Finite Group Databasebeta

← #479576

Subgroups — #479576

Order 512.

1109 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1109 conjugacy classes of subgroups — too many to draw. Only its 162 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×#2032C212yesmaximal, complemented
2#11390C211yesmaximal, characteristic, complemented
3#11287C213yesmaximal, complemented
4#11089C216yesmaximal, complemented
5#10980C213yesmaximal
6C2×#543C2211yescharacteristic
7C22×#121C2211yescharacteristic
8C22×#99C2216yes—
9C22×#99C2216yes—
10#2315C2213yes—
11#2032—28no—
12C2×#342C2213yes—
13C2×#342C2216yes—
14C2×#342C2213yes—
15C22⋊#95C2213yes—
16C22⋊#95C2213yes—
17C23×D4Q811yescharacteristic, complemented
18C23×D4C2312yes—
19C24×C4C2311yescharacteristic
20#543—43no—
21#543D414yescomplemented
22C2×C22⋊Q8—43no—
23C2×#121—43no—
24C2×#121—43no—
25C22×C4⋊C4Q811yescharacteristic, complemented
26C22×C22⋊C4Q812yescomplemented
27C22×C22⋊C4C2316yes—
28C22×C22⋊C4C2313yes—
29C22×C22⋊C4C2313yes—
30C2×#99—212no—
31C2×#99—212no—
32C2×#99—46no—
33C2×#99—46no—
34C2×#99—46no—
35C2×#99—46no—
36#385—46no—
37#380—43no—
38C2×#95—43no—
39C2×#95—43no—
40#342—212no—
41#342—26no—
42#342—43no—
43C25C2×Q812yes—
44C25C2411yescharacteristic
45C22×Q8—41no—
46C22×D4—22no—
47C22×D4—26no—
48C22×D4—26no—
49C22×D4—26no—
50C22×D4—26no—
51C22×D4—22no—
52C22×D4—43no—
53C22×D4—82no—
54C23×C4—23no—
55C23×C4—23no—
56C23×C4—23no—
57C23×C4—23no—
58C23×C4—26no—
59C23×C4—82no—
60C23×C4C2×Q812yes—
61C23×C4C2×Q811yescharacteristic
62C23×C4C2×Q811yescharacteristic
63C22⋊Q8—412no—
64C22⋊Q8—412no—
65#121—412no—
66#121—44no—
67C2×C4⋊C4—43no—
68C2×C4⋊C4—43no—
69C2×C4⋊C4—43no—
70C2×C4⋊C4—43no—
71C2×C22⋊C4—26no—
72C2×C22⋊C4—26no—
73C2×C22⋊C4—26no—
74C2×C22⋊C4—26no—
75C2×C22⋊C4—23no—
76C2×C22⋊C4—23no—
77C2×C22⋊C4—26no—
78C2×C22⋊C4—26no—
79C2×C22⋊C4—26no—
80C2×C22⋊C4—26no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—46no—
84C2×C22⋊C4C2×D416yes—
85C2×C22⋊C4#4116yes—
86#99—424no—
87#99—424no—
88#95—46no—
89#95—86no—
90#95—86no—
91C24—26no—
92C24—22no—
93C24—26no—
94C24—81no—
95C24—81no—
96C24—82no—
97C24—82no—
98C24C22×D411yescharacteristic
99C24C22×Q811yescharacteristic
100C24C2×#4113yes—
101C24H(5,𝔽2)13yes—
102C24H(5,𝔽2)13yes—
103C24H(5,𝔽2)13yes—
104C242−1+411yescharacteristic
105C2×Q8—44no—
106C2×Q8—161no—
107C2×Q8—161no—
108C2×D4—44no—
109C2×D4—46no—
110C2×D4—46no—
111C2×D4—46no—
112C2×D4—46no—
113C2×D4—46no—
114C2×D4—46no—
115C2×D4—46no—
116C2×D4—46no—
117C2×D4—42no—
118C2×D4—46no—
119C2×D4—86no—
120C2×D4—86no—
121C22×C4—23no—
122C22×C4—23no—
123C22×C4—26no—
124C22×C4—22no—
125C22×C4—26no—
126C22×C4—26no—
127C22×C4—26no—
128C22×C4—42no—
129C22×C4—46no—
130C22×C4—46no—
131C22×C4—43no—
132C22×C4—43no—
133C22×C4—43no—
134C22×C4—41no—
135C22×C4—43no—
136C22×C4—43no—
137C22×C4—46no—
138C22×C4—46no—
139C22×C4—46no—
140C22×C4—81no—
141C22×C4—83no—
142C22×C4—83no—
143C22×C4—83no—
144C22×C4—83no—
145C22×C4—86no—
146C22×C4—83no—
147C22×C4—83no—
148C22×C4—86no—
149C4⋊C4—44no—
150C4⋊C4—412no—
151C4⋊C4—163no—
152C4⋊C4—163no—
153C22⋊C4—212no—
154C22⋊C4—212no—
155C22⋊C4—48no—
156C22⋊C4—46no—
157C22⋊C4—46no—
158C22⋊C4—46no—
159C22⋊C4—46no—
160C22⋊C4—46no—
161C22⋊C4—46no—
162C23—43no—
163C23—43no—
164C23—41no—
165C23—43no—
166C23—43no—
167C23—41no—
168C23—42no—
169C23—46no—
170C23—46no—
171C23—86no—
172C23—86no—
173C23—81no—
174C23—83no—
175C23—83no—
176C23—81no—
177C23—162no—
178C23—166no—
179C23#53413yes—
180C23#53816yes—
181C23#53913yes—
182C23#54313yes—
183C23D4×Q811yescharacteristic
184C23#55013yes—
185C23#55413yes—
186C23#55513yes—
187C23#56013yes—
188C23H(3,𝔽4)13yes—
189C23H(3,𝔽4)11yescharacteristic
190C23#56213yes—
191Q8—164no—
192D4—88no—
193D4—86no—
194D4—86no—
195D4—82no—
196D4—82no—
197C2×C4—26no—
198C2×C4—43no—
199C2×C4—41no—
200C2×C4—43no—
201C2×C4—43no—
202C2×C4—46no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—46no—
206C2×C4—46no—
207C2×C4—46no—
208C2×C4—82no—
209C2×C4—86no—
210C2×C4—83no—
211C2×C4—83no—
212C2×C4—83no—
213C2×C4—83no—
214C2×C4—86no—
215C2×C4—86no—
216C2×C4—86no—
217C2×C4—86no—
218C2×C4—812no—
219C2×C4—86no—
220C2×C4—163no—
221C2×C4—163no—
222C2×C4—166no—
223C22—23no—
224C22—21no—
225C22—23no—
226C22—21no—
227C22—23no—
228C22—23no—
229C22—43no—
230C22—46no—
231C22—43no—
232C22—43no—
233C22—43no—
234C22—41no—
235C22—43no—
236C22—43no—
237C22—43no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—166no—
242C22—166no—
243C22#273213yes—
244C22#273313yes—
245C22#278413yes—
246C22#278511yescharacteristic
247C22#281211yescharacteristic
248C22#282413yes—
249C22#284011yescharacteristic
250C4—42no—
251C4—86no—
252C4—86no—
253C4—82no—
254C4—81no—
255C4—166no—
256C2—21no—
257C2—23no—
258C2—23no—
259C2—23no—
260C2—23no—
261C2—21no—
262C2—162no—
263C2#1721611yescharacteristic
264C2#3287311yescharacteristic
265C2#3287411yescharacteristic
subgroupG/H
center, socleC22#2840
commutator, Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2032C2semidirect
2. #11390C2semidirect
3. #11287C2semidirect
4. #11089C2semidirect
17. C23×D4Q8semidirect
21. #543D4semidirect
25. C22×C4⋊C4Q8semidirect
26. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20322
#113902
#112872
#110892
#109802