Finite Group Databasebeta

← #20200

Subgroups — #20200

Order 256.

310 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3374C211yesmaximal, characteristic, complemented
2#3368C211yesmaximal, characteristic
3#3359C211yesmaximal, characteristic, complemented
4#3278C211yesmaximal, characteristic
5#1719C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1649C211yesmaximal, characteristic, complemented
11#1648C211yesmaximal, characteristic, complemented
12#1642C211yesmaximal, characteristic
13#1626C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1584C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#557C2211yescharacteristic
18#557C2211yescharacteristic
19#550C2211yescharacteristic
20D4×Q8C2211yescharacteristic
21#500C2211yescharacteristic
22#492C2211yescharacteristic
23#491—21no—
24#490—21no—
25#487—21no—
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#486C2211yescharacteristic
29#486C2211yescharacteristic
30#479—21no—
31#478—21no—
32#475—21no—
33#475C2211yescharacteristic
34#475C2211yescharacteristic
35#474C2211yescharacteristic
36#468—21no—
37#462—21no—
38#462C2211yescharacteristic
39#462C2211yescharacteristic
40#461—21no—
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#460C2211yescharacteristic
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49C8×Q8C2211yescharacteristic
50#441—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16—21no—
54C4×SD16C2211yescharacteristic
55#436C2211yescharacteristic
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
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×Q16—21no—
72C2×Q16D411yescharacteristic
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D411yescharacteristic
79C2×M16—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127C2311yescharacteristic
88#125—21no—
89#125—21no—
90#125—41no—
91#125—41no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4C2311yescharacteristic
106#107—21no—
107#107—41no—
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—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
129#103D411yescharacteristic
130#103D411yescharacteristic
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—21no—
149C2×Q8—21no—
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—41no—
160C2×D4—41no—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—22no—
166Q16—42no—
167Q16—42no—
168SD16—41no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—81no—
174M16—42no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—41no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—82no—
227Q8AGL(1,ℤ/8)12yes—
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—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22D4211yescharacteristic
270C4—22no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4C2×AGL(1,ℤ/8)11yescharacteristic
282C4#57511yescharacteristic
283C4#57711yescharacteristic
284C4#57811yescharacteristic
285C2—81no—
286C2—161no—
287C2#326511yescharacteristic
288C2#327711yescharacteristic
289C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#550C22
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. #3374C2semidirect
3. #3359C2semidirect
5. #1719C2semidirect
6. #1700C2semidirect
7. #1699C2semidirect
8. #1686C2semidirect
9. #1682C2semidirect
10. #1649C2semidirect
11. #1648C2semidirect
13. #1626C2semidirect
14. #1592C2semidirect
15. #1584C2semidirect

Maximal subgroups

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

subgroupindex
#33742
#33682
#33592
#32782
#17192
#17002
#16992
#16862
#16822
#16492
#16482
#16422
#16262
#15922
#15842