Finite Group Databasebeta

← #28141

Subgroups — #28141

Order 256.

505 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 139 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×C8×D4C211yesmaximal, characteristic, complemented
3#2538C211yesmaximal, characteristic, complemented
4#2537C211yesmaximal, characteristic
5#2077C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1893C212yesmaximal, complemented
8#1892C212yesmaximal, complemented
9#1834C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C23×C8C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×#121—21no—
16C2×#121C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C4×SD16—24no—
21C8×D4—24no—
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2212yes—
28C2×C22⋊C8C2212yes—
29C2×C4×C8C2211yescharacteristic
30#396—22no—
31#394—22no—
32#390—21no—
33#389—21no—
34#384C2211yescharacteristic
35#384C2211yescharacteristic
36#380—21no—
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—
52C22×C8—24no—
53C22×C8—42no—
54C22×C8—42no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C22⋊Q8—24no—
59C22⋊Q8—42no—
60#121—24no—
61#121—42no—
62#121—44no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2312yes—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4D412yes—
86C2×C42—21no—
87C2×C42C2311yescharacteristic
88#106—22no—
89C4⋊C8—22no—
90#103—22no—
91#103—42no—
92#102—22no—
93#102—42no—
94C22⋊C8—24no—
95C4×C8D414yescomplemented
96#95—22no—
97#95—22no—
98C24—21no—
99C24C2×D412yes—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8#4112yes—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—82no—
116C2×D4SD1614yescomplemented
117C2×D4#4112yes—
118C22×C4—21no—
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—22no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136SD16—28no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—48no—
145C2×C8C2×D412yes—
146C2×C8C2×D412yes—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C22⋊C4—24no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—44no—
166C22⋊C4—42no—
167C22⋊C4—82no—
168C42—42no—
169C42C2×D412yes—
170C42#4112yes—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23C2×SD1614yes—
185C23C22×D411yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190D4—22no—
191D4—22no—
192D4—28no—
193D4—41no—
194D4—41no—
195D4—88no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—84no—
221C2×C4#12112yes—
222C2×C4#12112yes—
223C2×C4C2×SD1612yes—
224C2×C4#13512yes—
225C2×C4C22×D411yescharacteristic
226C2×C4C2×#4111yescharacteristic
227C2×C4C2×#4111yescharacteristic
228C2×C4C2×#4111yescharacteristic
229C2×C4H(5,𝔽2)11yescharacteristic
230C2×C4H(5,𝔽2)11yescharacteristic
231C8—22no—
232C8—22no—
233C8—42no—
234C22—24no—
235C22—24no—
236C22—24no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22C2×#12111yescharacteristic
249C22#54611yescharacteristic
250C22#54611yescharacteristic
251C22C22×SD1611yescharacteristic
252C22C2×#13511yescharacteristic
253C22#57611yescharacteristic
254C22#57811yescharacteristic
255C4—22no—
256C4—42no—
257C4—42no—
258C4—82no—
259C4—81no—
260C4—82no—
261C4—81no—
262C4#46514yes—
263C4#52511yescharacteristic
264C4#53511yescharacteristic
265C4#55311yescharacteristic
266C4#55311yescharacteristic
267C2—44no—
268C2—82no—
269C2#253511yescharacteristic
270C2C2×#46511yescharacteristic
271C2#306111yescharacteristic
272C2#329511yescharacteristic
273C2#329811yescharacteristic
274C2#330211yescharacteristic
275C2#330511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2537C2
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×C8×D4C2semidirect
3. #2538C2semidirect
5. #2077C2semidirect
6. #2000C2semidirect
7. #1893C2semidirect
8. #1892C2semidirect
9. #1834C2semidirect
10. #1803C2semidirect
11. #1802C2semidirect
95. C4×C8D4semidirect
116. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD162
C2×C8×D42
#25382
#25372
#20772
#20002
#18932
#18922
#18342
#18032
#18022