Finite Group Databasebeta

← #20282

Subgroups — #20282

Order 256.

331 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3396C211yesmaximal, characteristic, complemented
3#3373C211yesmaximal, characteristic, complemented
4#3278C211yesmaximal, characteristic, complemented
5#1721C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1688C211yesmaximal, characteristic, complemented
8#1687C211yesmaximal, characteristic, complemented
9#1666C211yesmaximal, characteristic, complemented
10#1640C211yesmaximal, characteristic, complemented
11#1627C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1620C211yesmaximal, characteristic, complemented
14#1593C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#559C2211yescharacteristic
18#559C2211yescharacteristic
19#550C2211yescharacteristic
20D4×Q8C2211yescharacteristic
21#496—21no—
22#495—21no—
23#493C2211yescharacteristic
24#493C2211yescharacteristic
25#493C2211yescharacteristic
26#492C2211yescharacteristic
27#489—21no—
28#486C2211yescharacteristic
29#486C2211yescharacteristic
30#476—21no—
31#475C2211yescharacteristic
32#468—21no—
33#463—21no—
34#463—21no—
35#463—21no—
36#463—21no—
37#462C2211yescharacteristic
38#460C2211yescharacteristic
39#460C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#459—21no—
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49C8×Q8C2211yescharacteristic
50#442—21no—
51#442—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16C2211yescharacteristic
54C4×D8—21no—
55#436C2211yescharacteristic
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#331C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69#137—44no—
70C2×Q16—21no—
71C2×Q16D411yescharacteristic
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D411yescharacteristic
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—21no—
84C2×D8—21no—
85C2×D8—41no—
86C2×M16—21no—
87C4⋊Q8—21no—
88C4⋊Q8C2311yescharacteristic
89#127—21no—
90#127C2311yescharacteristic
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#124—21no—
94#124—21no—
95#124—21no—
96#124—21no—
97#123—41no—
98#123—41no—
99C22⋊Q8—21no—
100C22⋊Q8—41no—
101#121—21no—
102#121—41no—
103C4×Q8C2311yescharacteristic
104C4×Q8C2311yescharacteristic
105C4×Q8C2311yescharacteristic
106C4×D4—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4C2311yescharacteristic
113#107D411yescharacteristic
114#106—21no—
115#106D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103D411yescharacteristic
126#103D411yescharacteristic
127#103D411yescharacteristic
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102—41no—
137#102D411yescharacteristic
138C22⋊C8—21no—
139#97—21no—
140#97—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—81no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4C2×D411yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171Q16—22no—
172Q16—42no—
173SD16—41no—
174SD16—42no—
175SD16—42no—
176SD16—81no—
177SD16—81no—
178SD16—81no—
179SD16—81no—
180D8—42no—
181D8—42no—
182D8—42no—
183D8—81no—
184D8—81no—
185D8—81no—
186D8—82no—
187M16—42no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42C2411yescharacteristic
220C23—21no—
221C23—41no—
222C23—41no—
223C23—41no—
224Q8—21no—
225Q8—21no—
226Q8—21no—
227Q8—21no—
228Q8—21no—
229Q8—21no—
230Q8—41no—
231Q8—41no—
232Q8—82no—
233D4—21no—
234D4—21no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—81no—
248D4—81no—
249D4—81no—
250D4—82no—
251D4—161no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—161no—
265C2×C4—161no—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C8—21no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8AGL(1,ℤ/8)12yes—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22D4211yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4C2×AGL(1,ℤ/8)11yescharacteristic
301C4#57511yescharacteristic
302C4#57611yescharacteristic
303C4#57711yescharacteristic
304C2—81no—
305C2—161no—
306C2—161no—
307C2—161no—
308C2#327111yescharacteristic
309C2#327411yescharacteristic
310C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3399C2semidirect
2. #3396C2semidirect
3. #3373C2semidirect
4. #3278C2semidirect
5. #1721C2semidirect
6. #1702C2semidirect
7. #1688C2semidirect
8. #1687C2semidirect
9. #1666C2semidirect
10. #1640C2semidirect
11. #1627C2semidirect
12. #1621C2semidirect
13. #1620C2semidirect
14. #1593C2semidirect
15. #1583C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33962
#33732
#32782
#17212
#17022
#16882
#16872
#16662
#16402
#16272
#16212
#16202
#15932
#15832