Finite Group Databasebeta

← #32987

Subgroups — #32987

Order 256.

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

normal subgroups only

This group has 679 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×#482C211yesmaximal, characteristic, complemented
4#2615C211yesmaximal, characteristic
5#1875C212yesmaximal, complemented
6#1874C212yesmaximal, complemented
7#1854C212yesmaximal, complemented
8#1853C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1780C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C2×#121C2211yescharacteristic
14C2×#121C2212yes—
15#520—24no—
16#520—24no—
17C2×C4×Q8—21no—
18C2×C4×D4—22no—
19C2×C4×D4—21no—
20C2×#117C2212yes—
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2212yes—
23C22×C4⋊C4C2211yescharacteristic
24#482—24no—
25#481—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36#393—21no—
37#392—21no—
38#392—22no—
39#380C2211yescharacteristic
40#380C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—24no—
49C22×Q8D411yescharacteristic
50C22×D4—21no—
51C22×D4—22no—
52C22×D4D411yescharacteristic
53C22×D4D412yes—
54C22×D4C2311yescharacteristic
55C23×C4—22no—
56C23×C4—21no—
57C23×C4C2311yescharacteristic
58C23×C4C2312yes—
59C22×C8C2311yescharacteristic
60C22×C8C2312yes—
61C22×C8C2311yescharacteristic
62#121—22no—
63#121—22no—
64#121—24no—
65#121—24no—
66C4×Q8—28no—
67C4×D4—216no—
68C4×D4—28no—
69#117—22no—
70#117—24no—
71#117—24no—
72#117D414yescomplemented
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4D414yescomplemented
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C42—22no—
91C2×C42—21no—
92#107—22no—
93#106—24no—
94#106—22no—
95#102—24no—
96#102—22no—
97#102—22no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100C22⋊C8—24no—
101C22⋊C8—24no—
102#95—21no—
103#95—22no—
104C24—21no—
105C24C2×D411yescharacteristic
106C24C2×D412yes—
107#41—28no—
108#41—28no—
109#41—216no—
110C2×Q8—22no—
111C2×Q8—24no—
112C2×D4—24no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—24no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—42no—
122C2×D4—44no—
123C2×D4—82no—
124C2×D4—82no—
125C2×D4—82no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—28no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—81no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4—82no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4#4114yes—
148C22×C4#4114yes—
149C22×C4C2411yescharacteristic
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C4⋊C4—28no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D414yes—
168C4⋊C4C2×D412yes—
169C22⋊C4—28no—
170C22⋊C4—24no—
171C22⋊C4—42no—
172C22⋊C4—44no—
173C22⋊C4—81no—
174C22⋊C4—82no—
175C42—28no—
176C42—24no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—41no—
188C23—42no—
189C23—41no—
190C23—42no—
191C23—41no—
192C23#12012yes—
193C23#12314yes—
194C23C22×D411yescharacteristic
195Q8—28no—
196D4—28no—
197D4—216no—
198D4—42no—
199D4—41no—
200D4—41no—
201D4—88no—
202D4—84no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—84no—
227C2×C4—88no—
228C2×C4#12012yes—
229C2×C4#12314yes—
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C2×C4C2×#4112yes—
233C2×C4C2×#4112yes—
234C8—42no—
235C8—41no—
236C8—41no—
237C22—24no—
238C22—24no—
239C22—22no—
240C22—24no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—82no—
250C22—82no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22C2×#12011yescharacteristic
255C22C2×#12312yes—
256C22#57611yescharacteristic
257C22#57712yes—
258C22#57711yescharacteristic
259C4—22no—
260C4—24no—
261C4—42no—
262C4—44no—
263C4—42no—
264C4—82no—
265C4—81no—
266C4#52511yescharacteristic
267C4#52512yes—
268C4#52511yescharacteristic
269C2—44no—
270C2—42no—
271C2—161no—
272C2#284911yescharacteristic
273C2#301011yescharacteristic
274C2#301211yescharacteristic
275C2#309612yes—
276C2#309912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ 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
3. C2×#482C2semidirect
5. #1875C2semidirect
6. #1874C2semidirect
7. #1854C2semidirect
8. #1853C2semidirect
9. #1825C2semidirect
10. #1823C2semidirect
11. #1780C2semidirect
72. #117D4semidirect
78. C2×C4⋊C4D4semidirect

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×#4822
#26152
#18752
#18742
#18542
#18532
#18252
#18232
#17802