Finite Group Databasebeta

← #6434719

Subgroups — #6434719

Order 512.

2981 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 2981 conjugacy classes of subgroups — too many to draw. Only its 610 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
1C2×#3458C211yesmaximal, characteristic, complemented
2#35097C213yesmaximal
3#29659C218yesmaximal, complemented
4#29658C218yesmaximal, complemented
5#23565C214yesmaximal, complemented
6#23468C214yesmaximal
7C2×#1820C213yesmaximal, complemented
8#3458—212no—
9#3458C2218yes—
10C2×#544—26no—
11C2×#543C2214yes—
12C2×#533—26no—
13C22×#123C2211yescharacteristic
14C22×#117C2213yes—
15#2853—26no—
16#2623—26no—
17#2498—212no—
18#2484C22112yes—
19#2482C22112yes—
20#2315C2216yes—
21#2308C2216yes—
22C2×#381C2213yes—
23#2062C22132yes—
24#2030C22116yes—
25#2029C22116yes—
26#1820C22124yes—
27C2×#342C22112yes—
28C23×D4C2311yescharacteristic
29C24×C4C2311yescharacteristic
30#544—248no—
31#544—224no—
32#544—224no—
33#543—224no—
34#543C23116yes—
35#533—248no—
36#533—248no—
37C2×C4⋊Q8—212no—
38C2×#125—212no—
39C2×#123—23no—
40C2×#123—23no—
41C2×#123—26no—
42C2×C22⋊Q8—224no—
43C2×C4×Q8—28no—
44C2×#117—26no—
45C2×#117—26no—
46C2×#117—212no—
47C2×#117D4112yes—
48C2×#117C23112yes—
49C22×C4⋊C4D416yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4C2311yescharacteristic
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2316yes—
54C22×C22⋊C4C2316yes—
55C22×C42D416yes—
56#400—212no—
57#399—212no—
58#398—212no—
59#397—212no—
60#388—224no—
61#387—224no—
62#386—212no—
63#386—212no—
64#385—212no—
65#385C23124yes—
66#381—23no—
67#381—23no—
68#381C23112yes—
69#381C23112yes—
70#380—212no—
71#379C23112yes—
72C2×#95—26no—
73#342C23148yes—
74C25C2411yescharacteristic
75C25C2411yescharacteristic
76C22×Q8—28no—
77C22×D4—23no—
78C22×D4—23no—
79C22×D4—28no—
80C23×C4—28no—
81C23×C4—26no—
82C23×C4C2×D413yes—
83C23×C4C2×D416yes—
84C23×C4C2×D413yes—
85C23×C4C2×D4112yes—
86C23×C4C2411yescharacteristic
87C23×C4C2412yes—
88C23×C4C2412yes—
89C4⋊Q8—248no—
90C4⋊Q8—248no—
91#125—296no—
92#123—41no—
93#123—43no—
94#123—46no—
95#123—46no—
96C22⋊Q8—296no—
97C22⋊Q8—296no—
98C4×Q8—264no—
99#117—224no—
100#117—224no—
101#117—224no—
102#117—224no—
103C2×C4⋊C4—224no—
104C2×C4⋊C4—212no—
105C2×C4⋊C4—212no—
106C2×C4⋊C4—26no—
107C2×C4⋊C4—26no—
108C2×C4⋊C4—224no—
109C2×C4⋊C4C2×D4112yes—
110C2×C4⋊C4C2×D4112yes—
111C2×C22⋊C4—212no—
112C2×C22⋊C4—212no—
113C2×C22⋊C4—224no—
114C2×C22⋊C4—23no—
115C2×C22⋊C4—23no—
116C2×C22⋊C4—26no—
117C2×C22⋊C4—26no—
118C2×C22⋊C4—26no—
119C2×C22⋊C4—224no—
120C2×C22⋊C4C2×D4112yes—
121C2×C22⋊C4C24112yes—
122C2×C22⋊C4C24112yes—
123C2×C42—26no—
124C2×C42—26no—
125C2×C42—212no—
126C2×C42C2×D4124yes—
127#95—212no—
128#95—212no—
129#95—224no—
130#95—224no—
131C24—23no—
132C24—24no—
133C24—82no—
134C24—82no—
135C24C22×D413yes—
136C24C22×D413yes—
137C242−1+418yes—
138C24C2511yescharacteristic
139C2×Q8—232no—
140C2×Q8—816no—
141C2×D4—43no—
142C2×D4—41no—
143C2×D4—412no—
144C2×D4—412no—
145C22×C4—212no—
146C22×C4—212no—
147C22×C4—224no—
148C22×C4—28no—
149C22×C4—212no—
150C22×C4—212no—
151C22×C4—212no—
152C22×C4—23no—
153C22×C4—26no—
154C22×C4—24no—
155C22×C4—412no—
156C22×C4—82no—
157C22×C4—812no—
158C22×C4—812no—
159C22×C4#12011yescharacteristic
160C22×C4#12013yes—
161C22×C4#12016yes—
162C22×C4#12016yes—
163C22×C4C22×D416yes—
164C22×C4C22×D416yes—
165C22×C4C22×D4124yes—
166C4⋊C4—296no—
167C4⋊C4—224no—
168C4⋊C4—224no—
169C4⋊C4—42no—
170C4⋊C4—46no—
171C4⋊C4—824no—
172C22⋊C4—224no—
173C22⋊C4—224no—
174C22⋊C4—224no—
175C22⋊C4—43no—
176C22⋊C4—41no—
177C22⋊C4—412no—
178C22⋊C4—42no—
179C22⋊C4—46no—
180C22⋊C4—824no—
181C42—224no—
182C42—224no—
183C23—46no—
184C23—46no—
185C23—42no—
186C23—412no—
187C23—41no—
188C23—46no—
189C23—88no—
190C23—164no—
191C23C2×#12013yes—
192C23C2×#12011yescharacteristic
193C23#536112yes—
194C23#536112yes—
195C23C23×D413yes—
196C23C2×2−1+414yes—
197Q8—832no—
198D4—86no—
199D4—82no—
200C2×C4—22no—
201C2×C4—26no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—212no—
205C2×C4—212no—
206C2×C4—224no—
207C2×C4—424no—
208C2×C4—424no—
209C2×C4—424no—
210C2×C4—412no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—412no—
214C2×C4—46no—
215C2×C4—812no—
216C2×C4—1612no—
217C2×C4C2×#12012yes—
218C2×C4C2×#12016yes—
219C22—22no—
220C22—26no—
221C22—26no—
222C22—412no—
223C22—44no—
224C22—412no—
225C22—812no—
226C22—81no—
227C22—82no—
228C22#261416yes—
229C22#261412yes—
230C22C22×#12011yescharacteristic
231C22C2×#53616yes—
232C4—22no—
233C4—26no—
234C4—424no—
235C4—812no—
236C4—81no—
237C4—82no—
238C2—26no—
239C2—22no—
240C2—26no—
241C2—161no—
242C2C2×#261411yescharacteristic
243C2#6039812yes—
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3458C2semidirect
3. #29659C2semidirect
4. #29658C2semidirect
5. #23565C2semidirect
7. C2×#1820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
#350972
#296592
#296582
#235652
#234682
C2×#18202