Finite Group Databasebeta

← #505682

Subgroups — #505682

Order 512.

544 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2239C212yesmaximal
2#21537C211yesmaximal, characteristic
3#20807C211yesmaximal, characteristic
4#13510C212yesmaximal, complemented
5#13488C212yesmaximal, complemented
6#13185C212yesmaximal, complemented
7#12875C212yesmaximal
8#12819C212yesmaximal, complemented
9#12786C211yesmaximal, characteristic
10C2×#470C2211yescharacteristic
11C2×C4×Q16C2211yescharacteristic
12C2×#430C2212yes—
13C22×#108—21no—
14#2905—21no—
15C2×#405C2211yescharacteristic
16#2323—21no—
17#2239—28no—
18C2×#365C2214yes—
19#2150C2212yes—
20C2×#362C2214yes—
21C2×#349C2212yes—
22#1971C2211yescharacteristic
23#1944C2211yescharacteristic
24#1933—21no—
25#1892—21no—
26#1849—21no—
27#1849C2212yes—
28#1848—21no—
29#1848C2212yes—
30#1809C2211yescharacteristic
31C4×#106C2211yescharacteristic
32#1763—21no—
33#1387C2214yes—
34#1383C2214yes—
35#1377C2212yes—
36C22×Q16C2311yescharacteristic
37C22×M16—22no—
38C22×M16C2311yescharacteristic
39C23×C8D411yescharacteristic
40C2×C22⋊Q8—21no—
41C2×C4×Q8—21no—
42C2×#117D411yescharacteristic
43C2×M32C2314yes—
44#470—22no—
45#470D414yes—
46C4×Q16—24no—
47#430—24no—
48#430—24no—
49C2×#108—22no—
50C2×#108—22no—
51C2×#108—22no—
52C2×#108—44no—
53C2×#108C2314yes—
54C2×#107C2311yescharacteristic
55C2×#106C2311yescharacteristic
56C2×#106C2311yescharacteristic
57#424—44no—
58#423—22no—
59#423—22no—
60C2×#103—21no—
61C2×#103—21no—
62C2×#103—21no—
63#405—24no—
64C2×#97D411yescharacteristic
65C2×#97C2311yescharacteristic
66C2×C4×C8D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#389—21no—
70#389—41no—
71#388—41no—
72#388—41no—
73#387—41no—
74#380—21no—
75C4×C4⋊C4—21no—
76#365—28no—
77C4⋊C16—24no—
78#362—28no—
79#349—24no—
80#340—21no—
81#340—21no—
82#340—21no—
83C22×Q8—21no—
84C23×C4C2×D411yescharacteristic
85C2×Q16—22no—
86C2×Q16—21no—
87C2×Q16—22no—
88C2×Q16C2×D412yes—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—24no—
92C2×M16—42no—
93C2×M16—44no—
94C2×M16—44no—
95C2×M16C2×D412yes—
96C2×M16#4112yes—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—42no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2411yescharacteristic
105C22⋊Q8—24no—
106C22⋊Q8—42no—
107C4×Q8—42no—
108C4×Q8—42no—
109#117—22no—
110#117—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4SD1611yescharacteristic
118C2×C4⋊C4SD1611yescharacteristic
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4SD1611yescharacteristic
122C2×C22⋊C4SD1611yescharacteristic
123C2×C42—41no—
124C2×C42—41no—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127M32—24no—
128M32—24no—
129C2×C16—24no—
130#108—24no—
131#108—24no—
132#108—44no—
133#108—44no—
134#108—44no—
135#107—22no—
136#106—21no—
137#106—22no—
138#106C2×D412yes—
139C4⋊C8—44no—
140C4⋊C8—44no—
141#103—24no—
142#103—42no—
143#103—42no—
144#97—22no—
145#97—22no—
146C4×C8—22no—
147C4×C8#4114yes—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153C24C2×SD1611yescharacteristic
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4#12112yes—
169C22×C4C2×SD1611yescharacteristic
170C22×C4C2×SD1611yescharacteristic
171C22×C4C2×SD1611yescharacteristic
172C22×C4C2×SD1611yescharacteristic
173C22×C4C2×SD1611yescharacteristic
174C22×C4#13712yes—
175C22×C4C22×D411yescharacteristic
176Q16—24no—
177Q16—42no—
178M16—24no—
179M16—24no—
180M16—44no—
181M16—44no—
182M16—84no—
183C2×C8—28no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—44no—
191C2×C8—84no—
192C2×C8#12112yes—
193C2×C8#12312yes—
194C2×C8AGL(1,ℤ/8)12yes—
195C2×C8C22×D411yescharacteristic
196C2×C8C2×#4112yes—
197C2×C8C2×#4111yescharacteristic
198C2×C82−1+412yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—44no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—82no—
209C4⋊C4—161no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—44no—
213C22⋊C4—161no—
214C42—21no—
215C42—21no—
216C42—82no—
217C42—161no—
218C42#12312yes—
219C42AGL(1,ℤ/8)12yes—
220C16—44no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23#46112yes—
226C23C22×SD1611yescharacteristic
227Q8—42no—
228Q8—42no—
229Q8—81no—
230Q8—81no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4#46112yes—
256C2×C4#48112yes—
257C2×C4#48112yes—
258C2×C4C2×#12111yescharacteristic
259C2×C4C2×#12311yescharacteristic
260C2×C4#53611yescharacteristic
261C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
262C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
263C2×C4C2×#13711yescharacteristic
264C8—22no—
265C8—22no—
266C8—22no—
267C8—84no—
268C8#54712yes—
269C8#55212yes—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22C2×#46111yescharacteristic
278C22C2×#48111yescharacteristic
279C22#322711yescharacteristic
280C4—22no—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—41no—
285C4—82no—
286C4—82no—
287C4—161no—
288C4—161no—
289C4#248611yescharacteristic
290C4#303811yescharacteristic
291C4#309211yescharacteristic
292C4#323911yescharacteristic
293C2—21no—
294C2—21no—
295C2—42no—
296C2#2764511yescharacteristic
297C2#3365811yescharacteristic
298C2#3365911yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13510C2semidirect
5. #13488C2semidirect
6. #13185C2semidirect
8. #12819C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22392
#215372
#208072
#135102
#134882
#131852
#128752
#128192
#127862