Finite Group Databasebeta

← #244476

Subgroups — #244476

Order 512.

458 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 122 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#22300C211yesmaximal, characteristic, complemented
2#22281C211yesmaximal, characteristic
3#10689C212yesmaximal, complemented
4#10627C212yesmaximal, complemented
5#10546C212yesmaximal
6#10372C212yesmaximal, complemented
7#8257C212yesmaximal, complemented
8#8251C212yesmaximal, complemented
9#8141C211yesmaximal, characteristic, complemented
10#3150C2211yescharacteristic
11#2931C2212yes—
12#2402C2211yescharacteristic
13#1963C2212yes—
14#1940C2212yes—
15#1939—22no—
16#1916—22no—
17#1876—22no—
18#1876—22no—
19#1843C2211yescharacteristic
20#1838C2211yescharacteristic
21C8×C4⋊C4C2212yes—
22#1746C2211yescharacteristic
23#1540C2212yes—
24#1540C2212yes—
25#1502C2212yes—
26#1502C2212yes—
27#1452C2211yescharacteristic
28#1450C2211yescharacteristic
29#1303C2212yes—
30#1303C2212yes—
31#1301C2212yes—
32#1301C2212yes—
33#1275C2212yes—
34#1274C2212yes—
35C2×#135—21no—
36C2×#135—21no—
37C2×#131—22no—
38#532C2312yes—
39#497—22no—
40#496—22no—
41#436—22no—
42#436—22no—
43C8×D4—22no—
44C8×D4—22no—
45#431C2312yes—
46#431C2312yes—
47#423C2312yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2312yes—
51C2×#104—22no—
52C2×#104—22no—
53#408—22no—
54C2×#97C2311yescharacteristic
55C2×C4×C8—22no—
56C2×C4×C8C2311yescharacteristic
57#400—22no—
58#398—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4C2312yes—
61#376C2311yescharacteristic
62#341—21no—
63#341—21no—
64#336—22no—
65#336—22no—
66#336D412yes—
67#336D412yes—
68#332—22no—
69#332—22no—
70#331—22no—
71#331—22no—
72#327—22no—
73#327—22no—
74#327—22no—
75#327—22no—
76#322—21no—
77#322—22no—
78C82—21no—
79C2×#41—22no—
80#135—82no—
81#135—82no—
82C2×Q16—21no—
83C2×Q16—21no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×D8—21no—
87C2×D8—21no—
88#131—82no—
89#131—82no—
90C2×M16—22no—
91C2×M16—42no—
92C22×C8—22no—
93C22×C8—42no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D412yes—
97C4⋊Q8#4112yes—
98#127#4112yes—
99#124—22no—
100#121—22no—
101#121—22no—
102C4×D4—22no—
103C4×D4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C42—22no—
108C2×C42C2×D412yes—
109C2×C42C2411yescharacteristic
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—82no—
117C4⋊C8—82no—
118#104—82no—
119#104—82no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#97—22no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—82no—
128C4×C8#4111yescharacteristic
129C4×C8#4112yes—
130C4×C8#4111yescharacteristic
131#95—22no—
132#95—22no—
133#95—22no—
134#95#4112yes—
135#95#4112yes—
136#41—42no—
137#41—42no—
138#41—42no—
139#41—42no—
140C2×Q8—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—42no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4C22×D411yescharacteristic
150C22×C4C2×#4112yes—
151Q16—81no—
152Q16—81no—
153SD16—82no—
154SD16—82no—
155D8—81no—
156D8—81no—
157M16—42no—
158M16—42no—
159M16—84no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—84no—
173C2×C8#12412yes—
174C2×C8#12412yes—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C42—22no—
183C42—22no—
184C42—42no—
185C42—42no—
186C42C2×#4112yes—
187C42C2×#4111yescharacteristic
188C42C2×#4111yescharacteristic
189C23—42no—
190C23#52511yescharacteristic
191Q8—42no—
192Q8—42no—
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—42no—
209C2×C4—42no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#52511yescharacteristic
213C2×C4C2×#12411yescharacteristic
214C2×C4#52912yes—
215C2×C4#52912yes—
216C2×C4#57612yes—
217C2×C4#57712yes—
218C8—22no—
219C8—22no—
220C8—41no—
221C8—41no—
222C8—82no—
223C8—82no—
224C8—82no—
225C8—82no—
226C22—22no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22#239811yescharacteristic
231C22#311912yes—
232C4—22no—
233C4—22no—
234C4—44no—
235C4—82no—
236C4—82no—
237C4—82no—
238C4#312211yescharacteristic
239C4#312311yescharacteristic
240C4#313612yes—
241C2—22no—
242C2—162no—
243C2#2383011yescharacteristic
244C2#2383312yes—
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22#2398
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
1. #22300C2semidirect
3. #10689C2semidirect
4. #10627C2semidirect
6. #10372C2semidirect
7. #8257C2semidirect
8. #8251C2semidirect
9. #8141C2semidirect

Maximal subgroups

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

subgroupindex
#223002
#222812
#106892
#106272
#105462
#103722
#82572
#82512
#81412