Finite Group Databasebeta

← #6434725

Subgroups — #6434725

Order 512.

2859 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 2859 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
1#62849C211yesmaximal, characteristic, complemented
2#35672C213yesmaximal
3#29659C218yesmaximal, complemented
4#29658C218yesmaximal
5#23566C214yesmaximal, complemented
6#23467C214yesmaximal
7#20691C213yesmaximal, complemented
8#3502—212no—
9#3502—212no—
10#3458C2214yes—
11#3458C2218yes—
12C22×C22⋊Q8C2211yescharacteristic
13C2×#519C2213yes—
14#2484C22112yes—
15#2482C22112yes—
16#2423—212no—
17#2421—212no—
18#2317C2216yes—
19#2310C2213yes—
20#2280C2216yes—
21#2062C22132yes—
22#2030C22116yes—
23#2029C22116yes—
24#1820C22124yes—
25#1740C22112yes—
26C23×Q8C2311yescharacteristic
27C24×C4C2311yescharacteristic
28Q82—216no—
29Q82—216no—
30#557—248no—
31#557—248no—
32#544—212no—
33#544—212no—
34#544—224no—
35#544—224no—
36#543C23116yes—
37#533—224no—
38#533—248no—
39C2×C22⋊Q8—212no—
40#519—224no—
41C2×C4×Q8D4112yes—
42C2×C4×Q8D4112yes—
43C2×#117C2316yes—
44C2×#117C2316yes—
45C2×#117C23112yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4C2312yes—
49#400—212no—
50#399—212no—
51#398—212no—
52#397—212no—
53#391—212no—
54#389—212no—
55#388—224no—
56#387—224no—
57#385C23124yes—
58#384—26no—
59#382—26no—
60#381C23112yes—
61#381C23112yes—
62#380—26no—
63#380—26no—
64#379C23112yes—
65C4×C22⋊C4—212no—
66#342C23148yes—
67C25C2411yescharacteristic
68C22×Q8—28no—
69C22×Q8C2×D4112yes—
70C23×C4—26no—
71C23×C4—28no—
72C23×C4C2411yescharacteristic
73C23×C4C2412yes—
74C23×C4C2412yes—
75C23×C4C2411yescharacteristic
76C4⋊Q8—248no—
77C4⋊Q8—248no—
78C4⋊Q8—224no—
79C4⋊Q8—224no—
80#125—296no—
81#125—248no—
82C22⋊Q8—248no—
83C22⋊Q8—248no—
84C22⋊Q8—416no—
85C4×Q8—232no—
86C4×Q8—232no—
87C4×Q8—216no—
88C4×Q8—216no—
89C4×Q8—248no—
90C4×Q8—248no—
91#117—248no—
92#117—224no—
93#117—224no—
94C2×C4⋊C4—26no—
95C2×C4⋊C4—212no—
96C2×C4⋊C4—26no—
97C2×C4⋊C4—224no—
98C2×C4⋊C4C2×D416yes—
99C2×C4⋊C4C2×D416yes—
100C2×C4⋊C4C2×D4112yes—
101C2×C4⋊C4C2×D4112yes—
102C2×C22⋊C4—212no—
103C2×C22⋊C4—212no—
104C2×C22⋊C4—212no—
105C2×C22⋊C4C24112yes—
106C2×C22⋊C4C24112yes—
107C2×C42—26no—
108C2×C42C2×D4112yes—
109C2×C42C2×D4124yes—
110#95—212no—
111#95—212no—
112#95—212no—
113#95—212no—
114#95—224no—
115C24—23no—
116C24—24no—
117C24—42no—
118C24—81no—
119C24C2511yescharacteristic
120C2×Q8—216no—
121C2×Q8—216no—
122C2×Q8—224no—
123C2×Q8—224no—
124C2×Q8#120116yes—
125C22×C4—28no—
126C22×C4—212no—
127C22×C4—23no—
128C22×C4—26no—
129C22×C4—24no—
130C22×C4—44no—
131C22×C4—412no—
132C22×C4—412no—
133C22×C4—82no—
134C22×C4C22×D4112yes—
135C22×C4C22×D4112yes—
136C22×C4C22×D4112yes—
137C22×C4C22×D416yes—
138C22×C42−1+418yes—
139C4⋊C4—248no—
140C4⋊C4—248no—
141C4⋊C4—224no—
142C4⋊C4—224no—
143C4⋊C4—212no—
144C4⋊C4—212no—
145C4⋊C4—44no—
146C4⋊C4—48no—
147C4⋊C4—412no—
148C4⋊C4—824no—
149C22⋊C4—224no—
150C22⋊C4—224no—
151C22⋊C4—424no—
152C22⋊C4—48no—
153C22⋊C4—812no—
154C42—248no—
155C42—224no—
156C42—412no—
157C23—26no—
158C23—41no—
159C23—41no—
160C23—46no—
161C23—43no—
162C23—41no—
163C23—46no—
164C23—88no—
165C23C23×D413yes—
166C23C2×2−1+414yes—
167Q8—232no—
168Q8—816no—
169Q8—816no—
170C2×C4—224no—
171C2×C4—212no—
172C2×C4—212no—
173C2×C4—412no—
174C2×C4—412no—
175C2×C4—412no—
176C2×C4—412no—
177C2×C4—412no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—412no—
181C2×C4—46no—
182C2×C4—88no—
183C2×C4C2×#12018yes—
184C2×C4C2×#12014yes—
185C2×C4#536112yes—
186C2×C4#536112yes—
187C22—23no—
188C22—23no—
189C22—44no—
190C22—412no—
191C22—42no—
192C22—86no—
193C22—81no—
194C22—81no—
195C22C22×#12011yescharacteristic
196C22C2×#53616yes—
197C4—28no—
198C4—412no—
199C4—412no—
200C4—812no—
201C4—81no—
202C4—82no—
203C4#261418yes—
204C2—26no—
205C2—82no—
206C2C2×#261411yescharacteristic
207C2#6040011yescharacteristic
208C2#6040211yescharacteristic
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×Q8 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62849C2semidirect
3. #29659C2semidirect
5. #23566C2semidirect
7. #20691C2semidirect

Maximal subgroups

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

subgroupindex
#628492
#356722
#296592
#296582
#235662
#234672
#206912