Finite Group Databasebeta

← #13103

Subgroups — #13103

Order 256.

323 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 61 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#3010C211yesmaximal, characteristic
2#2884C211yesmaximal, characteristic, complemented
3#2128C211yesmaximal, characteristic, complemented
4#2122C211yesmaximal, characteristic, complemented
5#1796C211yesmaximal, characteristic, complemented
6#1395C211yesmaximal, characteristic, complemented
7#1391C211yesmaximal, characteristic, complemented
8C2×2−1+4C2211yescharacteristic
9C2×AGL(1,ℤ/8)C411yescharacteristic, complemented
10C22×SD16C411yescharacteristic, complemented
11#535C2211yescharacteristic
12#519—21no—
13#449—21no—
14#449—21no—
15#448—21no—
16#448—21no—
17#421—21no—
18#419—21no—
19#419C411yescharacteristic, complemented
20#416C411yescharacteristic, complemented
21#413C2211yescharacteristic
22#410—21no—
23#410—22no—
24#410—22no—
25#410C2211yescharacteristic
26#410C2211yescharacteristic
27#410C2211yescharacteristic
28#408C2211yescharacteristic
29#353—22no—
30AΣL(1,𝔽16)—22no—
312−1+4—42no—
322−1+4—42no—
33C2×#41—21no—
34C2×#41—21no—
35C2×#41—21no—
36C2×#41C2×C411yescharacteristic
37C2×#41D411yescharacteristic
38C2×#41D411yescharacteristic
39C2×#41C2311yescharacteristic
40C22×Q8—21no—
41C22×Q8C2×C411yescharacteristic
42C22×Q8D411yescharacteristic
43C22×Q8D411yescharacteristic
44C22×D4C2×C411yescharacteristic
45AGL(1,ℤ/8)—42no—
46C2×SD16—21no—
47C2×SD16—21no—
48C2×SD16—21no—
49C2×SD16—42no—
50C2×D8—21no—
51C2×M16—21no—
52C2×M16C2×C411yescharacteristic
53C22×C8C2×C411yescharacteristic
54#127—21no—
55#124—21no—
56#121—21no—
57#121—21no—
58#120—21no—
59#120—21no—
60C4×Q8—42no—
61C4×Q8—42no—
62#117—24no—
63#117—21no—
64#117—21no—
65#117—21no—
66#117C2×C411yescharacteristic
67#104—42no—
68#103—21no—
69#103—21no—
70#103—21no—
71#103—42no—
72#102—21no—
73#101—21no—
74#100—21no—
75#99—24no—
76#99—21no—
77#99—21no—
78#99—21no—
79#99—21no—
80#99D412yes—
81#99D412yes—
82C22⋊C8—21no—
83C22⋊C8—21no—
84C24—21no—
85#41—22no—
86#41—22no—
87#41—41no—
88#41—41no—
89#41—41no—
90#41—41no—
91#41—41no—
92#41—41no—
93#41—44no—
94#41—44no—
95#41—44no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8—44no—
106C2×Q8C22⋊C411yescharacteristic
107C2×Q8C22⋊C411yescharacteristic
108C2×Q8C22⋊C411yescharacteristic
109C2×Q8#4111yescharacteristic
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4C22⋊C411yescharacteristic
120C2×D4C2×D411yescharacteristic
121C2×D4C2×D411yescharacteristic
122C2×D4#4111yescharacteristic
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4C22×C411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132SD16—42no—
133SD16—42no—
134SD16—42no—
135D8—42no—
136M16—41no—
137M16—41no—
138M16—41no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—41no—
142C2×C8—41no—
143C4⋊C4—21no—
144C4⋊C4—44no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C22⋊C4—21no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4—41no—
154C42—21no—
155C42—44no—
156C42—41no—
157C42—41no—
158C42—41no—
159C23—21no—
160C23—41no—
161C23—81no—
162C23—161no—
163C23C4×D411yescharacteristic
164C23#12011yescharacteristic
165C23#12111yescharacteristic
166Q8—24no—
167Q8—21no—
168Q8—21no—
169Q8—21no—
170Q8—21no—
171Q8—41no—
172Q8—42no—
173Q8—42no—
174Q8#9912yes—
175Q8#9912yes—
176D4—21no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—41no—
185D4—41no—
186D4—81no—
187D4—161no—
188D4—161no—
189C2×C4—24no—
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—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4C2×C22⋊C411yescharacteristic
209C2×C4C4×D411yescharacteristic
210C2×C4#12111yescharacteristic
211C2×C4#12311yescharacteristic
212C8—41no—
213C8—41no—
214C8—81no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—162no—
227C22#38611yescharacteristic
228C4—21no—
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—41no—
234C4—81no—
235C4—82no—
236C4—82no—
237C4—161no—
238C4C2×#9911yescharacteristic
239C4C2×#9911yescharacteristic
240C2—21no—
241C2—41no—
242C2—41no—
243C2—41no—
244C2—81no—
245C2—161no—
246C2#189811yescharacteristic
subgroupG/H
center, socleC2#1898
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#3010C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2884C2semidirect
3. #2128C2semidirect
4. #2122C2semidirect
5. #1796C2semidirect
6. #1395C2semidirect
7. #1391C2semidirect
9. C2×AGL(1,ℤ/8)C4semidirect
10. C22×SD16C4semidirect
19. #419C4semidirect
20. #416C4semidirect

Maximal subgroups

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

subgroupindex
#30102
#28842
#21282
#21222
#17962
#13952
#13912