Finite Group Databasebeta

← #19726

Subgroups — #19726

Order 256.

362 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3226C211yesmaximal, characteristic, complemented
2#3210C211yesmaximal, characteristic, complemented
3#3210C211yesmaximal, characteristic, complemented
4#3206C211yesmaximal, characteristic, complemented
5#3184C211yesmaximal, characteristic
6#3165C211yesmaximal, characteristic, complemented
7#1723C211yesmaximal, characteristic, complemented
8#1722C211yesmaximal, characteristic, complemented
9#1720C211yesmaximal, characteristic, complemented
10#1719C211yesmaximal, characteristic, complemented
11#1715C211yesmaximal, characteristic, complemented
12#1714C211yesmaximal, characteristic, complemented
13#1707C211yesmaximal, characteristic, complemented
14#1706C211yesmaximal, characteristic, complemented
15#1453C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#501—22no—
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#500C2211yescharacteristic
24#500C2211yescharacteristic
25#498C2211yescharacteristic
26#498C2211yescharacteristic
27#498C2211yescharacteristic
28#494C2211yescharacteristic
29#494C2211yescharacteristic
30#493C2211yescharacteristic
31#492C2211yescharacteristic
32#492C2211yescharacteristic
33#492C2211yescharacteristic
34#492C2211yescharacteristic
35#487C2211yescharacteristic
36#487C2211yescharacteristic
37#487C2211yescharacteristic
38#487C2211yescharacteristic
39#486C2211yescharacteristic
40#486C2211yescharacteristic
41#486C2211yescharacteristic
42#486C2211yescharacteristic
43#486C2211yescharacteristic
44#486C2211yescharacteristic
45#480—22no—
46#474—22no—
47#469—22no—
48#464—22no—
49#464—22no—
50#461—22no—
51#452—22no—
52#452—22no—
53#448—22no—
54#442—22no—
55#431C2211yescharacteristic
56#431C2211yescharacteristic
57#428—22no—
58C4×M16C2211yescharacteristic
59#322C2211yescharacteristic
60#322C2211yescharacteristic
61#322C2211yescharacteristic
62C82C2211yescharacteristic
63C2×#41—21no—
64C2×#41—21no—
65C22×Q8—21no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×D8—22no—
73C2×M16D412yes—
74C4⋊Q8—21no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#125—21no—
86#124—21no—
87#124—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107—22no—
107#107—22no—
108#106—22no—
109#106—22no—
110#106—22no—
111C4⋊C8D412yes—
112C4⋊C8D412yes—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123C22⋊C8D412yes—
124C22⋊C8D412yes—
125#97D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132#41—82no—
133#41—82no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—41no—
141C2×Q8—82no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—41no—
149C2×D4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156Q16—82no—
157Q16—82no—
158SD16—82no—
159SD16—82no—
160SD16—82no—
161SD16—82no—
162D8—44no—
163M16—24no—
164C2×C8C2×D412yes—
165C2×C8C2×D412yes—
166C2×C8C2×D412yes—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C2×C8C2×D412yes—
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—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C42—41no—
192C42C2×D411yescharacteristic
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2411yescharacteristic
196C23—41no—
197C23—41no—
198C23#12711yescharacteristic
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205Q8—82no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—82no—
213D4—82no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#12711yescharacteristic
227C2×C4#12711yescharacteristic
228C2×C4#12711yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C8—22no—
233C8—22no—
234C8—22no—
235C8—22no—
236C8—22no—
237C8#13714yes—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22C2×#12711yescharacteristic
248C4—21no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4C2×#13712yes—
258C4#57612yes—
259C4#57712yes—
260C2—41no—
261C2—161no—
262C2—161no—
263C2#315211yescharacteristic
264C2#315411yescharacteristic
265C2#315511yescharacteristic
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. #3226C2semidirect
2. #3210C2semidirect
3. #3210C2semidirect
4. #3206C2semidirect
6. #3165C2semidirect
7. #1723C2semidirect
8. #1722C2semidirect
9. #1720C2semidirect
10. #1719C2semidirect
11. #1715C2semidirect
12. #1714C2semidirect
13. #1707C2semidirect
14. #1706C2semidirect
15. #1453C2semidirect

Maximal subgroups

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

subgroupindex
#32262
#32102
#32062
#31842
#31652
#17232
#17222
#17202
#17192
#17152
#17142
#17072
#17062
#14532