Finite Group Databasebeta

← #19544

Subgroups — #19544

Order 256.

293 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3238C211yesmaximal, characteristic, complemented
2#3227C211yesmaximal, characteristic, complemented
3#3212C211yesmaximal, characteristic
4#2993C211yesmaximal, characteristic, complemented
5#1698C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1692C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1678C211yesmaximal, characteristic, complemented
10#1677C211yesmaximal, characteristic, complemented
11#1672C211yesmaximal, characteristic, complemented
12#1671C211yesmaximal, characteristic, complemented
13#1572C211yesmaximal, characteristic, complemented
14#1557C211yesmaximal, characteristic, complemented
15#1548C211yesmaximal, characteristic, complemented
16#532C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18#529C2211yescharacteristic
19#501—21no—
20#498C2211yescharacteristic
21#498C2211yescharacteristic
22#495—21no—
23#489—21no—
24#488—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#483—21no—
28#481—21no—
29#481—21no—
30#480—21no—
31#477C2211yescharacteristic
32#477C2211yescharacteristic
33#476—21no—
34#476—21no—
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#462C2211yescharacteristic
38#461C2211yescharacteristic
39#460C2211yescharacteristic
40#459C2211yescharacteristic
41#452—21no—
42#449—21no—
43#446C2211yescharacteristic
44C8×Q8C2211yescharacteristic
45#436—21no—
46#436C2211yescharacteristic
47#435—21no—
48C8×D4C2211yescharacteristic
49#433—21no—
50#432C2211yescharacteristic
51#431C2211yescharacteristic
52#431C2211yescharacteristic
53#424—21no—
54#423C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#333C2211yescharacteristic
62#333C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×D8—21no—
73C2×M16—21no—
74C2×M16—21no—
75C22×C8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127C2311yescharacteristic
83#126—21no—
84#125D411yescharacteristic
85#124—21no—
86#124D411yescharacteristic
87#123—21no—
88C22⋊Q8D411yescharacteristic
89#121—21no—
90#121—21no—
91#121D411yescharacteristic
92C4×Q8C2311yescharacteristic
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4C2311yescharacteristic
97#117—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C42C2311yescharacteristic
101#107—21no—
102#107—21no—
103#106—21no—
104#106—21no—
105#106—21no—
106#106—21no—
107#106—21no—
108#106—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8D411yescharacteristic
113C4⋊C8D411yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#102—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132C22⋊C8—21no—
133C22⋊C8—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136C22⋊C8D411yescharacteristic
137C22⋊C8D411yescharacteristic
138#97—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—81no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—82no—
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4#4111yescharacteristic
164Q16—81no—
165SD16—81no—
166SD16—81no—
167D8—81no—
168M16—81no—
169M16—81no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—81no—
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C42—21no—
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42#4111yescharacteristic
206C42C2411yescharacteristic
207C23—21no—
208C23—41no—
209C23#12111yescharacteristic
210Q8—21no—
211Q8—21no—
212Q8—41no—
213Q8—41no—
214Q8—81no—
215Q8—81no—
216Q8—82no—
217D4—21no—
218D4—21no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—81no—
224D4—81no—
225D4—82no—
226C2×C4—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—82no—
240C2×C4—82no—
241C2×C4#12111yescharacteristic
242C2×C4#12111yescharacteristic
243C2×C4#12111yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C8—21no—
248C8—21no—
249C8—21no—
250C8—21no—
251C8—41no—
252C8—41no—
253C8—41no—
254C8—41no—
255C8—41no—
256C8—41no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22C2×#12111yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—21no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4#57711yescharacteristic
278C4#57711yescharacteristic
279C4#57711yescharacteristic
280C4#57811yescharacteristic
281C2—41no—
282C2—81no—
283C2—161no—
284C2#304611yescharacteristic
285C2#305711yescharacteristic
286C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3212C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3238C2semidirect
2. #3227C2semidirect
4. #2993C2semidirect
5. #1698C2semidirect
6. #1697C2semidirect
7. #1692C2semidirect
8. #1691C2semidirect
9. #1678C2semidirect
10. #1677C2semidirect
11. #1672C2semidirect
12. #1671C2semidirect
13. #1572C2semidirect
14. #1557C2semidirect
15. #1548C2semidirect

Maximal subgroups

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

subgroupindex
#32382
#32272
#32122
#29932
#16982
#16972
#16922
#16912
#16782
#16772
#16722
#16712
#15722
#15572
#15482