Finite Group Databasebeta

← #18642

Subgroups — #18642

Order 256.

501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 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#3503C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic
5#1627C214yesmaximal, complemented
6#1626C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)—21no—
11Q82—21no—
12#557C2212yescomplemented
13#554—21no—
14#554—22no—
15D4×Q8—22no—
16D4×Q8C2212yescomplemented
17#544—22no—
18#544—21no—
19#533—21no—
20#533C2211yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#127C2211yescharacteristic
24#520—21no—
25C4×#41C2211yescharacteristic
26#492—21no—
27#486—22no—
28#486—21no—
29#486C2214yes—
30#460—21no—
31#460C2212yes—
32#460C2214yes—
33#459—21no—
34#459C2212yes—
35#450—22no—
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38#418—21no—
39C2×#102—22no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43Q8⋊C8C2214yescomplemented
44D4⋊C8C2212yescomplemented
45C2×#41D411yescharacteristic
46C22×Q8—22no—
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51AGL(1,ℤ/8)—44no—
52C2×SD16—22no—
53C2×SD16—24no—
54C2×D8—22no—
55C2×M16—21no—
56C22×C8—21no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—24no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D412yes—
66C4⋊Q8D411yescharacteristic
67C4⋊Q8C2312yes—
68#127—21no—
69#127—21no—
70#127—21no—
71#127C2312yes—
72#125—22no—
73#125—24no—
74#125—22no—
75#125—21no—
76C22⋊Q8—22no—
77C22⋊Q8—24no—
78C22⋊Q8—24no—
79C22⋊Q8—21no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8D412yes—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—21no—
87C4×Q8D411yescharacteristic
88C4×Q8C2311yescharacteristic
89C4×Q8C2312yes—
90C4×D4—24no—
91C4×D4—22no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4D411yescharacteristic
95C4×D4C2311yescharacteristic
96#117—22no—
97#117—21no—
98#117—21no—
99#117D411yescharacteristic
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4D412yes—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C4⋊C8—21no—
107C4⋊C8C2312yes—
108C4⋊C8C2312yes—
109#102—28no—
110#102—22no—
111#102—22no—
112#102—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117C24—41no—
118#41—22no—
119#41SD1614yescomplemented
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—82no—
125C2×Q8—84no—
126C2×Q8C2×D412yes—
127C2×Q8C2×D411yescharacteristic
128C2×D4—21no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—44no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—82no—
139C2×D4C2×D411yescharacteristic
140C22×C4—21no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148SD16—44no—
149SD16—84no—
150D8—44no—
151M16—42no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—42no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4C2×D414yes—
171C4⋊C4C2×D411yescharacteristic
172C4⋊C4C2×D411yescharacteristic
173C4⋊C4C2×D412yes—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C42—24no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—22no—
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2411yescharacteristic
188C23—21no—
189C23—42no—
190C23—82no—
191C23—82no—
192C23—82no—
193C23#12011yescharacteristic
194Q8—21no—
195Q8—24no—
196Q8—44no—
197Q8—42no—
198Q8—82no—
199Q8—84no—
200Q8C2×SD1612yes—
201D4—24no—
202D4—21no—
203D4—44no—
204D4—42no—
205D4—42no—
206D4—44no—
207D4—82no—
208D4—82no—
209D4—82no—
210D4C2×SD1612yes—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—84no—
224C2×C4#12011yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4C2×SD1612yes—
228C2×C4#13712yes—
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C8—42no—
233C8—42no—
234C8—42no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—82no—
241C22—82no—
242C22—82no—
243C22—164no—
244C22#44812yes—
245C22C2×#12011yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—41no—
250C4—42no—
251C4—44no—
252C4—41no—
253C4—42no—
254C4#44812yes—
255C4#45412yes—
256C4C22×SD1611yescharacteristic
257C4C2×#13711yescharacteristic
258C4#57612yes—
259C2—22no—
260C2—42no—
261C2—162no—
262C2C2×#44811yescharacteristic
263C2#300811yescharacteristic
264C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3038C2
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. #3503C2semidirect
2. #3208C2semidirect
3. C2×#486C2semidirect
5. #1627C2semidirect
6. #1626C2semidirect
7. #1549C2semidirect
8. #1493C2semidirect
9. #1476C2semidirect
12. #557C22semidirect
16. D4×Q8C22semidirect
41. #332C22semidirect
42. Q8⋊C8C22semidirect
43. Q8⋊C8C22semidirect
44. D4⋊C8C22semidirect
119. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#35032
#32082
C2×#4862
#30382
#16272
#16262
#15492
#14932
#14762