Finite Group Databasebeta

← #507763

Subgroups — #507763

Order 512.

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

normal subgroups only

This group has 437 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#21728C211yesmaximal, characteristic, complemented
2#21297C211yesmaximal, characteristic
3#13509C211yesmaximal, characteristic, complemented
4#13485C211yesmaximal, characteristic
5#13458C211yesmaximal, characteristic, complemented
6#13372C211yesmaximal, characteristic
7#13143C212yesmaximal
8#13119C212yesmaximal, complemented
9#13114C212yesmaximal, complemented
10#12943C212yesmaximal
11#12896C211yesmaximal, characteristic, complemented
12C2×#499C2211yescharacteristic
13#2943C2211yescharacteristic
14#2923—21no—
15C4×#125—21no—
16#2159C2212yes—
17C2×#365C2212yes—
18C2×#360C2212yes—
19#2139C2212yes—
20#1991—21no—
21#1975C2211yescharacteristic
22#1974C2211yescharacteristic
23#1946C2211yescharacteristic
24#1945C2211yescharacteristic
25#1944C2212yes—
26#1943—21no—
27#1938—21no—
28#1937—21no—
29#1846C2211yescharacteristic
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×#125—21no—
45C4×#41—21no—
46C2×M32C2312yes—
47C2×M32C2312yes—
48#499—24no—
49#442—21no—
50#441—21no—
51#440—22no—
52#433—42no—
53#432—42no—
54C2×#108C2311yescharacteristic
55C2×#108C2311yescharacteristic
56#427C2311yescharacteristic
57C2×#107—21no—
58C2×#107C2312yes—
59C2×#106—21no—
60C2×#106C2312yes—
61#424—21no—
62#424—21no—
63C2×#104—21no—
64C2×#104—21no—
65#416—21no—
66#408—21no—
67#408—21no—
68C4×M16—21no—
69C2×#97D411yescharacteristic
70C2×#97D411yescharacteristic
71C2×#97C2311yescharacteristic
72C2×#97C2311yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#389—22no—
76#387—42no—
77#382—21no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—42no—
80C43D411yescharacteristic
81#367—22no—
82#366—22no—
83#365—24no—
84#360—24no—
85#358—22no—
86#357—22no—
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
104#125—24no—
105#125—42no—
106C4×Q8—22no—
107C4×D4—22no—
108C4×D4—42no—
109#117—21no—
110#117—41no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C42—22no—
116C2×C42—41no—
117C2×C42—42no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121M32—22no—
122M32—22no—
123M32—22no—
124M32—22no—
125C2×C16—22no—
126C2×C16—22no—
127#108—22no—
128#108—22no—
129#107—24no—
130#107—22no—
131#107—22no—
132#107#4112yes—
133#106—22no—
134#106—21no—
135#106—42no—
136#106#4114yes—
137C4⋊C8—42no—
138C4⋊C8—42no—
139#104—42no—
140#104—42no—
141#103—22no—
142#102—22no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145#97—22no—
146#97—21no—
147#97—21no—
148#97—22no—
149#97—22no—
150#97—41no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—41no—
154#95—42no—
155#95—42no—
156#95—42no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161C2×Q8—21no—
162C2×D4—21no—
163C2×D4—41no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4AGL(1,ℤ/8)12yes—
171C22×C4C22×D411yescharacteristic
172Q16—22no—
173SD16—42no—
174D8—22no—
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8AGL(1,ℤ/8)12yes—
190C2×C8C2×#4111yescharacteristic
191C2×C8C2×#4112yes—
192C2×C8C2×#4111yescharacteristic
193C2×C82−1+412yes—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—48no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—84no—
202C22⋊C4—42no—
203C22⋊C4—41no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42—41no—
210C42—82no—
211C42—84no—
212C42#13512yes—
213C42#13712yes—
214C16—42no—
215C16—42no—
216C23—41no—
217C23C2×AGL(1,ℤ/8)11yescharacteristic
218Q8—41no—
219Q8—41no—
220D4—41no—
221D4—41no—
222D4—81no—
223D4—81no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—162no—
241C2×C4#52511yescharacteristic
242C2×C4#52511yescharacteristic
243C2×C4#53611yescharacteristic
244C2×C4C2×#13511yescharacteristic
245C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
246C2×C4C2×#13711yescharacteristic
247C8—22no—
248C8—41no—
249C8—41no—
250C8—81no—
251C8—81no—
252C8#55212yes—
253C8#55612yes—
254C22—22no—
255C22—82no—
256C22—81no—
257C22#310511yescharacteristic
258C22#310911yescharacteristic
259C22#323911yescharacteristic
260C4—42no—
261C4—42no—
262C4—42no—
263C4—84no—
264C4—82no—
265C4—81no—
266C4—162no—
267C4#259911yescharacteristic
268C4#310711yescharacteristic
269C4#311211yescharacteristic
270C4#322811yescharacteristic
271C2—21no—
272C2—21no—
273C2—161no—
274C2#2938911yescharacteristic
275C2#3369012yes—
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. #21728C2semidirect
3. #13509C2semidirect
5. #13458C2semidirect
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
#217282
#212972
#135092
#134852
#134582
#133722
#131432
#131192
#131142
#129432
#128962