Finite Group Databasebeta

← #24813

Subgroups — #24813

Order 256.

447 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#475C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic
4Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#2084C212yesmaximal, complemented
6#2065C212yesmaximal
7#1990C211yesmaximal, characteristic, complemented
8#1939C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1803C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#475—24no—
20C4×SD16—24no—
21C2×#106C2212yes—
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29C2×C4×C8C2211yescharacteristic
30#399—22no—
31#390C2212yes—
32#389—22no—
33#389C2212yes—
34#384—21no—
35#384C2211yescharacteristic
36#384C2212yes—
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16Q814yescomplemented
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C22⋊Q8—44no—
56C22⋊Q8—44no—
57C4×Q8—22no—
58C4×Q8—24no—
59C4×Q8—42no—
60C4×Q8D414yes—
61C4×D4—22no—
62C4×D4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C42—21no—
81C2×C42—22no—
82C2×C42C2312yes—
83C2×C42C2311yescharacteristic
84#106—22no—
85#106—24no—
86C4⋊C8—22no—
87C4⋊C8—42no—
88#103—22no—
89#103—24no—
90#102—22no—
91#102—42no—
92C4×C8—22no—
93#95—22no—
94#95D412yes—
95C24—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8C2×D412yes—
103C2×Q8C2×Q812yes—
104C2×Q8#4112yes—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4C2×Q812yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2411yescharacteristic
124SD16—28no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8C2×Q812yes—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4—42no—
148C4⋊C4—81no—
149C4⋊C4—82no—
150C4⋊C4SD1614yescomplemented
151C4⋊C4C2×D412yes—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C42—24no—
156C42—22no—
157C42—22no—
158C42—44no—
159C42C2×D412yes—
160C42#4112yes—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23C22×D411yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—22no—
172Q8—84no—
173Q8C22⋊Q814yescomplemented
174D4—22no—
175D4—22no—
176D4—41no—
177D4—41no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4#12112yes—
202C2×C4#12112yes—
203C2×C4C2×SD1614yes—
204C2×C4C2×SD1612yes—
205C2×C4AGL(1,ℤ/8)12yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C22×Q811yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)11yescharacteristic
211C2×C42−1+411yescharacteristic
212C8—22no—
213C8—41no—
214C8—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22C2×#12111yescharacteristic
223C22#54611yescharacteristic
224C22D4×Q811yescharacteristic
225C22C22×SD1611yescharacteristic
226C22C2×AGL(1,ℤ/8)11yescharacteristic
227C22#57711yescharacteristic
228C22#57811yescharacteristic
229C4—24no—
230C4—24no—
231C4—22no—
232C4—42no—
233C4—42no—
234C4—82no—
235C4—81no—
236C4—82no—
237C4#46014yes—
238C4C2×C22⋊Q811yescharacteristic
239C4#53711yescharacteristic
240C4#55211yescharacteristic
241C4#55411yescharacteristic
242C2—82no—
243C2#241711yescharacteristic
244C2C2×#46011yescharacteristic
245C2#304611yescharacteristic
246C2#329811yescharacteristic
247C2#333211yescharacteristic
248C2Q8×SD1611yescharacteristic
249C2#338611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#475C2semidirect
4. Q8×C4⋊C4C2semidirect
5. #2084C2semidirect
7. #1990C2semidirect
8. #1939C2semidirect
10. #1803C2semidirect
49. C2×SD16Q8semidirect
150. C4⋊C4SD16semidirect
173. Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×C4×SD162
#25422
Q8×C4⋊C42
#20842
#20652
#19902
#19392
#18802
#18032
#18022