Finite Group Databasebeta

← #28257

Subgroups — #28257

Order 256.

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

normal subgroups only

This group has 519 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×#460C211yesmaximal, characteristic, complemented
2C2×#463C212yesmaximal, complemented
3C2×#459C211yesmaximal, characteristic
4#2328C211yesmaximal, characteristic, complemented
5#2326C211yesmaximal, characteristic, complemented
6#2055C212yesmaximal, complemented
7#2054C212yesmaximal, complemented
8#1843C211yesmaximal, characteristic, complemented
9#1805C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1802C212yesmaximal, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×C4×D4C2211yescharacteristic
19#463—28no—
20#460—24no—
21#459—24no—
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2212yes—
26C2×#103C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#390C2212yes—
33#388—22no—
34#386—22no—
35#386—22no—
36#385—22no—
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39#376C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42C22×Q8C2311yescharacteristic
43C22×D4—21no—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C22×D4C2312yes—
47C23×C4—21no—
48C23×C4—22no—
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16—42no—
52C2×D8—24no—
53C2×D8—24no—
54C2×D8—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57#127—22no—
58#127D414yes—
59#124—22no—
60#124D414yescomplemented
61C4×Q8—22no—
62C4×Q8—22no—
63C4×D4—22no—
64C4×D4—24no—
65C4×D4—24no—
66C4×D4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82C4⋊C8—22no—
83C4⋊C8—22no—
84C4⋊C8—24no—
85C4⋊C8—42no—
86#103—24no—
87#102—24no—
88#102—24no—
89#102—24no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95D412yes—
94#95D412yes—
95C24—21no—
96C24—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8C2×D412yes—
103C2×D4—24no—
104C2×D4—24no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—82no—
116C2×D4—82no—
117C2×D4C2×D412yes—
118C2×D4C2×D414yes—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2411yescharacteristic
133SD16—48no—
134D8—48no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C22⋊C4—28no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C22⋊C4—82no—
155C42—21no—
156C42—21no—
157C42—42no—
158C42C2×D412yes—
159C42C2×D412yes—
160C42#4112yes—
161C42#4112yes—
162C23—22no—
163C23—24no—
164C23—41no—
165C23—42no—
166C23—41no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23C22×D411yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—41no—
178Q8—41no—
179D4—22no—
180D4—24no—
181D4—24no—
182D4—22no—
183D4—41no—
184D4—42no—
185D4—41no—
186D4—42no—
187D4—84no—
188D4—84no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—84no—
207C2×C4—88no—
208C2×C4#12112yes—
209C2×C4#12114yes—
210C2×C4#12112yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4H(5,𝔽2)11yescharacteristic
216C2×C42−1+411yescharacteristic
217C8—42no—
218C8—42no—
219C22—44no—
220C22—44no—
221C22—44no—
222C22—42no—
223C22—44no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22—81no—
228C22—81no—
229C22—82no—
230C22—82no—
231C22C2×#12111yescharacteristic
232C22C2×#12111yescharacteristic
233C22#53711yescharacteristic
234C22#57612yes—
235C22#57712yes—
236C4—22no—
237C4—22no—
238C4—24no—
239C4—42no—
240C4—42no—
241C4—81no—
242C4—81no—
243C4—82no—
244C4D4211yescharacteristic
245C4#54711yescharacteristic
246C4#54711yescharacteristic
247C4#55511yescharacteristic
248C2—84no—
249C2—82no—
250C2#253711yescharacteristic
251C2#304111yescharacteristic
252C2#304211yescharacteristic
253C2#304512yes—
254C2#325712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#460C2semidirect
2. C2×#463C2semidirect
4. #2328C2semidirect
5. #2326C2semidirect
6. #2055C2semidirect
7. #2054C2semidirect
8. #1843C2semidirect
9. #1805C2semidirect
10. #1804C2semidirect
11. #1802C2semidirect
60. #124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4602
C2×#4632
C2×#4592
#23282
#23262
#20552
#20542
#18432
#18052
#18042
#18022