Finite Group Databasebeta

← #33855

Subgroups — #33855

Order 256.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3417C211yesmaximal, characteristic, complemented
2#3416C211yesmaximal, characteristic, complemented
3#3293C211yesmaximal, characteristic, complemented
4#3290C211yesmaximal, characteristic, complemented
5#2251C211yesmaximal, characteristic, complemented
6#2223C211yesmaximal, characteristic, complemented
7#2222C211yesmaximal, characteristic, complemented
8#2214C211yesmaximal, characteristic, complemented
9#2212C211yesmaximal, characteristic, complemented
10#2195C211yesmaximal, characteristic, complemented
11#2195C211yesmaximal, characteristic, complemented
12#2194C211yesmaximal, characteristic, complemented
13#2193C211yesmaximal, characteristic, complemented
14#2179C211yesmaximal, characteristic, complemented
15#2171C211yesmaximal, characteristic, complemented
16#578—21no—
17#578C2211yescharacteristic
18#578C2211yescharacteristic
19#578C2211yescharacteristic, complemented
20#577—21no—
21#576C2211yescharacteristic
22#510—21no—
23#510—21no—
24#510—21no—
25#510C2211yescharacteristic
26#510C2211yescharacteristic
27#510C2211yescharacteristic, complemented
28#509—21no—
29#509C2211yescharacteristic
30#508—21no—
31#508—21no—
32#508C2211yescharacteristic
33C2×Q32—21no—
34C2×Q32C2211yescharacteristic, complemented
35C2×Q32C2211yescharacteristic, complemented
36C2×SD32—21no—
37C2×SD32C2211yescharacteristic
38C2×SD32C2211yescharacteristic
39#504C2211yescharacteristic
40#504C2211yescharacteristic, complemented
41#495C2211yescharacteristic
42#494C2211yescharacteristic, complemented
43#473—21no—
44#473C2211yescharacteristic
45#473C2211yescharacteristic
46#473C2211yescharacteristic, complemented
47#472C2211yescharacteristic
48#471—21no—
49#456—21no—
50#456—21no—
51#455—21no—
52#454—21no—
53#443C2211yescharacteristic
54#443C2211yescharacteristic
55#443C2211yescharacteristic, complemented
56#368C2211yescharacteristic, complemented
57#364C2211yescharacteristic, complemented
58#362C2211yescharacteristic, complemented
59#362C2211yescharacteristic, complemented
60#362C2211yescharacteristic, complemented
61#361C2211yescharacteristic
62#359C2211yescharacteristic
63#359C2211yescharacteristic, complemented
64#350C2211yescharacteristic
65#350C2211yescharacteristic, complemented
66#346C2211yescharacteristic, complemented
672−1+4—21no—
682−1+4—21no—
692−1+4—21no—
70H(5,𝔽2)—21no—
71#137—21no—
72#137—21no—
73#137—21no—
74#137—21no—
75#137—21no—
76#137—21no—
77#137—41no—
78#137—41no—
79#137—41no—
80#137—41no—
81#137—41no—
82#137—41no—
83AGL(1,ℤ/8)—21no—
84AGL(1,ℤ/8)—21no—
85AGL(1,ℤ/8)—41no—
86AGL(1,ℤ/8)—41no—
87#135—21no—
88#135—21no—
89#135—21no—
90#135—21no—
91#135—41no—
92#135—41no—
93#135D411yescharacteristic
94#135C2311yescharacteristic
95#135C2311yescharacteristic
96C2×Q16—21no—
97C2×Q16—21no—
98C2×Q16—21no—
99C2×Q16—21no—
100C2×Q16—41no—
101C2×Q16C2311yescharacteristic
102C2×Q16C2311yescharacteristic
103C2×Q16C2311yescharacteristic
104C2×SD16—21no—
105C2×SD16—41no—
106C2×D8—21no—
107C2×D8C2311yescharacteristic
108#131—21no—
109#131—21no—
110#131C2311yescharacteristic
111#131C2311yescharacteristic
112C4⋊Q8—21no—
113#124—21no—
114Q32—21no—
115Q32—21no—
116Q32—21no—
117Q32—21no—
118Q32—21no—
119Q32—41no—
120Q32D411yescharacteristic
121SD32—21no—
122SD32—21no—
123SD32—21no—
124SD32—21no—
125SD32—21no—
126SD32—21no—
127SD32—41no—
128SD32D411yescharacteristic
129SD32D411yescharacteristic
130D16—21no—
131D16D411yescharacteristic
132M32—21no—
133M32—21no—
134M32C2311yescharacteristic
135M32C2311yescharacteristic
136C2×C16—21no—
137C2×C16—21no—
138C2×C16C2311yescharacteristic
139C2×C16C2311yescharacteristic
140#108D411yescharacteristic
141#108C2311yescharacteristic
142#108C2311yescharacteristic
143#104—21no—
144#104—21no—
145#104D411yescharacteristic
146#104D411yescharacteristic
147#101—21no—
148#101—21no—
149#101—21no—
150#100—21no—
151C4×C8C2311yescharacteristic
152#41—21no—
153#41—21no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162#41—81no—
163#41—81no—
164#41—81no—
165#41C2×D411yescharacteristic
166#41C2×D411yescharacteristic
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—81no—
180Q16—21no—
181Q16—21no—
182Q16—21no—
183Q16—21no—
184Q16—21no—
185Q16—21no—
186Q16—41no—
187Q16—41no—
188Q16—41no—
189Q16—41no—
190Q16—41no—
191Q16—41no—
192Q16—41no—
193Q16—41no—
194Q16D811yescharacteristic
195Q16C2×D411yescharacteristic
196Q16C2×D411yescharacteristic
197SD16—21no—
198SD16—21no—
199SD16—41no—
200SD16—41no—
201SD16—41no—
202SD16—41no—
203SD16—41no—
204SD16—41no—
205SD16—41no—
206SD16—41no—
207SD16—41no—
208SD16—41no—
209SD16D811yescharacteristic
210SD16D811yescharacteristic
211D8—21no—
212D8—21no—
213D8—41no—
214D8—41no—
215D8D811yescharacteristic
216D8C2×D411yescharacteristic
217D8C2×D411yescharacteristic
218M16—21no—
219M16—21no—
220M16—21no—
221M16—21no—
222M16—41no—
223M16—41no—
224M16C2×D411yescharacteristic
225M16C2×D411yescharacteristic
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8C2×D411yescharacteristic
231C2×C8C2411yescharacteristic
232C4⋊C4—81no—
233C22⋊C4—81no—
234C42C2×D411yescharacteristic
235C16—21no—
236C16—21no—
237C16—21no—
238C16C2×D411yescharacteristic
239C16C2×D411yescharacteristic
240C23—41no—
241C23—81no—
242Q8—21no—
243Q8—21no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—81no—
251Q8—81no—
252Q8—81no—
253Q8C2×D811yescharacteristic
254Q8C2×D811yescharacteristic
255D4—21no—
256D4—21no—
257D4—21no—
258D4—21no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—81no—
264D4—81no—
265D4—81no—
266D4—81no—
267D4—81no—
268D4C2×D811yescharacteristic
269D4C2×D811yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4C22×D411yescharacteristic
286C8—21no—
287C8—21no—
288C8—41no—
289C8—41no—
290C8C2×D811yescharacteristic
291C8C2×D811yescharacteristic
292C8C22×D411yescharacteristic
293C8H(5,𝔽2)11yescharacteristic
294C22—21no—
295C22—21no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22—161no—
300C22—321no—
301C22#57611yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4D4211yescharacteristic
311C4C22×D811yescharacteristic
312C2—21no—
313C2—41no—
314C2—41no—
315C2—81no—
316C2—81no—
317C2—161no—
318C2D4×D811yescharacteristic
subgroupG/H
center, socleC2D4×D8
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3417C2semidirect
2. #3416C2semidirect
3. #3293C2semidirect
4. #3290C2semidirect
5. #2251C2semidirect
6. #2223C2semidirect
7. #2222C2semidirect
8. #2214C2semidirect
9. #2212C2semidirect
10. #2195C2semidirect
11. #2195C2semidirect
12. #2194C2semidirect
13. #2193C2semidirect
14. #2179C2semidirect
15. #2171C2semidirect
19. #578C22semidirect
27. #510C22semidirect
34. C2×Q32C22semidirect
35. C2×Q32C22semidirect
40. #504C22semidirect
42. #494C22semidirect
46. #473C22semidirect
55. #443C22semidirect
56. #368C22semidirect
57. #364C22semidirect
58. #362C22semidirect
59. #362C22semidirect
60. #362C22semidirect
63. #359C22semidirect
65. #350C22semidirect
66. #346C22semidirect

Maximal subgroups

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

subgroupindex
#34172
#34162
#32932
#32902
#22512
#22232
#22222
#22142
#22122
#21952
#21942
#21932
#21792
#21712