Finite Group Databasebeta

← #20070

Subgroups — #20070

Order 256.

333 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3373C211yesmaximal, characteristic, complemented
3#3298C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1681C211yesmaximal, characteristic, complemented
9#1667C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1640C211yesmaximal, characteristic, complemented
12#1627C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1585C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#559C2211yescharacteristic
18#550C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#547C2211yescharacteristic
21#496—21no—
22#493C2211yescharacteristic
23#492C2211yescharacteristic
24#489—21no—
25#487C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#483—21no—
30#476—21no—
31#475C2211yescharacteristic
32#475C2211yescharacteristic
33#465—21no—
34#465—21no—
35#463—21no—
36#463—21no—
37#462C2211yescharacteristic
38#461C2211yescharacteristic
39#460C2211yescharacteristic
40#459—21no—
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46#448—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49#442—21no—
50#442—21no—
51C4×SD16C2211yescharacteristic
52C4×SD16C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C8×D4C2211yescharacteristic
55C2×#103—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C22×Q8—21no—
70#135—44no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#124—21no—
92#124—21no—
93#123—41no—
94#123—41no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97C22⋊Q8—41no—
98#121—21no—
99#121—21no—
100#121—41no—
101C4×Q8C2311yescharacteristic
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C2×C4⋊C4—21no—
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103D411yescharacteristic
127#103D411yescharacteristic
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102—41no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×Q8—81no—
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×D4—41no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173Q16—42no—
174SD16—22no—
175SD16—41no—
176SD16—42no—
177SD16—81no—
178SD16—81no—
179SD16—81no—
180D8—42no—
181D8—42no—
182D8—42no—
183D8—81no—
184D8—81no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—42no—
191C2×C8—81no—
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223Q8—21no—
224Q8—21no—
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—82no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—81no—
244D4—81no—
245D4—82no—
246D4—82no—
247D4—161no—
248D4AGL(1,ℤ/8)12yes—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—161no—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22D4211yescharacteristic
290C4—21no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4C2×AGL(1,ℤ/8)11yescharacteristic
300C4#57511yescharacteristic
301C4#57611yescharacteristic
302C4#57711yescharacteristic
303C2—42no—
304C2—81no—
305C2—161no—
306C2—161no—
307C2#326511yescharacteristic
308C2#326811yescharacteristic
309C2#329011yescharacteristic
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. #3373C2semidirect
3. #3298C2semidirect
4. #3284C2semidirect
5. #1717C2semidirect
6. #1700C2semidirect
7. #1686C2semidirect
8. #1681C2semidirect
9. #1667C2semidirect
10. #1642C2semidirect
11. #1640C2semidirect
12. #1627C2semidirect
13. #1625C2semidirect
14. #1585C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33732
#32982
#32842
#17172
#17002
#16862
#16812
#16672
#16422
#16402
#16272
#16252
#15852