Finite Group Databasebeta

← #507762

Subgroups — #507762

Order 512.

463 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 122 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#21727C211yesmaximal, characteristic, complemented
2#21297C211yesmaximal, characteristic
3#13508C211yesmaximal, characteristic, complemented
4#13484C211yesmaximal, characteristic
5#13457C211yesmaximal, characteristic, complemented
6#13371C211yesmaximal, characteristic
7#13143C212yesmaximal
8#13119C212yesmaximal, complemented
9#13114C212yesmaximal, complemented
10#12943C212yesmaximal
11#12896C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13#2943C2211yescharacteristic
14#2923—21no—
15C4×C4⋊Q8—21no—
16#2159C2212yes—
17C2×#365C2212yes—
18C2×#360C2212yes—
19#2139C2212yes—
20#1990—21no—
21#1975C2211yescharacteristic
22#1974C2211yescharacteristic
23#1946C2211yescharacteristic
24#1945C2211yescharacteristic
25#1944—21no—
26#1944C2212yes—
27#1938—21no—
28#1937—21no—
29#1845C2211yescharacteristic
30#1778C2211yescharacteristic
31#1776—21no—
32#1776C2212yes—
33C4×#104—21no—
34C4×#97C2211yescharacteristic
35#1387C2212yes—
36#1387C2212yes—
37#1386C2212yes—
38#1383C2212yes—
39#1377C2212yes—
40#1376C2211yescharacteristic
41#1376C2211yescharacteristic
42#1375C2212yes—
43C2×#135C2311yescharacteristic
44C2×C4⋊Q8—21no—
45C4×#41—21no—
46C2×C4×Q8—41no—
47C2×M32C2312yes—
48C2×M32C2312yes—
49#498—24no—
50#442—21no—
51#441—21no—
52#440—21no—
53#440—21no—
54#433—42no—
55#432—42no—
56C2×#108C2311yescharacteristic
57C2×#108C2311yescharacteristic
58#427C2311yescharacteristic
59C2×#107—21no—
60C2×#107C2312yes—
61C2×#106—21no—
62C2×#106C2312yes—
63#424—21no—
64#424—21no—
65C2×#104—21no—
66C2×#104—21no—
67#416—21no—
68#408—21no—
69#408—21no—
70C4×M16—21no—
71C2×#97D411yescharacteristic
72C2×#97D411yescharacteristic
73C2×#97C2311yescharacteristic
74C2×#97C2311yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—41no—
78#389—22no—
79#389—41no—
80#384—21no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—41no—
83C43D411yescharacteristic
84#367—22no—
85#366—22no—
86#365—24no—
87#360—24no—
88#358—22no—
89#357—22no—
90#341—21no—
91#340—22no—
92#340—22no—
93#339—21no—
94#339—21no—
95C2×#41—21no—
96C22×Q8—41no—
97#135—22no—
98#135—22no—
99C2×Q16#4111yescharacteristic
100C2×SD16—21no—
101C2×D8#4111yescharacteristic
102C2×M16—21no—
103C2×M16—21no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2411yescharacteristic
108C4⋊Q8—24no—
109C4⋊Q8—42no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8—84no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—41no—
116C4×D4—41no—
117#117—21no—
118#117—41no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42—42no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132M32—22no—
133M32—22no—
134M32—22no—
135M32—22no—
136C2×C16—22no—
137C2×C16—22no—
138#108—22no—
139#108—22no—
140#107—22no—
141#107—22no—
142#107—42no—
143#107#4111yescharacteristic
144#107#4111yescharacteristic
145#106—24no—
146#106—22no—
147#106—21no—
148#106#4114yes—
149C4⋊C8—42no—
150C4⋊C8—42no—
151#104—42no—
152#104—42no—
153#103—21no—
154#103—21no—
155#102—21no—
156#102—21no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159#97—22no—
160#97—21no—
161#97—21no—
162#97—22no—
163#97—22no—
164#97—41no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—41no—
168#95—42no—
169#95—42no—
170#41—41no—
171#41—41no—
172#41—41no—
173#41—41no—
174C2×Q8—21no—
175C2×Q8—44no—
176C2×Q8—82no—
177C2×Q8—82no—
178C2×D4—21no—
179C2×D4—41no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4AGL(1,ℤ/8)11yescharacteristic
188C22×C4AGL(1,ℤ/8)11yescharacteristic
189C22×C4C22×D411yescharacteristic
190Q16—22no—
191SD16—42no—
192D8—22no—
193M16—41no—
194M16—41no—
195M16—41no—
196M16—41no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8AGL(1,ℤ/8)12yes—
208C2×C8C2×#4111yescharacteristic
209C2×C8C2×#4112yes—
210C2×C8C2×#4111yescharacteristic
211C2×C82−1+412yes—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—24no—
215C4⋊C4—44no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—42no—
232C42—42no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—84no—
237C42#13512yes—
238C42#13712yes—
239C16—42no—
240C16—42no—
241C23—41no—
242C23C2×AGL(1,ℤ/8)11yescharacteristic
243Q8—41no—
244Q8—41no—
245Q8—88no—
246D4—41no—
247D4—41no—
248D4—81no—
249D4—81no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—161no—
272C2×C4—161no—
273C2×C4#52511yescharacteristic
274C2×C4#52511yescharacteristic
275C2×C4#53611yescharacteristic
276C2×C4C2×#13511yescharacteristic
277C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
278C2×C4C2×#13711yescharacteristic
279C8—22no—
280C8—41no—
281C8—41no—
282C8—81no—
283C8—81no—
284C8#55212yes—
285C8#55612yes—
286C22—21no—
287C22—21no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22#310511yescharacteristic
292C22#310911yescharacteristic
293C22#323911yescharacteristic
294C4—24no—
295C4—42no—
296C4—41no—
297C4—41no—
298C4—84no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—162no—
303C4#259911yescharacteristic
304C4#310711yescharacteristic
305C4#311211yescharacteristic
306C4#322811yescharacteristic
307C2—21no—
308C2—21no—
309C2—161no—
310C2#2938911yescharacteristic
311C2#3368811yescharacteristic
312C2#3368911yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#135C23
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
1. #21727C2semidirect
3. #13508C2semidirect
5. #13457C2semidirect
8. #13119C2semidirect
9. #13114C2semidirect
11. #12896C2semidirect

Maximal subgroups

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

subgroupindex
#217272
#212972
#135082
#134842
#134572
#133712
#131432
#131192
#131142
#129432
#128962