Finite Group Databasebeta

← #507994

Subgroups — #507994

Order 512.

411 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 120 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#33260C211yesmaximal, characteristic, complemented
2#21296C211yesmaximal, characteristic
3#13543C211yesmaximal, characteristic, complemented
4#13533C211yesmaximal, characteristic
5#13509C211yesmaximal, characteristic, complemented
6#13485C211yesmaximal, characteristic
7#13174C212yesmaximal, complemented
8#13153C212yesmaximal
9#13114C212yesmaximal, complemented
10#12943C212yesmaximal
11#12596C211yesmaximal, characteristic, complemented
12#2943C2211yescharacteristic
13#2922—21no—
14#2647—21no—
15#2159C2212yes—
16C2×#365C2212yes—
17C2×#360C2212yes—
18#2139C2212yes—
19C2×#347C2212yes—
20#1975C2211yescharacteristic
21#1974C2211yescharacteristic
22#1946C2211yescharacteristic
23#1945C2211yescharacteristic
24#1944C2212yes—
25#1943C2212yes—
26#1941—21no—
27#1941—21no—
28#1938—21no—
29#1937—21no—
30#1846C2211yescharacteristic
31#1844C2211yescharacteristic
32#1836—21no—
33#1778C2211yescharacteristic
34#1759—21no—
35#1726C2211yescharacteristic
36#1387C2212yes—
37#1387C2212yes—
38#1386C2212yes—
39#1383C2212yes—
40#1377C2212yes—
41#1376C2211yescharacteristic
42#1376C2211yescharacteristic
43C2×#135C2311yescharacteristic
44C4×#41—21no—
45C2×M32C2312yes—
46C2×M32C2312yes—
47#442—21no—
48#441—21no—
49#440—22no—
50#433—42no—
51#431—42no—
52C2×#108C2311yescharacteristic
53C2×#108C2311yescharacteristic
54#427C2311yescharacteristic
55C2×#107C2312yes—
56C2×#106—21no—
57C2×#106—21no—
58C2×#106C2312yes—
59#424—21no—
60#424—21no—
61C2×#104—21no—
62C2×#104—21no—
63#416—21no—
64#408—21no—
65#408—21no—
66C4×M16—21no—
67C2×#97D411yescharacteristic
68C2×#97C2311yescharacteristic
69C2×#97C2311yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#400—42no—
74#398—42no—
75#389—22no—
76#389—22no—
77#384—21no—
78#382—21no—
79#376D411yescharacteristic
80#367—22no—
81#366—22no—
82#365—24no—
83#360—24no—
84#358—22no—
85#357—22no—
86#347—24no—
87#341—21no—
88#340—22no—
89#340—22no—
90#339—21no—
91#339—21no—
92C2×#41—21no—
93#135—22no—
94#135—22no—
95C2×Q16#4111yescharacteristic
96C2×SD16—21no—
97C2×D8#4111yescharacteristic
98C2×M16—21no—
99C2×M16—21no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4×Q8—22no—
105C4×D4—22no—
106C4×D4—42no—
107#117—21no—
108#117—41no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C42—41no—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118M32—22no—
119M32—22no—
120M32—22no—
121M32#4114yes—
122C2×C16—22no—
123C2×C16—22no—
124#108—22no—
125#108—22no—
126#107—22no—
127#107—22no—
128#107#4112yes—
129#106—21no—
130#106—22no—
131#106—22no—
132#106—42no—
133#106—42no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136#104—42no—
137#104—42no—
138#103—22no—
139#102—22no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142#97—22no—
143#97—21no—
144#97—21no—
145#97—22no—
146#97—41no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—41no—
150C4×C8—41no—
151#95—22no—
152#95—42no—
153#95—42no—
154#95—42no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8—21no—
160C2×D4—21no—
161C2×D4—41no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4AGL(1,ℤ/8)12yes—
169C22×C4C22×D411yescharacteristic
170Q16—22no—
171SD16—42no—
172D8—22no—
173M16—41no—
174M16—41no—
175M16—41no—
176M16—41no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8AGL(1,ℤ/8)12yes—
188C2×C8C2×#4111yescharacteristic
189C2×C8C2×#4111yescharacteristic
190C2×C8C2×#4112yes—
191C2×C82−1+412yes—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—162no—
199C4⋊C4—162no—
200C4⋊C4—162no—
201C22⋊C4—42no—
202C22⋊C4—41no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42—41no—
210C42#13712yes—
211C16—24no—
212C16—42no—
213C23—41no—
214C23C2×AGL(1,ℤ/8)11yescharacteristic
215Q8—41no—
216Q8—41no—
217D4—41no—
218D4—41no—
219D4—81no—
220D4—81no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—84no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#52511yescharacteristic
239C2×C4#53211yescharacteristic
240C2×C4#53611yescharacteristic
241C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
242C2×C4C2×#13711yescharacteristic
243C2×C4#57511yescharacteristic
244C8—22no—
245C8—41no—
246C8—41no—
247C8—81no—
248C8—81no—
249C8#55212yes—
250C8#55212yes—
251C22—22no—
252C22—81no—
253C22—82no—
254C22#310911yescharacteristic
255C22#317111yescharacteristic
256C22#323911yescharacteristic
257C4—42no—
258C4—42no—
259C4—81no—
260C4—81no—
261C4—82no—
262C4—162no—
263C4—162no—
264C4#261111yescharacteristic
265C4#311611yescharacteristic
266C4#318211yescharacteristic
267C4#323411yescharacteristic
268C2—21no—
269C2—21no—
270C2—161no—
271C2#2958011yescharacteristic
272C2#3369112yes—
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#135C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33260C2semidirect
3. #13543C2semidirect
5. #13509C2semidirect
7. #13174C2semidirect
9. #13114C2semidirect
11. #12596C2semidirect

Maximal subgroups

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

subgroupindex
#332602
#212962
#135432
#135332
#135092
#134852
#131742
#131532
#131142
#129432
#125962