Finite Group Databasebeta

← #19462

Subgroups — #19462

Order 256.

340 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 109 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#3227C211yesmaximal, characteristic, complemented
2#3109C211yesmaximal, characteristic
3C2×#461C211yesmaximal, characteristic, complemented
4#2974C211yesmaximal, characteristic, complemented
5#1701C212yesmaximal, complemented
6#1691C212yesmaximal, complemented
7#1681C212yesmaximal, complemented
8#1671C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic, complemented
10#1556C211yesmaximal, characteristic, complemented
11#1551C211yesmaximal, characteristic, complemented
12C22×SD16—21no—
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#498C2212yescomplemented
17#486C2212yescomplemented
18#481—21no—
19#481—21no—
20#481—21no—
21#476—22no—
22#476C2211yescharacteristic
23#476C2211yescharacteristic
24#461—21no—
25#461—21no—
26#461C2212yes—
27#461C2212yes—
28#459C2211yescharacteristic
29#459C2211yescharacteristic
30#447—21no—
31#442—21no—
32#442—21no—
33#436C2212yes—
34#435—22no—
35C8×D4C2212yes—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#431C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×#103—21no—
41#407—21no—
42C2×C22⋊C8—21no—
43#335C2212yescomplemented
44#334C2212yescomplemented
45#333C2212yescomplemented
46#332C2212yescomplemented
47#332C2212yescomplemented
48#331C2212yescomplemented
49C2×#41—21no—
50C22×Q8—21no—
51C22×D4D411yescharacteristic
52C23×C4—21no—
53C2×SD16—22no—
54C2×SD16—22no—
55C2×SD16—44no—
56C2×D8—21no—
57C2×D8—21no—
58C2×M16—22no—
59C22×C8—21no—
60C22×C8—22no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8C2312yes—
65C4⋊Q8C2311yescharacteristic
66#127C2311yescharacteristic
67#124—21no—
68#121—21no—
69#121—21no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4D411yescharacteristic
75C4×D4D411yescharacteristic
76C4×D4C2311yescharacteristic
77C4×D4C2311yescharacteristic
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C42C2311yescharacteristic
83#106—22no—
84#106—22no—
85#106—21no—
86#106—21no—
87C4⋊C8—21no—
88C4⋊C8D412yes—
89C4⋊C8C2312yes—
90C4⋊C8C2312yes—
91C4⋊C8C2311yescharacteristic
92C4⋊C8C2311yescharacteristic
93#103—22no—
94#103—22no—
95#102—21no—
96#102—21no—
97#102—22no—
98#102—22no—
99#102—21no—
100#102—21no—
101C22⋊C8—21no—
102C22⋊C8—21no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105C22⋊C8D411yescharacteristic
106C22⋊C8D411yescharacteristic
107#97—21no—
108#97—21no—
109C4×C8C2312yes—
110C24—21no—
111#41—82no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—41no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4C2×D411yescharacteristic
125C2×D4C2×D411yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—82no—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4#4111yescharacteristic
137SD16—44no—
138SD16—44no—
139D8—42no—
140D8—42no—
141M16—82no—
142C2×C8—22no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8C2×D412yes—
149C2×C8C2×D412yes—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4C2×D411yescharacteristic
158C4⋊C4C2×D411yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C42C2×D411yescharacteristic
166C42C2×D411yescharacteristic
167C42#4111yescharacteristic
168C42C2411yescharacteristic
169C23—21no—
170C23—21no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—81no—
175C23#12111yescharacteristic
176Q8—42no—
177Q8—42no—
178Q8—81no—
179Q8—82no—
180D4—21no—
181D4—21no—
182D4—21no—
183D4—42no—
184D4—41no—
185D4—41no—
186D4—42no—
187D4—81no—
188D4—82no—
189D4AGL(1,ℤ/8)12yes—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—81no—
204C2×C4—82no—
205C2×C4#12111yescharacteristic
206C2×C4#12111yescharacteristic
207C2×C4#12111yescharacteristic
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C8—22no—
212C8—22no—
213C8—42no—
214C8—41no—
215C8—41no—
216C8—42no—
217C22—21no—
218C22—21no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—84no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22#47112yes—
230C22C2×#12111yescharacteristic
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—41no—
236C4—41no—
237C4—82no—
238C4—81no—
239C4C2×AGL(1,ℤ/8)11yescharacteristic
240C4#57511yescharacteristic
241C4#57712yes—
242C2—22no—
243C2—42no—
244C2—81no—
245C2—161no—
246C2#304211yescharacteristic
247C2#305511yescharacteristic
248C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3109C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3227C2semidirect
3. C2×#461C2semidirect
4. #2974C2semidirect
5. #1701C2semidirect
6. #1691C2semidirect
7. #1681C2semidirect
8. #1671C2semidirect
9. #1573C2semidirect
10. #1556C2semidirect
11. #1551C2semidirect
16. #498C22semidirect
17. #486C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. #332C22semidirect
48. #331C22semidirect

Maximal subgroups

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

subgroupindex
#32272
#31092
C2×#4612
#29742
#17012
#16912
#16812
#16712
#15732
#15562
#15512