Finite Group Databasebeta

← #20079

Subgroups — #20079

Order 256.

332 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3375C211yesmaximal, characteristic, complemented
3#3362C211yesmaximal, characteristic, complemented
4#3287C211yesmaximal, characteristic, complemented
5#1721C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1688C211yesmaximal, characteristic, complemented
8#1677C211yesmaximal, characteristic, complemented
9#1668C211yesmaximal, characteristic, complemented
10#1631C211yesmaximal, characteristic, complemented
11#1623C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1620C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#559C2211yescharacteristic
18Q82C2211yescharacteristic
19#550C2211yescharacteristic
20#550C2211yescharacteristic
21#496—21no—
22#493C2211yescharacteristic
23#493C2211yescharacteristic
24#493C2211yescharacteristic
25#492C2211yescharacteristic
26#489—21no—
27#486—21no—
28#486C2211yescharacteristic
29#486C2211yescharacteristic
30#477C2211yescharacteristic
31#476—21no—
32#476C2211yescharacteristic
33#474—21no—
34#466—21no—
35#464—21no—
36#463—21no—
37#460—21no—
38#460C2211yescharacteristic
39#460C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#459—21no—
44#459C2211yescharacteristic
45#449—21no—
46#449—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49C8×Q8C2211yescharacteristic
50#442—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16—21no—
54C4×SD16C2211yescharacteristic
55C8×D4C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#331C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16D411yescharacteristic
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—41no—
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—41no—
84C22×C8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—41no—
88C4⋊Q8C2311yescharacteristic
89#127—21no—
90#127—21no—
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#127C2311yescharacteristic
94#124—21no—
95#124—21no—
96#123—41no—
97#121—21no—
98#121—21no—
99#121—41no—
100#121—41no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8C2311yescharacteristic
104C4×Q8C2311yescharacteristic
105C4×Q8C2311yescharacteristic
106C4×D4—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4C2311yescharacteristic
111#107—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#103D411yescharacteristic
124#103D411yescharacteristic
125#103D411yescharacteristic
126#102—21no—
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—21no—
136#102—21no—
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—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
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—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172Q16—22no—
173SD16—41no—
174SD16—42no—
175SD16—42no—
176SD16—42no—
177SD16—42no—
178SD16—82no—
179SD16—81no—
180D8—42no—
181D8—42no—
182D8—81no—
183D8—81no—
184D8—82no—
185D8—81no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—42no—
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—41no—
202C4⋊C4—41no—
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42C2411yescharacteristic
218C23—21no—
219C23—41no—
220C23—41no—
221C23—41no—
222Q8—21no—
223Q8—21no—
224Q8—21no—
225Q8—21no—
226Q8—21no—
227Q8—81no—
228Q8—161no—
229Q8AGL(1,ℤ/8)12yes—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—82no—
245D4—81no—
246D4—81no—
247D4—81no—
248D4—82no—
249D4—161no—
250D4—161no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—161no—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C8—22no—
268C8—21no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22D4211yescharacteristic
289C4—22no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4C2×AGL(1,ℤ/8)11yescharacteristic
299C4#57511yescharacteristic
300C4#57711yescharacteristic
301C4#57711yescharacteristic
302C2—81no—
303C2—161no—
304C2—161no—
305C2—161no—
306C2#326511yescharacteristic
307C2#327111yescharacteristic
308C2#329211yescharacteristic
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. #3375C2semidirect
3. #3362C2semidirect
4. #3287C2semidirect
5. #1721C2semidirect
6. #1698C2semidirect
7. #1688C2semidirect
8. #1677C2semidirect
9. #1668C2semidirect
10. #1631C2semidirect
11. #1623C2semidirect
12. #1621C2semidirect
13. #1620C2semidirect
14. #1587C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33752
#33622
#32872
#17212
#16982
#16882
#16772
#16682
#16312
#16232
#16212
#16202
#15872
#15852