Finite Group Databasebeta

← #18903

Subgroups — #18903

Order 256.

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

normal subgroups only

This group has 603 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#3227C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic, complemented
4C2×#474C211yesmaximal, characteristic, complemented
5#1654C214yesmaximal, complemented
6#1639C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11#547—21no—
12#547C2212yes—
13#546—22no—
14#546—22no—
15D42—21no—
16D42C2212yes—
17#537—21no—
18#534—22no—
19#532C2211yescharacteristic
20#532C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic, complemented
22C2×#121—22no—
23C2×C4×D4C2211yescharacteristic
24#498C2212yescomplemented
25#492—21no—
26#487—21no—
27#486C2212yescomplemented
28#481—22no—
29#476—22no—
30#474—22no—
31#474C2214yescomplemented
32#461C2214yescomplemented
33#448—22no—
34#431C2211yescharacteristic, complemented
35#431C2211yescharacteristic, complemented
36C2×#107—21no—
37C2×C4⋊C8C2211yescharacteristic, complemented
38C2×#102—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—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8D411yescharacteristic
62C4⋊Q8C2312yes—
63C4⋊Q8C2311yescharacteristic
64#127D411yescharacteristic
65#127C2311yescharacteristic
66#124—21no—
67#124—21no—
68#123—24no—
69#123—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—24no—
73#121—24no—
74#121—22no—
75#121—22no—
76#121—21no—
77#121—21no—
78#121—22no—
79#121—22no—
80#121D412yes—
81#121D412yes—
82#120—24no—
83#120—24no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4D412yes—
89C4×D4D412yes—
90C4×D4C2312yes—
91C4×D4C2312yes—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C22⋊C4—24no—
97C2×C22⋊C4—21no—
98C2×C42C2311yescharacteristic
99#107—24no—
100#106—24no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8C2312yes—
104C4⋊C8C2312yes—
105#103—24no—
106#102—24no—
107#102—24no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110C4×C8—21no—
111C4×C8C2312yes—
112C24—21no—
113C24—24no—
114#41—44no—
115#41—48no—
116#41—44no—
117#41—42no—
118#41—42no—
119#41—42no—
120#41—42no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—41no—
125C2×Q8—82no—
126C2×D4—24no—
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—24no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—44no—
138C2×D4—48no—
139C2×D4—416no—
140C2×D4—44no—
141C2×D4—84no—
142C2×D4SD1614yescomplemented
143C2×D4C2×D412yes—
144C2×D4C2×D412yes—
145C2×D4C2×D412yes—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—41no—
153C22×C4—82no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157SD16—84no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—42no—
162C4⋊C4—24no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2411yescharacteristic
185C23—22no—
186C23—24no—
187C23—22no—
188C23—22no—
189C23—48no—
190C23—81no—
191C23—81no—
192C23—84no—
193C23—84no—
194C23#12011yescharacteristic
195Q8—41no—
196Q8—41no—
197Q8—42no—
198Q8—44no—
199Q8—82no—
200D4—22no—
201D4—24no—
202D4—24no—
203D4—42no—
204D4—48no—
205D4—48no—
206D4—44no—
207D4—44no—
208D4—41no—
209D4—41no—
210D4—42no—
211D4—42no—
212D4—81no—
213D4—84no—
214D4C2×SD1614yes—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—48no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4C2×SD1612yes—
233C2×C4AGL(1,ℤ/8)12yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C8—42no—
238C8—42no—
239C8—42no—
240C22—21no—
241C22—21no—
242C22—24no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—84no—
252C22—168no—
253C22#45012yes—
254C22C2×#12011yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—82no—
262C4#45012yes—
263C4C22×SD1611yescharacteristic
264C4C2×AGL(1,ℤ/8)11yescharacteristic
265C4#57612yes—
266C2—22no—
267C2—44no—
268C2—82no—
269C2—82no—
270C2C2×#45011yescharacteristic
271C2#300811yescharacteristic
272C2#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. #3227C2semidirect
3. #3206C2semidirect
4. C2×#474C2semidirect
5. #1654C2semidirect
6. #1639C2semidirect
7. #1540C2semidirect
8. #1480C2semidirect
9. #1478C2semidirect
21. C2×C4⋊Q8C22semidirect
24. #498C22semidirect
27. #486C22semidirect
31. #474C22semidirect
32. #461C22semidirect
34. #431C22semidirect
35. #431C22semidirect
37. C2×C4⋊C8C22semidirect
39. #332C22semidirect
40. #331C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect
142. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#34962
#32272
#32062
C2×#4742
#16542
#16392
#15402
#14802
#14782