal, characteristic, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1809C212yesmaximal, complemented
11#1808C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#123—21no—
15C2×C4×D4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#435—24no—
19C2×#107C2212yes—
20C2×#106—24no—
21C2×#106C2212yes—
22C2×#106C2212yes—
23C2×C4⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#400—21no—
28#399—21no—
29#398—21no—
30#397—21no—
31#389C2212yes—
32#389C2212yes—
33#386—21no—
34#385—21no—
35#381—21no—
36#380C2211yescharacteristic
37#380C2211yescharacteristic
38#380C2212yes—
39#380C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×D4D411yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C2×M16—22no—
51C2×M16—24no—
52C22×C8—24no—
53C22×C8D414yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58#123—44no—
59C4×D4—22no—
60C4×D4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4D411yescharacteristic
79C2×C22⋊C4D411yescharacteristic
80C2×C42C2311yescharacteristic
81#107—24no—
82#106—24no—
83#106—24no—
84C4⋊C8—22no—
85C22⋊C8—22no—
86C22⋊C8—22no—
87#97—22no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94C24C2×D411yescharacteristic
95C24C2×D411yescharacteristic
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—82no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4#4112yes—
120C22×C4C2411yescharacteristic
121M16—44no—
122C2×C8—22no—
123C2×C8—28no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8C2×D412yes—
129C2×C8C2×D412yes—
130C2×C8#4114yes—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—82no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—44no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—82no—
150C42—21no—
151C42—21no—
152C23—22no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23—22no—
159C23#12112yes—
160C23AGL(1,ℤ/8)12yes—
161C23#13712yes—
162C23C22×D411yescharacteristic
163D4—44no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—84no—
191C2×C4—84no—
192C2×C4#12112yes—
193C2×C4C22×D411yescharacteristic
194C2×C4C2×#4112yes—
195C2×C4C2×#4111yescharacteristic
196C2×C42−1+412yes—
197C8—22no—
198C8—41no—
199C8—41no—
200C8#12314yescomplemented
201C22—24no—
202C22—24no—
203C22—22no—
204C22—22no—
205C22—42no—
206C22—42no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22#46814yes—
211C22C2×#12111yescharacteristic
212C22#54712yes—
213C22C2×AGL(1,ℤ/8)11yescharacteristic
214C22C2×#13711yescharacteristic
215C22#57712yes—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C4—82no—
222C4—82no—
223C4—82no—
224C4C2×#12311yescharacteristic
225C4#53611yescharacteristic
226C4#55212yes—
227C2—24no—
228C2—81no—
229C2#248611yescharacteristic
230C2C2×#46811yescharacteristic
231C2#305711yescharacteristic
232C2#334412yes—
233C2#334512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#106C2semidirect
5. #2088C2semidirect
6. #2077C2semidirect
7. #1943C2semidirect
8. #1824C2semidirect
9. #1823C2semidirect
10. #1809C2semidirect
11. #1808C2semidirect
200. C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C22×#1062
#24872
#24862
#20882
#20772
#19432
#18242
#18232
#18092
#18082