Finite Group Databasebeta

← #19915

Subgroups — #19915

Order 256.

330 conjugacy classes of subgroups.

The subgroup lattice has 306 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#3330C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic, complemented
5#1708C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1642C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15C8×D8C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#492C2211yescharacteristic
23#490—21no—
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#484—21no—
29#483—21no—
30#481—21no—
31#478—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#474C2211yescharacteristic
36#468—21no—
37#465—21no—
38#465—21no—
39#462C2211yescharacteristic
40#461C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46#449—21no—
47#446C2211yescharacteristic
48#442—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C2×#106—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70#135—44no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×D8—21no—
77C2×D8—21no—
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#125—21no—
88#123—41no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98C4×Q8C2311yescharacteristic
99C4×D4—21no—
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—41no—
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—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150#41—81no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172Q16—42no—
173SD16—22no—
174SD16—42no—
175SD16—81no—
176SD16—81no—
177D8—22no—
178D8—42no—
179D8—42no—
180M16—81no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—21no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219C23—41no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—81no—
237D4—82no—
238D4—82no—
239D4—82no—
240D4—161no—
241D4#13512yescomplemented
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—161no—
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C8—21no—
264C8—21no—
265C8—21no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8AGL(1,ℤ/8)12yes—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22D4211yescharacteristic
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4C2×#13511yescharacteristic
295C4C2×AGL(1,ℤ/8)11yescharacteristic
296C4#57711yescharacteristic
297C4#57711yescharacteristic
298C2—42no—
299C2—81no—
300C2—81no—
301C2—161no—
302C2#327211yescharacteristic
303C2#328211yescharacteristic
304C2#329011yescharacteristic
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. #3330C2semidirect
4. #3284C2semidirect
5. #1708C2semidirect
6. #1700C2semidirect
7. #1697C2semidirect
8. #1671C2semidirect
9. #1665C2semidirect
10. #1644C2semidirect
11. #1642C2semidirect
12. #1635C2semidirect
13. #1625C2semidirect
14. #1590C2semidirect
15. C8×D8C2semidirect
241. D4#135semidirect

Maximal subgroups

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

subgroupindex
#33662
#33352
#33302
#32842
#17082
#17002
#16972
#16712
#16652
#16442
#16422
#16352
#16252
#15902
C8×D82