Finite Group Databasebeta

← #18923

Subgroups — #18923

Order 256.

561 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 129 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#3497C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic, complemented
4C2×#475C211yesmaximal, characteristic, complemented
5#1651C214yesmaximal, complemented
6#1642C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C2×2−1+4—21no—
11#550—21no—
12#550C2212yes—
13D4×Q8—21no—
14D4×Q8C2212yes—
15#548—22no—
16#548—22no—
17#537—22no—
18#537—21no—
19#532—22no—
20#532C2211yescharacteristic
21#532C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24#498C2212yes—
25#492—21no—
26#487—21no—
27#486C2212yes—
28#481—22no—
29#476—22no—
30#475—22no—
31#475C2214yes—
32#461C2214yes—
33#448—22no—
34#431C2211yescharacteristic
35#431C2211yescharacteristic
36C2×#106—21no—
37C2×C4⋊C8C2211yescharacteristic
38C2×#103—21no—
39#332C2212yes—
40#331C2212yes—
41Q8⋊C8C2214yes—
42D4⋊C8C2214yes—
432−1+4—44no—
442−1+4—44no—
45C2×#41—24no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41—21no—
49C2×#41—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—21no—
53C22×Q8D411yescharacteristic
54C2×SD16—24no—
55C22×C8—21no—
56C4⋊Q8—22no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8C2312yes—
62C4⋊Q8C2311yescharacteristic
63#127—22no—
64#127D411yescharacteristic
65#127C2311yescharacteristic
66#124—24no—
67#124—21no—
68#124—21no—
69C22⋊Q8—22no—
70C22⋊Q8—24no—
71C22⋊Q8—22no—
72C22⋊Q8—24no—
73#121—24no—
74#121—22no—
75#121—24no—
76#121—24no—
77#121—21no—
78#121—21no—
79#121—22no—
80#121D412yes—
81#121D412yes—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8D412yes—
85C4×Q8C2312yes—
86C4×D4—24no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4D412yes—
92C4×D4C2312yes—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D411yescharacteristic
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99#106—24no—
100#106—24no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8C2312yes—
104C4⋊C8C2312yes—
105#103—24no—
106#103—24no—
107#102—24no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110C4×C8—21no—
111C4×C8C2312yes—
112#41—416no—
113#41—44no—
114#41—48no—
115#41—44no—
116#41—42no—
117#41—42no—
118#41—42no—
119#41—42no—
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—44no—
127C2×Q8—48no—
128C2×Q8—44no—
129C2×Q8—41no—
130C2×Q8—82no—
131C2×Q8SD1614yescomplemented
132C2×Q8C2×D412yes—
133C2×D4—24no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×D412yes—
142C2×D4C2×D412yes—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153SD16—84no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—42no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C42—22no—
178C42—22no—
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C42C2×D411yescharacteristic
182C42C2411yescharacteristic
183C23—22no—
184C23—22no—
185C23#12011yescharacteristic
186Q8—22no—
187Q8—24no—
188Q8—48no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—44no—
193Q8—82no—
194Q8C2×SD1614yes—
195D4—24no—
196D4—42no—
197D4—48no—
198D4—44no—
199D4—44no—
200D4—41no—
201D4—41no—
202D4—42no—
203D4—42no—
204D4—81no—
205D4—84no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—48no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—82no—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4C2×SD1612yes—
225C2×C4AGL(1,ℤ/8)12yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C8—42no—
230C8—42no—
231C8—42no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22#45012yes—
241C22C2×#12011yescharacteristic
242C4—24no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4#45012yes—
251C4C22×SD1611yescharacteristic
252C4C2×AGL(1,ℤ/8)11yescharacteristic
253C4#57712yes—
254C2—22no—
255C2—82no—
256C2—82no—
257C2C2×#45011yescharacteristic
258C2#301011yescharacteristic
259C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3497C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3227C2semidirect
3. #3206C2semidirect
4. C2×#475C2semidirect
5. #1651C2semidirect
6. #1642C2semidirect
7. #1540C2semidirect
8. #1480C2semidirect
9. #1479C2semidirect
131. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#34972
#32272
#32062
C2×#4752
#16512
#16422
#15402
#14802
#14792