Finite Group Databasebeta

← #20193

Subgroups — #20193

Order 256.

336 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3372C211yesmaximal, characteristic, complemented
2#3368C211yesmaximal, characteristic, complemented
3#3295C211yesmaximal, characteristic, complemented
4#3274C211yesmaximal, characteristic
5#1717C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1698C211yesmaximal, characteristic, complemented
8#1688C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1626C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1582C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#557C2211yescharacteristic, complemented
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20D42C2211yescharacteristic
21#493C2211yescharacteristic
22#492C2211yescharacteristic, complemented
23#491—21no—
24#490—21no—
25#487C2211yescharacteristic, complemented
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic, complemented
29#481—21no—
30#479—21no—
31#478—21no—
32#477C2211yescharacteristic, complemented
33#476C2211yescharacteristic
34#474C2211yescharacteristic
35#469—21no—
36#465—21no—
37#461—21no—
38#461C2211yescharacteristic
39#460—21no—
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#460C2211yescharacteristic, complemented
43#459C2211yescharacteristic
44#450—21no—
45#449—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#446C2211yescharacteristic, complemented
49#440—21no—
50#440—21no—
51C4×SD16C2211yescharacteristic
52C4×SD16C2211yescharacteristic
53#436C2211yescharacteristic
54C8×D4C2211yescharacteristic
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70AGL(1,ℤ/8)—44no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—41no—
76C2×SD16D411yescharacteristic
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×M16—21no—
82C22×C8—21no—
83C4⋊Q8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#125—21no—
90#125—21no—
91#125—41no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98#120—41no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8C2311yescharacteristic
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#106—21no—
111#106D411yescharacteristic
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103D411yescharacteristic
124#103D411yescharacteristic
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—21no—
135#102—41no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
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—81no—
165C2×D4—81no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172SD16—22no—
173SD16—41no—
174SD16—42no—
175SD16—42no—
176SD16—42no—
177SD16—81no—
178SD16—82no—
179D8—42no—
180D8—42no—
181D8—81no—
182M16—42no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—81no—
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42C2411yescharacteristic
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23—82no—
222C23—82no—
223C23—161no—
224Q8—21no—
225Q8—21no—
226Q8—21no—
227Q8—21no—
228Q8—41no—
229Q8—41no—
230Q8—81no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—81no—
243D4—81no—
244D4—82no—
245D4—82no—
246D4—82no—
247D4—161no—
248D4AGL(1,ℤ/8)12yes—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—161no—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—21no—
267C8—21no—
268C8—21no—
269C8—21no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—41no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—164no—
288C22D4211yescharacteristic
289C4—21no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4C2×AGL(1,ℤ/8)11yescharacteristic
299C4#57511yescharacteristic
300C4#57711yescharacteristic
301C4#57711yescharacteristic
302C2—42no—
303C2—81no—
304C2—161no—
305C2—161no—
306C2#326511yescharacteristic
307C2#327511yescharacteristic
308C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3274C2
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. #3372C2semidirect
2. #3368C2semidirect
3. #3295C2semidirect
5. #1717C2semidirect
6. #1703C2semidirect
7. #1698C2semidirect
8. #1688C2semidirect
9. #1681C2semidirect
10. #1639C2semidirect
11. #1631C2semidirect
12. #1626C2semidirect
13. #1622C2semidirect
14. #1592C2semidirect
15. #1582C2semidirect
17. #557C22semidirect
22. #492C22semidirect
25. #487C22semidirect
28. #486C22semidirect
32. #477C22semidirect
42. #460C22semidirect
47. #446C22semidirect
48. #446C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊C8C22semidirect
63. Q8⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

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

subgroupindex
#33722
#33682
#32952
#32742
#17172
#17032
#16982
#16882
#16812
#16392
#16312
#16262
#16222
#15922
#15822