Finite Group Databasebeta

← #33057

Subgroups — #33057

Order 256.

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

normal subgroups only

This group has 470 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
1C2×#489C211yesmaximal, characteristic, complemented
2C2×C4×M16C211yesmaximal, characteristic, complemented
3#2857C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic
5#1993C211yesmaximal, characteristic, complemented
6#1959C211yesmaximal, characteristic, complemented
7#1934C212yesmaximal, complemented
8#1933C212yesmaximal, complemented
9#1895C212yesmaximal
10#1894C212yesmaximal, complemented
11#1836C211yesmaximal, characteristic, complemented
12C22×M16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#489—24no—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C4×M16—24no—
25C2×#97C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×C4×C8C2212yes—
28#391—21no—
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#389—21no—
32#389C2212yes—
33#388—21no—
34#388—21no—
35#386—21no—
36#386—21no—
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39#380—21no—
40#380C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×M16—24no—
53C2×M16—24no—
54C2×M16—24no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#124—22no—
60#124—22no—
61C22⋊Q8—24no—
62C22⋊Q8—44no—
63#121—24no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86#103—24no—
87#103—24no—
88#102—24no—
89#102—24no—
90#97—22no—
91#97D414yescomplemented
92C4×C8—24no—
93#95—22no—
94#95—22no—
95#95—21no—
96#95—21no—
97#95—21no—
98#95—21no—
99C24—21no—
100C24C2×D411yescharacteristic
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4#4114yes—
135C22×C4C2411yescharacteristic
136M16—216no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8C2×D414yes—
142C2×C8#4114yes—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C22⋊C4—44no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—21no—
164C42—21no—
165C42C2×D412yes—
166C42#4112yes—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23#12412yes—
181C23C22×D411yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—84no—
185D4—42no—
186D4—42no—
187D4—84no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4#12112yes—
218C2×C4#12112yes—
219C2×C4#12412yes—
220C2×C4AGL(1,ℤ/8)12yes—
221C2×C4#13712yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4112yes—
224C2×C4C2×#4112yes—
225C2×C4C2×#4111yescharacteristic
226C8—24no—
227C8—24no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22C2×#12111yescharacteristic
243C22C2×#12411yescharacteristic
244C22#53211yescharacteristic
245C22C2×AGL(1,ℤ/8)11yescharacteristic
246C22C2×#13711yescharacteristic
247C22#57511yescharacteristic
248C22#57511yescharacteristic
249C4—22no—
250C4—22no—
251C4—22no—
252C4—22no—
253C4—22no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4—82no—
258C4—82no—
259C4#52512yes—
260C4#52912yes—
261C2—42no—
262C2—161no—
263C2#285011yescharacteristic
264C2#305511yescharacteristic
265C2#305611yescharacteristic
266C2#314011yescharacteristic
267C2#314111yescharacteristic
268C2#317111yescharacteristic
269C2#317211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#489C2semidirect
2. C2×C4×M16C2semidirect
3. #2857C2semidirect
5. #1993C2semidirect
6. #1959C2semidirect
7. #1934C2semidirect
8. #1933C2semidirect
10. #1894C2semidirect
11. #1836C2semidirect
91. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×C4×M162
#28572
#28502
#19932
#19592
#19342
#19332
#18952
#18942
#18362