Finite Group Databasebeta

← #19871

Subgroups — #19871

Order 256.

316 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3396C211yesmaximal, characteristic, complemented
3#3394C211yesmaximal, characteristic, complemented
4#3304C211yesmaximal, characteristic, complemented
5#1721C211yesmaximal, characteristic, complemented
6#1686C211yesmaximal, characteristic, complemented
7#1682C211yesmaximal, characteristic, complemented
8#1668C211yesmaximal, characteristic, complemented
9#1667C211yesmaximal, characteristic, complemented
10#1640C211yesmaximal, characteristic, complemented
11#1628C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1620C211yesmaximal, characteristic, complemented
14#1584C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#559C2211yescharacteristic
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20#496—21no—
21#495—21no—
22#495—21no—
23#493C2211yescharacteristic
24#493C2211yescharacteristic
25#493C2211yescharacteristic
26#492C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#482—21no—
30#475C2211yescharacteristic
31#465—21no—
32#464—21no—
33#463—21no—
34#463—21no—
35#463—21no—
36#463—21no—
37#462C2211yescharacteristic
38#460C2211yescharacteristic
39#460C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#459—21no—
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#442—21no—
50C4×Q16C2211yescharacteristic
51C4×SD16—21no—
52C4×SD16C2211yescharacteristic
53C4×D8—21no—
54C8×D4C2211yescharacteristic
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#332C2211yescharacteristic
59#331C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16D411yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—41no—
84C22×C8—21no—
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#127C2311yescharacteristic
90#126—41no—
91#124—21no—
92#124—21no—
93#124—21no—
94#124—21no—
95#124—21no—
96#123—41no—
97#123—41no—
98C22⋊Q8—21no—
99#121—21no—
100#121—41no—
101C4×Q8—21no—
102C4×Q8C2311yescharacteristic
103C4×Q8C2311yescharacteristic
104C4×Q8C2311yescharacteristic
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4C2311yescharacteristic
111#117—21no—
112#107D411yescharacteristic
113#106—21no—
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103D411yescharacteristic
125#103D411yescharacteristic
126#103D411yescharacteristic
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—41no—
136#102D411yescharacteristic
137C22⋊C8—21no—
138#97—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8C2×D411yescharacteristic
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
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—
165C22×C4—41no—
166Q16—41no—
167Q16—81no—
168SD16—22no—
169SD16—42no—
170SD16—81no—
171SD16—81no—
172SD16—81no—
173SD16—81no—
174SD16—81no—
175D8—42no—
176D8—42no—
177D8—81no—
178D8—81no—
179D8—81no—
180D8—81no—
181D8—82no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—81no—
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42C2411yescharacteristic
217C23—21no—
218C23—41no—
219C23—41no—
220C23—41no—
221Q8—21no—
222Q8—21no—
223Q8—21no—
224Q8—21no—
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—81no—
244D4—81no—
245D4—81no—
246D4—82no—
247D4—161no—
248C2×C4—21no—
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—82no—
259C2×C4—161no—
260C2×C4—161no—
261C2×C4C22×D411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8#13512yes—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22D4211yescharacteristic
286C4—21no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4C2×#13511yescharacteristic
296C4#57511yescharacteristic
297C4#57611yescharacteristic
298C4#57611yescharacteristic
299C2—81no—
300C2—161no—
301C2—161no—
302C2—161no—
303C2#326811yescharacteristic
304C2#328711yescharacteristic
305C2#328811yescharacteristic
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. #3396C2semidirect
3. #3394C2semidirect
4. #3304C2semidirect
5. #1721C2semidirect
6. #1686C2semidirect
7. #1682C2semidirect
8. #1668C2semidirect
9. #1667C2semidirect
10. #1640C2semidirect
11. #1628C2semidirect
12. #1621C2semidirect
13. #1620C2semidirect
14. #1584C2semidirect
15. SD16⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33962
#33942
#33042
#17212
#16862
#16822
#16682
#16672
#16402
#16282
#16212
#16202
#15842
SD16⋊C82