Finite Group Databasebeta

← #24643

Subgroups — #24643

Order 256.

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

normal subgroups only

This group has 448 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×#461C211yesmaximal, characteristic, complemented
2C2×#442C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic, complemented
4#2357C211yesmaximal, characteristic, complemented
5#2055C212yesmaximal, complemented
6#2012C212yesmaximal
7#1983C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1927C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×D4C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#461—24no—
19#442—24no—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29C2×#97C2211yescharacteristic
30#397—21no—
31#397—21no—
32#391C2212yes—
33#390C2212yes—
34#388—22no—
35#382C2211yescharacteristic
36#381—21no—
37#380—22no—
38C4×C4⋊C4C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40C4×C22⋊C4—21no—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C2×SD16—24no—
49C2×SD16—42no—
50C2×SD16—42no—
51C2×D8—22no—
52C2×D8—22no—
53C2×D8—22no—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#124—24no—
60#124—42no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4D414yes—
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—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C42—22no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80#106—22no—
81C4⋊C8—22no—
82C4⋊C8—42no—
83#103—24no—
84#103—24no—
85#103—42no—
86#102—22no—
87#102—22no—
88#102—42no—
89#97—22no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—21no—
95#95D411yescharacteristic
96#95D411yescharacteristic
97C24—21no—
98C24—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C2×D412yes—
116C2×D4#4112yes—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—82no—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130SD16—48no—
131D8—28no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8#4112yes—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4C2×D412yes—
150C22⋊C4—24no—
151C22⋊C4—44no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C22⋊C4—42no—
155C22⋊C4—82no—
156C42—22no—
157C42—22no—
158C42—82no—
159C42C2×D412yes—
160C42#4112yes—
161C23—22no—
162C23—22no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23C22×D411yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176D4—22no—
177D4—22no—
178D4—22no—
179D4—22no—
180D4—41no—
181D4—41no—
182D4—41no—
183D4—41no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—84no—
203C2×C4—84no—
204C2×C4#12112yes—
205C2×C4#12112yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4111yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)11yescharacteristic
211C2×C4H(5,𝔽2)11yescharacteristic
212C8—22no—
213C8—41no—
214C8—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—44no—
220C22—82no—
221C22—82no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22C2×#12111yescharacteristic
226C22#54611yescharacteristic
227C22#55011yescharacteristic
228C22#57512yes—
229C22#57712yes—
230C4—22no—
231C4—22no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—82no—
236C4—82no—
237C4—81no—
238C4#52511yescharacteristic
239C4#53511yescharacteristic
240C4#55311yescharacteristic
241C4#55511yescharacteristic
242C2—82no—
243C2—82no—
244C2#241411yescharacteristic
245C2#304011yescharacteristic
246C2#304211yescharacteristic
247C2#330712yes—
248C2#340112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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×#461C2semidirect
3. #2547C2semidirect
4. #2357C2semidirect
5. #2055C2semidirect
7. #1983C2semidirect
8. #1960C2semidirect
9. #1940C2semidirect
10. #1927C2semidirect
11. #1804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×#4422
#25472
#23572
#20552
#20122
#19832
#19602
#19402
#19272
#18042