Finite Group Databasebeta

← #33015

Subgroups — #33015

Order 256.

629 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 143 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×#519C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic
3C2×#448C211yesmaximal, characteristic
4#2624C211yesmaximal, characteristic
5#1880C212yesmaximal, complemented
6#1880C212yesmaximal, complemented
7#1865C212yesmaximal, complemented
8#1863C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2212yes—
14C2×#121C2211yescharacteristic
15#519—24no—
16#519—24no—
17C2×C4×Q8—21no—
18C2×C4×Q8—21no—
19C2×C4×Q8C2212yes—
20C2×C4×Q8C2212yes—
21C2×#117—22no—
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#482—24no—
25#448—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#399—21no—
36#397—21no—
37#394—22no—
38#390C2212yes—
39#389C2212yes—
40#380C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—24no—
46C22×Q8—24no—
47C22×Q8—21no—
48C22×Q8—22no—
49C22×Q8D414yes—
50C22×Q8C2312yes—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×SD16—24no—
56C2×SD16—24no—
57C2×SD16—24no—
58C22×C8C2312yes—
59C22×C8C2311yescharacteristic
60C22×C8C2311yescharacteristic
61#121—22no—
62#121—22no—
63C4×Q8—28no—
64C4×Q8—28no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—24no—
69#117—216no—
70#117—22no—
71#117—22no—
72#117—22no—
73#117D414yes—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4D412yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—21no—
92C2×C42C2312yes—
93#107—22no—
94#106—22no—
95#103—24no—
96#103—24no—
97#103—24no—
98#102—22no—
99#102—22no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102#95—21no—
103#95—22no—
104C24—21no—
105C24C2×D411yescharacteristic
106C2×Q8—28no—
107C2×Q8—28no—
108C2×Q8—28no—
109C2×Q8—216no—
110C2×Q8—24no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8C2×D414yes—
115C2×Q8#4114yes—
116C2×Q8#4114yes—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—42no—
122C2×D4—82no—
123C22×C4—28no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4C2×D412yes—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138SD16—48no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—28no—
147C4⋊C4—28no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—82no—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D412yes—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—42no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C22⋊C4—81no—
165C42—28no—
166C42—28no—
167C42—24no—
168C42—22no—
169C42—22no—
170C23—22no—
171C23—21no—
172C23—22no—
173C23—21no—
174C23—41no—
175C23—42no—
176C23—41no—
177C23—42no—
178C23—41no—
179C23#12012yes—
180C23C22×D411yescharacteristic
181Q8—216no—
182Q8—24no—
183Q8—24no—
184Q8—24no—
185Q8#12318yes—
186D4—42no—
187D4—41no—
188D4—41no—
189D4—84no—
190C2×C4—216no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4#12012yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4112yes—
218C2×C4C2×#4112yes—
219C8—42no—
220C8—41no—
221C8—41no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—81no—
228C22—82no—
229C22—82no—
230C22—81no—
231C22—81no—
232C22C2×#12011yescharacteristic
233C22#52512yes—
234C22#57712yes—
235C22#57711yescharacteristic
236C22#57811yescharacteristic
237C4—28no—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—82no—
244C4—81no—
245C4C2×#12312yes—
246C4#52511yescharacteristic
247C4#52511yescharacteristic
248C2—42no—
249C2—161no—
250C2#284911yescharacteristic
251C2#301011yescharacteristic
252C2#301311yescharacteristic
253C2#312112yes—
254C2#312612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#519C2semidirect
5. #1880C2semidirect
6. #1880C2semidirect
7. #1865C2semidirect
8. #1863C2semidirect
9. #1825C2semidirect
10. #1818C2semidirect
11. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5192
C2×#4822
C2×#4482
#26242
#18802
#18652
#18632
#18252
#18182
#17872