Finite Group Databasebeta

← #20195

Subgroups — #20195

Order 256.

361 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#3366C211yesmaximal, characteristic, complemented
2#3335C211yesmaximal, characteristic, complemented
3#3274C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1667C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1629C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C2×AGL(1,ℤ/8)—21no—
18#554C2211yescharacteristic, complemented
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21D42C2211yescharacteristic
22#498C2211yescharacteristic, complemented
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#490—21no—
27#486C2211yescharacteristic, complemented
28#486C2211yescharacteristic, complemented
29#483—21no—
30#481—21no—
31#478—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic, complemented
35#469—21no—
36#468—21no—
37#465—21no—
38#461C2211yescharacteristic
39#460C2211yescharacteristic
40#459—21no—
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic, complemented
44#450—21no—
45#449—21no—
46#449—21no—
47#447—21no—
48#446C2211yescharacteristic, complemented
49#442—21no—
50C4×SD16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54C8×D4C2211yescharacteristic, complemented
55C2×#106—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, 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—41no—
78C2×SD16D411yescharacteristic
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—21no—
84C2×D8D411yescharacteristic
85C2×M16—21no—
86C2×M16—21no—
87C22×C8—21no—
88C4⋊Q8C2311yescharacteristic
89C4⋊Q8C2311yescharacteristic
90#127—21no—
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#125—21no—
94#123—41no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97C22⋊Q8—41no—
98#121—21no—
99#121—21no—
100#121—21no—
101#121—21no—
102#121—41no—
103#120—41no—
104C4×Q8C2311yescharacteristic
105C4×D4—21no—
106C4×D4—21no—
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C4×D4C2311yescharacteristic
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106D411yescharacteristic
119C4⋊C8—21no—
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#103—21no—
126#103D411yescharacteristic
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—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—
177C22×C4—41no—
178SD16—22no—
179SD16—42no—
180SD16—42no—
181SD16—42no—
182SD16—81no—
183SD16—82no—
184D8—22no—
185D8—42no—
186D8—42no—
187D8—42no—
188D8—42no—
189D8—81no—
190M16—42no—
191M16—42no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—42no—
198C2×C8C2×D411yescharacteristic
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
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—
221C42—21no—
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—81no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—81no—
250D4—81no—
251D4—82no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4—161no—
256D4AGL(1,ℤ/8)12yescomplemented
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—161no—
273C2×C4C22×D411yescharacteristic
274C2×C4C22×D411yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C8—21no—
277C8—21no—
278C8—21no—
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—41no—
284C8AGL(1,ℤ/8)12yescomplemented
285C22—22no—
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
311C4C2×AGL(1,ℤ/8)11yescharacteristic
312C4#57711yescharacteristic
313C4#57711yescharacteristic
314C2—42no—
315C2—81no—
316C2—81no—
317C2—161no—
318C2—161no—
319C2#326511yescharacteristic
320C2#327211yescharacteristic
321C2#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. #3366C2semidirect
2. #3335C2semidirect
3. #3274C2semidirect
4. #3272C2semidirect
5. #1715C2semidirect
6. #1697C2semidirect
7. #1691C2semidirect
8. #1686C2semidirect
9. #1667C2semidirect
10. #1642C2semidirect
11. #1629C2semidirect
12. #1625C2semidirect
13. #1622C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect
18. #554C22semidirect
22. #498C22semidirect
27. #486C22semidirect
28. #486C22semidirect
34. #475C22semidirect
43. #459C22semidirect
48. #446C22semidirect
54. C8×D4C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #333C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect
256. D4AGL(1,ℤ/8)semidirect
284. 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
#33662
#33352
#32742
#32722
#17152
#16972
#16912
#16862
#16672
#16422
#16292
#16252
#16222
#15922
#15812