Finite Group Databasebeta

← #28264

Subgroups — #28264

Order 256.

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

normal subgroups only

This group has 425 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×C8×Q8C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2582C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic
5#2066C212yesmaximal, complemented
6#2065C212yesmaximal
7#1968C212yesmaximal
8#1941C212yesmaximal, complemented
9#1844C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1802C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17C8×Q8—24no—
18C4×SD16—24no—
19C2×#106C2212yes—
20C2×#106C2211yescharacteristic
21C2×#106C2212yes—
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2212yes—
28C2×C4×C8C2212yes—
29C2×C4×C8C2211yescharacteristic
30#398—22no—
31#396—22no—
32#391—21no—
33#391C2212yes—
34#390C2212yes—
35#389—21no—
36#389C2212yes—
37#389C2212yes—
38#384C2211yescharacteristic
39#384C2211yescharacteristic
40#380—21no—
41#340C2212yes—
42#340C2212yes—
43C22×Q8D411yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16Q814yescomplemented
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22⋊Q8—44no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42C2312yes—
78C2×C42C2311yescharacteristic
79#106—22no—
80#106—24no—
81#106—42no—
82C4⋊C8—24no—
83C4⋊C8—22no—
84#103—22no—
85#103—42no—
86#102—22no—
87#102—42no—
88C4×C8—24no—
89C4×C8D414yes—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94C24—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8SD1614yescomplemented
103C2×Q8C2×Q812yes—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4C2×Q812yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123SD16—28no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8C2×D412yes—
130C2×C8C2×D412yes—
131C2×C8C2×Q812yes—
132C2×C8#4112yes—
133C4⋊C4—24no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—81no—
148C4⋊C4—81no—
149C4⋊C4—82no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C22⋊C4—82no—
153C42—22no—
154C42—22no—
155C42—42no—
156C42C2×D412yes—
157C42#4112yes—
158C23—22no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23C22×D411yescharacteristic
166Q8—28no—
167Q8—22no—
168Q8—22no—
169Q8—41no—
170Q8—41no—
171D4—22no—
172D4—22no—
173D4—41no—
174D4—41no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—84no—
198C2×C4#12112yes—
199C2×C4#12112yes—
200C2×C4C2×SD1614yes—
201C2×C4C2×SD1612yes—
202C2×C4#13512yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C22×Q811yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C42−1+411yescharacteristic
209C8—22no—
210C8—42no—
211C8C22⋊Q814yescomplemented
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22C2×#12111yescharacteristic
220C22D4×Q811yescharacteristic
221C22#55011yescharacteristic
222C22C22×SD1611yescharacteristic
223C22C2×#13511yescharacteristic
224C22#57711yescharacteristic
225C22#57711yescharacteristic
226C4—24no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—82no—
232C4—81no—
233C4—81no—
234C4#46514yes—
235C4C2×C22⋊Q811yescharacteristic
236C4#53711yescharacteristic
237C4#55411yescharacteristic
238C4#55611yescharacteristic
239C2—82no—
240C2#253811yescharacteristic
241C2C2×#46511yescharacteristic
242C2#306011yescharacteristic
243C2Q8×SD1611yescharacteristic
244C2#338111yescharacteristic
245C2#339211yescharacteristic
246C2#339411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C8×Q8C2semidirect
3. #2582C2semidirect
5. #2066C2semidirect
8. #1941C2semidirect
9. #1844C2semidirect
10. #1803C2semidirect
49. C2×SD16Q8semidirect
102. C2×Q8SD16semidirect
211. C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C8×Q82
C2×C4×SD162
#25822
#25382
#20662
#20652
#19682
#19412
#18442
#18032
#18022