Finite Group Databasebeta

← #507707

Subgroups — #507707

Order 512.

476 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 130 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#33253C211yesmaximal, characteristic
2#21736C211yesmaximal, characteristic, complemented
3#13488C211yesmaximal, characteristic, complemented
4#13487C211yesmaximal, characteristic
5#13368C211yesmaximal, characteristic, complemented
6#13367C211yesmaximal, characteristic
7#13273C212yesmaximal, complemented
8#13260C212yesmaximal
9#13155C212yesmaximal
10#13154C212yesmaximal, complemented
11#12659C211yesmaximal, characteristic, complemented
12#3162C2211yescharacteristic
13C2×#499C2211yescharacteristic
14C2×#495C2211yescharacteristic
15C2×#488—21no—
16#2961—21no—
17#2861—21no—
18C2×#364C2212yes—
19C2×#362C2212yes—
20C2×#361C2212yes—
21#2141C2212yes—
22#2140C2212yes—
23C2×#347C2212yes—
24#1994—21no—
25#1973—21no—
26#1973—21no—
27#1972C2211yescharacteristic
28#1972C2211yescharacteristic
29#1971C2211yescharacteristic
30#1971C2211yescharacteristic
31#1946C2212yes—
32#1945—21no—
33#1849C2211yescharacteristic
34#1773—21no—
35#1772C2212yes—
36C4×#97C2211yescharacteristic
37#1386C2212yes—
38#1386C2212yes—
39#1383C2212yes—
40#1383C2212yes—
41#1377C2211yescharacteristic
42#1377C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46#533—21no—
47C2×#125—21no—
48C2×#124—21no—
49C2×M32C2312yes—
50C2×M32C2312yes—
51#501—24no—
52#499—24no—
53#495—24no—
54#488—44no—
55#446—44no—
56C2×#108—21no—
57C2×#108C2312yes—
58#427—21no—
59#427C2312yes—
60C2×#107—21no—
61C2×#106C2311yescharacteristic
62C2×#106C2311yescharacteristic
63#424—22no—
64#423—21no—
65C2×#103—21no—
66C2×#103—21no—
67C2×#102—21no—
68C2×#102—21no—
69#405—22no—
70C2×#97D411yescharacteristic
71C2×#97D411yescharacteristic
72C2×#97C2311yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—21no—
77#391—21no—
78#390—21no—
79#390—21no—
80#388—44no—
81C43D411yescharacteristic
82#364—24no—
83#362—42no—
84#361—42no—
85#358—24no—
86#357—24no—
87#347—42no—
88#339—22no—
89#339—22no—
90C22×Q8—21no—
91C22×D4—21no—
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—21no—
95C2×Q16—21no—
96C2×SD16—42no—
97C2×SD16—42no—
98C2×SD16—42no—
99C2×D8—22no—
100C2×D8—22no—
101C2×D8—21no—
102C2×D8—21no—
103C2×M16—22no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2411yescharacteristic
108C4⋊Q8—22no—
109#125—22no—
110#125—42no—
111#125—42no—
112#124—42no—
113#124—42no—
114C22⋊Q8—42no—
115C4×Q8—42no—
116#117—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C42—22no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127M32—22no—
128M32—22no—
129M32—22no—
130M32—22no—
131C2×C16—22no—
132C2×C16—22no—
133#108—22no—
134#108—22no—
135#108—42no—
136#107—22no—
137#107—42no—
138#107#4114yes—
139#106—22no—
140#106—22no—
141#106—22no—
142#106—22no—
143C4⋊C8—42no—
144C4⋊C8—44no—
145#103—42no—
146#103—42no—
147#102—42no—
148#102—42no—
149#97—22no—
150#97—22no—
151#97—22no—
152#97—21no—
153#97—21no—
154#97—42no—
155C4×C8—22no—
156C4×C8—42no—
157C4×C8#4114yes—
158#95—42no—
159#95—42no—
160C24—41no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4AGL(1,ℤ/8)12yes—
180C22×C4C22×D411yescharacteristic
181Q16—42no—
182Q16—42no—
183SD16—84no—
184D8—42no—
185D8—42no—
186M16—42no—
187M16—42no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—42no—
194C2×C8#12412yes—
195C2×C8AGL(1,ℤ/8)12yes—
196C2×C8AGL(1,ℤ/8)12yes—
197C2×C8#13712yes—
198C2×C8C2×#4112yes—
199C2×C8C2×#4112yes—
200C2×C8H(5,𝔽2)11yescharacteristic
201C2×C82−1+411yescharacteristic
202C4⋊C4—24no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—84no—
210C22⋊C4—42no—
211C22⋊C4—84no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42—82no—
221C42#12412yes—
222C16—42no—
223C16—42no—
224C23—82no—
225C23—82no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23C2×AGL(1,ℤ/8)11yescharacteristic
230Q8—81no—
231Q8—81no—
232Q8—81no—
233Q8—81no—
234Q8—162no—
235D4—81no—
236D4—81no—
237D4—81no—
238D4—81no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4#52511yescharacteristic
254C2×C4C2×#12411yescharacteristic
255C2×C4#53711yescharacteristic
256C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
257C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
258C2×C4C2×#13711yescharacteristic
259C8—22no—
260C8—41no—
261C8—82no—
262C8#48914yes—
263C8#55612yes—
264C8#55912yes—
265C22—22no—
266C22—161no—
267C22—161no—
268C22—162no—
269C22—162no—
270C22—161no—
271C22#310911yescharacteristic
272C22#313811yescharacteristic
273C22#325411yescharacteristic
274C4—42no—
275C4—42no—
276C4—81no—
277C4—84no—
278C4—161no—
279C4—161no—
280C4—161no—
281C4#259211yescharacteristic
282C4#311211yescharacteristic
283C4C2×#48911yescharacteristic
284C4#325311yescharacteristic
285C2—21no—
286C2—21no—
287C2—321no—
288C2#2935411yescharacteristic
289C2#3369012yes—
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21736C2semidirect
3. #13488C2semidirect
5. #13368C2semidirect
7. #13273C2semidirect
10. #13154C2semidirect
11. #12659C2semidirect

Maximal subgroups

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

subgroupindex
#332532
#217362
#134882
#134872
#133682
#133672
#132732
#132602
#131552
#131542
#126592