Finite Group Databasebeta

← #29237

Subgroups — #29237

Order 256.

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

normal subgroups only

This group has 463 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×#389C211yesmaximal, characteristic, complemented
3#2087C212yesmaximal, complemented
4#2065C212yesmaximal, complemented
5#1919C212yesmaximal, complemented
6#1874C211yesmaximal, characteristic
7#1874C211yesmaximal, characteristic
8#1862C212yesmaximal, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1852C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×#121C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4—21no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26#390C2211yescharacteristic
27#389—21no—
28#389—21no—
29#389C2212yes—
30#389C2211yescharacteristic
31#389C2212yes—
32#388—21no—
33#386—21no—
34#380C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—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—42no—
56C4×D4—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42—21no—
76C2×C42—42no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#107—42no—
82#106—24no—
83C4⋊C8—42no—
84#102—24no—
85#102—42no—
86C22⋊C8—22no—
87C22⋊C8—22no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—21no—
98C2×D4—22no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4C2×Q812yes—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×Q812yes—
130C22×C4#4112yes—
131C22×C4C2411yescharacteristic
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4C2×Q812yes—
150C4⋊C4#4114yes—
151C22⋊C4—24no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C42—22no—
157C42—22no—
158C42—41no—
159C42—41no—
160C42—42no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23C22⋊Q812yes—
173C23#13512yes—
174C23AGL(1,ℤ/8)12yes—
175C23C22×D411yescharacteristic
176D4—24no—
177D4—48no—
178D4—42no—
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—22no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—48no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4C22⋊Q812yes—
214C2×C4AGL(1,ℤ/8)12yes—
215C2×C4#13712yes—
216C2×C4C22×Q811yescharacteristic
217C2×C4C2×#4111yescharacteristic
218C2×C4C2×#4112yes—
219C2×C4H(5,𝔽2)12yes—
220C8—42no—
221C8—42no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#47814yes—
236C22C2×C22⋊Q811yescharacteristic
237C22#52511yescharacteristic
238C22#53411yescharacteristic
239C22C2×#13511yescharacteristic
240C22C2×AGL(1,ℤ/8)11yescharacteristic
241C22C2×AGL(1,ℤ/8)11yescharacteristic
242C22C2×#13711yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—42no—
247C4—44no—
248C4—41no—
249C4—41no—
250C4—81no—
251C4—81no—
252C4—82no—
253C4#55312yes—
254C4#55412yes—
255C2—24no—
256C2—82no—
257C2#257911yescharacteristic
258C2C2×#47811yescharacteristic
259C2#307911yescharacteristic
260C2#310511yescharacteristic
261C2#311211yescharacteristic
262C2#319011yescharacteristic
263C2#319211yescharacteristic
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×#389C2semidirect
3. #2087C2semidirect
4. #2065C2semidirect
5. #1919C2semidirect
8. #1862C2semidirect
9. #1853C2semidirect
10. #1852C2semidirect
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×#3892
#20872
#20652
#19192
#18742
#18622
#18532
#18522
#17972