Finite Group Databasebeta

← #20071

Subgroups — #20071

Order 256.

319 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3372C211yesmaximal, characteristic, complemented
2#3368C211yesmaximal, characteristic, complemented
3#3335C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic
5#1714C211yesmaximal, characteristic, complemented
6#1692C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1648C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1585C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#557C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#492C2211yescharacteristic
24#492C2211yescharacteristic
25#491—21no—
26#490—21no—
27#487C2211yescharacteristic
28#486C2211yescharacteristic
29#483—21no—
30#481—21no—
31#479—21no—
32#478—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#465—21no—
38#465—21no—
39#462C2211yescharacteristic
40#461—21no—
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#449—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49#440—21no—
50#440—21no—
51C4×SD16C2211yescharacteristic
52C4×SD16C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C8×D4C2211yescharacteristic
55C2×#106—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—41no—
76C2×SD16D411yescharacteristic
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#125—21no—
88#125—21no—
89#125—41no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#121—41no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—41no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106D411yescharacteristic
115#106D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
129#103D411yescharacteristic
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
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—21no—
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—42no—
168SD16—22no—
169SD16—41no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—81no—
174SD16—82no—
175D8—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—41no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—81no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—81no—
234D4—82no—
235D4—82no—
236D4—161no—
237D4AGL(1,ℤ/8)12yes—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—161no—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4H(5,𝔽2)11yescharacteristic
257C8—22no—
258C8—21no—
259C8—21no—
260C8—21no—
261C8—21no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C22—22no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22D4211yescharacteristic
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4C2×AGL(1,ℤ/8)11yescharacteristic
287C4#57511yescharacteristic
288C4#57711yescharacteristic
289C4#57811yescharacteristic
290C2—42no—
291C2—81no—
292C2—161no—
293C2#326511yescharacteristic
294C2#326911yescharacteristic
295C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3284C2
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. #3372C2semidirect
2. #3368C2semidirect
3. #3335C2semidirect
5. #1714C2semidirect
6. #1692C2semidirect
7. #1686C2semidirect
8. #1683C2semidirect
9. #1681C2semidirect
10. #1648C2semidirect
11. #1647C2semidirect
12. #1631C2semidirect
13. #1622C2semidirect
14. #1585C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33722
#33682
#33352
#32842
#17142
#16922
#16862
#16832
#16812
#16482
#16472
#16312
#16222
#15852