Finite Group Databasebeta

← #129204

Subgroups — #129204

Order 512.

1362 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 134 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#60065C211yesmaximal, characteristic
2#12682C211yesmaximal, characteristic, complemented
3#12680C211yesmaximal, characteristic, complemented
4#11419C214yesmaximal, complemented
5C2×#585C411yescharacteristic
6C2×#568C411yescharacteristic
7C22×#131C2211yescharacteristic
8#2880—28no—
9#2879—22no—
10#2879C412yes—
11C2×#408—22no—
12C2×#408C2212yes—
13C2×#349—22no—
14#2112C2212yescomplemented
15C2×C22⋊C16—22no—
16C2×C22⋊C16C2212yescomplemented
17#1314—28no—
18#585—44no—
19#585—44no—
20C2×2−1+4—21no—
21C2×2−1+4—22no—
22C2×H(5,𝔽2)—21no—
23C2×H(5,𝔽2)—22no—
24C2×H(5,𝔽2)C814yes—
25C22×#41—22no—
26C22×#41—24no—
27C22×#41C2×C411yescharacteristic
28C22×#41C2×C412yes—
29#568—44no—
30C2×#131—24no—
31C2×#131—24no—
32C2×#131—44no—
33C22×M16—22no—
34C22×M16C2×C412yes—
35C22×M16C2×C411yescharacteristic
36C22×M16D412yes—
37C23×C8D412yes—
38C23×C8C2311yescharacteristic
39C2×M32—22no—
40C22×C16—22no—
41#408—28no—
42#408—28no—
43#408—24no—
44#408—24no—
45#408—22no—
46#408—22no—
47#408—24no—
48#407—28no—
49#407—24no—
50C2×C22⋊C8—28no—
51C2×C22⋊C8D418yes—
52#349—44no—
53C22⋊C16—24no—
54C22⋊C16—24no—
55C22⋊C16—44no—
562−1+4—48no—
572−1+4—416no—
58H(5,𝔽2)—48no—
59H(5,𝔽2)—48no—
60H(5,𝔽2)—48no—
61H(5,𝔽2)—416no—
62C2×#41—24no—
63C2×#41—24no—
64C2×#41—24no—
65C2×#41—24no—
66C2×#41—24no—
67C2×#41—28no—
68C2×#41—24no—
69C2×#41—24no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—22no—
74C2×#41—22no—
75C2×#41—22no—
76C2×#41—44no—
77C2×#41—44no—
78C2×#41—44no—
79C2×#41—44no—
80C2×#41—416no—
81C2×#41—416no—
82C2×#41—44no—
83C2×#41—44no—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8M1611yescharacteristic
87C22×Q8M1612yes—
88C22×D4—22no—
89C22×D4—24no—
90C22×D4—28no—
91C22×D4—24no—
92C22×D4C2×C812yes—
93C22×D4C2×C814yes—
94C22×D4M1611yescharacteristic
95C22×D4M1612yes—
96C23×C4—24no—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C414yes—
99C23×C4C22×C411yescharacteristic
100#131—416no—
101#131—416no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—44no—
111C2×M16—44no—
112C2×M16—44no—
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—24no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8C22⋊C412yes—
119C22×C8C22⋊C414yes—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D414yes—
123M32—44no—
124C2×C16—44no—
125C2×C16—42no—
126C2×C16—42no—
127C22⋊C8—216no—
128C22⋊C8—28no—
129C22⋊C8—28no—
130C24—24no—
131C24C22⋊C812yes—
132C24C22⋊C814yes—
133C24C22×C811yescharacteristic
134#41—28no—
135#41—28no—
136#41—216no—
137#41—48no—
138#41—44no—
139#41—48no—
140#41—44no—
141#41—48no—
142#41—48no—
143#41—48no—
144#41—48no—
145#41—48no—
146#41—416no—
147#41—432no—
148#41—416no—
149#41—48no—
150#41—44no—
151#41—44no—
152C2×Q8—24no—
153C2×Q8—24no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—416no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—416no—
176C2×D4—432no—
177C2×D4—48no—
178C2×D4—48no—
179C2×D4—48no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4C22⋊C812yes—
195C22×C4C22⋊C814yes—
196C22×C4#9914yes—
197C22×C4#10014yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×M1611yescharacteristic
201C22×C4C2×M1612yes—
202M16—48no—
203M16—44no—
204M16—48no—
205M16—44no—
206M16—44no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—44no—
214C2×C8—48no—
215C2×C8—48no—
216C2×C8—44no—
217C2×C8#12012yes—
218C2×C8#12012yes—
219C16—82no—
220C16—82no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—44no—
227C23—48no—
228C23—44no—
229C23—48no—
230C23—88no—
231C23#32314yes—
232C23C2×C22⋊C812yes—
233C23C2×C22⋊C811yescharacteristic
234Q8—28no—
235Q8—48no—
236Q8—44no—
237Q8—44no—
238D4—216no—
239D4—28no—
240D4—48no—
241D4—44no—
242D4—416no—
243D4—48no—
244D4—48no—
245D4—44no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—416no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4#32314yes—
264C2×C4#37911yescharacteristic
265C2×C4#40712yes—
266C2×C4#40711yescharacteristic
267C2×C4C2×#9912yes—
268C2×C4C2×#10012yes—
269C8—22no—
270C8—22no—
271C8—42no—
272C8—44no—
273C22—22no—
274C22—24no—
275C22—22no—
276C22—24no—
277C22—44no—
278C22—44no—
279C22—44no—
280C22—48no—
281C22—82no—
282C22—82no—
283C22—164no—
284C22—168no—
285C22C2×#32312yes—
286C22#177911yescharacteristic
287C4—24no—
288C4—22no—
289C4—44no—
290C4#146212yes—
291C4#178111yescharacteristic
292C4#178311yescharacteristic
293C2—22no—
294C2—42no—
295C2—44no—
296C2—84no—
297C2#829811yescharacteristic
298C2#1268412yes—
subgroupG/H
centerC2×C4#379
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1779
prime radicalC2×#585C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12682C2semidirect
3. #12680C2semidirect
4. #11419C2semidirect
14. #2112C22semidirect
16. C2×C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#600652
#126822
#126802
#114192