Finite Group Databasebeta

← #33013

Subgroups — #33013

Order 256.

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

normal subgroups only

This group has 667 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×#520C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic
4#2624C211yesmaximal, characteristic, complemented
5#1880C212yesmaximal, complemented
6#1879C212yesmaximal, complemented
7#1865C212yesmaximal, complemented
8#1864C212yesmaximal, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1788C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C22×D8C2211yescharacteristic
15C2×#121C2211yescharacteristic
16#520—24no—
17#520—24no—
18C2×C4×Q8C2212yes—
19C2×C4×D4—22no—
20C2×C4×D4C2212yes—
21C2×#117—21no—
22C2×#117C2211yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2211yescharacteristic
25#481—24no—
26#449—24no—
27C2×#106C2212yes—
28C2×#103C2211yescharacteristic
29C2×#103C2212yes—
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
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—
45C2×#41—24no—
46C2×#41—24no—
47C2×#41—22no—
48C2×#41D414yes—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4C2311yescharacteristic
58C2×SD16—22no—
59C2×SD16—24no—
60C2×D8—22no—
61C2×D8—24no—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2312yes—
65#121—22no—
66#121—22no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×D4—216no—
70C4×D4—24no—
71C4×D4—24no—
72#117—28no—
73#117—22no—
74#117—22no—
75C2×C4⋊C4—28no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D414yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4D412yes—
92C2×C42—22no—
93C2×C42C2312yes—
94#106—24no—
95#103—22no—
96#103—24no—
97#102—22no—
98#102—24no—
99#102—24no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102#95—21no—
103#95—22no—
104C24—21no—
105C24—21no—
106C24C2×D411yescharacteristic
107#41—216no—
108#41—28no—
109#41—28no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8C2×D412yes—
113C2×Q8#4114yes—
114C2×D4—28no—
115C2×D4—24no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—42no—
123C2×D4—82no—
124C2×D4C2×D412yes—
125C2×D4#4114yes—
126C22×C4—28no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—28no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4C2×D412yes—
141C22×C4C2×D412yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D412yes—
145C22×C4C2411yescharacteristic
146SD16—44no—
147D8—44no—
148C2×C8—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—28no—
156C4⋊C4—22no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—82no—
165C4⋊C4C2×D414yes—
166C22⋊C4—28no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—82no—
170C22⋊C4—82no—
171C22⋊C4—81no—
172C42—28no—
173C42—22no—
174C42—22no—
175C23—22no—
176C23—22no—
177C23—24no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23—42no—
186C23—41no—
187C23—42no—
188C23—41no—
189C23#12012yes—
190C23C22×D411yescharacteristic
191Q8—22no—
192Q8—24no—
193Q8#12314yes—
194D4—216no—
195D4—22no—
196D4—24no—
197D4—41no—
198D4—41no—
199D4—42no—
200D4—84no—
201D4#12314yes—
202C2×C4—216no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—88no—
225C2×C4#12012yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4112yes—
229C2×C4C2×#4112yes—
230C8—41no—
231C8—41no—
232C8—42no—
233C22—22no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22—42no—
238C22—44no—
239C22—42no—
240C22—44no—
241C22—42no—
242C22—42no—
243C22—81no—
244C22—82no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22C2×#12011yescharacteristic
249C22#52512yes—
250C22#57611yescharacteristic
251C22#57712yes—
252C22#57711yescharacteristic
253C4—22no—
254C4—24no—
255C4—42no—
256C4—42no—
257C4—44no—
258C4—82no—
259C4—81no—
260C4C2×#12311yescharacteristic
261C4C2×#12311yescharacteristic
262C4#52512yes—
263C2—42no—
264C2—44no—
265C2—161no—
266C2#284911yescharacteristic
267C2#301011yescharacteristic
268C2#301211yescharacteristic
269C2#311912yes—
270C2#312512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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×#520C2semidirect
2. C2×#481C2semidirect
4. #2624C2semidirect
5. #1880C2semidirect
6. #1879C2semidirect
7. #1865C2semidirect
8. #1864C2semidirect
9. #1823C2semidirect
10. #1818C2semidirect
11. #1788C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4812
C2×#4492
#26242
#18802
#18792
#18652
#18642
#18232
#18182
#17882