Finite Group Databasebeta

← #18626

Subgroups — #18626

Order 256.

544 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 123 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#3511C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3C2×#492C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic, complemented
5#1624C214yesmaximal, complemented
6#1622C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)—21no—
11C22×SD16—22no—
12#556—21no—
13#555C2212yescomplemented
14#554C2212yescomplemented
15#553—22no—
16#548—22no—
17#547—21no—
18#542—22no—
19#537—21no—
20#533—21no—
21#532—21no—
22#532C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×#127C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#493—21no—
28#492—22no—
29#492C2214yes—
30#486—21no—
31#460—21no—
32#460C2212yes—
33#459—21no—
34#459C2214yes—
35#459C2212yes—
36#447—22no—
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#418—21no—
40C2×C4×C8C2211yescharacteristic
41#333C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43D4⋊C8C2214yescomplemented
44D4⋊C8C2212yescomplemented
45C2×#41—22no—
46C2×#41—21no—
47C2×#41D411yescharacteristic
48C22×Q8D412yes—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52AGL(1,ℤ/8)—44no—
53C2×SD16—28no—
54C2×SD16—22no—
55C2×SD16—48no—
56C2×D8—22no—
57C2×D8—24no—
58C2×M16—21no—
59C22×C8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8D412yes—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8C2312yes—
65#127—21no—
66#127—21no—
67#127—21no—
68#127D411yescharacteristic
69#127C2312yes—
70#126—24no—
71#125—22no—
72#124—22no—
73#124—24no—
74#124—21no—
75#123—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80#121—22no—
81#121—24no—
82#121—22no—
83#121—22no—
84#121—21no—
85#121D412yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8D411yescharacteristic
89C4×Q8C2311yescharacteristic
90C4×D4—22no—
91C4×D4—24no—
92C4×D4—22no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4D411yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2312yes—
99#117—22no—
100#117—22no—
101#117—21no—
102#117D411yescharacteristic
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8C2312yes—
110C4⋊C8C2312yes—
111#102—22no—
112#102—22no—
113#102—24no—
114C22⋊C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C2312yes—
118C24—41no—
119#41—22no—
120#41—22no—
121#41—42no—
122#41—42no—
123#41—84no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—44no—
127C2×Q8C2×D411yescharacteristic
128C2×Q8C2×D414yes—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—44no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—82no—
144C2×D4C2×D412yes—
145C2×D4C2×D411yescharacteristic
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155SD16—416no—
156SD16—44no—
157D8—44no—
158D8—84no—
159M16—42no—
160C2×C8—24no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4C2×D411yescharacteristic
174C4⋊C4C2×D411yescharacteristic
175C4⋊C4C2×D412yes—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—22no—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C42—24no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2411yescharacteristic
192C23—21no—
193C23—22no—
194C23—42no—
195C23—82no—
196C23—82no—
197C23—82no—
198C23#12011yescharacteristic
199Q8—21no—
200Q8—21no—
201Q8—44no—
202Q8—42no—
203Q8—44no—
204D4—21no—
205D4—21no—
206D4—24no—
207D4—44no—
208D4—42no—
209D4—44no—
210D4—42no—
211D4—42no—
212D4—44no—
213D4—42no—
214D4—42no—
215D4—82no—
216D4—84no—
217D4—82no—
218D4—82no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4#12011yescharacteristic
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4#13512yes—
237C2×C4AGL(1,ℤ/8)12yes—
238C2×C4C22×D411yescharacteristic
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C8—24no—
242C8—42no—
243C8—42no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—42no—
250C22—41no—
251C22—42no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—164no—
256C22#44912yes—
257C22C2×#12011yescharacteristic
258C4—21no—
259C4—21no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—41no—
264C4—44no—
265C4#44912yes—
266C4#45312yes—
267C4C2×#13511yescharacteristic
268C4C2×AGL(1,ℤ/8)11yescharacteristic
269C4#57712yes—
270C2—22no—
271C2—81no—
272C2—82no—
273C2—162no—
274C2C2×#44911yescharacteristic
275C2#301011yescharacteristic
276C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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. #3511C2semidirect
2. #3205C2semidirect
3. C2×#492C2semidirect
4. #3038C2semidirect
5. #1624C2semidirect
6. #1622C2semidirect
7. #1548C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
13. #555C22semidirect
14. #554C22semidirect
41. #333C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35112
#32052
C2×#4922
#30382
#16242
#16222
#15482
#14922
#14762