Finite Group Databasebeta

← #28147

Subgroups — #28147

Order 256.

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

normal subgroups only

This group has 471 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×C4×SD16C211yesmaximal, characteristic, complemented
2C2×#436C211yesmaximal, characteristic, complemented
3#2536C211yesmaximal, characteristic, complemented
4#2535C211yesmaximal, characteristic
5#2077C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1895C212yesmaximal, complemented
8#1894C212yesmaximal, complemented
9#1835C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×M16C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×#121C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C4×SD16—24no—
20#436—24no—
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28C2×C4×C8C2211yescharacteristic
29#398—21no—
30#397—21no—
31#394—21no—
32#392—21no—
33#388—22no—
34#386—22no—
35#382C2211yescharacteristic
36#382C2211yescharacteristic
37#380C2212yes—
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—22no—
45C22×D4D411yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4C2312yes—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×M16—24no—
53C2×M16—42no—
54C2×M16—42no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C22⋊Q8—24no—
59C22⋊Q8—42no—
60#121—24no—
61#121—42no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×D4—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2312yes—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4D412yes—
81C2×C42—21no—
82C2×C42C2311yescharacteristic
83#106—22no—
84C4⋊C8—22no—
85#103—22no—
86#103—42no—
87#102—22no—
88#102—42no—
89C22⋊C8—24no—
90C4×C8D414yescomplemented
91#95—22no—
92#95—22no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97C24—21no—
98C24C2×D412yes—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8#4112yes—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—42no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—82no—
115C2×D4#4112yes—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—82no—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133SD16—28no—
134M16—48no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8C2×D412yes—
143C2×C8C2×D412yes—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C22⋊C4—24no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—42no—
160C22⋊C4—82no—
161C42—42no—
162C42C2×D412yes—
163C42#4112yes—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23C22×D411yescharacteristic
177Q8—22no—
178Q8—22no—
179Q8—41no—
180Q8—41no—
181D4—22no—
182D4—22no—
183D4—44no—
184D4—41no—
185D4—41no—
186D4—88no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4#12112yes—
211C2×C4#12112yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C4H(5,𝔽2)11yescharacteristic
217C2×C4H(5,𝔽2)11yescharacteristic
218C8—22no—
219C8—22no—
220C8—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—44no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—81no—
231C22—81no—
232C22—82no—
233C22C2×#12111yescharacteristic
234C22#54611yescharacteristic
235C22#54611yescharacteristic
236C22#57512yes—
237C22#57712yes—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—81no—
244C4—82no—
245C4—82no—
246C4#52511yescharacteristic
247C4#53511yescharacteristic
248C4#55311yescharacteristic
249C4#55311yescharacteristic
250C2—82no—
251C2—82no—
252C2#253511yescharacteristic
253C2#305411yescharacteristic
254C2#305711yescharacteristic
255C2#332312yes—
256C2#332612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
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
1. C2×C4×SD16C2semidirect
2. C2×#436C2semidirect
3. #2536C2semidirect
5. #2077C2semidirect
6. #2000C2semidirect
7. #1895C2semidirect
8. #1894C2semidirect
9. #1835C2semidirect
10. #1805C2semidirect
11. #1804C2semidirect
90. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD162
C2×#4362
#25362
#25352
#20772
#20002
#18952
#18942
#18352
#18052
#18042