Finite Group Databasebeta

← #27684

Subgroups — #27684

Order 256.

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

normal subgroups only

This group has 441 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
1C4×#121C211yesmaximal, characteristic
2C2×#382C211yesmaximal, characteristic, complemented
3#2101C212yesmaximal, complemented
4#2092C212yesmaximal, complemented
5#1949C212yesmaximal, complemented
6#1874C211yesmaximal, characteristic
7#1874C211yesmaximal, characteristic
8#1853C211yesmaximal, characteristic, complemented
9#1852C211yesmaximal, characteristic, complemented
10#1804C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×#121C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25#390C2211yescharacteristic
26#389C2211yescharacteristic
27#388—21no—
28#386—21no—
29#382—21no—
30#382—21no—
31#382C2212yes—
32#382C2212yes—
33#380C2211yescharacteristic
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36C2×#95—21no—
37C2×#95—21no—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53#121—22no—
54#121Q814yes—
55C4×D4—24no—
56C4×D4—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—24no—
73C2×C42—21no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80C4⋊C8—24no—
81#102—42no—
82#102—42no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92C24—21no—
93C24C2×D411yescharacteristic
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—22no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4C2×Q812yes—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×Q812yes—
129C22×C4#4112yes—
130C22×C4C2411yescharacteristic
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4C2×Q812yes—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—42no—
157C42#4114yes—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—22no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23C22⋊Q812yes—
170C23#13512yes—
171C23AGL(1,ℤ/8)12yes—
172C23C22×D411yescharacteristic
173D4—24no—
174D4—48no—
175D4—42no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—84no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4C22⋊Q812yes—
211C2×C4#12312yes—
212C2×C4#12312yes—
213C2×C4C22×Q811yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4112yes—
216C2×C4H(5,𝔽2)12yes—
217C8—42no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22#47814yes—
233C22C2×C22⋊Q811yescharacteristic
234C22C2×#12311yescharacteristic
235C22#53511yescharacteristic
236C22C2×#13511yescharacteristic
237C22C2×AGL(1,ℤ/8)11yescharacteristic
238C22#57511yescharacteristic
239C22#57511yescharacteristic
240C4—22no—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—81no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4#55312yes—
251C4#55412yes—
252C2—24no—
253C2—82no—
254C2#248711yescharacteristic
255C2C2×#47811yescharacteristic
256C2#308011yescharacteristic
257C2#309111yescharacteristic
258C2#309411yescharacteristic
259C2#321111yescharacteristic
260C2#321311yescharacteristic
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×#382C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #1949C2semidirect
8. #1853C2semidirect
9. #1852C2semidirect
10. #1804C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
C2×#3822
#21012
#20922
#19492
#18742
#18532
#18522
#18042
#17972