Finite Group Databasebeta

← #20187

Subgroups — #20187

Order 256.

304 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3385C211yesmaximal, characteristic, complemented
2#3362C211yesmaximal, characteristic, complemented
3#3332C211yesmaximal, characteristic
4#3331C211yesmaximal, characteristic, complemented
5#1719C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1695C211yesmaximal, characteristic, complemented
9#1684C211yesmaximal, characteristic, complemented
10#1645C211yesmaximal, characteristic, complemented
11#1635C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16Q82C2211yescharacteristic
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#492C2211yescharacteristic
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#486—21no—
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#483—21no—
29#481—21no—
30#479—21no—
31#477C2211yescharacteristic
32#476—21no—
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#475C2211yescharacteristic
37#474—21no—
38#470—21no—
39#469—21no—
40#462C2211yescharacteristic
41#460—21no—
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#440—21no—
49C4×SD16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#417—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16—41no—
72C2×SD16D411yescharacteristic
73C2×SD16D411yescharacteristic
74C2×D8—21no—
75C2×M16—21no—
76C2×M16—21no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—41no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127—21no—
84#127C2311yescharacteristic
85#125—21no—
86#123—41no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#121—41no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8C2311yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×D4—21no—
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#106—21no—
107#106—21no—
108#106—21no—
109#106—21no—
110#106D411yescharacteristic
111#106D411yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—41no—
124#103D411yescharacteristic
125#103D411yescharacteristic
126#102—21no—
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#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—81no—
168SD16—22no—
169SD16—41no—
170SD16—42no—
171SD16—82no—
172D8—81no—
173M16—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
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—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—41no—
213Q8—21no—
214Q8—21no—
215Q8—21no—
216Q8—41no—
217Q8—41no—
218Q8—81no—
219Q8—81no—
220Q8—161no—
221Q8#13712yes—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—82no—
231D4—81no—
232D4—82no—
233D4—161no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—161no—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C8—21no—
254C8—21no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22D4211yescharacteristic
272C4—22no—
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4C2×#13711yescharacteristic
283C4#57511yescharacteristic
284C4#57711yescharacteristic
285C4#57711yescharacteristic
286C2—81no—
287C2—81no—
288C2—161no—
289C2#326611yescharacteristic
290C2#327511yescharacteristic
291C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3332C2
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. #3385C2semidirect
2. #3362C2semidirect
4. #3331C2semidirect
5. #1719C2semidirect
6. #1702C2semidirect
7. #1697C2semidirect
8. #1695C2semidirect
9. #1684C2semidirect
10. #1645C2semidirect
11. #1635C2semidirect
12. #1631C2semidirect
13. #1622C2semidirect
14. #1592C2semidirect
15. #1583C2semidirect

Maximal subgroups

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

subgroupindex
#33852
#33622
#33322
#33312
#17192
#17022
#16972
#16952
#16842
#16452
#16352
#16312
#16222
#15922
#15832