Finite Group Databasebeta

← #19695

Subgroups — #19695

Order 256.

373 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3210C211yesmaximal, characteristic, complemented
2#3207C211yesmaximal, characteristic, complemented
3#3184C211yesmaximal, characteristic, complemented
4#3182C211yesmaximal, characteristic, complemented
5#3169C211yesmaximal, characteristic, complemented
6#3131C211yesmaximal, characteristic, complemented
7#1722C211yesmaximal, characteristic, complemented
8#1721C211yesmaximal, characteristic, complemented
9#1720C211yesmaximal, characteristic, complemented
10#1719C211yesmaximal, characteristic, complemented
11#1718C211yesmaximal, characteristic, complemented
12#1717C211yesmaximal, characteristic, complemented
13#1717C211yesmaximal, characteristic, complemented
14#1717C211yesmaximal, characteristic, complemented
15#1451C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19#532C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#494C2211yescharacteristic
24#493C2211yescharacteristic
25#493C2211yescharacteristic
26#493C2211yescharacteristic
27#492C2211yescharacteristic
28#492C2211yescharacteristic
29#492C2211yescharacteristic
30#488—22no—
31#487C2211yescharacteristic
32#487C2211yescharacteristic
33#487C2211yescharacteristic
34#487C2211yescharacteristic
35#487C2211yescharacteristic
36#486C2211yescharacteristic
37#486C2211yescharacteristic
38#486C2211yescharacteristic
39#486C2211yescharacteristic
40#486C2211yescharacteristic
41#486C2211yescharacteristic
42#486C2211yescharacteristic
43#486C2211yescharacteristic
44#486C2211yescharacteristic
45#469—22no—
46#468—22no—
47#466—22no—
48#464—22no—
49#463—22no—
50#452—22no—
51#449—22no—
52#442—22no—
53#440—22no—
54C4×D8—22no—
55#431C2211yescharacteristic
56#416—22no—
57C4×M16C2211yescharacteristic
58C2×C4×C8C2211yescharacteristic
59#322C2211yescharacteristic
60#322C2211yescharacteristic
61#322C2211yescharacteristic
62#322C2211yescharacteristic
63C2×#41—21no—
64C2×#41—21no—
65C2×#41—21no—
66C2×Q16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—22no—
73C2×M16D412yes—
74C22×C8D412yes—
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#127C2311yescharacteristic
81#127C2311yescharacteristic
82#127C2311yescharacteristic
83#125—21no—
84#124—21no—
85#124—21no—
86#124—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103#117—21no—
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—22no—
107#106—22no—
108C4⋊C8D412yes—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#102—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—
134#41—82no—
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—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158Q16—82no—
159SD16—44no—
160SD16—82no—
161SD16—82no—
162D8—44no—
163D8—44no—
164D8—82no—
165M16—42no—
166C2×C8—42no—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C2×C8C2×D412yes—
172C2×C8C2×D412yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C42—41no—
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2411yescharacteristic
197C23—41no—
198C23—41no—
199C23—41no—
200C23#12711yescharacteristic
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205Q8—42no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—82no—
216D4—82no—
217D4—82no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4#12711yescharacteristic
230C2×C4#12711yescharacteristic
231C2×C4#12711yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C8—22no—
236C8—22no—
237C8—22no—
238C8—22no—
239C8—22no—
240C8—22no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22C2×#12711yescharacteristic
254C4—21no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—81no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4#57512yes—
263C4#57512yes—
264C4#57612yes—
265C2—41no—
266C2—161no—
267C2—161no—
268C2—161no—
269C2#315011yescharacteristic
270C2#315311yescharacteristic
271C2#315511yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
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. #3210C2semidirect
2. #3207C2semidirect
3. #3184C2semidirect
4. #3182C2semidirect
5. #3169C2semidirect
6. #3131C2semidirect
7. #1722C2semidirect
8. #1721C2semidirect
9. #1720C2semidirect
10. #1719C2semidirect
11. #1718C2semidirect
12. #1717C2semidirect
13. #1717C2semidirect
14. #1717C2semidirect
15. #1451C2semidirect

Maximal subgroups

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

subgroupindex
#32102
#32072
#31842
#31822
#31692
#31312
#17222
#17212
#17202
#17192
#17182
#17172
#14512