Finite Group Databasebeta

← #20205

Subgroups — #20205

Order 256.

319 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3375C211yesmaximal, characteristic, complemented
2#3373C211yesmaximal, characteristic, complemented
3#3362C211yesmaximal, characteristic, complemented
4#3278C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1645C211yesmaximal, characteristic, complemented
11#1637C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic
13#1620C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1584C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#559C2211yescharacteristic
18Q82C2211yescharacteristic
19#550C2211yescharacteristic
20D4×Q8C2211yescharacteristic
21#493C2211yescharacteristic
22#492C2211yescharacteristic
23#489—21no—
24#489—21no—
25#487C2211yescharacteristic
26#486—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#486C2211yescharacteristic
30#477C2211yescharacteristic
31#476—21no—
32#476—21no—
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#474—21no—
36#468—21no—
37#464—21no—
38#463—21no—
39#462C2211yescharacteristic
40#460—21no—
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49C8×Q8C2211yescharacteristic
50#442—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16—21no—
54C4×SD16C2211yescharacteristic
55#436C2211yescharacteristic
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
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×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—41no—
80C2×M16—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—41no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127—21no—
88#127C2311yescharacteristic
89#127C2311yescharacteristic
90#124—21no—
91#124—21no—
92#123—41no—
93C22⋊Q8—21no—
94C22⋊Q8—41no—
95#121—21no—
96#121—41no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107#107—21no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106D411yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—41no—
123#103D411yescharacteristic
124#103D411yescharacteristic
125#103D411yescharacteristic
126#102—21no—
127#102—21no—
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—21no—
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—21no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8C2×D411yescharacteristic
154C2×Q8C2×D411yescharacteristic
155C2×Q8C2×D411yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169Q16—22no—
170Q16—42no—
171SD16—41no—
172SD16—42no—
173SD16—42no—
174SD16—42no—
175SD16—42no—
176SD16—81no—
177D8—42no—
178D8—82no—
179M16—42no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—41no—
216C23—41no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—82no—
226Q8—161no—
227Q8AGL(1,ℤ/8)12yes—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—82no—
239D4—81no—
240D4—81no—
241D4—161no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—161no—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C8—21no—
260C8—21no—
261C8—21no—
262C8—21no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22D4211yescharacteristic
278C4—22no—
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—81no—
287C4—81no—
288C4C2×AGL(1,ℤ/8)11yescharacteristic
289C4#57511yescharacteristic
290C4#57711yescharacteristic
291C4#57811yescharacteristic
292C2—81no—
293C2—161no—
294C2—161no—
295C2#326511yescharacteristic
296C2#327711yescharacteristic
297C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#1621C2
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. #3375C2semidirect
2. #3373C2semidirect
3. #3362C2semidirect
4. #3278C2semidirect
5. #1717C2semidirect
6. #1698C2semidirect
7. #1697C2semidirect
8. #1686C2semidirect
9. #1682C2semidirect
10. #1645C2semidirect
11. #1637C2semidirect
13. #1620C2semidirect
14. #1592C2semidirect
15. #1584C2semidirect

Maximal subgroups

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

subgroupindex
#33752
#33732
#33622
#32782
#17172
#16982
#16972
#16862
#16822
#16452
#16372
#16212
#16202
#15922
#15842