Finite Group Databasebeta

← #508515

Subgroups — #508515

Order 512.

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

normal subgroups only

This group has 479 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#33133C211yesmaximal, characteristic
2#20807C211yesmaximal, characteristic
3#13541C212yesmaximal, complemented
4#13536C212yesmaximal, complemented
5#13510C212yesmaximal, complemented
6#13488C212yesmaximal, complemented
7#13003C212yesmaximal
8#12928C212yesmaximal
9#12847C211yesmaximal, characteristic
10C2×#467C2211yescharacteristic
11C2×#430—21no—
12C22×#108C2212yes—
13#2905—21no—
14C2×#405C2211yescharacteristic
15#2625—21no—
16C2×#365C2214yes—
17C2×#362C2214yes—
18#2112C2212yes—
19#1971C2211yescharacteristic
20#1971C2211yescharacteristic
21#1944C2211yescharacteristic
22#1944C2211yescharacteristic
23#1933—21no—
24#1933—21no—
25#1894—21no—
26#1849C2212yes—
27#1849C2212yes—
28#1848—21no—
29#1848—21no—
30#1807C2211yescharacteristic
31#1387C2214yes—
32#1383C2214yes—
33#1377C2212yes—
34#1377C2212yes—
35C22×Q16C2311yescharacteristic
36C22×M16—22no—
37C22×M16D411yescharacteristic
38C23×C8C2311yescharacteristic
39C2×C22⋊Q8—21no—
40C2×#117D411yescharacteristic
41C2×M32C2314yes—
42#467—24no—
43#430—44no—
44C2×#108—22no—
45C2×#108—22no—
46C2×#108—22no—
47C2×#108—44no—
48C2×#108C2314yes—
49C2×#107C2311yescharacteristic
50C2×#106C2311yescharacteristic
51C2×#106C2311yescharacteristic
52#424—44no—
53#423—22no—
54#423—22no—
55C2×#103—21no—
56C2×#103—21no—
57C2×#103—21no—
58#405—24no—
59C2×#97C2311yescharacteristic
60C2×#97C2311yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63#400—41no—
64#398—41no—
65#397—41no—
66#397—41no—
67#391—21no—
68#391—21no—
69#389—21no—
70#389—21no—
71#380—21no—
72#365—44no—
73#362—44no—
74C22⋊C16—24no—
75#340—21no—
76#340—21no—
77#340—21no—
78C22×Q8—21no—
79C23×C4C2×D411yescharacteristic
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—24no—
87C2×M16—42no—
88C2×M16—44no—
89C2×M16—44no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—42no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8#4114yes—
97C22×C8C2411yescharacteristic
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100#117—22no—
101#117—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4SD1611yescharacteristic
109C2×C4⋊C4SD1611yescharacteristic
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4SD1611yescharacteristic
113C2×C22⋊C4SD1611yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116M32—24no—
117M32—24no—
118C2×C16—24no—
119#108—24no—
120#108—24no—
121#108—44no—
122#108—44no—
123#108—44no—
124#107—22no—
125#106—22no—
126#106—22no—
127C4⋊C8—44no—
128C4⋊C8—44no—
129#103—42no—
130#103—42no—
131#103—42no—
132#97—22no—
133#97—22no—
134C4×C8—22no—
135C4×C8—22no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—41no—
142#95—41no—
143C24C2×SD1611yescharacteristic
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4#12312yes—
159C22×C4C2×SD1611yescharacteristic
160C22×C4C2×SD1611yescharacteristic
161C22×C4C2×SD1611yescharacteristic
162C22×C4C2×SD1611yescharacteristic
163C22×C4C2×SD1611yescharacteristic
164C22×C4AGL(1,ℤ/8)12yes—
165C22×C4C22×D411yescharacteristic
166Q16—44no—
167M16—28no—
168M16—44no—
169M16—44no—
170M16—84no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—28no—
178C2×C8—44no—
179C2×C8—84no—
180C2×C8#12312yes—
181C2×C8AGL(1,ℤ/8)12yes—
182C2×C8#13712yes—
183C2×C8C2×#4112yes—
184C2×C8H(5,𝔽2)12yes—
185C2×C82−1+412yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—82no—
195C4⋊C4—161no—
196C4⋊C4—161no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—82no—
200C22⋊C4—161no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C16—44no—
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23#48112yes—
211C23C22×SD1611yescharacteristic
212Q8—81no—
213Q8—81no—
214Q8—81no—
215Q8—81no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4#48112yes—
243C2×C4C2×#12311yescharacteristic
244C2×C4#53511yescharacteristic
245C2×C4#53611yescharacteristic
246C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
247C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
248C2×C4C2×#13711yescharacteristic
249C8—22no—
250C8—22no—
251C8—22no—
252C8—84no—
253C8#53914yes—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—81no—
260C22—81no—
261C22C2×#48111yescharacteristic
262C22#320611yescharacteristic
263C22#322711yescharacteristic
264C4—22no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—161no—
270C4—161no—
271C4—161no—
272C4—161no—
273C4#262311yescharacteristic
274C4#309211yescharacteristic
275C4#321711yescharacteristic
276C4#323811yescharacteristic
277C2—21no—
278C2—21no—
279C2—42no—
280C2#2990711yescharacteristic
281C2#3365811yescharacteristic
282C2#3365911yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13541C2semidirect
4. #13536C2semidirect
5. #13510C2semidirect
6. #13488C2semidirect

Maximal subgroups

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

subgroupindex
#331332
#208072
#135412
#135362
#135102
#134882
#130032
#129282
#128472