Finite Group Databasebeta

← #18902

Subgroups — #18902

Order 256.

613 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 129 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#3496C211yesmaximal, characteristic
2#3226C211yesmaximal, characteristic, complemented
3#3226C211yesmaximal, characteristic, complemented
4C2×#461C211yesmaximal, characteristic, complemented
5#1647C214yesmaximal, complemented
6#1639C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11C22×SD16—21no—
12#547—21no—
13#547C2212yes—
14#546—22no—
15#546—22no—
16D42—21no—
17D42C2212yes—
18#537—21no—
19#534—22no—
20#532C2211yescharacteristic
21#532C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic, complemented
23C2×#121—22no—
24C2×C4×D4C2211yescharacteristic
25#500—21no—
26#500C2212yescomplemented
27#486—21no—
28#486C2212yescomplemented
29#480—22no—
30#480—22no—
31#474—22no—
32#474C2214yescomplemented
33#461—22no—
34#461C2214yescomplemented
35#431C2211yescharacteristic, complemented
36#431C2211yescharacteristic, complemented
37C2×C4⋊C8C2211yescharacteristic, complemented
38C2×#103—21no—
39#332C2212yescomplemented
40#331C2212yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43H(5,𝔽2)—44no—
44H(5,𝔽2)—44no—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—21no—
48C2×#41—21no—
49C22×Q8—21no—
50C22×D4—24no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4D411yescharacteristic
55C23×C4—21no—
56C2×SD16—24no—
57C2×SD16—44no—
58C22×C8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8D411yescharacteristic
63C4⋊Q8C2312yes—
64C4⋊Q8C2311yescharacteristic
65#127D411yescharacteristic
66#127C2311yescharacteristic
67#124—21no—
68#124—21no—
69#123—24no—
70#123—24no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73#121—24no—
74#121—24no—
75#121—22no—
76#121—22no—
77#121—21no—
78#121—21no—
79#121—22no—
80#121—22no—
81#121D412yes—
82#121D412yes—
83#120—24no—
84#120—24no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4D412yes—
90C4×D4D412yes—
91C4×D4C2312yes—
92C4×D4C2312yes—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D411yescharacteristic
97C2×C22⋊C4—24no—
98C2×C22⋊C4—21no—
99C2×C42C2311yescharacteristic
100#107—24no—
101#107—24no—
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8C2312yes—
105C4⋊C8C2312yes—
106#103—24no—
107#102—24no—
108#102—24no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C4×C8—21no—
112C4×C8C2312yes—
113C24—21no—
114C24—24no—
115#41—44no—
116#41—48no—
117#41—44no—
118#41—42no—
119#41—42no—
120#41—42no—
121#41—42no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—41no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×D4—24no—
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—24no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—44no—
140C2×D4—48no—
141C2×D4—416no—
142C2×D4—44no—
143C2×D4—84no—
144C2×D4D814yes—
145C2×D4C2×D412yes—
146C2×D4C2×D412yes—
147C2×D4C2×D412yes—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—82no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159SD16—48no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C4⋊C4—24no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2411yescharacteristic
187C23—22no—
188C23—24no—
189C23—22no—
190C23—22no—
191C23—48no—
192C23—81no—
193C23—81no—
194C23—84no—
195C23—84no—
196C23#12011yescharacteristic
197Q8—41no—
198Q8—41no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—82no—
203D4—22no—
204D4—24no—
205D4—24no—
206D4—42no—
207D4—48no—
208D4—48no—
209D4—44no—
210D4—44no—
211D4—41no—
212D4—41no—
213D4—42no—
214D4—42no—
215D4—81no—
216D4—84no—
217D4C2×D814yes—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—48no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4#12011yescharacteristic
233C2×C4#12011yescharacteristic
234C2×C4#12011yescharacteristic
235C2×C4C2×D812yes—
236C2×C4AGL(1,ℤ/8)12yes—
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4C22×D411yescharacteristic
240C8—42no—
241C8—42no—
242C8—42no—
243C22—21no—
244C22—21no—
245C22—24no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—84no—
255C22—168no—
256C22#44712yes—
257C22C2×#12011yescharacteristic
258C4—21no—
259C4—21no—
260C4—21no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—82no—
265C4#44712yes—
266C4C22×D811yescharacteristic
267C4C2×AGL(1,ℤ/8)11yescharacteristic
268C4#57712yes—
269C2—22no—
270C2—44no—
271C2—82no—
272C2—82no—
273C2C2×#44711yescharacteristic
274C2#300911yescharacteristic
275C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3496C2
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
2. #3226C2semidirect
3. #3226C2semidirect
4. C2×#461C2semidirect
5. #1647C2semidirect
6. #1639C2semidirect
7. #1540C2semidirect
8. #1480C2semidirect
9. #1478C2semidirect
22. C2×C4⋊Q8C22semidirect
26. #500C22semidirect
28. #486C22semidirect
32. #474C22semidirect
34. #461C22semidirect
35. #431C22semidirect
36. #431C22semidirect
37. C2×C4⋊C8C22semidirect
39. #332C22semidirect
40. #331C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#32262
C2×#4612
#16472
#16392
#15402
#14802
#14782