Finite Group Databasebeta

← #20051

Subgroups — #20051

Order 256.

296 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3392C211yesmaximal, characteristic
2#3387C211yesmaximal, characteristic, complemented
3#3369C211yesmaximal, characteristic, complemented
4#3304C211yesmaximal, characteristic
5#1714C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1684C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1648C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic
13#1623C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16Q82C2211yescharacteristic
17#557C2211yescharacteristic
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#492—21no—
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#489—21no—
27#487C2211yescharacteristic
28#486C2211yescharacteristic
29#482—21no—
30#479—21no—
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#475—21no—
34#475—21no—
35#475C2211yescharacteristic
36#475C2211yescharacteristic
37#465—21no—
38#464—21no—
39#462C2211yescharacteristic
40#461—21no—
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#460C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#441—21no—
50#441—21no—
51C4×Q16C2211yescharacteristic
52C4×SD16—21no—
53C4×SD16C2211yescharacteristic
54C8×D4C2211yescharacteristic
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16D411yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—41no—
76C2×SD16D411yescharacteristic
77C22×C8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—41no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#126—41no—
88#125—21no—
89#125—41no—
90#124—21no—
91C22⋊Q8—21no—
92#121—21no—
93#121—41no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104#117—21no—
105#107—21no—
106#107—41no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
129#103D411yescharacteristic
130#103D411yescharacteristic
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102D411yescharacteristic
137C22⋊C8—21no—
138#97—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—41no—
161Q16—22no—
162Q16—42no—
163SD16—41no—
164SD16—42no—
165SD16—81no—
166SD16—82no—
167SD16—81no—
168SD16—81no—
169SD16—81no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—81no—
176C2×C8C2×D411yescharacteristic
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C4⋊C4—21no—
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—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—41no—
207C42—41no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—41no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—81no—
222Q8—161no—
223Q8#13712yes—
224D4—21no—
225D4—21no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—82no—
231D4—81no—
232D4—161no—
233C2×C4—21no—
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—82no—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C8—22no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—21no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22D4211yescharacteristic
265C4—22no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4C2×#13711yescharacteristic
277C4#57511yescharacteristic
278C4#57711yescharacteristic
279C4#57711yescharacteristic
280C2—81no—
281C2—161no—
282C2#326611yescharacteristic
283C2#327111yescharacteristic
284C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#550C22
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
2. #3387C2semidirect
3. #3369C2semidirect
5. #1714C2semidirect
6. #1702C2semidirect
7. #1686C2semidirect
8. #1684C2semidirect
9. #1671C2semidirect
10. #1650C2semidirect
11. #1648C2semidirect
13. #1623C2semidirect
14. #1587C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33922
#33872
#33692
#33042
#17142
#17022
#16862
#16842
#16712
#16502
#16482
#16312
#16232
#15872
#15852