Finite Group Databasebeta

← #24645

Subgroups — #24645

Order 256.

424 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 123 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×#462C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic
4#2356C211yesmaximal, characteristic, complemented
5#2056C212yesmaximal, complemented
6#2015C212yesmaximal
7#1983C212yesmaximal
8#1960C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1928C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×SD16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#462—24no—
19#440—24no—
20C2×#107C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29C2×#97C2211yescharacteristic
30#397—21no—
31#397—21no—
32#391C2212yes—
33#391C2212yes—
34#387—22no—
35#383—21no—
36#382C2211yescharacteristic
37#380—22no—
38C4×C4⋊C4C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40#376—21no—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×Q16—24no—
48C2×Q16—42no—
49C2×Q16—42no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×SD16—22no—
53C22×C8C2312yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58#124—24no—
59#124—42no—
60C4×Q8—22no—
61C4×Q8D414yes—
62C4×D4—22no—
63C4×D4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D412yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42—22no—
76C2×C42C2312yes—
77C2×C42C2311yescharacteristic
78#107—22no—
79C4⋊C8—22no—
80C4⋊C8—42no—
81#103—22no—
82#103—24no—
83#103—24no—
84#103—42no—
85#102—22no—
86#102—42no—
87#97—22no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—21no—
94#95—22no—
95#95D411yescharacteristic
96#95D411yescharacteristic
97C24—21no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8C2×D412yes—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4#4112yes—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128Q16—48no—
129SD16—28no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8#4112yes—
139C4⋊C4—24no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4C2×D412yes—
149C22⋊C4—44no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C42—24no—
153C42—22no—
154C42—22no—
155C42—44no—
156C42C2×D412yes—
157C42#4112yes—
158C23—22no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23C22×D411yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—41no—
167Q8—41no—
168Q8—42no—
169Q8—42no—
170Q8—42no—
171Q8—42no—
172D4—22no—
173D4—22no—
174D4—41no—
175D4—41no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—84no—
198C2×C4#12112yes—
199C2×C4#12112yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C4H(5,𝔽2)11yescharacteristic
206C8—22no—
207C8—41no—
208C8—42no—
209C22—42no—
210C22—44no—
211C22—82no—
212C22—81no—
213C22—81no—
214C22C2×#12111yescharacteristic
215C22#54611yescharacteristic
216C22#55011yescharacteristic
217C22#57512yes—
218C22#57812yes—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—82no—
226C4—82no—
227C4—81no—
228C4#52511yescharacteristic
229C4#53511yescharacteristic
230C4#55311yescharacteristic
231C4#55511yescharacteristic
232C2—82no—
233C2#241411yescharacteristic
234C2#304011yescharacteristic
235C2#304411yescharacteristic
236C2#331512yes—
237C2#340212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#462C2semidirect
4. #2356C2semidirect
5. #2056C2semidirect
8. #1960C2semidirect
9. #1940C2semidirect
10. #1928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×#4402
#25472
#23562
#20562
#20152
#19832
#19602
#19402
#19282
#18042