Finite Group Databasebeta

← #18381

Subgroups — #18381

Order 256.

633 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 127 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#3187C212yesmaximal, complemented
3#2564C211yesmaximal, characteristic, complemented
4#1616C212yesmaximal, complemented
5#1613C212yesmaximal, complemented
6#1604C212yesmaximal, complemented
7#1601C212yesmaximal, complemented
8C2×#327C212yesmaximal, complemented
9#1459C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11#547—22no—
12#546—24no—
13D42—22no—
14#537C2212yes—
15#534—21no—
16#534C2212yes—
17#532—21no—
18C2×C22⋊Q8C2211yescharacteristic, complemented
19C2×#121—21no—
20C2×#121C2212yes—
21C2×C4×D4C2211yescharacteristic
22#483—22no—
23#481C2214yescomplemented
24#465—24no—
25#450—22no—
26#448C2214yescomplemented
27C2×C22⋊C8C2212yescomplemented
28#400C2211yescharacteristic, complemented
29#399C2211yescharacteristic, complemented
30#395C2211yescharacteristic, complemented
31#393C2211yescharacteristic, complemented
32#387—21no—
33#386—21no—
34#385—21no—
35#380—21no—
36C2×#95C2211yescharacteristic, complemented
37#327C2214yescomplemented
38#327C2214yescomplemented
39#324C2212yescomplemented
40#323C2212yescomplemented
41H(5,𝔽2)—44no—
42H(5,𝔽2)—44no—
43C2×#41—24no—
44C2×#41—22no—
45C22×Q8C2311yescharacteristic
46C22×D4—24no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4D411yescharacteristic
51C22×D4D412yes—
52C23×C4—21no—
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C2×SD16—24no—
56C22×C8—22no—
57C4⋊Q8—21no—
58#127—21no—
59#124—22no—
60#123—24no—
61#123—24no—
62C22⋊Q8—24no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65#121—22no—
66#121—22no—
67#121—24no—
68#121—22no—
69#121—22no—
70#121—22no—
71#121—22no—
72#121D414yes—
73#121C2314yes—
74#120—24no—
75#120—24no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D412yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4D411yescharacteristic
93C2×C22⋊C4C2311yescharacteristic
94C2×C42D411yescharacteristic
95#106—24no—
96#103—24no—
97#102—24no—
98C22⋊C8—22no—
99C22⋊C8C2314yes—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95C2311yescharacteristic
106#95C2311yescharacteristic
107C24—22no—
108C24—22no—
109C24C2×D411yescharacteristic
110C24C2×D411yescharacteristic
111#41—44no—
112#41—416no—
113#41—44no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—81no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—416no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—82no—
132C2×D4—82no—
133C2×D4SD1614yescomplemented
134C2×D4C2×D414yes—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D412yes—
154C22×C4C2×D412yes—
155C22×C4C2411yescharacteristic
156SD16—84no—
157C2×C8—24no—
158C2×C8—82no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4C2×D414yes—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—22no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—81no—
178C42—21no—
179C42—21no—
180C23—21no—
181C23—21no—
182C23—24no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—44no—
188C23—44no—
189C23—82no—
190C23—82no—
191C23—82no—
192C23—82no—
193C23#12012yes—
194C23C2×SD1612yes—
195C23C2×SD1612yes—
196C23AGL(1,ℤ/8)12yes—
197C23C22×D411yescharacteristic
198Q8—44no—
199Q8—44no—
200D4—28no—
201D4—44no—
202D4—44no—
203D4—44no—
204D4—48no—
205D4—44no—
206D4—44no—
207D4—48no—
208D4—82no—
209D4—82no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—48no—
222C2×C4—41no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—84no—
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4C2×SD1612yes—
242C2×C4C22×D412yes—
243C8—44no—
244C22—21no—
245C22—22no—
246C22—21no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—81no—
258C22—164no—
259C22—164no—
260C22#45014yes—
261C22C2×#12011yescharacteristic
262C22C22×SD1611yescharacteristic
263C22C2×AGL(1,ℤ/8)11yescharacteristic
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—44no—
270C4—41no—
271C4—41no—
272C4—81no—
273C4—81no—
274C4—81no—
275C2—21no—
276C2—21no—
277C2—24no—
278C2—42no—
279C2—84no—
280C2C2×#45011yescharacteristic
281C2#301811yescharacteristic
282C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3187C2semidirect
3. #2564C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1604C2semidirect
7. #1601C2semidirect
8. C2×#327C2semidirect
9. #1459C2semidirect
18. C2×C22⋊Q8C22semidirect
23. #481C22semidirect
26. #448C22semidirect
27. C2×C22⋊C8C22semidirect
28. #400C22semidirect
29. #399C22semidirect
30. #395C22semidirect
31. #393C22semidirect
36. C2×#95C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect
40. #323C22semidirect
133. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#34962
#31872
#25642
#16162
#16132
#16042
#16012
C2×#3272
#14592