Finite Group Databasebeta

← #18599

Subgroups — #18599

Order 256.

595 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 117 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#3235C211yesmaximal, characteristic, complemented
3#3182C211yesmaximal, characteristic, complemented
4#3101C211yesmaximal, characteristic, complemented
5#1647C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1635C212yesmaximal, complemented
8#1629C212yesmaximal, complemented
9#1553C211yesmaximal, characteristic, complemented
10#1490C211yesmaximal, characteristic, complemented
11#1484C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13#547—21no—
14#547C2212yes—
15#546—22no—
16#546—22no—
17D42—21no—
18D42C2212yes—
19#537—21no—
20#534—22no—
21#533C2211yescharacteristic, complemented
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24C2×#121—22no—
25C2×C4×D4C2211yescharacteristic
26#501—21no—
27#500C2211yescharacteristic, complemented
28#492C2211yescharacteristic, complemented
29#490—21no—
30#487C2211yescharacteristic, complemented
31#486C2211yescharacteristic, complemented
32#483—21no—
33#482—22no—
34#476C2212yescomplemented
35#476C2212yescomplemented
36#474C2212yescomplemented
37#468—22no—
38#461C2212yescomplemented
39#450—21no—
40#440—22no—
41C4×SD16—22no—
42#432C2211yescharacteristic, complemented
43#431C2211yescharacteristic, complemented
44C4×M16C2211yescharacteristic, complemented
45#332C2212yescomplemented
46#331C2212yescomplemented
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51H(5,𝔽2)—48no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—21no—
55C2×#41—21no—
56C22×D4—24no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—22no—
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C2×SD16—22no—
63C2×SD16—22no—
64C2×M16—21no—
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68C4⋊Q8C2311yescharacteristic
69#127D411yescharacteristic
70#127C2311yescharacteristic
71#125—21no—
72#124—21no—
73#124—21no—
74#123—24no—
75#123—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80#121—24no—
81#121—24no—
82#121—22no—
83#121—22no—
84#121—21no—
85#121—21no—
86#121—22no—
87#121—22no—
88#121D412yes—
89#121D412yes—
90#120—24no—
91#120—24no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4D412yes—
99C4×D4D412yes—
100C4×D4C2312yes—
101C4×D4C2312yes—
102#117—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4D411yescharacteristic
105C2×C22⋊C4—24no—
106C2×C22⋊C4—21no—
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#106—22no—
111#106—22no—
112C4⋊C8C2312yes—
113C4⋊C8C2312yes—
114#103—22no—
115#103—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120C22⋊C8—21no—
121C22⋊C8—21no—
122#97—21no—
123#97—21no—
124C4×C8—21no—
125C4×C8C2311yescharacteristic
126C4×C8C2311yescharacteristic
127C24—21no—
128C24—24no—
129#41—44no—
130#41—44no—
131#41—48no—
132#41—44no—
133#41—44no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—48no—
150C2×D4—44no—
151C2×D4—416no—
152C2×D4—44no—
153C2×D4—84no—
154C2×D4C2×D412yes—
155C2×D4C2×D412yes—
156C2×D4C2×D412yes—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—82no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167SD16—44no—
168SD16—44no—
169M16—42no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—21no—
173C2×C8—22no—
174C4⋊C4—24no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—41no—
196C42—41no—
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2411yescharacteristic
201C23—24no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—48no—
206C23—81no—
207C23—81no—
208C23—84no—
209C23—84no—
210C23#12011yescharacteristic
211Q8—42no—
212Q8—42no—
213Q8—42no—
214Q8—42no—
215D4—24no—
216D4—22no—
217D4—24no—
218D4—42no—
219D4—42no—
220D4—44no—
221D4—44no—
222D4—48no—
223D4—48no—
224D4—42no—
225D4—42no—
226D4—81no—
227D4—84no—
228D4AGL(1,ℤ/8)14yes—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C8—22no—
251C8—41no—
252C8—42no—
253C8—42no—
254C22—21no—
255C22—24no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—84no—
267C22—168no—
268C22C2×#12011yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4C2×AGL(1,ℤ/8)12yes—
279C4#57712yes—
280C2—41no—
281C2—44no—
282C2—82no—
283C2—82no—
284C2#300311yescharacteristic
285C2#300911yescharacteristic
286C2#301811yescharacteristic
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. #3235C2semidirect
3. #3182C2semidirect
4. #3101C2semidirect
5. #1647C2semidirect
6. #1639C2semidirect
7. #1635C2semidirect
8. #1629C2semidirect
9. #1553C2semidirect
10. #1490C2semidirect
11. #1484C2semidirect
21. #533C22semidirect
27. #500C22semidirect
28. #492C22semidirect
30. #487C22semidirect
31. #486C22semidirect
34. #476C22semidirect
35. #476C22semidirect
36. #474C22semidirect
38. #461C22semidirect
42. #432C22semidirect
43. #431C22semidirect
44. C4×M16C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. D4⋊C8C22semidirect
48. D4⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#32352
#31822
#31012
#16472
#16392
#16352
#16292
#15532
#14902
#14842