Finite Group Databasebeta

← #20227

Subgroups — #20227

Order 256.

312 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3367C211yesmaximal, characteristic, complemented
2#3354C211yesmaximal, characteristic
3#3340C211yesmaximal, characteristic, complemented
4#3276C211yesmaximal, characteristic, complemented
5#1716C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1692C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1646C211yesmaximal, characteristic, complemented
13#1637C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#556C2211yescharacteristic
18#555C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#489—21no—
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#485—21no—
29#482—21no—
30#482—21no—
31#481—21no—
32#477—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#474C2211yescharacteristic
36#474C2211yescharacteristic
37#474C2211yescharacteristic
38#470—21no—
39#469—21no—
40#465—21no—
41#463—21no—
42#462C2211yescharacteristic
43#461C2211yescharacteristic
44#461C2211yescharacteristic
45#452—21no—
46#451—21no—
47#446C2211yescharacteristic
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55#427—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69#137—44no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#126—41no—
86#124—21no—
87#124—41no—
88#123—41no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93#121—21no—
94#121—21no—
95#121—21no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104#117—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—41no—
117#106D411yescharacteristic
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—41no—
130#103D411yescharacteristic
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
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—81no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8C2×D411yescharacteristic
155C2×D4—41no—
156C2×D4—41no—
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C2×D4C2×D411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—42no—
166Q16—42no—
167Q16—81no—
168SD16—22no—
169SD16—42no—
170SD16—42no—
171D8—41no—
172M16—42no—
173M16—42no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—42no—
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216Q8—21no—
217Q8—21no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—82no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—82no—
233D4—161no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8AGL(1,ℤ/8)12yes—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22D4211yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4C2×AGL(1,ℤ/8)11yescharacteristic
284C4#57511yescharacteristic
285C4#57711yescharacteristic
286C4#57811yescharacteristic
287C2—81no—
288C2—81no—
289C2—81no—
290C2#326911yescharacteristic
291C2#327211yescharacteristic
292C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3354C2
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. #3367C2semidirect
3. #3340C2semidirect
4. #3276C2semidirect
5. #1716C2semidirect
6. #1699C2semidirect
7. #1692C2semidirect
8. #1691C2semidirect
9. #1669C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1646C2semidirect
13. #1637C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect

Maximal subgroups

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

subgroupindex
#33672
#33542
#33402
#32762
#17162
#16992
#16922
#16912
#16692
#16542
#16522
#16462
#16372
#15922
#15812