Finite Group Databasebeta

← #507995

Subgroups — #507995

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#33261C211yesmaximal, characteristic, complemented
2#21296C211yesmaximal, characteristic
3#13542C211yesmaximal, characteristic, complemented
4#13534C211yesmaximal, characteristic
5#13508C211yesmaximal, characteristic, complemented
6#13484C211yesmaximal, 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#1942—21no—
27#1942—21no—
28#1938—21no—
29#1937—21no—
30#1846C2211yescharacteristic
31#1845C2211yescharacteristic
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—21no—
50#440—21no—
51#433—42no—
52#431—42no—
53C2×#108C2311yescharacteristic
54C2×#108C2311yescharacteristic
55#427C2311yescharacteristic
56C2×#107—21no—
57C2×#107—21no—
58C2×#107C2312yes—
59C2×#106C2312yes—
60#424—21no—
61#424—21no—
62C2×#104—21no—
63C2×#104—21no—
64#416—21no—
65#408—21no—
66#408—21no—
67C4×M16—21no—
68C2×#97D411yescharacteristic
69C2×#97C2311yescharacteristic
70C2×#97C2311yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#400—42no—
75#398—42no—
76#389—22no—
77#389—22no—
78#384—21no—
79#382—21no—
80#376D411yescharacteristic
81#367—22no—
82#366—22no—
83#365—24no—
84#360—24no—
85#358—22no—
86#357—22no—
87#347—24no—
88#341—21no—
89#340—22no—
90#340—22no—
91#339—21no—
92#339—21no—
93C2×#41—21no—
94#135—22no—
95#135—22no—
96C2×Q16#4111yescharacteristic
97C2×SD16—21no—
98C2×D8#4111yescharacteristic
99C2×M16—21no—
100C2×M16—21no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2411yescharacteristic
105C4×Q8—21no—
106C4×Q8—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—41no—
110C4×D4—41no—
111#117—21no—
112#117—41no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C42—41no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122M32—22no—
123M32—22no—
124M32—22no—
125M32#4114yes—
126C2×C16—22no—
127C2×C16—22no—
128#108—22no—
129#108—22no—
130#107—22no—
131#107—22no—
132#107—42no—
133#107—42no—
134#107#4111yescharacteristic
135#107#4111yescharacteristic
136#106—21no—
137#106—22no—
138#106—22no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141#104—42no—
142#104—42no—
143#103—21no—
144#103—21no—
145#102—21no—
146#102—21no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149#97—22no—
150#97—21no—
151#97—21no—
152#97—22no—
153#97—41no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8—41no—
158#95—21no—
159#95—21no—
160#95—42no—
161#95—42no—
162#95—42no—
163#41—41no—
164#41—41no—
165#41—41no—
166#41—41no—
167C2×Q8—21no—
168C2×D4—21no—
169C2×D4—41no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4AGL(1,ℤ/8)11yescharacteristic
178C22×C4AGL(1,ℤ/8)11yescharacteristic
179C22×C4C22×D411yescharacteristic
180Q16—22no—
181SD16—42no—
182D8—22no—
183M16—41no—
184M16—41no—
185M16—41no—
186M16—41no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8AGL(1,ℤ/8)12yes—
198C2×C8C2×#4111yescharacteristic
199C2×C8C2×#4111yescharacteristic
200C2×C8C2×#4112yes—
201C2×C82−1+412yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—162no—
211C4⋊C4—162no—
212C4⋊C4—162no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42#13712yes—
225C16—24no—
226C16—42no—
227C23—41no—
228C23C2×AGL(1,ℤ/8)11yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—41no—
232D4—41no—
233D4—81no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#52511yescharacteristic
259C2×C4#53211yescharacteristic
260C2×C4#53611yescharacteristic
261C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
262C2×C4C2×#13711yescharacteristic
263C2×C4#57511yescharacteristic
264C8—22no—
265C8—41no—
266C8—41no—
267C8—81no—
268C8—81no—
269C8#55212yes—
270C8#55212yes—
271C22—21no—
272C22—21no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22#310911yescharacteristic
277C22#317111yescharacteristic
278C22#323911yescharacteristic
279C4—42no—
280C4—41no—
281C4—41no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—162no—
287C4—162no—
288C4#261111yescharacteristic
289C4#311611yescharacteristic
290C4#318211yescharacteristic
291C4#323411yescharacteristic
292C2—21no—
293C2—21no—
294C2—161no—
295C2#2958011yescharacteristic
296C2#3369211yescharacteristic
297C2#3369311yescharacteristic
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. #33261C2semidirect
3. #13542C2semidirect
5. #13508C2semidirect
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
#332612
#212962
#135422
#135342
#135082
#134842
#131742
#131532
#131142
#129432
#125962