Finite Group Databasebeta

← #20252

Subgroups — #20252

Order 256.

290 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3393C211yesmaximal, characteristic
2#3384C211yesmaximal, characteristic
3#3356C211yesmaximal, characteristic, complemented
4#3328C211yesmaximal, characteristic, complemented
5#1716C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1690C211yesmaximal, characteristic, complemented
9#1672C211yesmaximal, characteristic, complemented
10#1652C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1643C211yesmaximal, characteristic, complemented
13#1638C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic
15#1584C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#501—21no—
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#495—21no—
25#494C2211yescharacteristic
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#485—21no—
29#485—21no—
30#483—21no—
31#478—21no—
32#477C2211yescharacteristic
33#476—21no—
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#475C2211yescharacteristic
37#474C2211yescharacteristic
38#470—21no—
39#470—21no—
40#464—21no—
41#464—21no—
42#462C2211yescharacteristic
43#461C2211yescharacteristic
44#461C2211yescharacteristic
45#451—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#442—21no—
49C4×Q16—21no—
50C4×Q16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#428—21no—
55#419—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C22×Q8—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16D411yescharacteristic
72C2×SD16—21no—
73C2×SD16—21no—
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C2×M16—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#126—41no—
83#126—41no—
84#125—21no—
85#124—21no—
86#124—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8C2311yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×D4—21no—
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102#117—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107D411yescharacteristic
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
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#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—41no—
131#103D411yescharacteristic
132#103D411yescharacteristic
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159Q16—41no—
160Q16—42no—
161Q16—81no—
162Q16—81no—
163SD16—81no—
164SD16—81no—
165D8—22no—
166M16—42no—
167M16—81no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—21no—
205C42—21no—
206C42—41no—
207C42—41no—
208C42C2411yescharacteristic
209C23—21no—
210C23—21no—
211Q8—21no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—81no—
222Q8—82no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—82no—
242C2×C4—161no—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C8—21no—
247C8—21no—
248C8—21no—
249C8—41no—
250C8—41no—
251C8—41no—
252C8—41no—
253C8—41no—
254C8#13712yes—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22D4211yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4C2×#13711yescharacteristic
274C4#57511yescharacteristic
275C4#57711yescharacteristic
276C4#57711yescharacteristic
277C2—81no—
278C2—81no—
279C2#326711yescharacteristic
280C2#327811yescharacteristic
281C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalC2×D8D4
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
3. #3356C2semidirect
4. #3328C2semidirect
5. #1716C2semidirect
6. #1699C2semidirect
7. #1691C2semidirect
8. #1690C2semidirect
9. #1672C2semidirect
10. #1652C2semidirect
11. #1651C2semidirect
12. #1643C2semidirect
13. #1638C2semidirect
15. #1584C2semidirect

Maximal subgroups

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

subgroupindex
#33932
#33842
#33562
#33282
#17162
#16992
#16912
#16902
#16722
#16522
#16512
#16432
#16382
#15942
#15842