Finite Group Databasebeta

← #19742

Subgroups — #19742

Order 256.

369 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3227C211yesmaximal, characteristic, complemented
2#3226C211yesmaximal, characteristic, complemented
3#3207C211yesmaximal, characteristic, complemented
4#3207C211yesmaximal, characteristic, complemented
5#3207C211yesmaximal, characteristic, complemented
6#3206C211yesmaximal, characteristic, complemented
7#1721C211yesmaximal, characteristic, complemented
8#1719C211yesmaximal, characteristic, complemented
9#1718C211yesmaximal, characteristic, complemented
10#1717C211yesmaximal, characteristic, complemented
11#1715C211yesmaximal, characteristic, complemented
12#1714C211yesmaximal, characteristic, complemented
13#1708C211yesmaximal, characteristic, complemented
14#1706C211yesmaximal, characteristic, complemented
15#1454C211yesmaximal, characteristic, complemented
16#532C2211yescharacteristic
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#493C2211yescharacteristic
25#493C2211yescharacteristic
26#493C2211yescharacteristic
27#492C2211yescharacteristic
28#492C2211yescharacteristic
29#492C2211yescharacteristic
30#492C2211yescharacteristic
31#492C2211yescharacteristic
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#481—22no—
45#480—22no—
46#476—22no—
47#474—22no—
48#463—22no—
49#463—22no—
50#463—22no—
51#461—22no—
52#449—22no—
53#449—22no—
54#449—22no—
55#448—22no—
56#431C2211yescharacteristic
57#431C2211yescharacteristic
58#431C2211yescharacteristic
59#322C2211yescharacteristic
60#322C2211yescharacteristic
61#322C2211yescharacteristic
62C82C2211yescharacteristic
63C2×#41—21no—
64C2×#41—21no—
65C2×#41—21no—
66C22×Q8—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#127C2311yescharacteristic
86#124—21no—
87#124—21no—
88#124—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#106—22no—
107#106—22no—
108C4⋊C8D412yes—
109C4⋊C8D412yes—
110C4⋊C8D412yes—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123C22⋊C8D412yes—
124C22⋊C8D412yes—
125C22⋊C8D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132#41—82no—
133#41—82no—
134#41—82no—
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—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161SD16—82no—
162SD16—82no—
163SD16—82no—
164SD16—82no—
165SD16—82no—
166D8—82no—
167D8—82no—
168D8—82no—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C2×C8C2×D412yes—
172C2×C8C2×D412yes—
173C2×C8C2×D412yes—
174C2×C8C2×D412yes—
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—21no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—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—
206Q8—82no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—82no—
217D4—82no—
218D4—82no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4#12711yescharacteristic
232C2×C4#12711yescharacteristic
233C2×C4#12711yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C8—22no—
238C8—22no—
239C8—22no—
240C8—22no—
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22C2×#12711yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4#57612yes—
265C4#57712yes—
266C4#57712yes—
267C2—41no—
268C2—161no—
269C2—161no—
270C2—161no—
271C2#315411yescharacteristic
272C2#315411yescharacteristic
273C2#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. #3227C2semidirect
2. #3226C2semidirect
3. #3207C2semidirect
4. #3207C2semidirect
5. #3207C2semidirect
6. #3206C2semidirect
7. #1721C2semidirect
8. #1719C2semidirect
9. #1718C2semidirect
10. #1717C2semidirect
11. #1715C2semidirect
12. #1714C2semidirect
13. #1708C2semidirect
14. #1706C2semidirect
15. #1454C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#32262
#32072
#32062
#17212
#17192
#17182
#17172
#17152
#17142
#17082
#17062
#14542