Finite Group Databasebeta

← #20158

Subgroups — #20158

Order 256.

377 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3354C211yesmaximal, characteristic, complemented
2#3325C211yesmaximal, characteristic, complemented
3D4×D8C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1687C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1629C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C22×D8—21no—
18#555C2211yescharacteristic, complemented
19#547C2211yescharacteristic, complemented
20D42C2211yescharacteristic
21D42C2211yescharacteristic
22#498C2211yescharacteristic, complemented
23#493C2211yescharacteristic
24#492C2211yescharacteristic, complemented
25#492C2211yescharacteristic, complemented
26#486C2211yescharacteristic, complemented
27#486C2211yescharacteristic, complemented
28#482—21no—
29#482—21no—
30#481—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic, complemented
33#474C2211yescharacteristic, complemented
34#469—21no—
35#469—21no—
36#468—21no—
37#466—21no—
38#461C2211yescharacteristic
39#459C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic, complemented
42#459C2211yescharacteristic, complemented
43#450—21no—
44#449—21no—
45#447—21no—
46#447—21no—
47#447—21no—
48C4×D8C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic, complemented
51#436C2211yescharacteristic, complemented
52#436C2211yescharacteristic, complemented
53C8×D4C2211yescharacteristic
54#428—21no—
55#418—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72AGL(1,ℤ/8)—44no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—41no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—44no—
84C2×D8D411yescharacteristic
85C2×D8D411yescharacteristic
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—21no—
89C22×C8—21no—
90C4⋊Q8C2311yescharacteristic
91C4⋊Q8C2311yescharacteristic
92#127C2311yescharacteristic
93#127C2311yescharacteristic
94#124—41no—
95#123—41no—
96C22⋊Q8—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#121—21no—
102#121—21no—
103#121—21no—
104#120—41no—
105#120—41no—
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C4×D4C2311yescharacteristic
110#117—21no—
111#117—21no—
112C2×C4⋊C4—21no—
113#107—21no—
114#107D411yescharacteristic
115#107D411yescharacteristic
116#106—21no—
117#106—21no—
118#106—21no—
119#106—21no—
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#102—21no—
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—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138#102D411yescharacteristic
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C4×C8C2311yescharacteristic
146C24—41no—
147C24—41no—
148#41—42no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4C2×D411yescharacteristic
173C2×D4C2×D411yescharacteristic
174C2×D4C2×D411yescharacteristic
175C2×D4C2×D411yescharacteristic
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181SD16—42no—
182SD16—42no—
183SD16—81no—
184SD16—82no—
185D8—22no—
186D8—22no—
187D8—42no—
188D8—42no—
189D8—42no—
190D8—42no—
191D8—81no—
192D8—81no—
193M16—42no—
194M16—42no—
195M16—81no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—42no—
202C2×C8C2×D411yescharacteristic
203C2×C8C2×D411yescharacteristic
204C2×C8C2×D411yescharacteristic
205C2×C8C2×D411yescharacteristic
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—41no—
211C4⋊C4C2×D411yescharacteristic
212C4⋊C4C2×D411yescharacteristic
213C4⋊C4C2×D411yescharacteristic
214C4⋊C4C2×D411yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C42—41no—
225C42C2411yescharacteristic
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—82no—
236C23—161no—
237C23—161no—
238Q8—41no—
239Q8—41no—
240Q8—81no—
241D4—21no—
242D4—21no—
243D4—21no—
244D4—21no—
245D4—21no—
246D4—21no—
247D4—21no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—81no—
257D4—81no—
258D4—82no—
259D4—82no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4—161no—
264D4AGL(1,ℤ/8)12yes—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—82no—
280C2×C4C22×D411yescharacteristic
281C2×C4C22×D411yescharacteristic
282C2×C4H(5,𝔽2)11yescharacteristic
283C8—21no—
284C8—21no—
285C8—21no—
286C8—41no—
287C8—41no—
288C8—41no—
289C8—41no—
290C8—41no—
291C8AGL(1,ℤ/8)12yescomplemented
292C22—22no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—164no—
311C22—164no—
312C22D4211yescharacteristic
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4C2×AGL(1,ℤ/8)11yescharacteristic
321C4C2×AGL(1,ℤ/8)11yescharacteristic
322C4#57611yescharacteristic
323C4#57711yescharacteristic
324C2—42no—
325C2—81no—
326C2—81no—
327C2—81no—
328C2—161no—
329C2—161no—
330C2#326411yescharacteristic
331C2#327211yescharacteristic
332C2#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. #3354C2semidirect
2. #3325C2semidirect
3. D4×D8C2semidirect
4. #3272C2semidirect
5. #1715C2semidirect
6. #1697C2semidirect
7. #1687C2semidirect
8. #1685C2semidirect
9. #1671C2semidirect
10. #1639C2semidirect
11. #1629C2semidirect
12. #1625C2semidirect
13. #1624C2semidirect
14. #1594C2semidirect
15. #1581C2semidirect
18. #555C22semidirect
19. #547C22semidirect
22. #498C22semidirect
24. #492C22semidirect
25. #492C22semidirect
26. #486C22semidirect
27. #486C22semidirect
32. #476C22semidirect
33. #474C22semidirect
41. #459C22semidirect
42. #459C22semidirect
50. #436C22semidirect
51. #436C22semidirect
52. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #331C22semidirect
61. #331C22semidirect
63. D4⋊C8C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect
291. 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
#33542
#33252
D4×D82
#32722
#17152
#16972
#16872
#16852
#16712
#16392
#16292
#16252
#16242
#15942
#15812