Finite Group Databasebeta

← #17630

Subgroups — #17630

Order 256.

533 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 183 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#3506C211yesmaximal, characteristic
2#2975C211yesmaximal, characteristic, complemented
3#2349C211yesmaximal, characteristic, complemented
4#1518C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1513C214yesmaximal, complemented
7#1504C214yesmaximal, complemented
8#1503C211yesmaximal, characteristic, complemented
9#1502C211yesmaximal, characteristic, complemented
10#559—22no—
11#556C412yescomplemented
12#555C412yescomplemented
13#553—22no—
14#550C412yescomplemented
15#547C412yescomplemented
16#538—22no—
17#535—21no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20#525C2214yes—
21#520—21no—
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#436—22no—
25#436—22no—
26#431C2211yescharacteristic
27#431C2211yescharacteristic
28#407C2214yes—
29#389—21no—
30#385—22no—
31#382C2211yescharacteristic
32#380C2212yes—
33C4×C4⋊C4C2211yescharacteristic
34C2×#95C2212yes—
35#328C2212yes—
36#328C2212yes—
37#328C2214yes—
38#327C2212yes—
39#327C2212yes—
40#327C2214yes—
41C2×#41—21no—
42C2×#41C2×C414yescomplemented
43C22×D4D411yescharacteristic
44C23×C4—22no—
45C23×C4C2312yes—
46C2×M16—24no—
47C4⋊Q8C2×C411yescharacteristic, complemented
48C4⋊Q8C2×C411yescharacteristic, complemented
49C4⋊Q8C2×C411yescharacteristic, complemented
50#127—22no—
51#127C2×C411yescharacteristic, complemented
52#126—24no—
53#125—21no—
54#124—22no—
55#124—24no—
56#124—21no—
57#123—24no—
58#123—24no—
59C22⋊Q8—21no—
60C22⋊Q8C2×C414yescomplemented
61C22⋊Q8C2312yes—
62#121—22no—
63#121—22no—
64#121—21no—
65#121—24no—
66#121C2×C414yescomplemented
67#121C2312yes—
68#120—24no—
69C4×Q8—22no—
70C4×Q8—21no—
71C4×Q8C2×C412yescomplemented
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—21no—
76C4×D4—44no—
77C4×D4C2×C412yes—
78C4×D4C2×C412yes—
79C4×D4C2×C412yescomplemented
80C4×D4D412yes—
81C4×D4D412yes—
82#117—21no—
83#117—22no—
84#117C2×C414yescomplemented
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4D411yescharacteristic
90C2×C22⋊C4—22no—
91C2×C22⋊C4D412yes—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94C4⋊C8—21no—
95C4⋊C8—21no—
96C22⋊C8—24no—
97C22⋊C8C2312yes—
98C22⋊C8C2312yes—
99C4×C8—21no—
100C4×C8—21no—
101#95—22no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2312yes—
105#95C2311yescharacteristic
106C24C2×D412yes—
107#41—42no—
108#41—42no—
109#41—84no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8C22×C412yes—
113C2×D4—22no—
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4C22⋊C412yes—
121C2×D4C22⋊C412yes—
122C2×D4C22×C412yes—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4C22×C414yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D414yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138M16—84no—
139C2×C8—22no—
140C2×C8—22no—
141C4⋊C4—21no—
142C4⋊C4—24no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—82no—
151C4⋊C4C22⋊C412yes—
152C4⋊C4C22⋊C412yes—
153C4⋊C4C22×C412yes—
154C4⋊C4C22×C412yes—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—22no—
158C22⋊C4—24no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—44no—
162C22⋊C4C22⋊C414yes—
163C22⋊C4C22⋊C414yes—
164C42—24no—
165C42—22no—
166C42—42no—
167C42C22×C411yescharacteristic
168C42C22×C411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C23—22no—
172C23—22no—
173C23—22no—
174C23—42no—
175C23—42no—
176C23C2×C22⋊C414yes—
177C23C22×D411yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—81no—
181D4—24no—
182D4—24no—
183D4—42no—
184D4—42no—
185D4—42no—
186D4—42no—
187D4—81no—
188D4—84no—
189D4—84no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—41no—
198C2×C4—44no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4C2×C22⋊C414yes—
211C2×C4C2×C22⋊C411yescharacteristic
212C2×C4C2×C22⋊C411yescharacteristic
213C2×C4C2×C22⋊C411yescharacteristic
214C2×C4C2×C22⋊C411yescharacteristic
215C2×C4AGL(1,ℤ/8)12yes—
216C2×C4#13712yes—
217C2×C4C23×C411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C8—42no—
220C8—42no—
221C22—21no—
222C22—21no—
223C22—24no—
224C22—24no—
225C22—24no—
226C22—44no—
227C22—44no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22C22×C22⋊C411yescharacteristic
232C22C2×AGL(1,ℤ/8)11yescharacteristic
233C22C2×#13711yescharacteristic
234C4—21no—
235C4—21no—
236C4—24no—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—81no—
244C4—82no—
245C4#41712yes—
246C4#41712yes—
247C2—21no—
248C2—21no—
249C2—44no—
250C2—82no—
251C2#288311yescharacteristic
252C2C2×#41711yescharacteristic
253C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3506C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2975C2semidirect
3. #2349C2semidirect
4. #1518C2semidirect
5. #1517C2semidirect
6. #1513C2semidirect
7. #1504C2semidirect
8. #1503C2semidirect
9. #1502C2semidirect
11. #556C4semidirect
12. #555C4semidirect
14. #550C4semidirect
15. #547C4semidirect
42. C2×#41C2×C4semidirect
47. C4⋊Q8C2×C4semidirect
48. C4⋊Q8C2×C4semidirect
49. C4⋊Q8C2×C4semidirect
51. #127C2×C4semidirect
60. C22⋊Q8C2×C4semidirect
66. #121C2×C4semidirect
71. C4×Q8C2×C4semidirect
79. C4×D4C2×C4semidirect
84. #117C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35062
#29752
#23492
#15182
#15172
#15132
#15042
#15032
#15022