Finite Group Databasebeta

← #24578

Subgroups — #24578

Order 256.

448 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 139 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
2C22×#106C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic
4#2345C211yesmaximal, characteristic, complemented
5#2084C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#1948C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic, complemented
11#1823C211yesmaximal, characteristic, complemented
12C23×C8C2211yescharacteristic
13C2×#123—21no—
14C2×#123—21no—
15C2×#121C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#481—24no—
19C2×#106—24no—
20C2×#106C2212yes—
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2212yes—
24C2×C4⋊C8C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#399—22no—
30#390C2212yes—
31#389C2212yes—
32#387—22no—
33#384C2212yes—
34#380—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C22×C8—22no—
49C22×C8—24no—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53#123—44no—
54#123—44no—
55#121—22no—
56#121—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D414yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4D411yescharacteristic
76C2×C22⋊C4D411yescharacteristic
77C2×C42—22no—
78C2×C42C2312yes—
79#106—24no—
80#106—24no—
81#106—22no—
82#106—22no—
83C4⋊C8—24no—
84#102—24no—
85#102—22no—
86#102—22no—
87C22⋊C8—22no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95D412yes—
92C24—21no—
93C24C2×D411yescharacteristic
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—22no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—42no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4#4112yes—
118C22×C4C2411yescharacteristic
119C2×C8—28no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8#4114yes—
127C4⋊C4—24no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—44no—
139C4⋊C4—82no—
140C4⋊C4C2×D412yes—
141C4⋊C4C2×D412yes—
142C22⋊C4—22no—
143C22⋊C4—44no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C22⋊C4SD1614yes—
148C42—22no—
149C42—22no—
150C42—82no—
151C23—22no—
152C23—22no—
153C23—41no—
154C23—42no—
155C23—41no—
156C23—41no—
157C23—42no—
158C23#12112yes—
159C23C2×SD1612yes—
160C23AGL(1,ℤ/8)12yes—
161C23C22×D411yescharacteristic
162D4—41no—
163D4—41no—
164D4—42no—
165D4—84no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—84no—
189C2×C4—84no—
190C2×C4#12112yes—
191C2×C4C2×SD1614yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4112yes—
194C2×C4C2×#4111yescharacteristic
195C2×C42−1+412yes—
196C8—24no—
197C8—41no—
198C8—41no—
199C22—24no—
200C22—24no—
201C22—22no—
202C22—22no—
203C22—82no—
204C22—82no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22#46014yes—
209C22C2×#12111yescharacteristic
210C22#54712yes—
211C22C22×SD1611yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22#57712yes—
214C4—24no—
215C4—22no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—82no—
220C4—82no—
221C4—81no—
222C4#52511yescharacteristic
223C4#53611yescharacteristic
224C4#55212yes—
225C2—24no—
226C2—161no—
227C2#241311yescharacteristic
228C2C2×#46011yescharacteristic
229C2#304211yescharacteristic
230C2#333512yes—
231C2#334512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#106C2semidirect
4. #2345C2semidirect
5. #2084C2semidirect
6. #2065C2semidirect
7. #1948C2semidirect
8. #1926C2semidirect
10. #1850C2semidirect
11. #1823C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C22×#1062
#25012
#23452
#20842
#20652
#19482
#19262
#18932
#18502
#18232