Finite Group Databasebeta

← #6434724

Subgroups — #6434724

Order 512.

3135 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3135 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#62843C211yesmaximal, characteristic, complemented
2#35540C213yesmaximal
3#29657C218yesmaximal, complemented
4#29656C218yesmaximal, complemented
5#23566C214yesmaximal, complemented
6#23467C214yesmaximal, complemented
7#20690C213yesmaximal, complemented
8#3501—212no—
9#3501—212no—
10#3458C2214yes—
11C2×#543C2218yes—
12C22×#121C2211yescharacteristic
13C2×#518C2213yes—
14#2483C22112yes—
15#2422—212no—
16#2420—212no—
17#2405C22112yes—
18#2317C2216yes—
19#2309C2213yes—
20#2280C2216yes—
21#2062C22132yes—
22#2030C22116yes—
23#2029C22116yes—
24#1819C22124yes—
25#1740C22112yes—
26C23×D4C2312yes—
27C23×D4C2311yescharacteristic
28C24×C4C2311yescharacteristic
29#554—248no—
30#554—248no—
31D4×Q8—216no—
32D4×Q8—216no—
33#544—212no—
34#544—212no—
35#543—248no—
36#543C23116yes—
37#533—224no—
38C2×#123—212no—
39C2×C22⋊Q8—224no—
40C2×C22⋊Q8—224no—
41C2×#121—212no—
42#518—224no—
43C2×C4×D4D4112yes—
44C2×C4×D4D4112yes—
45C2×#117C2316yes—
46C2×#117C2316yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C23112yes—
50#397—224no—
51#394—224no—
52#390—212no—
53#389—212no—
54#386—212no—
55#385—212no—
56#385C23124yes—
57#384—26no—
58#382—26no—
59#381—212no—
60#381C23112yes—
61#380—26no—
62#380—26no—
63#379C23112yes—
64#379C23112yes—
65C4×C22⋊C4—212no—
66#342C23148yes—
67C25C2411yescharacteristic
68C25C2412yes—
69C22×Q8—28no—
70C22×Q8—28no—
71C22×D4—212no—
72C22×D4—28no—
73C22×D4—84no—
74C22×D4C2×D4112yes—
75C23×C4—26no—
76C23×C4—28no—
77C23×C4C2×D4112yes—
78C23×C4C2×D4112yes—
79C23×C4C2411yescharacteristic
80C23×C4C2412yes—
81C23×C4C2411yescharacteristic
82C4⋊Q8—224no—
83C4⋊Q8—224no—
84#125—248no—
85#123—824no—
86C22⋊Q8—296no—
87C22⋊Q8—296no—
88C22⋊Q8—248no—
89C22⋊Q8—248no—
90#121—416no—
91C4×Q8—216no—
92C4×Q8—216no—
93C4×D4—248no—
94C4×D4—248no—
95#117—224no—
96#117—224no—
97C2×C4⋊C4—224no—
98C2×C4⋊C4—224no—
99C2×C4⋊C4—26no—
100C2×C4⋊C4—26no—
101C2×C4⋊C4—224no—
102C2×C4⋊C4C2×D416yes—
103C2×C4⋊C4C2×D416yes—
104C2×C22⋊C4—248no—
105C2×C22⋊C4—212no—
106C2×C22⋊C4—212no—
107C2×C22⋊C4—212no—
108C2×C22⋊C4—212no—
109C2×C22⋊C4—212no—
110C2×C22⋊C4C2×D4124yes—
111C2×C22⋊C4C24112yes—
112C2×C22⋊C4C24112yes—
113C2×C42—26no—
114C2×C42C2×D4112yes—
115#95—212no—
116#95—212no—
117#95—224no—
118C24—28no—
119C24—23no—
120C24—24no—
121C24—42no—
122C24—81no—
123C24—82no—
124C24—82no—
125C24C22×D4112yes—
126C24C2511yescharacteristic
127C2×Q8—216no—
128C2×Q8—216no—
129C2×Q8—168no—
130C2×Q8—168no—
131C2×D4—224no—
132C2×D4—224no—
133C2×D4—424no—
134C2×D4—48no—
135C2×D4—812no—
136C2×D4#120116yes—
137C22×C4—224no—
138C22×C4—224no—
139C22×C4—23no—
140C22×C4—212no—
141C22×C4—26no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—412no—
145C22×C4—412no—
146C22×C4—82no—
147C22×C4—812no—
148C22×C4—812no—
149C22×C4C22×D4112yes—
150C22×C4C22×D4112yes—
151C22×C4C22×D416yes—
152C22×C42−1+418yes—
153C4⋊C4—248no—
154C4⋊C4—248no—
155C4⋊C4—212no—
156C4⋊C4—212no—
157C4⋊C4—44no—
158C4⋊C4—412no—
159C4⋊C4—824no—
160C22⋊C4—248no—
161C22⋊C4—224no—
162C22⋊C4—224no—
163C22⋊C4—424no—
164C22⋊C4—48no—
165C22⋊C4—812no—
166C22⋊C4—812no—
167C22⋊C4—812no—
168C42—224no—
169C42—412no—
170C23—224no—
171C23—26no—
172C23—41no—
173C23—41no—
174C23—412no—
175C23—412no—
176C23—46no—
177C23—43no—
178C23—41no—
179C23—46no—
180C23—88no—
181C23—168no—
182C23C2×#12018yes—
183C23C23×D413yes—
184C23C2×2−1+414yes—
185Q8—816no—
186Q8—816no—
187D4—232no—
188D4—816no—
189C2×C4—212no—
190C2×C4—212no—
191C2×C4—412no—
192C2×C4—412no—
193C2×C4—412no—
194C2×C4—412no—
195C2×C4—412no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—412no—
199C2×C4—46no—
200C2×C4—824no—
201C2×C4—824no—
202C2×C4—88no—
203C2×C4C2×#12014yes—
204C2×C4#536112yes—
205C2×C4#536112yes—
206C22—23no—
207C22—23no—
208C22—28no—
209C22—28no—
210C22—28no—
211C22—44no—
212C22—412no—
213C22—42no—
214C22—824no—
215C22—86no—
216C22—81no—
217C22—81no—
218C22C22×#12011yescharacteristic
219C22C2×#53616yes—
220C4—412no—
221C4—412no—
222C4—812no—
223C4—81no—
224C4—82no—
225C4#261418yes—
226C2—26no—
227C2—48no—
228C2—82no—
229C2C2×#261411yescharacteristic
230C2#6039911yescharacteristic
231C2#6040111yescharacteristic
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×D4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62843C2semidirect
3. #29657C2semidirect
4. #29656C2semidirect
5. #23566C2semidirect
6. #23467C2semidirect
7. #20690C2semidirect

Maximal subgroups

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

subgroupindex
#628432
#355402
#296572
#296562
#235662
#234672
#206902