Finite Group Databasebeta

← #18622

Subgroups — #18622

Order 256.

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

normal subgroups only

This group has 533 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×#492C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic
5#1623C214yesmaximal, complemented
6#1622C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)—21no—
11C22×SD16—22no—
12Q82C2212yescomplemented
13#557—21no—
14#554—22no—
15#554C2212yescomplemented
16D4×Q8—22no—
17D4×Q8—21no—
18#544—22no—
19#544—21no—
20#533—21no—
21#533C2211yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×#127C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#492—22no—
28#492—21no—
29#492C2214yes—
30#486—21no—
31#460—21no—
32#460C2214yes—
33#460C2212yes—
34#459—21no—
35#459C2212yes—
36#450—22no—
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#418—21no—
40C2×C4×C8C2211yescharacteristic
41#333C2214yescomplemented
42Q8⋊C8C2214yescomplemented
43Q8⋊C8C2212yescomplemented
44D4⋊C8C2212yescomplemented
45C2×#41D411yescharacteristic
46C22×Q8—22no—
47C22×Q8D412yes—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51AGL(1,ℤ/8)—44no—
52C2×SD16—28no—
53C2×SD16—22no—
54C2×SD16—24no—
55C2×SD16—48no—
56C2×D8—22no—
57C2×M16—21no—
58C22×C8—21no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D412yes—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2312yes—
70#127—21no—
71#127—21no—
72#127—21no—
73#127C2312yes—
74#125—22no—
75#125—24no—
76#125—22no—
77#125—21no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80C22⋊Q8—24no—
81C22⋊Q8—22no—
82C22⋊Q8—24no—
83C22⋊Q8—21no—
84C22⋊Q8D412yes—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—21no—
89C4×Q8D411yescharacteristic
90C4×Q8C2311yescharacteristic
91C4×Q8C2312yes—
92C4×D4—24no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4D411yescharacteristic
97C4×D4C2311yescharacteristic
98#117—22no—
99#117—21no—
100#117—21no—
101#117D411yescharacteristic
102C2×C4⋊C4—22no—
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#41SD1614yescomplemented
121C2×Q8—24no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—44no—
125C2×Q8—84no—
126C2×Q8C2×D412yes—
127C2×Q8C2×D411yescharacteristic
128C2×Q8C2×D414yes—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—44no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—82no—
140C2×D4C2×D411yescharacteristic
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149SD16—416no—
150SD16—44no—
151SD16—84no—
152D8—44no—
153M16—42no—
154C2×C8—24no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D411yescharacteristic
175C4⋊C4C2×D411yescharacteristic
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C42—24no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—22no—
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2411yescharacteristic
190C23—21no—
191C23—42no—
192C23—82no—
193C23—82no—
194C23—82no—
195C23#12011yescharacteristic
196Q8—21no—
197Q8—24no—
198Q8—44no—
199Q8—44no—
200Q8—42no—
201Q8—84no—
202Q8C2×SD1612yescomplemented
203D4—24no—
204D4—21no—
205D4—44no—
206D4—42no—
207D4—42no—
208D4—44no—
209D4—82no—
210D4—82no—
211D4—82no—
212D4C2×SD1612yescomplemented
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—84no—
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4C2×SD1612yes—
230C2×C4#13712yes—
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C8—24no—
235C8—42no—
236C8—42no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22—164no—
246C22#44812yes—
247C22C2×#12011yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—44no—
255C4—42no—
256C4#44812yes—
257C4#45312yes—
258C4C22×SD1611yescharacteristic
259C4C2×#13711yescharacteristic
260C4#57712yes—
261C2—22no—
262C2—42no—
263C2—162no—
264C2C2×#44811yescharacteristic
265C2#301011yescharacteristic
266C2C2×#45311yescharacteristic
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×#492C2semidirect
5. #1623C2semidirect
6. #1622C2semidirect
7. #1548C2semidirect
8. #1493C2semidirect
9. #1476C2semidirect
12. Q82C22semidirect
15. #554C22semidirect
41. #333C22semidirect
42. Q8⋊C8C22semidirect
43. Q8⋊C8C22semidirect
44. D4⋊C8C22semidirect
120. #41SD16semidirect
202. Q8C2×SD16semidirect
212. D4C2×SD16semidirect

Maximal subgroups

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

subgroupindex
#35032
#32082
C2×#4922
#30382
#16232
#16222
#15482
#14932
#14762