Finite Group Databasebeta

← #20316

Subgroups — #20316

Order 256.

314 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3367C211yesmaximal, characteristic, complemented
2#3366C211yesmaximal, characteristic
3#3356C211yesmaximal, characteristic, complemented
4#3278C211yesmaximal, characteristic, complemented
5#1708C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1698C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1685C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1642C211yesmaximal, characteristic, complemented
12#1637C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1591C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20D4×Q8C2211yescharacteristic
21#498C2211yescharacteristic
22#492C2211yescharacteristic
23#490—21no—
24#489—21no—
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#485—21no—
30#483—21no—
31#478—21no—
32#477—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#470—21no—
38#468—21no—
39#463—21no—
40#462C2211yescharacteristic
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#459—21no—
44#459C2211yescharacteristic
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49#442—21no—
50#440—21no—
51C4×Q16C2211yescharacteristic
52C4×D8C2211yescharacteristic
53#436C2211yescharacteristic
54#436C2211yescharacteristic
55#428—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
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×D8—21no—
77C2×D8D411yescharacteristic
78C2×M16—21no—
79C2×M16—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#126—41no—
87#125—21no—
88#124—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94#121—21no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×Q8C2311yescharacteristic
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107D411yescharacteristic
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
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#103—41no—
127#103D411yescharacteristic
128#103D411yescharacteristic
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
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—81no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8C2×D411yescharacteristic
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
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—22no—
168Q16—42no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—81no—
173D8—41no—
174D8—42no—
175M16—42no—
176M16—42no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—21no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—82no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4—161no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—161no—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8AGL(1,ℤ/8)12yes—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22D4211yescharacteristic
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4C2×AGL(1,ℤ/8)11yescharacteristic
286C4#57511yescharacteristic
287C4#57711yescharacteristic
288C4#57811yescharacteristic
289C2—81no—
290C2—81no—
291C2—161no—
292C2#327211yescharacteristic
293C2#327711yescharacteristic
294C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3366C2
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. #3367C2semidirect
3. #3356C2semidirect
4. #3278C2semidirect
5. #1708C2semidirect
6. #1700C2semidirect
7. #1698C2semidirect
8. #1691C2semidirect
9. #1685C2semidirect
10. #1644C2semidirect
11. #1642C2semidirect
12. #1637C2semidirect
13. #1628C2semidirect
14. #1591C2semidirect
15. #1589C2semidirect

Maximal subgroups

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

subgroupindex
#33672
#33662
#33562
#32782
#17082
#17002
#16982
#16912
#16852
#16442
#16422
#16372
#16282
#15912
#15892