Finite Group Databasebeta

← #17621

Subgroups — #17621

Order 256.

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

normal subgroups only

This group has 495 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#3501C211yesmaximal, characteristic
2#2973C211yesmaximal, characteristic
3#2353C211yesmaximal, characteristic
4#1520C212yesmaximal, complemented
5#1515C212yesmaximal, complemented
6#1511C212yesmaximal, complemented
7#1508C212yesmaximal, complemented
8#1506C212yesmaximal, complemented
9C2×#328C212yesmaximal, complemented
10#554—24no—
11#554C414yescomplemented
12D4×Q8C414yescomplemented
13#544—21no—
14#543—21no—
15#543—21no—
16#533C2212yes—
17C2×C22⋊Q8C2212yes—
18C2×C22⋊Q8C2212yes—
19#518—21no—
20C2×C4×D4C2211yescharacteristic
21#435—24no—
22#432C2212yes—
23#407C2212yes—
24C2×C22⋊C8C2212yes—
25#388—21no—
26#387—21no—
27#386—21no—
28#383C2211yescharacteristic
29#381C2211yescharacteristic
30#380C2211yescharacteristic
31C4×C22⋊C4—21no—
32C4×C22⋊C4C2211yescharacteristic
33#376C2211yescharacteristic
34C2×#95C2211yescharacteristic
35#328C2214yes—
36#328C2214yes—
37#328C2214yes—
38#328C2214yes—
39C22×Q8C2×C412yes—
40C22×Q8C2×C412yescomplemented
41C22×D4D411yescharacteristic
42C23×C4—21no—
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C2×M16—22no—
46C22×C8—22no—
47C4⋊Q8—24no—
48C4⋊Q8C2×C412yescomplemented
49C4⋊Q8C2×C412yescomplemented
50#125—24no—
51#125—22no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8C2×C414yes—
62C22⋊Q8C2×C414yescomplemented
63C22⋊Q8C2314yes—
64C4×Q8—22no—
65C4×Q8C2×C414yescomplemented
66C4×D4—24no—
67C4×D4—24no—
68C4×D4C2×C414yes—
69C4×D4D414yes—
70#117—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2×C412yes—
75C2×C4⋊C4C2×C412yescomplemented
76C2×C4⋊C4D411yescharacteristic
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4D411yescharacteristic
83C2×C22⋊C4D411yescharacteristic
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C4⋊C8—22no—
88C22⋊C8—22no—
89C22⋊C8—22no—
90C22⋊C8C2314yes—
91#97—22no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97#95C2311yescharacteristic
98#95C2311yescharacteristic
99#95C2311yescharacteristic
100#95C2311yescharacteristic
101C24C2×D411yescharacteristic
102C24C2×D411yescharacteristic
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—82no—
107C2×Q8—82no—
108C2×Q8C22×C414yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4C22⋊C414yes—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4—81no—
125C22×C4C22×C412yes—
126C22×C4C22×C412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2411yescharacteristic
134M16—82no—
135C2×C8—24no—
136C2×C8—82no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—81no—
147C4⋊C4C22⋊C414yes—
148C4⋊C4C22×C414yes—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—81no—
155C22⋊C4C22⋊C414yes—
156C22⋊C4C22⋊C414yes—
157C42—24no—
158C42—42no—
159C42—81no—
160C42C22×C412yes—
161C42C2×D412yes—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23C2×C22⋊C412yes—
169C23C2×C22⋊C412yes—
170C23#13712yes—
171C23#13712yes—
172C23C22×D411yescharacteristic
173Q8—44no—
174Q8—82no—
175Q8—82no—
176Q8—82no—
177D4—28no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—84no—
207C2×C4C2×C22⋊C412yes—
208C2×C4C2×C22⋊C412yes—
209C2×C4C2×C22⋊C412yes—
210C2×C4C2×C22⋊C412yes—
211C2×C4C23×C411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C8—44no—
214C22—21no—
215C22—22no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22—81no—
224C22—81no—
225C22#41914yes—
226C22C22×C22⋊C411yescharacteristic
227C22C2×#13711yescharacteristic
228C22C2×#13711yescharacteristic
229C4—21no—
230C4—21no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—44no—
237C4—44no—
238C4—81no—
239C4—81no—
240C4—81no—
241C2—21no—
242C2—21no—
243C2—24no—
244C2—42no—
245C2#288311yescharacteristic
246C2C2×#41911yescharacteristic
247C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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
4. #1520C2semidirect
5. #1515C2semidirect
6. #1511C2semidirect
7. #1508C2semidirect
8. #1506C2semidirect
9. C2×#328C2semidirect
11. #554C4semidirect
12. D4×Q8C4semidirect
40. C22×Q8C2×C4semidirect
48. C4⋊Q8C2×C4semidirect
49. C4⋊Q8C2×C4semidirect
62. C22⋊Q8C2×C4semidirect
65. C4×Q8C2×C4semidirect
75. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35012
#29732
#23532
#15202
#15152
#15112
#15082
#15062
C2×#3282