Finite Group Databasebeta

← #20037

Subgroups — #20037

Order 256.

335 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3358C211yesmaximal, characteristic, complemented
3#3304C211yesmaximal, characteristic, complemented
4#3278C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1666C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1652C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1621C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#556C2211yescharacteristic
18#550C2211yescharacteristic
19#550C2211yescharacteristic
20D4×Q8C2211yescharacteristic
21#498C2211yescharacteristic
22#496—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#486—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#482—21no—
30#475C2211yescharacteristic
31#474—21no—
32#474C2211yescharacteristic
33#468—21no—
34#465—21no—
35#463—21no—
36#462C2211yescharacteristic
37#461C2211yescharacteristic
38#460—21no—
39#460C2211yescharacteristic
40#460C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#440—21no—
50C4×Q16C2211yescharacteristic
51C4×D8—21no—
52C4×D8C2211yescharacteristic
53#436C2211yescharacteristic
54C8×D4C2211yescharacteristic
55#416—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C22×Q8—21no—
70#137—44no—
71C2×Q16—21no—
72C2×Q16D411yescharacteristic
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—41no—
83C2×D8D411yescharacteristic
84C2×M16—21no—
85C22×C8—21no—
86C4⋊Q8—21no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127—21no—
90#127—21no—
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#126—41no—
94#124—21no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97C22⋊Q8—41no—
98#121—21no—
99#121—21no—
100#121—41no—
101#121—41no—
102C4×Q8—21no—
103C4×Q8C2311yescharacteristic
104C4×Q8C2311yescharacteristic
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108C4×D4C2311yescharacteristic
109C4×D4C2311yescharacteristic
110#117—21no—
111#107—21no—
112#107D411yescharacteristic
113#107D411yescharacteristic
114#106—21no—
115#106—41no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103D411yescharacteristic
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—21no—
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×Q8—41no—
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8C2×D411yescharacteristic
155C2×Q8C2×D411yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173Q16—22no—
174Q16—42no—
175SD16—42no—
176SD16—42no—
177SD16—42no—
178SD16—81no—
179SD16—81no—
180SD16—81no—
181SD16—81no—
182D8—41no—
183D8—42no—
184D8—82no—
185D8—81no—
186M16—42no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—81no—
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C42—41no—
219C42C2411yescharacteristic
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224Q8—21no—
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—82no—
232Q8AGL(1,ℤ/8)12yes—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—82no—
246D4—81no—
247D4—81no—
248D4—82no—
249D4—161no—
250D4—161no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C8—22no—
270C8—21no—
271C8—21no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22D4211yescharacteristic
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4C2×AGL(1,ℤ/8)11yescharacteristic
301C4#57511yescharacteristic
302C4#57611yescharacteristic
303C4#57711yescharacteristic
304C2—81no—
305C2—81no—
306C2—161no—
307C2—161no—
308C2#326411yescharacteristic
309C2#327111yescharacteristic
310C2#329111yescharacteristic
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. #3397C2semidirect
2. #3358C2semidirect
3. #3304C2semidirect
4. #3278C2semidirect
5. #1715C2semidirect
6. #1702C2semidirect
7. #1701C2semidirect
8. #1666C2semidirect
9. #1665C2semidirect
10. #1652C2semidirect
11. #1651C2semidirect
12. #1621C2semidirect
13. #1621C2semidirect
14. #1587C2semidirect
15. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33582
#33042
#32782
#17152
#17022
#17012
#16662
#16652
#16522
#16512
#16212
#15872
#15862