Finite Group Databasebeta

← #24228

Subgroups — #24228

Order 256.

715 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 135 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#3037C212yesmaximal, complemented
3C2×#455C212yesmaximal, complemented
4#2326C211yesmaximal, characteristic, complemented
5#2039C212yesmaximal, complemented
6#2007C212yesmaximal, complemented
7#1963C212yesmaximal, complemented
8#1800C211yesmaximal, characteristic, complemented
9#1792C212yesmaximal, complemented
10C2×H(5,𝔽2)C2212yes—
11C2×#137C2214yescomplemented
12#547—22no—
13#546—24no—
14D42—22no—
15#537C2212yes—
16#534—22no—
17#532C2212yes—
18C2×#124C2211yescharacteristic
19C2×#121—22no—
20C2×C4×Q8C2211yescharacteristic, complemented
21C2×C4×D4C2211yescharacteristic
22#464—24no—
23#462—22no—
24#460—22no—
25#455—28no—
26#423C2212yescomplemented
27C2×#104C2214yescomplemented
28C2×#104C2214yescomplemented
29#417C2214yescomplemented
30C2×#100C2212yes—
31#391C2211yescharacteristic, complemented
32#390C2211yescharacteristic
33#388—22no—
34#385—22no—
35#384C2211yescharacteristic, complemented
36#376C2211yescharacteristic, complemented
37#343C2212yescomplemented
38H(5,𝔽2)—24no—
39H(5,𝔽2)—24no—
40H(5,𝔽2)—44no—
41C2×#41—24no—
42C2×#41C2314yes—
43C22×Q8C2311yescharacteristic
44C22×D4—24no—
45C22×D4—22no—
46C22×D4—24no—
47C22×D4C2311yescharacteristic
48C23×C4D412yes—
49#137—28no—
50#137—44no—
51C2×Q16—24no—
52C2×SD16—24no—
53C2×M16C2314yes—
54C4⋊Q8D412yes—
55#127D412yes—
56#124—22no—
57#124D414yes—
58#124D414yescomplemented
59#123—28no—
60C22⋊Q8—24no—
61#121—24no—
62#121—24no—
63#121—24no—
64#121—22no—
65#121—24no—
66#121—22no—
67#120—28no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4D412yes—
83C2×C42—21no—
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87C4⋊C8—22no—
88C4⋊C8—22no—
89#104—28no—
90#104—44no—
91#103—24no—
92#102—24no—
93#100—24no—
94#95—21no—
95#95—21no—
96#95—22no—
97C24—24no—
98C24C2×D412yes—
99#41—24no—
100#41—24no—
101#41—24no—
102#41—24no—
103#41—48no—
104#41—48no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8C2×D414yes—
109C2×Q8C2×D412yescomplemented
110C2×D4—28no—
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—48no—
118C2×D4—48no—
119C2×D4—48no—
120C2×D4—48no—
121C2×D4—84no—
122C2×D4C2×D414yes—
123C2×D4C2×D412yes—
124C2×D4#4112yes—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—82no—
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138Q16—48no—
139SD16—48no—
140M16—24no—
141M16—24no—
142C2×C8—24no—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—82no—
151C22⋊C4—28no—
152C22⋊C4—28no—
153C22⋊C4—24no—
154C22⋊C4—24no—
155C22⋊C4—24no—
156C22⋊C4—82no—
157C42—21no—
158C42—22no—
159C42—42no—
160C42C2×D412yes—
161C42C2×D412yes—
162C42C2×D412yescomplemented
163C23—22no—
164C23—22no—
165C23—24no—
166C23—42no—
167C23—42no—
168C23—48no—
169C23—84no—
170C23—84no—
171C23#12114yes—
172C23C22×D411yescharacteristic
173Q8—24no—
174Q8—24no—
175Q8—24no—
176Q8—42no—
177D4—24no—
178D4—24no—
179D4—24no—
180D4—24no—
181D4—44no—
182D4—416no—
183D4—44no—
184D4—44no—
185D4—82no—
186D4—84no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—48no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—82no—
208C2×C4#12012yes—
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D412yes—
213C2×C4C2×#4111yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C8—44no—
216C22—21no—
217C22—21no—
218C22—24no—
219C22—44no—
220C22—42no—
221C22—42no—
222C22—44no—
223C22—44no—
224C22—44no—
225C22—82no—
226C22—168no—
227C22C2×#12011yescharacteristic
228C22C2×#12111yescharacteristic
229C22#53511yescharacteristic
230C4—21no—
231C4—21no—
232C4—22no—
233C4—42no—
234C4—44no—
235C4—42no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4D4212yes—
240C4#54612yes—
241C2—21no—
242C2—21no—
243C2—44no—
244C2—84no—
245C2#240811yescharacteristic
246C2#301911yescharacteristic
247C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3037C2semidirect
3. C2×#455C2semidirect
4. #2326C2semidirect
5. #2039C2semidirect
6. #2007C2semidirect
7. #1963C2semidirect
8. #1800C2semidirect
9. #1792C2semidirect
11. C2×#137C22semidirect
20. C2×C4×Q8C22semidirect
26. #423C22semidirect
27. C2×#104C22semidirect
28. C2×#104C22semidirect
29. #417C22semidirect
31. #391C22semidirect
35. #384C22semidirect
36. #376C22semidirect
37. #343C22semidirect
58. #124D4semidirect
109. C2×Q8C2×D4semidirect
162. C42C2×D4semidirect

Maximal subgroups

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

subgroupindex
#34962
#30372
C2×#4552
#23262
#20392
#20072
#19632
#18002
#17922