Finite Group Databasebeta

← #19650

Subgroups — #19650

Order 256.

360 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3228C211yesmaximal, characteristic, complemented
2#3210C211yesmaximal, characteristic, complemented
3#3185C211yesmaximal, characteristic, complemented
4#3184C211yesmaximal, characteristic
5#3167C211yesmaximal, characteristic, complemented
6#3131C211yesmaximal, characteristic, complemented
7#1720C211yesmaximal, characteristic, complemented
8#1719C211yesmaximal, characteristic, complemented
9#1716C211yesmaximal, characteristic, complemented
10#1715C211yesmaximal, characteristic, complemented
11#1711C211yesmaximal, characteristic, complemented
12#1709C211yesmaximal, characteristic, complemented
13#1708C211yesmaximal, characteristic, complemented
14#1707C211yesmaximal, characteristic, complemented
15C8×M16C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19#532C2211yescharacteristic
20#500C2211yescharacteristic
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#498C2211yescharacteristic
25#494C2211yescharacteristic
26#494C2211yescharacteristic
27#493C2211yescharacteristic
28#492C2211yescharacteristic
29#492C2211yescharacteristic
30#488—22no—
31#487C2211yescharacteristic
32#487C2211yescharacteristic
33#487C2211yescharacteristic
34#487C2211yescharacteristic
35#487C2211yescharacteristic
36#487C2211yescharacteristic
37#486C2211yescharacteristic
38#486C2211yescharacteristic
39#486C2211yescharacteristic
40#486C2211yescharacteristic
41#486C2211yescharacteristic
42#486C2211yescharacteristic
43#486C2211yescharacteristic
44#486C2211yescharacteristic
45#482—22no—
46#478—22no—
47#470—22no—
48#469—22no—
49#465—22no—
50#464—22no—
51#452—22no—
52#442—22no—
53#441—22no—
54C4×SD16—22no—
55#431C2211yescharacteristic
56#416—22no—
57C4×M16C2211yescharacteristic
58C2×C4×C8C2211yescharacteristic
59#322C2211yescharacteristic
60#322C2211yescharacteristic
61C82C2211yescharacteristic
62C82C2211yescharacteristic
63C2×#41—21no—
64C2×#41—21no—
65C2×Q16—22no—
66C2×Q16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×D8—22no—
70C2×M16D412yes—
71C22×C8D412yes—
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2311yescharacteristic
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78#127C2311yescharacteristic
79#127C2311yescharacteristic
80#125—21no—
81#125—21no—
82#124—21no—
83#124—21no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100#117—21no—
101#117—21no—
102C2×C42C2311yescharacteristic
103#107—22no—
104#106—22no—
105#106—22no—
106#106—22no—
107#106—22no—
108C4⋊C8D412yes—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123C22⋊C8D412yes—
124#97D412yes—
125C4×C8D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132#41—44no—
133#41—82no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—41no—
147C2×D4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153Q16—44no—
154Q16—82no—
155SD16—44no—
156SD16—82no—
157D8—44no—
158M16—24no—
159C2×C8—42no—
160C2×C8C2×D412yes—
161C2×C8C2×D412yes—
162C2×C8C2×D412yes—
163C2×C8C2×D412yes—
164C2×C8C2×D412yes—
165C2×C8C2×D412yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C42—41no—
189C42—41no—
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C42C2411yescharacteristic
194C23—41no—
195C23—41no—
196C23#12711yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—42no—
203D4—42no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—82no—
210D4—82no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4#12711yescharacteristic
223C2×C4#12711yescharacteristic
224C2×C4#12711yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C8—22no—
229C8—22no—
230C8—22no—
231C8—22no—
232C8—22no—
233C8#13514yes—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22C2×#12711yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—81no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4C2×#13512yes—
254C4#57512yes—
255C4#57712yes—
256C2—41no—
257C2—161no—
258C2—161no—
259C2#314711yescharacteristic
260C2#315311yescharacteristic
261C2#315411yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
prime radical#3184C2
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
1. #3228C2semidirect
2. #3210C2semidirect
3. #3185C2semidirect
5. #3167C2semidirect
6. #3131C2semidirect
7. #1720C2semidirect
8. #1719C2semidirect
9. #1716C2semidirect
10. #1715C2semidirect
11. #1711C2semidirect
12. #1709C2semidirect
13. #1708C2semidirect
14. #1707C2semidirect
15. C8×M16C2semidirect

Maximal subgroups

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

subgroupindex
#32282
#32102
#31852
#31842
#31672
#31312
#17202
#17192
#17162
#17152
#17112
#17092
#17082
#17072
C8×M162