Finite Group Databasebeta

← #391942

Subgroups — #391942

Order 512.

1275 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1275 conjugacy classes of subgroups — too many to draw. Only its 242 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×#2643C211yesmaximal, characteristic
2#14353C211yesmaximal, characteristic, complemented
3#14349C211yesmaximal, characteristic, complemented
4#11203C216yesmaximal, complemented
5#11043C216yesmaximal
6C22×#99C2214yes—
7#2643—24no—
8#2643—24no—
9C2×#401C411yescharacteristic, complemented
10C2×#400C413yescomplemented
11C2×#397C413yescomplemented
12C2×#396C411yescharacteristic, complemented
13C2×#386C2213yes—
14C2×#385C2213yes—
15C2×#380C2216yes—
16C22×#95C2211yescharacteristic
17C2×#342C22112yes—
18#1444—24no—
19#1443—24no—
20C22⋊#95C2216yes—
21C23×D4C2311yescharacteristic
22C24×C4C2313yes—
23C22×C4⋊C4—26no—
24C22×C4⋊C4C2×C413yes—
25C22×C4⋊C4C2×C413yes—
26C22×C22⋊C4C2×C413yes—
27C22×C22⋊C4C2×C413yes—
28C22×C22⋊C4C2314yes—
29C22×C22⋊C4C2314yes—
30C22×C22⋊C4C2313yes—
31C2×#99—412no—
32#401—24no—
33#400—212no—
34#397—212no—
35#396—24no—
36#386—26no—
37#386—26no—
38#385—26no—
39#385—26no—
40#380—412no—
41C2×#95—26no—
42C2×#95—26no—
43C2×#95—26no—
44C2×#95C2×C413yes—
45C2×#95C2×C413yes—
46C2×#95C2×C413yes—
47C2×#95C2×C413yes—
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C413yes—
50C2×#95D414yes—
51C2×#95D414yes—
52#342—412no—
53C25C22×C411yescharacteristic
54C25C2411yescharacteristic
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—26no—
58C22×D4—26no—
59C23×C4—26no—
60C23×C4—26no—
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—83no—
64C23×C4—83no—
65C23×C4C22×C411yescharacteristic
66C23×C4C22×C413yes—
67C23×C4C22×C413yes—
68C23×C4C22×C413yes—
69C23×C4C22×C413yes—
70C23×C4C2×D416yes—
71C23×C4C2×D416yes—
72C2×C4⋊C4—26no—
73C2×C4⋊C4—26no—
74C2×C4⋊C4—412no—
75C2×C4⋊C4—86no—
76C2×C4⋊C4—86no—
77C2×C4⋊C4—83no—
78C2×C4⋊C4—83no—
79C2×C22⋊C4—23no—
80C2×C22⋊C4—23no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—212no—
84C2×C22⋊C4—212no—
85C2×C22⋊C4—212no—
86C2×C22⋊C4—212no—
87C2×C22⋊C4—26no—
88C2×C22⋊C4—26no—
89C2×C22⋊C4—83no—
90C2×C22⋊C4—83no—
91#99—416no—
92#95—26no—
93#95—26no—
94#95—26no—
95#95—26no—
96#95—412no—
97#95—86no—
98#95—86no—
99#95C22⋊C414yes—
100#95C22⋊C4112yes—
101C24—21no—
102C24—26no—
103C24—43no—
104C24—41no—
105C24—43no—
106C24—41no—
107C24C23×C411yescharacteristic
108C24C22×D411yescharacteristic
109C24C22×D411yescharacteristic
110C24H(5,𝔽2)16yes—
111C242−1+416yes—
112C2×D4—43no—
113C2×D4—43no—
114C2×D4—46no—
115C2×D4—44no—
116C2×D4—412no—
117C22×C4—26no—
118C22×C4—26no—
119C22×C4—26no—
120C22×C4—26no—
121C22×C4—212no—
122C22×C4—21no—
123C22×C4—23no—
124C22×C4—23no—
125C22×C4—26no—
126C22×C4—26no—
127C22×C4—26no—
128C22×C4—46no—
129C22×C4—46no—
130C22×C4—46no—
131C22×C4—46no—
132C22×C4—43no—
133C22×C4—43no—
134C22×C4—43no—
135C22×C4—43no—
136C22×C4—43no—
137C22×C4—43no—
138C22×C4—43no—
139C22×C4—43no—
140C22×C4—412no—
141C22×C4—412no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—412no—
145C22×C4—412no—
146C22×C4—812no—
147C22×C4—812no—
148C22×C4C2×C22⋊C416yes—
149C22×C4C2×C22⋊C416yes—
150C4⋊C4—86no—
151C4⋊C4—86no—
152C4⋊C4—824no—
153C22⋊C4—43no—
154C22⋊C4—43no—
155C22⋊C4—46no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C22⋊C4—412no—
159C22⋊C4—412no—
160C22⋊C4—86no—
161C22⋊C4—86no—
162C23—23no—
163C23—23no—
164C23—23no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—46no—
169C23—46no—
170C23—46no—
171C23—46no—
172C23—43no—
173C23—43no—
174C23—41no—
175C23—44no—
176C23—86no—
177C23—86no—
178C23C22×C22⋊C411yescharacteristic
179C23#52016yes—
180C23#53513yes—
181C23#53613yes—
182C23#53716yes—
183C23#56014yes—
184C23#563112yes—
185D4—81no—
186D4—83no—
187D4—84no—
188C2×C4—26no—
189C2×C4—26no—
190C2×C4—26no—
191C2×C4—26no—
192C2×C4—26no—
193C2×C4—26no—
194C2×C4—412no—
195C2×C4—412no—
196C2×C4—43no—
197C2×C4—43no—
198C2×C4—43no—
199C2×C4—43no—
200C2×C4—43no—
201C2×C4—43no—
202C2×C4—43no—
203C2×C4—44no—
204C2×C4—812no—
205C2×C4—86no—
206C2×C4—86no—
207C2×C4—86no—
208C2×C4—86no—
209C2×C4—812no—
210C2×C4—812no—
211C2×C4—812no—
212C2×C4—812no—
213C2×C4—84no—
214C2×C4—84no—
215C22—23no—
216C22—23no—
217C22—26no—
218C22—26no—
219C22—23no—
220C22—26no—
221C22—26no—
222C22—23no—
223C22—44no—
224C22—412no—
225C22—81no—
226C22—83no—
227C22—83no—
228C22—164no—
229C22#231113yes—
230C22#238314yes—
231C22#281811yescharacteristic
232C22#282113yes—
233C22#282314yes—
234C22#288313yes—
235C22#288411yescharacteristic
236C4—46no—
237C4—46no—
238C4—81no—
239C4—83no—
240C4—83no—
241C4—164no—
242C4—164no—
243C2—21no—
244C2—24no—
245C2—24no—
246C2—23no—
247C2—161no—
248C2#1466511yescharacteristic
249C2#2187613yes—
250C2#2187713yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14353C2semidirect
3. #14349C2semidirect
4. #11203C2semidirect
9. C2×#401C4semidirect
10. C2×#400C4semidirect
11. C2×#397C4semidirect
12. C2×#396C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26432
#143532
#143492
#112032
#110432