Finite Group Databasebeta

← #20303

Subgroups — #20303

Order 256.

350 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3332C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1706C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1685C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1619C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#556C2211yescharacteristic
18#550C2211yescharacteristic, complemented
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#500C2211yescharacteristic, complemented
22#496—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic, complemented
26#486C2211yescharacteristic
27#486C2211yescharacteristic, complemented
28#482—21no—
29#481—21no—
30#475C2211yescharacteristic, complemented
31#474C2211yescharacteristic
32#469—21no—
33#468—21no—
34#468—21no—
35#463—21no—
36#461C2211yescharacteristic
37#461C2211yescharacteristic, complemented
38#460—21no—
39#460C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#450—21no—
45#449—21no—
46#449—21no—
47#446C2211yescharacteristic, complemented
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic, complemented
54#418—21no—
55#417—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66C82C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C22×D4—21no—
70C22×D4—21no—
71AGL(1,ℤ/8)—44no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16D411yescharacteristic
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—41no—
84C2×D8D411yescharacteristic
85C2×M16—21no—
86C2×M16—21no—
87C2×M16—21no—
88C4⋊Q8C2311yescharacteristic
89C4⋊Q8C2311yescharacteristic
90#127—21no—
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#126—41no—
94#124—21no—
95#123—41no—
96C22⋊Q8—21no—
97C22⋊Q8—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#121—21no—
102#121—41no—
103#120—41no—
104C4×Q8—21no—
105C4×Q8C2311yescharacteristic
106C4×D4—21no—
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C4×D4C2311yescharacteristic
110#117—21no—
111C2×C4⋊C4—21no—
112#107—21no—
113#107—41no—
114#107D411yescharacteristic
115#106—21no—
116#106D411yescharacteristic
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
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#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C2×D411yescharacteristic
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×D4—41no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4C2×D411yescharacteristic
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176SD16—41no—
177SD16—42no—
178SD16—42no—
179SD16—42no—
180SD16—81no—
181SD16—81no—
182SD16—81no—
183SD16—81no—
184D8—22no—
185D8—42no—
186D8—82no—
187D8—81no—
188D8—81no—
189M16—42no—
190M16—81no—
191M16—81no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8C2×D411yescharacteristic
198C2×C8C2×D411yescharacteristic
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C42—21no—
221C42—41no—
222C42C2411yescharacteristic
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—82no—
229C23—82no—
230C23—161no—
231Q8—21no—
232Q8—21no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236Q8—41no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—81no—
252D4—81no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—82no—
257D4—161no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4C22×D411yescharacteristic
274C2×C4C22×D411yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C8—22no—
277C8—21no—
278C8—21no—
279C8—21no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—41no—
284C8AGL(1,ℤ/8)12yes—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—164no—
301C22D4211yescharacteristic
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4C2×AGL(1,ℤ/8)11yescharacteristic
311C4#57511yescharacteristic
312C4#57611yescharacteristic
313C4#57611yescharacteristic
314C2—81no—
315C2—81no—
316C2—81no—
317C2—161no—
318C2—161no—
319C2#327311yescharacteristic
320C2#327411yescharacteristic
321C2#329111yescharacteristic
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. #3397C2semidirect
2. #3332C2semidirect
3. #3324C2semidirect
4. #3272C2semidirect
5. #1706C2semidirect
6. #1701C2semidirect
7. #1701C2semidirect
8. #1686C2semidirect
9. #1685C2semidirect
10. #1650C2semidirect
11. #1647C2semidirect
12. #1621C2semidirect
13. #1619C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect
18. #550C22semidirect
21. #500C22semidirect
25. #492C22semidirect
27. #486C22semidirect
30. #475C22semidirect
37. #461C22semidirect
47. #446C22semidirect
53. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. C82C22semidirect

Maximal subgroups

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

subgroupindex
#33972
#33322
#33242
#32722
#17062
#17012
#16862
#16852
#16502
#16472
#16212
#16192
#15902
#15892