Finite Group Databasebeta

← #18935

Subgroups — #18935

Order 256.

493 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 113 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#3549C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3235C211yesmaximal, characteristic, complemented
4#3210C211yesmaximal, characteristic, complemented
5#1646C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1634C212yesmaximal, complemented
8#1629C212yesmaximal, complemented
9#1551C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic, complemented
11#1480C211yesmaximal, characteristic, complemented
12H(3,𝔽4)—22no—
13#555—21no—
14#555—21no—
15#555C2212yes—
16D42—21no—
17D42—21no—
18D42C2212yes—
19#535—22no—
20#535—22no—
21#535—22no—
22#533C2211yescharacteristic, complemented
23#532C2211yescharacteristic
24#532C2211yescharacteristic
25#532C2211yescharacteristic
26#501—21no—
27#494C2211yescharacteristic, complemented
28#492C2211yescharacteristic
29#490—21no—
30#487C2211yescharacteristic, complemented
31#486C2211yescharacteristic
32#482—22no—
33#476C2212yescomplemented
34#476C2212yescomplemented
35#475—21no—
36#474C2212yescomplemented
37#465—22no—
38#464—22no—
39#461C2212yescomplemented
40#460—21no—
41#452—22no—
42#432C2211yescharacteristic, complemented
43#431C2211yescharacteristic, complemented
44C2×C4⋊C8C2211yescharacteristic, complemented
45#333C2212yescomplemented
46#331C2212yes—
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51C2×#41—21no—
52C2×#41—21no—
53C2×#41D412yes—
54C22×D4—22no—
55C22×D4—22no—
56C22×D4—22no—
57C2×Q16—22no—
58C2×SD16—22no—
59C2×SD16—22no—
60C22×C8—21no—
61C4⋊Q8D411yescharacteristic
62C4⋊Q8D411yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66#127—24no—
67#127D411yescharacteristic
68#127D411yescharacteristic
69#127C2311yescharacteristic
70#125—21no—
71#124—24no—
72#124—24no—
73#124—24no—
74#124—21no—
75#124—21no—
76#124D412yes—
77C22⋊Q8—21no—
78C22⋊Q8—21no—
79#121—22no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121—21no—
84#121—21no—
85#121—21no—
86#121—21no—
87#121—22no—
88#121—22no—
89#121D412yes—
90#121D412yes—
91#120—24no—
92#120—24no—
93#120—24no—
94#120—24no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4C2312yes—
102C4×D4C2312yes—
103#117—22no—
104#117—22no—
105#117—22no—
106#117—21no—
107C2×C42C2311yescharacteristic
108#107—22no—
109#106—22no—
110#106—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C24—24no—
127#41—81no—
128#41—82no—
129#41—81no—
130#41—82no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8C2×D412yes—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—82no—
149C2×D4—84no—
150C2×D4—82no—
151C2×D4—84no—
152C2×D4C2×D412yes—
153C2×D4C2×D412yes—
154C2×D4C2×D412yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161Q16—82no—
162SD16—82no—
163SD16—82no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—81no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—41no—
191C42—24no—
192C42—41no—
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2411yescharacteristic
197C23—22no—
198C23—22no—
199C23—22no—
200C23—84no—
201C23—84no—
202C23—84no—
203C23#12011yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—41no—
207Q8—41no—
208Q8—81no—
209Q8—81no—
210D4—22no—
211D4—24no—
212D4—22no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—41no—
217D4—42no—
218D4—41no—
219D4—81no—
220D4—81no—
221D4—82no—
222D4—82no—
223D4—82no—
224D4—84no—
225D4—84no—
226D4—84no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C8—41no—
247C8—41no—
248C8—42no—
249C8—42no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—168no—
263C22C2×#12011yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—81no—
272C4—81no—
273C4#57611yescharacteristic
274C4#57712yes—
275C4#57711yescharacteristic
276C2—41no—
277C2—82no—
278C2—82no—
279C2—82no—
280C2#300911yescharacteristic
281C2#301211yescharacteristic
282C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3549C2
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. #3246C2semidirect
3. #3235C2semidirect
4. #3210C2semidirect
5. #1646C2semidirect
6. #1639C2semidirect
7. #1634C2semidirect
8. #1629C2semidirect
9. #1551C2semidirect
10. #1496C2semidirect
11. #1480C2semidirect
22. #533C22semidirect
27. #494C22semidirect
30. #487C22semidirect
33. #476C22semidirect
34. #476C22semidirect
36. #474C22semidirect
39. #461C22semidirect
42. #432C22semidirect
43. #431C22semidirect
44. C2×C4⋊C8C22semidirect
45. #333C22semidirect
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
#35492
#32462
#32352
#32102
#16462
#16392
#16342
#16292
#15512
#14962
#14802