Finite Group Databasebeta

← #19480

Subgroups — #19480

Order 256.

332 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3207C211yesmaximal, characteristic, complemented
2#3115C211yesmaximal, characteristic, complemented
3#3035C211yesmaximal, characteristic, complemented
4#2990C211yesmaximal, characteristic, complemented
5#1702C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1688C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1679C211yesmaximal, characteristic, complemented
11#1667C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13#1573C211yesmaximal, characteristic, complemented
14#1557C211yesmaximal, characteristic, complemented
15#1547C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19C4×#41C2211yescharacteristic
20#493C2211yescharacteristic
21#492C2211yescharacteristic
22#487C2211yescharacteristic
23#486C2211yescharacteristic
24#482—21no—
25#475C2211yescharacteristic
26#474C2211yescharacteristic
27#464—21no—
28#463—21no—
29#463—21no—
30#463—21no—
31#462C2211yescharacteristic
32#461C2211yescharacteristic
33#460C2211yescharacteristic
34#460C2211yescharacteristic
35#459C2211yescharacteristic
36#459C2211yescharacteristic
37#449—21no—
38#449—21no—
39#449—21no—
40#446C2211yescharacteristic
41#446C2211yescharacteristic
42#442—21no—
43#440—21no—
44#435—21no—
45#435—21no—
46C8×D4C2211yescharacteristic
47C8×D4C2211yescharacteristic
48#432—21no—
49#432C2211yescharacteristic
50#431C2211yescharacteristic
51#423C2211yescharacteristic
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2211yescharacteristic
54#416—21no—
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#331C2211yescharacteristic
64#331C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41D411yescharacteristic
70#135—44no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×M16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#124—21no—
90#124—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95C4×Q8D411yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4D411yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106#117D411yescharacteristic
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109#107—21no—
110#106—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8D411yescharacteristic
116C4⋊C8D411yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8D411yescharacteristic
139C22⋊C8D411yescharacteristic
140#97—21no—
141#97—21no—
142#97—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—22no—
146#41—41no—
147#41—42no—
148#41—42no—
149#41—81no—
150#41—81no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C2×D411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4#4111yescharacteristic
172Q16—42no—
173SD16—42no—
174SD16—42no—
175SD16—42no—
176SD16—81no—
177SD16—81no—
178D8—42no—
179D8—42no—
180D8—81no—
181D8—81no—
182M16—81no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—42no—
190C2×C8—81no—
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⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42C2×D411yescharacteristic
215C42C2×D411yescharacteristic
216C42#4111yescharacteristic
217C42C2411yescharacteristic
218C23—21no—
219C23—41no—
220C23—41no—
221C23#12111yescharacteristic
222Q8—21no—
223Q8—21no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228D4—21no—
229D4—42no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—81no—
239D4—81no—
240D4—82no—
241D4—82no—
242D4AGL(1,ℤ/8)12yes—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4#12111yescharacteristic
260C2×C4#12111yescharacteristic
261C2×C4#12111yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4C22×D411yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C8—21no—
266C8—21no—
267C8—21no—
268C8—21no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—41no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—21no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22C2×#12111yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4C2×AGL(1,ℤ/8)11yescharacteristic
298C4#57511yescharacteristic
299C4#57611yescharacteristic
300C4#57711yescharacteristic
301C2—41no—
302C2—42no—
303C2—161no—
304C2—161no—
305C2#304511yescharacteristic
306C2#305511yescharacteristic
307C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3207C2semidirect
2. #3115C2semidirect
3. #3035C2semidirect
4. #2990C2semidirect
5. #1702C2semidirect
6. #1700C2semidirect
7. #1688C2semidirect
8. #1686C2semidirect
9. #1681C2semidirect
10. #1679C2semidirect
11. #1667C2semidirect
12. #1665C2semidirect
13. #1573C2semidirect
14. #1557C2semidirect
15. #1547C2semidirect

Maximal subgroups

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

subgroupindex
#32072
#31152
#30352
#29902
#17022
#17002
#16882
#16862
#16812
#16792
#16672
#16652
#15732
#15572
#15472