Finite Group Databasebeta

← #331063

Subgroups — #331063

Order 512.

476 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 110 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#30234C211yesmaximal, characteristic, complemented
2#22283C211yesmaximal, characteristic
3#10636C212yesmaximal, complemented
4#10368C211yesmaximal, characteristic, complemented
5#10366C211yesmaximal, characteristic, complemented
6#10287C211yesmaximal, characteristic, complemented
7#10285C211yesmaximal, characteristic, complemented
8#10221C212yesmaximal
9#9289C212yesmaximal, complemented
10#9287C212yesmaximal, complemented
11#8802C211yesmaximal, characteristic, complemented
12#3227C2212yes—
13#3206C2212yes—
14#3150C2211yescharacteristic
15#2932C2211yescharacteristic
16#2932C2211yescharacteristic
17#2864C2211yescharacteristic
18#2862C2211yescharacteristic
19#2101—22no—
20#2101—22no—
21#1993C2212yes—
22#1993C2212yes—
23#1962C2211yescharacteristic
24#1962C2211yescharacteristic
25#1876—22no—
26#1876—22no—
27#1838C2211yescharacteristic
28#1745C2211yescharacteristic
29#1549C2212yes—
30#1548C2212yes—
31#1517C2212yes—
32#1517C2212yes—
33#1286C2212yes—
34#1286C2212yes—
35#1281C2212yes—
36#1276C2211yescharacteristic
37#1276C2211yescharacteristic
38#1275C2212yes—
39C2×#135—22no—
40C2×#131—21no—
41C2×#131—21no—
42#532C2311yescharacteristic
43#532C2311yescharacteristic
44C2×C4⋊Q8C2312yes—
45#498—22no—
46#497—21no—
47#497—21no—
48#496—21no—
49#496—21no—
50#492—22no—
51#487—22no—
52#486—22no—
53#481—42no—
54#476—42no—
55#461—42no—
56#448—42no—
57#435—42no—
58#435—42no—
59#431C2312yes—
60#431C2312yes—
61#423C2311yescharacteristic
62#423C2311yescharacteristic
63C2×#104—22no—
64C2×#104—22no—
65#408—21no—
66#408—21no—
67C2×#97C2311yescharacteristic
68C2×#97C2311yescharacteristic
69#391—42no—
70#389—22no—
71#387—42no—
72#382—22no—
73#382C2312yes—
74C43C2311yescharacteristic
75#341—22no—
76#340—22no—
77#340—22no—
78#340—24no—
79#336D412yes—
80#336D412yes—
81#333—42no—
82#332—42no—
83#327—42no—
84#327—42no—
85C2×#41—21no—
86C2×#41—21no—
87C22×Q8—22no—
88#135—82no—
89#135—82no—
90C2×Q16—22no—
91C2×SD16—22no—
92C2×SD16—22no—
93C2×SD16—44no—
94C2×D8—22no—
95#131—82no—
96#131—82no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—41no—
100C2×M16—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D412yes—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8#4111yescharacteristic
111C4⋊Q8#4111yescharacteristic
112#127#4111yescharacteristic
113#127#4111yescharacteristic
114#124—21no—
115#124—21no—
116#121—22no—
117#121—22no—
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C42—22no—
125C2×C42C2×D412yes—
126C2×C42C2411yescharacteristic
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132#104—82no—
133#104—82no—
134#103—44no—
135#102—44no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140#97—21no—
141#97—21no—
142#97—22no—
143C4×C8—22no—
144C4×C8—22no—
145#95—22no—
146#95—22no—
147#95—44no—
148#95—42no—
149#41—42no—
150#41—42no—
151#41—42no—
152#41—42no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—84no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—41no—
163C2×D4—41no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4C22×D411yescharacteristic
170C22×C4H(5,𝔽2)12yes—
171Q16—82no—
172SD16—82no—
173SD16—82no—
174SD16—164no—
175D8—82no—
176M16—42no—
177M16—42no—
178M16—82no—
179M16—82no—
180C2×C8—22no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—82no—
192C2×C8—82no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—84no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C42—22no—
205C42—22no—
206C42—82no—
207C42C2×#4111yescharacteristic
208C42C2×#4111yescharacteristic
209C42H(5,𝔽2)12yes—
210C23—41no—
211C23—41no—
212C23#53511yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—84no—
216Q8—162no—
217D4—42no—
218D4—42no—
219D4—82no—
220D4—82no—
221D4—82no—
222D4—82no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4#52511yescharacteristic
238C2×C4#53511yescharacteristic
239C2×C4#53812yes—
240C2×C4#53812yes—
241C2×C4#57712yes—
242C2×C4#57712yes—
243C8—42no—
244C8—41no—
245C8—41no—
246C8—82no—
247C8—82no—
248C8—81no—
249C8—81no—
250C22—22no—
251C22—81no—
252C22—81no—
253C22—82no—
254C22—82no—
255C22#263111yescharacteristic
256C22#322112yes—
257C4—22no—
258C4—22no—
259C4—82no—
260C4—81no—
261C4—81no—
262C4—162no—
263C4—162no—
264C4#312312yes—
265C4#322211yescharacteristic
266C4#322211yescharacteristic
267C2—22no—
268C2—161no—
269C2—161no—
270C2#3004911yescharacteristic
271C2#3005612yes—
subgroupG/H
center, socleC22#2631
commutator, Frattini, cosocleC2×C42C24
prime radical#3150C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30234C2semidirect
3. #10636C2semidirect
4. #10368C2semidirect
5. #10366C2semidirect
6. #10287C2semidirect
7. #10285C2semidirect
9. #9289C2semidirect
10. #9287C2semidirect
11. #8802C2semidirect

Maximal subgroups

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

subgroupindex
#302342
#222832
#106362
#103682
#103662
#102872
#102852
#102212
#92892
#92872
#88022