Finite Group Databasebeta

← #326581

Subgroups — #326581

Order 512.

464 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 126 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#22308C211yesmaximal, characteristic, complemented
2#22282C211yesmaximal, characteristic
3#10665C212yesmaximal, complemented
4#10370C212yesmaximal, complemented
5#9750C212yesmaximal
6#9629C212yesmaximal, complemented
7#8880C212yesmaximal, complemented
8#8878C212yesmaximal, complemented
9#8783C211yesmaximal, characteristic, complemented
10#3147C2211yescharacteristic
11#2932C2212yes—
12#2399C2211yescharacteristic
13#1984—22no—
14#1963C2212yes—
15#1940C2212yes—
16#1916—22no—
17#1876—22no—
18#1876—22no—
19#1837C2211yescharacteristic
20#1832C2211yescharacteristic
21C8×C4⋊C4C2212yes—
22#1746C2211yescharacteristic
23#1571C2212yes—
24#1570C2212yes—
25#1564C2212yes—
26#1502C2212yes—
27#1502C2212yes—
28#1480C2212yes—
29#1480C2212yes—
30#1303C2212yes—
31#1303C2212yes—
32#1281C2211yescharacteristic
33#1281C2211yescharacteristic
34#1274C2212yes—
35C2×#135—22no—
36C2×#131—22no—
37#532C2312yes—
38C2×C4×Q8—21no—
39#495—22no—
40#494Q811yescharacteristic, complemented
41#493Q811yescharacteristic, complemented
42#492Q812yescomplemented
43#435—42no—
44#435—42no—
45#431C2312yes—
46#431C2312yes—
47#423C2312yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×#104—22no—
51C2×#104—22no—
52#408—22no—
53C2×#97C2311yescharacteristic
54C2×C4×C8—22no—
55C2×C4×C8C2311yescharacteristic
56#398—22no—
57#395—22no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4C2312yes—
60#376C2311yescharacteristic
61#341—22no—
62#336—22no—
63#336—22no—
64#336—21no—
65#336—21no—
66#336D412yes—
67#336D412yes—
68#335—22no—
69#335—42no—
70#334—22no—
71#334—42no—
72#327—22no—
73#327—22no—
74#327—22no—
75#327—22no—
76D4⋊C8—42no—
77D4⋊C8—42no—
78C2×#41—22no—
79C22×Q8—21no—
80#135—82no—
81#135—82no—
82C2×Q16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×D8—22no—
86#131—82no—
87#131—82no—
88C2×M16—22no—
89C2×M16—42no—
90C22×C8—22no—
91C22×C8—42no—
92C22×C8C2×D412yes—
93C22×C8C2×D412yes—
94C4⋊Q8C2×Q812yes—
95#127C2×Q812yes—
96#124—22no—
97#121—22no—
98#121—22no—
99C4×Q8—82no—
100C4×D4—22no—
101C4×D4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C42—22no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2411yescharacteristic
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—82no—
117#104—82no—
118#104—82no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—44no—
122#97—22no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8—82no—
126C4×C8C2×Q811yescharacteristic
127C4×C8C2×Q811yescharacteristic
128C4×C8#4112yes—
129#95—21no—
130#95—21no—
131#95—22no—
132#95#4112yes—
133#95#4112yes—
134#41—42no—
135#41—42no—
136#41—42no—
137#41—42no—
138C2×Q8—22no—
139C2×Q8—44no—
140C2×Q8—81no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—42no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4C22×D411yescharacteristic
151C22×C4C2×#4112yes—
152Q16—82no—
153SD16—82no—
154SD16—82no—
155D8—82no—
156M16—42no—
157M16—42no—
158M16—84no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—84no—
171C2×C8C22⋊Q812yes—
172C2×C8C22⋊Q812yes—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—22no—
182C42—22no—
183C42—42no—
184C42—42no—
185C42C22×Q811yescharacteristic
186C42C2×#4111yescharacteristic
187C42H(5,𝔽2)12yes—
188C23—42no—
189C23#52511yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—84no—
193D4—42no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—82no—
198D4—82no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4C2×C22⋊Q811yescharacteristic
214C2×C4#53511yescharacteristic
215C2×C4#55412yes—
216C2×C4#55912yes—
217C2×C4#57612yes—
218C2×C4#57712yes—
219C8—22no—
220C8—22no—
221C8—42no—
222C8—82no—
223C8—82no—
224C8—82no—
225C8—82no—
226C22—22no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22#258011yescharacteristic
231C22#312211yescharacteristic
232C22#312611yescharacteristic
233C4—22no—
234C4—22no—
235C4—44no—
236C4—82no—
237C4—81no—
238C4—81no—
239C4—82no—
240C4#308112yes—
241C4#322412yes—
242C2—21no—
243C2—21no—
244C2—162no—
245C2#2926311yescharacteristic
246C2#2927012yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radical#3147C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22308C2semidirect
3. #10665C2semidirect
4. #10370C2semidirect
6. #9629C2semidirect
7. #8880C2semidirect
8. #8878C2semidirect
9. #8783C2semidirect
40. #494Q8semidirect
41. #493Q8semidirect
42. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#223082
#222822
#106652
#103702
#97502
#96292
#88802
#88782
#87832