Finite Group Databasebeta

← #33096

Subgroups — #33096

Order 256.

460 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#2854C211yesmaximal, characteristic
2C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#1926C212yesmaximal, complemented
4#1925C212yesmaximal, complemented
5#1921C212yesmaximal, complemented
6#1920C212yesmaximal, complemented
7#1874C211yesmaximal, characteristic
8#1874C211yesmaximal, characteristic
9#1853C211yesmaximal, characteristic, complemented
10#1852C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic, complemented
12C2×#123—21no—
13C2×#121C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42—21no—
17C22×C42C2211yescharacteristic
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C22⋊C8C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24#390C2211yescharacteristic
25#390C2211yescharacteristic
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#388—21no—
29#388—21no—
30#386—21no—
31#380C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2212yes—
35C4×C4⋊C4C2212yes—
36C2×#95—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52#123—44no—
53#121—22no—
54#121—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—44no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#107—24no—
82#106—24no—
83#102—24no—
84#102—24no—
85C22⋊C8—22no—
86C22⋊C8—22no—
87#95—22no—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92C24—21no—
93C24C2×D411yescharacteristic
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—42no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4#4112yes—
126C22×C4#4112yes—
127C22×C4C2411yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4#4114yes—
145C4⋊C4#4114yes—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—44no—
159C42—44no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23#12312yes—
172C23#13512yes—
173C23AGL(1,ℤ/8)12yes—
174C23C22×D411yescharacteristic
175D4—42no—
176D4—42no—
177D4—84no—
178C2×C4—28no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—48no—
196C2×C4—48no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4#12312yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4#13712yes—
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C4C2×#4112yes—
217C2×C4C2×#4112yes—
218C8—42no—
219C8—42no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22#48214yes—
234C22C2×#12311yescharacteristic
235C22#52511yescharacteristic
236C22#52511yescharacteristic
237C22C2×#13511yescharacteristic
238C22C2×AGL(1,ℤ/8)11yescharacteristic
239C22C2×AGL(1,ℤ/8)11yescharacteristic
240C22C2×#13711yescharacteristic
241C4—22no—
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—44no—
247C4—44no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4#52912yes—
252C4#52912yes—
253C2—24no—
254C2—161no—
255C2#285311yescharacteristic
256C2C2×#48211yescharacteristic
257C2#309211yescharacteristic
258C2#310511yescharacteristic
259C2#310811yescharacteristic
260C2#310911yescharacteristic
261C2#311211yescharacteristic
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
2. C2×C4×C4⋊C4C2semidirect
3. #1926C2semidirect
4. #1925C2semidirect
5. #1921C2semidirect
6. #1920C2semidirect
9. #1853C2semidirect
10. #1852C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28542
C2×C4×C4⋊C42
#19262
#19252
#19212
#19202
#18742
#18532
#18522
#18172