Finite Group Databasebeta

← #19852

Subgroups — #19852

Order 256.

330 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3361C211yesmaximal, characteristic, complemented
3#3335C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1692C211yesmaximal, characteristic, complemented
7#1677C211yesmaximal, characteristic, complemented
8#1667C211yesmaximal, characteristic, complemented
9#1666C211yesmaximal, characteristic, complemented
10#1640C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1582C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#559C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#496—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#488—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#483—21no—
30#481—21no—
31#478—21no—
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#465—21no—
36#465—21no—
37#463—21no—
38#463—21no—
39#462C2211yescharacteristic
40#460C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#449—21no—
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#442—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52C4×D8—21no—
53C8×D4C2211yescharacteristic
54C8×D4C2211yescharacteristic
55C2×#106—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—41no—
75C2×SD16D411yescharacteristic
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#125—21no—
90#124—21no—
91#124—21no—
92#123—41no—
93#123—41no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96C22⋊Q8—41no—
97#121—21no—
98#121—21no—
99#121—41no—
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110#107—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106D411yescharacteristic
116#106D411yescharacteristic
117C4⋊C8—21no—
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103—21no—
126#103D411yescharacteristic
127#103D411yescharacteristic
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—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170Q16—42no—
171SD16—22no—
172SD16—22no—
173SD16—42no—
174SD16—81no—
175SD16—81no—
176SD16—82no—
177D8—42no—
178D8—42no—
179D8—42no—
180D8—81no—
181D8—81no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—42no—
188C2×C8—42no—
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⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—21no—
215C42—21no—
216C42C2411yescharacteristic
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221Q8—21no—
222Q8—21no—
223Q8—21no—
224Q8—21no—
225Q8—41no—
226Q8—41no—
227Q8—81no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—81no—
240D4—81no—
241D4—82no—
242D4—82no—
243D4—161no—
244D4AGL(1,ℤ/8)12yes—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—161no—
260C2×C4—161no—
261C2×C4C22×D411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8#13512yescomplemented
273C22—22no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22D4211yescharacteristic
286C4—21no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4C2×#13511yescharacteristic
296C4C2×AGL(1,ℤ/8)11yescharacteristic
297C4#57711yescharacteristic
298C4#57711yescharacteristic
299C2—42no—
300C2—81no—
301C2—161no—
302C2—161no—
303C2#326511yescharacteristic
304C2#328411yescharacteristic
305C2#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. #3399C2semidirect
2. #3361C2semidirect
3. #3335C2semidirect
4. #3284C2semidirect
5. #1715C2semidirect
6. #1692C2semidirect
7. #1677C2semidirect
8. #1667C2semidirect
9. #1666C2semidirect
10. #1640C2semidirect
11. #1631C2semidirect
12. #1625C2semidirect
13. #1622C2semidirect
14. #1582C2semidirect
15. SD16⋊C8C2semidirect
272. C8#135semidirect

Maximal subgroups

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

subgroupindex
#33992
#33612
#33352
#32842
#17152
#16922
#16772
#16672
#16662
#16402
#16312
#16252
#16222
#15822
SD16⋊C82