Finite Group Databasebeta

← #24531

Subgroups — #24531

Order 256.

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

normal subgroups only

This group has 612 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×#447C211yesmaximal, characteristic
2C22×#102C211yesmaximal, characteristic, complemented
3#2504C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic, complemented
5#2073C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1879C212yesmaximal, complemented
9#1850C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1804C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2211yescharacteristic
14C22×D8C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×C4×D4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20#447—24no—
21C2×C4⋊C8C2212yescomplemented
22C2×#102—24no—
23C2×#102C2212yes—
24C2×#102C2211yescharacteristic
25C2×#102C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2212yescomplemented
28C2×C22⋊C8C2211yescharacteristic
29#397—22no—
30#390C2212yes—
31#390C2212yescomplemented
32#386—21no—
33#386—21no—
34#386—22no—
35#382C2212yescomplemented
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#121—22no—
66#121—22no—
67C4×D4—24no—
68C4×D4—42no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4D411yescharacteristic
84C2×C22⋊C4D411yescharacteristic
85C2×C42C2312yes—
86C4⋊C8—42no—
87#102—24no—
88#102—24no—
89#102—22no—
90#102—22no—
91#102—42no—
92C22⋊C8—22no—
93#95—21no—
94#95—22no—
95#95—22no—
96#95D412yes—
97C24—21no—
98C24—21no—
99C24—24no—
100C24—21no—
101C24—44no—
102C24—42no—
103C24—42no—
104C24—82no—
105C24C2×D411yescharacteristic
106C2×D4—28no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—44no—
118C2×D4—48no—
119C2×D4—44no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4C2×D412yes—
124C2×D4C2×D412yes—
125C2×D4#4114yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4C2×D412yes—
137C22×C4C2×D412yes—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D412yes—
140C22×C4#4112yes—
141C22×C4C2411yescharacteristic
142D8—44no—
143D8—44no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—44no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—44no—
162C22⋊C4—44no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—82no—
166C22⋊C4—82no—
167C22⋊C4—82no—
168C42—22no—
169C42—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—22no—
175C23—48no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—41no—
182C23—42no—
183C23—48no—
184C23—41no—
185C23—42no—
186C23—41no—
187C23—41no—
188C23—42no—
189C23—84no—
190C23—88no—
191C23—88no—
192C23#12112yes—
193C23#13512yes—
194C23AGL(1,ℤ/8)12yes—
195C23C22×D411yescharacteristic
196D4—22no—
197D4—22no—
198D4—22no—
199D4—24no—
200D4—48no—
201D4—41no—
202D4—41no—
203D4—42no—
204D4—42no—
205D4—84no—
206D4#12314yes—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—88no—
227C2×C4—84no—
228C2×C4#12112yes—
229C2×C4C22×D411yescharacteristic
230C2×C4C2×#4112yes—
231C2×C4C2×#4111yescharacteristic
232C2×C4H(5,𝔽2)12yes—
233C8—41no—
234C8—42no—
235C8—41no—
236C22—22no—
237C22—22no—
238C22—24no—
239C22—24no—
240C22—24no—
241C22—42no—
242C22—42no—
243C22—44no—
244C22—44no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—816no—
249C22—82no—
250C22—82no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—82no—
255C22—82no—
256C22#46314yes—
257C22C2×#12111yescharacteristic
258C22#54612yes—
259C22C2×#13511yescharacteristic
260C22C2×AGL(1,ℤ/8)11yescharacteristic
261C22#57612yes—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—81no—
266C4—82no—
267C4—82no—
268C4C2×#12311yescharacteristic
269C4#53411yescharacteristic
270C4#55112yes—
271C2—24no—
272C2—44no—
273C2—82no—
274C2—161no—
275C2#241211yescharacteristic
276C2C2×#46311yescharacteristic
277C2#304111yescharacteristic
278C2#329712yes—
279C2#331812yes—
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. #2504C2semidirect
4. D4×C22⋊C4C2semidirect
5. #2073C2semidirect
6. #2054C2semidirect
7. #1893C2semidirect
8. #1879C2semidirect
9. #1850C2semidirect
10. #1824C2semidirect
11. #1804C2semidirect
21. C2×C4⋊C8C22semidirect
27. C2×#102C22semidirect
31. #390C22semidirect
35. #382C22semidirect
39. #340C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C22×#1022
#25042
D4×C22⋊C42
#20732
#20542
#18932
#18792
#18502
#18242
#18042