Finite Group Databasebeta

← #20309

Subgroups — #20309

Order 256.

360 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3274C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1706C211yesmaximal, characteristic, complemented
6#1693C211yesmaximal, characteristic, complemented
7#1687C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1685C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1628C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C2×AGL(1,ℤ/8)—21no—
18#556C2211yescharacteristic, complemented
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21D42C2211yescharacteristic
22#500C2211yescharacteristic, complemented
23#496—21no—
24#493C2211yescharacteristic
25#492C2211yescharacteristic
26#492C2211yescharacteristic, complemented
27#486C2211yescharacteristic
28#486C2211yescharacteristic, complemented
29#484—21no—
30#482—21no—
31#475C2211yescharacteristic, complemented
32#474C2211yescharacteristic
33#469—21no—
34#468—21no—
35#468—21no—
36#463—21no—
37#461C2211yescharacteristic
38#461C2211yescharacteristic
39#460—21no—
40#460C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic, complemented
44#450—21no—
45#449—21no—
46#449—21no—
47#447—21no—
48#446C2211yescharacteristic, complemented
49#440—21no—
50C4×SD16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#436C2211yescharacteristic, complemented
55#428—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—
72AGL(1,ℤ/8)—44no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16D411yescharacteristic
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—21no—
84C2×D8—21no—
85C2×D8—21no—
86C2×D8D411yescharacteristic
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—21no—
90C4⋊Q8C2311yescharacteristic
91C4⋊Q8C2311yescharacteristic
92#127—21no—
93#127C2311yescharacteristic
94#127C2311yescharacteristic
95#126—41no—
96#124—21no—
97#123—41no—
98C22⋊Q8—21no—
99C22⋊Q8—21no—
100#121—21no—
101#121—21no—
102#121—21no—
103#121—21no—
104#121—41no—
105#120—41no—
106C4×Q8—21no—
107C4×Q8C2311yescharacteristic
108C4×D4—21no—
109C4×D4C2311yescharacteristic
110C4×D4C2311yescharacteristic
111C4×D4C2311yescharacteristic
112#117—21no—
113C2×C4⋊C4—21no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107D411yescharacteristic
118#106—21no—
119#106D411yescharacteristic
120C4⋊C8—21no—
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102—41no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138#102D411yescharacteristic
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142#97—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C4×C8C2311yescharacteristic
146C24—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—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C2×D4C2×D411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177SD16—22no—
178SD16—42no—
179SD16—42no—
180SD16—42no—
181SD16—42no—
182SD16—81no—
183SD16—81no—
184SD16—81no—
185D8—22no—
186D8—42no—
187D8—42no—
188D8—42no—
189D8—81no—
190D8—81no—
191M16—42no—
192M16—42no—
193M16—42no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C4⋊C4C2×D411yescharacteristic
213C4⋊C4C2×D411yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C42—21no—
223C42—41no—
224C42C2411yescharacteristic
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—82no—
231C23—82no—
232C23—161no—
233Q8—21no—
234Q8—21no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—21no—
243D4—21no—
244D4—21no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—81no—
255D4—82no—
256D4—82no—
257D4—82no—
258D4—82no—
259D4—161no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C8—21no—
279C8—21no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—41no—
284C8—41no—
285C8AGL(1,ℤ/8)12yescomplemented
286C8AGL(1,ℤ/8)12yescomplemented
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—164no—
303C22D4211yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4C2×AGL(1,ℤ/8)11yescharacteristic
313C4C2×AGL(1,ℤ/8)11yescharacteristic
314C4#57611yescharacteristic
315C4#57711yescharacteristic
316C2—81no—
317C2—81no—
318C2—81no—
319C2—161no—
320C2—161no—
321C2#327211yescharacteristic
322C2#327411yescharacteristic
323C2#329011yescharacteristic
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. #3353C2semidirect
3. #3274C2semidirect
4. #3272C2semidirect
5. #1706C2semidirect
6. #1693C2semidirect
7. #1687C2semidirect
8. #1686C2semidirect
9. #1685C2semidirect
10. #1642C2semidirect
11. #1639C2semidirect
12. #1628C2semidirect
13. #1625C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect
18. #556C22semidirect
22. #500C22semidirect
26. #492C22semidirect
28. #486C22semidirect
31. #475C22semidirect
43. #459C22semidirect
48. #446C22semidirect
54. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. C82C22semidirect
285. C8AGL(1,ℤ/8)semidirect
286. C8AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#33972
#33532
#32742
#32722
#17062
#16932
#16872
#16862
#16852
#16422
#16392
#16282
#16252
#15902
#15892