Finite Group Databasebeta

← #32720

Subgroups — #32720

Order 256.

445 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#2856C211yesmaximal, characteristic
2#2855C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2096C211yesmaximal, characteristic, complemented
5#2065C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2034C212yesmaximal, complemented
9#2014C211yesmaximal, characteristic, complemented
10#2000C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×#127C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#121—21no—
16C2×#121—21no—
17C2×#121C2212yes—
18C22×C42C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×C22⋊C8C2212yes—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#390—21no—
29#390—21no—
30#390C2212yes—
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#387—21no—
34#387—21no—
35#385—21no—
36#384C2211yescharacteristic
37#381—21no—
38#380C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×D4—22no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C2×SD16—42no—
52C2×SD16—42no—
53C2×SD16—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56#127—42no—
57C22⋊Q8—42no—
58#121—42no—
59#121—42no—
60#121—44no—
61#121—44no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—42no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#106—42no—
82C4⋊C8—42no—
83#103—42no—
84#102—42no—
85#102—42no—
86#102—42no—
87C22⋊C8—42no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95C24—22no—
96C24C2×D411yescharacteristic
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—42no—
115C2×D4—42no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4SD1614yescomplemented
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2411yescharacteristic
137SD16—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—44no—
160C42—22no—
161C42—22no—
162C42—41no—
163C42—41no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23C2×SD1612yes—
178C23C22×D411yescharacteristic
179Q8—42no—
180Q8—42no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—84no—
186D4—84no—
187D4—88no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×SD1612yes—
220C2×C4C2×SD1612yes—
221C2×C4AGL(1,ℤ/8)12yes—
222C2×C4H(5,𝔽2)11yescharacteristic
223C2×C4H(5,𝔽2)12yes—
224C2×C4H(5,𝔽2)12yes—
225C2×C42−1+411yescharacteristic
226C8—42no—
227C8—42no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—82no—
237C22—82no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22#53411yescharacteristic
243C22#53511yescharacteristic
244C22#53711yescharacteristic
245C22C22×SD1611yescharacteristic
246C22C2×AGL(1,ℤ/8)11yescharacteristic
247C22#57511yescharacteristic
248C22#57511yescharacteristic
249C4—22no—
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4—81no—
257C4—81no—
258C4—81no—
259C4#56012yes—
260C4#56212yes—
261C2—42no—
262C2—162no—
263C2#281311yescharacteristic
264C2#318711yescharacteristic
265C2#319411yescharacteristic
266C2#320811yescharacteristic
267C2#321911yescharacteristic
268C2#324611yescharacteristic
269C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2855C2semidirect
3. #2099C2semidirect
4. #2096C2semidirect
5. #2065C2semidirect
6. #2055C2semidirect
7. #2041C2semidirect
8. #2034C2semidirect
9. #2014C2semidirect
10. #2000C2semidirect
11. #1797C2semidirect
130. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#28562
#28552
#20992
#20962
#20652
#20552
#20412
#20342
#20142
#20002
#17972