Finite Group Databasebeta

← #508598

Subgroups — #508598

Order 512.

499 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 142 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#33207C211yesmaximal, characteristic
2C2×#1945C211yesmaximal, characteristic
3#13543C212yesmaximal, complemented
4#13521C214yesmaximal, complemented
5#13457C212yesmaximal, complemented
6#12928C212yesmaximal
7#12870C212yesmaximal
8#12788C211yesmaximal, characteristic
9C2×#498C2211yescharacteristic
10C22×#108C2211yescharacteristic
11C2×#427C2211yescharacteristic
12C2×#424—21no—
13C22×C4×C8C2211yescharacteristic
14#2857—21no—
15C2×#365C2214yes—
16C2×#364C2212yes—
17#2112C2212yes—
18#1990—21no—
19#1945—21no—
20#1945—21no—
21#1945C2212yes—
22#1945C2212yes—
23#1941—21no—
24#1941C2212yes—
25#1844C2211yescharacteristic
26#1807—21no—
27#1807C2212yes—
28C2×#339—21no—
29#1386C2214yes—
30#1386C2214yes—
31#1383C2214yes—
32#1376C2212yes—
33C22×M16—21no—
34C23×C8D411yescharacteristic
35C23×C8C2311yescharacteristic
36C2×C4⋊Q8—21no—
37C2×C22⋊Q8—41no—
38C2×#117—21no—
39C22×C42D411yescharacteristic
40C2×M32C2314yes—
41#498—42no—
42C2×#108—21no—
43C2×#108—42no—
44C2×#108C2312yes—
45#427—21no—
46#427—21no—
47#427Q812yescomplemented
48#427C2312yes—
49C2×#107Q811yescharacteristic, complemented
50C2×#107Q811yescharacteristic, complemented
51C2×#106—21no—
52C2×#106—21no—
53C2×#106—21no—
54C2×#106C2312yes—
55C2×#106C2312yes—
56#424—22no—
57#424—22no—
58#424—42no—
59C2×C4⋊C8—42no—
60C2×C4×C8—42no—
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—41no—
66#389—22no—
67#389—41no—
68#384—21no—
69#380—22no—
70#380—41no—
71#365—44no—
72#364—42no—
73C22⋊C16—22no—
74C22⋊C16—22no—
75#340—22no—
76#340—22no—
77#339—22no—
78#339—22no—
79C22×Q8—41no—
80C23×C4—21no—
81C23×C4C2×D411yescharacteristic
82C2×M16—22no—
83C2×M16—41no—
84C2×M16—42no—
85C2×M16—42no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8SD1612yes—
92C22×C8SD1612yes—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×Q811yescharacteristic
97C22×C8C2×Q811yescharacteristic
98C22×C8#4111yescharacteristic
99C22×C8#4111yescharacteristic
100C22×C8C2411yescharacteristic
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C22⋊Q8—84no—
104#117—22no—
105#117—22no—
106#117—41no—
107#117—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124M32—24no—
125M32—24no—
126C2×C16—24no—
127#108—24no—
128#108—42no—
129#107—21no—
130#107—21no—
131#107C2×Q812yes—
132#107C2×Q812yes—
133#106—22no—
134#106—22no—
135#106—24no—
136#106—24no—
137#106—42no—
138#106—42no—
139C4⋊C8—44no—
140C4⋊C8—44no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—41no—
144C4×C8—41no—
145#95—42no—
146#95—42no—
147C24AGL(1,ℤ/8)11yescharacteristic
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—82no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—24no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4C22⋊Q811yescharacteristic
165C22×C4C22⋊Q811yescharacteristic
166C22×C4C2×SD1611yescharacteristic
167C22×C4C2×SD1611yescharacteristic
168C22×C4AGL(1,ℤ/8)11yescharacteristic
169C22×C4C22×D411yescharacteristic
170M16—42no—
171M16—42no—
172M16—82no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—82no—
187C2×C8C22⋊Q811yescharacteristic
188C2×C8C22⋊Q811yescharacteristic
189C2×C8C2×SD1612yes—
190C2×C8C2×SD1612yes—
191C2×C8C22×Q811yescharacteristic
192C2×C8C2×#4111yescharacteristic
193C2×C8H(5,𝔽2)12yes—
194C2×C82−1+412yes—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—82no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—84no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42#13712yes—
217C16—44no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—41no—
222C23#47611yescharacteristic
223C23#47611yescharacteristic
224C23C2×AGL(1,ℤ/8)11yescharacteristic
225Q8—164no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—48no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#47611yescharacteristic
254C2×C4#47611yescharacteristic
255C2×C4C2×C22⋊Q811yescharacteristic
256C2×C4#53411yescharacteristic
257C2×C4#53611yescharacteristic
258C2×C4C22×SD1611yescharacteristic
259C2×C4C2×#13711yescharacteristic
260C2×C4#57511yescharacteristic
261C8—21no—
262C8—21no—
263C8—22no—
264C8—22no—
265C8—82no—
266C8#53912yes—
267C8#54412yes—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#223012yes—
278C22C2×#47611yescharacteristic
279C22#319311yescharacteristic
280C22#323411yescharacteristic
281C4—24no—
282C4—21no—
283C4—21no—
284C4—42no—
285C4—82no—
286C4—82no—
287C4—162no—
288C4—162no—
289C4#262711yescharacteristic
290C4#307811yescharacteristic
291C4#318711yescharacteristic
292C4#322911yescharacteristic
293C2—21no—
294C2—21no—
295C2—22no—
296C2—41no—
297C2#2996811yescharacteristic
298C2C2×#223011yescharacteristic
299C2#3363711yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13543C2semidirect
4. #13521C2semidirect
5. #13457C2semidirect
47. #427Q8semidirect
49. C2×#107Q8semidirect
50. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
#332072
C2×#19452
#135432
#135212
#134572
#129282
#128702
#127882