Finite Group Databasebeta

← #331089

Subgroups — #331089

Order 512.

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

normal subgroups only

This group has 443 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#22237C211yesmaximal, characteristic
3#10633C212yesmaximal, complemented
4#10368C211yesmaximal, characteristic
5#10367C211yesmaximal, characteristic, complemented
6#10295C211yesmaximal, characteristic, complemented
7#10287C211yesmaximal, characteristic
8#10222C212yesmaximal
9#9289C212yesmaximal
10#9288C212yesmaximal, complemented
11#8802C211yesmaximal, characteristic, complemented
12#3230C2212yes—
13#3206C2212yes—
14#3135C2211yescharacteristic
15#2961C2211yescharacteristic
16#2932C2211yescharacteristic
17#2864C2211yescharacteristic
18#2862C2211yescharacteristic
19#2101—22no—
20#2101—22no—
21#1994C2211yescharacteristic
22#1993C2212yes—
23#1993C2212yes—
24#1962C2211yescharacteristic
25#1929—22no—
26#1866—22no—
27#1849C2211yescharacteristic
28#1745C2211yescharacteristic
29#1549C2212yes—
30#1548C2212yes—
31#1518C2212yes—
32#1517C2212yes—
33#1286C2212yes—
34#1286C2212yes—
35#1281C2212yes—
36#1276C2211yescharacteristic
37#1276C2211yescharacteristic
38#1275C2212yes—
39C2×#131—21no—
40#533C2311yescharacteristic
41#532C2311yescharacteristic
42C2×C4⋊Q8C2312yes—
43#500—22no—
44#492—22no—
45#491—21no—
46#490—21no—
47#489—22no—
48#487—22no—
49#487—22no—
50#484—42no—
51#477—42no—
52#461—42no—
53#448—42no—
54#446—42no—
55#435—42no—
56#431C2312yes—
57#431C2312yes—
58C2×#108—22no—
59#424—21no—
60#423C2311yescharacteristic
61#423C2311yescharacteristic
62C2×#104—22no—
63#416—22no—
64#408—21no—
65#405—21no—
66C2×#97C2311yescharacteristic
67C2×#97C2311yescharacteristic
68#391—42no—
69#389—22no—
70#387—42no—
71#382—22no—
72#382C2312yes—
73C43C2311yescharacteristic
74#340—22no—
75#340—22no—
76#340—24no—
77#339—22no—
78#336D412yes—
79#336D412yes—
80#333—42no—
81#332—42no—
82#328—42no—
83#327—42no—
84C2×#41—21no—
85C22×Q8—22no—
86C2×SD16—44no—
87#131—82no—
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—41no—
91C22×C8—41no—
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8#4111yescharacteristic
100C4⋊Q8#4112yes—
101#127#4111yescharacteristic
102#125—21no—
103#124—21no—
104C22⋊Q8—22no—
105#121—22no—
106C4×Q8—22no—
107C4×D4—22no—
108#117—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—42no—
113C2×C42—22no—
114C2×C42C2×D412yes—
115C2×C42C2411yescharacteristic
116#108—82no—
117#107—44no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—42no—
123#104—82no—
124#103—24no—
125#103—44no—
126#103—44no—
127#102—24no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—42no—
131#97—21no—
132#97—21no—
133#97—22no—
134#97—41no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—41no—
138#95—22no—
139#95—22no—
140#95—44no—
141#95—42no—
142#41—42no—
143#41—42no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—84no—
149C2×D4—21no—
150C2×D4—22no—
151C2×D4—41no—
152C22×C4—22no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4C22×D411yescharacteristic
157C22×C4H(5,𝔽2)12yes—
158SD16—164no—
159M16—42no—
160M16—42no—
161M16—82no—
162C2×C8—22no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—82no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—84no—
185C22⋊C4—41no—
186C22⋊C4—42no—
187C42—22no—
188C42—22no—
189C42—41no—
190C42—82no—
191C42C2×#4111yescharacteristic
192C42C2×#4111yescharacteristic
193C42H(5,𝔽2)12yes—
194C23—41no—
195C23#53511yescharacteristic
196Q8—42no—
197Q8—82no—
198Q8—84no—
199Q8—162no—
200D4—42no—
201D4—82no—
202D4—82no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4#52511yescharacteristic
216C2×C4#53511yescharacteristic
217C2×C4#53812yes—
218C2×C4#53812yes—
219C2×C4#57712yes—
220C2×C4#57812yes—
221C8—42no—
222C8—41no—
223C8—41no—
224C8—82no—
225C8—82no—
226C8—81no—
227C8—81no—
228C22—22no—
229C22—81no—
230C22—82no—
231C22#263111yescharacteristic
232C22#322512yes—
233C4—22no—
234C4—22no—
235C4—82no—
236C4—81no—
237C4—82no—
238C4—162no—
239C4—162no—
240C4#312012yes—
241C4#322211yescharacteristic
242C4#322311yescharacteristic
243C2—22no—
244C2—161no—
245C2#3005411yescharacteristic
246C2#3005612yes—
subgroupG/H
center, socleC22#2631
commutator, Frattini, cosocleC2×C42C24
prime radical#532C23
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. #10633C2semidirect
5. #10367C2semidirect
6. #10295C2semidirect
10. #9288C2semidirect
11. #8802C2semidirect

Maximal subgroups

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

subgroupindex
#302342
#222372
#106332
#103682
#103672
#102952
#102872
#102222
#92892
#92882
#88022