Finite Group Databasebeta

← #492696

Subgroups — #492696

Order 512.

399 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 118 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#33923C212yesmaximal, complemented
2#20020C211yesmaximal, characteristic
3#20006C211yesmaximal, characteristic
4#12523C211yesmaximal, characteristic, complemented
5#12456C212yesmaximal, complemented
6#12448C211yesmaximal, characteristic, complemented
7#12446C211yesmaximal, characteristic, complemented
8#12397C212yesmaximal, complemented
9#12394C212yesmaximal, complemented
10C8×Q32C211yesmaximal, characteristic, complemented
11#12039C211yesmaximal, characteristic
12#3393—21no—
13#3380—21no—
14#3342C2212yes—
15#3338C2212yes—
16#2254C2212yes—
17#2254C2212yes—
18#2237—22no—
19#2235—22no—
20#2227C2212yes—
21#2225C2212yes—
22#2219—22no—
23C4×Q32C2211yescharacteristic
24#2169C2212yes—
25#2159—22no—
26#1713C2211yescharacteristic
27#1713C2211yescharacteristic
28#1676—21no—
29#1675C2212yes—
30#1673C2212yes—
31#1670—21no—
32#1650—22no—
33#1632—22no—
34C8×Q16C2211yescharacteristic
35C8×Q16C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41#1332C2211yescharacteristic
42#1332C2211yescharacteristic
43D4⋊C16C2212yes—
44C8×C16C2211yescharacteristic
45#556—22no—
46#554—22no—
47C2×Q32D411yescharacteristic
48C2×M32—22no—
49#500—21no—
50#500—21no—
51#500C2312yes—
52#500C2312yes—
53#499—41no—
54#495—41no—
55#494—21no—
56#494—21no—
57#485—42no—
58#484—42no—
59#478—42no—
60#477—22no—
61#476—22no—
62#475—22no—
63#467—22no—
64#467—22no—
65#464—42no—
66#461—22no—
67C8×Q8—21no—
68C8×Q8—21no—
69C4×Q16—41no—
70C4×Q16C2311yescharacteristic
71C4×Q16C2311yescharacteristic
72C4×D8—41no—
73C4×D8D411yescharacteristic
74C8×D4C2312yes—
75#427—22no—
76C2×#107—22no—
77#367—22no—
78#367—22no—
79#367—22no—
80#367—42no—
81#366—22no—
82#366D411yescharacteristic
83C4⋊C16C2312yes—
84#358—22no—
85#358—22no—
86#358D411yescharacteristic
87#358D411yescharacteristic
88C22⋊C16—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#332—22no—
96Q8⋊C8—21no—
97Q8⋊C8—21no—
98D4⋊C8D412yes—
99C82C2311yescharacteristic
100C2×Q16—41no—
101C2×Q16—41no—
102C2×Q16—81no—
103C2×Q16C2×D411yescharacteristic
104C2×Q16C2×D411yescharacteristic
105C2×SD16—42no—
106C2×D8D811yescharacteristic
107C22×C8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110#126—82no—
111#125—42no—
112#124—42no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C22⋊Q8—82no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—42no—
119C4×D4—42no—
120C4×D4C2×D412yes—
121#117—42no—
122C2×C4⋊C4—42no—
123Q32—22no—
124M32—44no—
125C2×C16—21no—
126C2×C16—22no—
127C2×C16C2×D411yescharacteristic
128C2×C16C2×D411yescharacteristic
129#107—22no—
130#107—22no—
131#107—22no—
132#107—22no—
133#107—42no—
134#107—41no—
135#107—41no—
136#107—81no—
137#107D811yescharacteristic
138#107C2×D411yescharacteristic
139#107C2×D411yescharacteristic
140#106—42no—
141#106—42no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8C2×D412yes—
147#103—21no—
148#103—21no—
149#103—42no—
150#103—42no—
151#103—42no—
152#102—42no—
153#102D812yes—
154C22⋊C8—22no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8C2×D411yescharacteristic
158C4×C8C2×D411yescharacteristic
159C4×C8C2411yescharacteristic
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×D4C2×D812yes—
165C22×C4—22no—
166C22×C4—82no—
167Q16—21no—
168Q16—21no—
169Q16—21no—
170Q16—21no—
171Q16—82no—
172Q16—161no—
173Q16—162no—
174SD16—162no—
175D8—22no—
176C2×C8—21no—
177C2×C8—22no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8C2×D811yescharacteristic
183C2×C8C2×D811yescharacteristic
184C2×C8C22×D411yescharacteristic
185C2×C8H(5,𝔽2)11yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4C2×D812yes—
198C22⋊C4—22no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C42—41no—
203C42—41no—
204C42—82no—
205C42C22×D411yescharacteristic
206C16—22no—
207C16—21no—
208C16—42no—
209C16#13512yes—
210C23—22no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—82no—
216Q8—82no—
217Q8—162no—
218D4—22no—
219D4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—322no—
230C2×C4D4211yescharacteristic
231C2×C4C22×D811yescharacteristic
232C2×C4#57611yescharacteristic
233C8—22no—
234C8—21no—
235C8—21no—
236C8—42no—
237C8—81no—
238C8—81no—
239C8#51012yes—
240C8C2×#13511yescharacteristic
241C8#57811yescharacteristic
242C22—42no—
243C22—42no—
244C22—42no—
245C22D4×D811yescharacteristic
246C4—21no—
247C4—42no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—162no—
253C4—162no—
254C4#328311yescharacteristic
255C4#329311yescharacteristic
256C4C2×#51011yescharacteristic
257C4#341611yescharacteristic
258C2—82no—
259C2#1997111yescharacteristic
260C2#3384411yescharacteristic
261C2#3385611yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D8D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33923C2semidirect
4. #12523C2semidirect
5. #12456C2semidirect
6. #12448C2semidirect
7. #12446C2semidirect
8. #12397C2semidirect
9. #12394C2semidirect
10. C8×Q32C2semidirect

Maximal subgroups

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

subgroupindex
#339232
#200202
#200062
#125232
#124562
#124482
#124462
#123972
#123942
C8×Q322
#120392