Finite Group Databasebeta

← #24526

Subgroups — #24526

Order 256.

640 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 143 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×#447C211yesmaximal, characteristic
2C22×#102C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic, complemented
5#2071C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1879C212yesmaximal, complemented
9#1850C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1802C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2211yescharacteristic
14C22×D8C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×#123—21no—
17C2×#123—21no—
18C2×#121C2211yescharacteristic
19C2×C4×D4C2212yes—
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4—21no—
22#447—24no—
23C2×C4⋊C8C2212yescomplemented
24C2×#102—24no—
25C2×#102C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2212yes—
28C2×#102C2211yescharacteristic
29C2×#102C2212yescomplemented
30C2×C22⋊C8C2211yescharacteristic
31#399—22no—
32#390C2212yes—
33#390C2212yescomplemented
34#386—22no—
35#384C2212yescomplemented
36#380C2211yescharacteristic
37#379—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yescomplemented
40#340C2212yescomplemented
41#340C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×D4—24no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—42no—
49C22×D4D414yes—
50C22×D4C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—22no—
55C23×C4C2311yescharacteristic
56C2×D8—22no—
57C2×D8—22no—
58C2×D8—24no—
59C2×D8—42no—
60C22×C8—22no—
61C22×C8—42no—
62C22×C8C2312yes—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#123—44no—
66#123—44no—
67#121—22no—
68#121—22no—
69C4×D4—24no—
70C4×D4—42no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—44no—
87C2×C22⋊C4D411yescharacteristic
88C2×C22⋊C4D411yescharacteristic
89C2×C42C2312yes—
90C4⋊C8—42no—
91#102—24no—
92#102—24no—
93#102—22no—
94#102—22no—
95#102—42no—
96C22⋊C8—22no—
97#95—21no—
98#95D412yes—
99C24—21no—
100C24—21no—
101C24—24no—
102C24—21no—
103C24—44no—
104C24—42no—
105C24—42no—
106C24—82no—
107C24C2×D411yescharacteristic
108C2×D4—28no—
109C2×D4—24no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—44no—
120C2×D4—48no—
121C2×D4—44no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4C2×D412yes—
126C2×D4C2×D412yes—
127C2×D4#4114yes—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4—82no—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4#4112yes—
144C22×C4C2411yescharacteristic
145D8—44no—
146D8—44no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—44no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C22⋊C4—22no—
165C22⋊C4—44no—
166C22⋊C4—44no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—44no—
170C22⋊C4—44no—
171C22⋊C4—82no—
172C22⋊C4D814yescomplemented
173C42—22no—
174C42—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—22no—
180C23—48no—
181C23—41no—
182C23—41no—
183C23—42no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—48no—
189C23—41no—
190C23—42no—
191C23—41no—
192C23—41no—
193C23—42no—
194C23—84no—
195C23—88no—
196C23—88no—
197C23#12112yes—
198C23C2×D812yes—
199C23AGL(1,ℤ/8)12yes—
200C23C22×D411yescharacteristic
201D4—22no—
202D4—22no—
203D4—22no—
204D4—24no—
205D4—48no—
206D4—41no—
207D4—41no—
208D4—42no—
209D4—42no—
210D4—84no—
211D4#12314yescomplemented
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—88no—
231C2×C4—84no—
232C2×C4#12112yes—
233C2×C4C2×D814yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C2×#4112yes—
236C2×C4C2×#4111yescharacteristic
237C2×C4H(5,𝔽2)12yes—
238C8—41no—
239C8—42no—
240C8—41no—
241C22—22no—
242C22—24no—
243C22—22no—
244C22—24no—
245C22—24no—
246C22—42no—
247C22—44no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—816no—
254C22—82no—
255C22—82no—
256C22—81no—
257C22—81no—
258C22—82no—
259C22—81no—
260C22—82no—
261C22#45914yes—
262C22C2×#12111yescharacteristic
263C22#54612yes—
264C22C22×D811yescharacteristic
265C22C2×AGL(1,ℤ/8)11yescharacteristic
266C22#57712yes—
267C4—22no—
268C4—24no—
269C4—42no—
270C4—82no—
271C4—81no—
272C4—82no—
273C4C2×#12311yescharacteristic
274C4#53411yescharacteristic
275C4#55112yes—
276C2—24no—
277C2—44no—
278C2—82no—
279C2—161no—
280C2#241211yescharacteristic
281C2C2×#45911yescharacteristic
282C2#304211yescharacteristic
283C2#329412yes—
284C2#330612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#102C2semidirect
3. #2501C2semidirect
4. D4×C22⋊C4C2semidirect
5. #2071C2semidirect
6. #2054C2semidirect
7. #1893C2semidirect
8. #1879C2semidirect
9. #1850C2semidirect
10. #1823C2semidirect
11. #1802C2semidirect
23. C2×C4⋊C8C22semidirect
29. C2×#102C22semidirect
33. #390C22semidirect
35. #384C22semidirect
39. #340C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #336C22semidirect
172. C22⋊C4D8semidirect
211. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C22×#1022
#25012
D4×C22⋊C42
#20712
#20542
#18932
#18792
#18502
#18232
#18022