Finite Group Databasebeta

← #244188

Subgroups — #244188

Order 512.

594 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22281C211yesmaximal, characteristic
2#22243C211yesmaximal, characteristic, complemented
3#17936C212yesmaximal, complemented
4#17929C212yesmaximal, complemented
5#10665C212yesmaximal, complemented
6#10664C212yesmaximal, complemented
7#10034C212yesmaximal
8#10014C212yesmaximal, complemented
9#8141C211yesmaximal, characteristic, complemented
10#3150C2211yescharacteristic
11#3103C2212yes—
12#3103C2212yes—
13C2×#446C2212yes—
14#2931C2212yes—
15#2399C2211yescharacteristic
16#1985C2212yes—
17#1963C2212yes—
18#1917—24no—
19#1876—24no—
20#1838C2211yescharacteristic
21#1832C2211yescharacteristic
22#1746C2211yescharacteristic
23#1592—24no—
24#1590—24no—
25#1585—24no—
26#1582—24no—
27#1571C2212yes—
28#1570C2212yes—
29#1487C2212yes—
30#1487C2212yes—
31#1480C2212yes—
32#1480C2212yes—
33#1452C2211yescharacteristic
34#1450C2211yescharacteristic
35#1275C2212yes—
36#1274C2212yes—
37C2×#135—22no—
38C2×#131—22no—
39#532C2312yes—
40C2×C4×Q8C2312yes—
41#497—22no—
42#496—22no—
43#481—22no—
44#481—22no—
45#475—22no—
46#475—22no—
47#460—22no—
48#460—22no—
49#448—22no—
50#448—22no—
51#446—22no—
52#446—22no—
53#446—22no—
54#446—22no—
55C4×SD16—24no—
56C4×SD16—24no—
57#436—22no—
58#436—22no—
59C8×D4—22no—
60C8×D4—22no—
61#431C2312yes—
62#431C2312yes—
63#423C2312yes—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×#104—24no—
67#408—22no—
68C2×#97—22no—
69C2×#97C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#398—22no—
72#395—22no—
73C4×C4⋊C4—22no—
74#376C2311yescharacteristic
75#341—22no—
76#336—24no—
77#336—22no—
78#336D414yes—
79#335—24no—
80#334—24no—
81Q8⋊C8—24no—
82Q8⋊C8—24no—
83D4⋊C8—24no—
84D4⋊C8—24no—
85#322D412yes—
86#322D414yes—
87C82D412yes—
88C2×#41—22no—
89C22×Q8#4112yes—
90#135—84no—
91C2×Q16—22no—
92C2×SD16—24no—
93C2×SD16—24no—
94C2×SD16—24no—
95C2×D8—22no—
96#131—84no—
97C2×M16—22no—
98C2×M16—42no—
99C22×C8—22no—
100C22×C8—42no—
101C22×C8C2×D412yes—
102C22×C8C2×D412yes—
103C4⋊Q8#4112yes—
104#127#4112yes—
105#124—22no—
106#121—22no—
107#121—22no—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×D4—22no—
113C4×D4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4#4112yes—
117C2×C42—22no—
118C2×C42C2×D412yes—
119C2×C42C2411yescharacteristic
120#106—24no—
121#106—24no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—42no—
129C4⋊C8—42no—
130#104—84no—
131#103—24no—
132#103—24no—
133#102—24no—
134#102—24no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139#97—22no—
140#97—42no—
141#97—42no—
142C4×C8C2×D414yes—
143C4×C8C2×D414yes—
144C4×C8C2×D411yescharacteristic
145C4×C8C2×D411yescharacteristic
146C4×C8C2×D412yes—
147#95—22no—
148#95—22no—
149#95—22no—
150#41—44no—
151#41—44no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—82no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—42no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4C22×D411yescharacteristic
165C22×C4C2×#4112yes—
166Q16—82no—
167SD16—84no—
168SD16—84no—
169SD16—84no—
170D8—82no—
171M16—44no—
172M16—84no—
173C2×C8—24no—
174C2×C8—24no—
175C2×C8—24no—
176C2×C8—22no—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—84no—
182C2×C8#12712yes—
183C2×C8#12712yes—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—84no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C42—24no—
194C42—42no—
195C42—42no—
196C42C22×D411yescharacteristic
197C42C22×D411yescharacteristic
198C42C2×#4112yes—
199C23—42no—
200C23#52511yescharacteristic
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205Q8—44no—
206D4—44no—
207D4—44no—
208D4—44no—
209D4—84no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#52512yes—
222C2×C4#52512yes—
223C2×C4#52511yescharacteristic
224C2×C4C2×#12711yescharacteristic
225C2×C4#57714yes—
226C8—22no—
227C8—22no—
228C8—44no—
229C8—44no—
230C8—42no—
231C8—82no—
232C8—82no—
233C22—22no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22#239711yescharacteristic
238C22#312612yes—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—82no—
243C4—82no—
244C4—82no—
245C4#312312yes—
246C4#315412yes—
247C2—22no—
248C2—162no—
249C2#2379512yes—
250C2#2381111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutator, Frattini, cosocleC2×C42C24
socleC22#2397
prime radical#3150C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22243C2semidirect
3. #17936C2semidirect
4. #17929C2semidirect
5. #10665C2semidirect
6. #10664C2semidirect
8. #10014C2semidirect
9. #8141C2semidirect

Maximal subgroups

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

subgroupindex
#222812
#222432
#179362
#179292
#106652
#106642
#100342
#100142
#81412