Finite Group Databasebeta

← #314641

Subgroups — #314641

Order 512.

541 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 118 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#21927C211yesmaximal, characteristic, complemented
2#21817C211yesmaximal, characteristic
3#10667C211yesmaximal, characteristic, complemented
4#10666C211yesmaximal, characteristic, complemented
5#10629C211yesmaximal, characteristic, complemented
6#10629C211yesmaximal, characteristic, complemented
7#9582C212yesmaximal, complemented
8#9581C212yesmaximal, complemented
9#8863C212yesmaximal, complemented
10#8732C211yesmaximal, characteristic, complemented
11#8409C212yesmaximal
12#3449C2211yescharacteristic
13#2921C2211yescharacteristic
14#2298C2211yescharacteristic
15#1980—22no—
16#1963C2211yescharacteristic
17#1963C2211yescharacteristic
18#1962C2211yescharacteristic
19#1962C2211yescharacteristic
20#1924—22no—
21#1917—22no—
22#1843C2211yescharacteristic, complemented
23#1832C2211yescharacteristic, complemented
24#1798C2212yes—
25#1789—22no—
26#1767C2212yescomplemented
27#1571C2212yescomplemented
28#1570C2212yescomplemented
29#1540C2212yes—
30#1540C2212yes—
31#1480C2212yes—
32#1480C2212yes—
33#1458C2212yes—
34#1301C2212yescomplemented
35#1301C2212yescomplemented
36#1276C2212yescomplemented
37#1274C2211yescharacteristic, complemented
38#1274C2211yescharacteristic, complemented
39C22×#41—21no—
40#537—22no—
41#536—21no—
42#536—21no—
43#535—21no—
44#535—21no—
45#534—22no—
46#532—21no—
47#532C2311yescharacteristic
48#532C2311yescharacteristic
49#525—42no—
50C2×#117C2311yescharacteristic
51#433—42no—
52#431C2312yes—
53#431C2312yes—
54#423C2311yescharacteristic
55#423C2311yescharacteristic
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#104—22no—
60C2×#104—22no—
61C2×#104—22no—
62C2×#104—22no—
63#407—21no—
64C2×C22⋊C8—22no—
65C4×M16—21no—
66C2×#97—22no—
67#388—22no—
68#387—22no—
69C4×C4⋊C4D412yes—
70C4×C22⋊C4D412yes—
71#376C2311yescharacteristic
72C43C2311yescharacteristic
73#336—22no—
74#336—22no—
75#336—22no—
76#335—42no—
77#334—42no—
78#332—22no—
79#332—22no—
80#331—22no—
81#331—22no—
82D4⋊C8—42no—
83D4⋊C8—42no—
84#324—22no—
85#323—22no—
86C2×#41—21no—
87C2×#41—21no—
88C2×#41—21no—
89C2×#41—21no—
90C2×#41—41no—
91C2×#41—41no—
92C2×#41—82no—
93C2×#41—82no—
94C22×Q8—21no—
95C22×D4—21no—
96C22×D4—41no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C2×M16—21no—
100C2×M16—21no—
101C22×C8—22no—
102C22×C8—22no—
103C4⋊Q8—21no—
104C4⋊Q8#4111yescharacteristic
105C4⋊Q8#4111yescharacteristic
106#127—21no—
107#127#4111yescharacteristic
108#127#4111yescharacteristic
109#124—21no—
110#124—21no—
111#124—21no—
112#124—21no—
113#123—42no—
114#123—42no—
115C22⋊Q8—42no—
116C22⋊Q8—42no—
117#121—22no—
118#121—22no—
119#121—42no—
120#121—42no—
121#121—42no—
122#120—42no—
123#120—42no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—42no—
127#117—21no—
128#117—21no—
129#117—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4#4112yes—
132C2×C22⋊C4—22no—
133C2×C22⋊C4#4112yes—
134C2×C42—22no—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42C2411yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—82no—
148#104—82no—
149#104—82no—
150#104—82no—
151#104—82no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—42no—
155C22⋊C8—82no—
156#97—41no—
157#97—82no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—41no—
161#95—22no—
162#95—22no—
163#95—22no—
164C24—41no—
165C24#12311yescharacteristic
166#41—42no—
167#41—42no—
168#41—42no—
169#41—42no—
170#41—84no—
171#41—82no—
172#41—82no—
173#41—82no—
174#41—82no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—82no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—82no—
190C2×D4—84no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4#12311yescharacteristic
205C22×C4#12312yes—
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4112yes—
208M16—42no—
209M16—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—82no—
233C42—21no—
234C42—21no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—82no—
240C42C2×#4111yescharacteristic
241C42C2×#4111yescharacteristic
242C42H(5,𝔽2)12yes—
243C23—21no—
244C23—22no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—82no—
249C23—162no—
250C23C2×#12311yescharacteristic
251Q8—42no—
252Q8—42no—
253Q8—82no—
254D4—42no—
255D4—42no—
256D4—82no—
257D4—82no—
258D4—82no—
259D4—82no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4—162no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4#52511yescharacteristic
288C2×C4#53511yescharacteristic
289C2×C4#55112yes—
290C2×C4#55112yes—
291C2×C4#57612yes—
292C2×C4#57712yes—
293C8—82no—
294C8—82no—
295C8—82no—
296C8—82no—
297C8—81no—
298C8—81no—
299C22—22no—
300C22—41no—
301C22—42no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22—82no—
306C22—82no—
307C22—164no—
308C22#248211yescharacteristic
309C22#309612yes—
310C4—22no—
311C4—22no—
312C4—41no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—81no—
317C4—81no—
318C4#311812yes—
319C4#322011yescharacteristic
320C4#322211yescharacteristic
321C2—22no—
322C2—81no—
323C2—161no—
324C2—161no—
325C2#2759411yescharacteristic
326C2#2759912yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2482
prime radical#3449C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21927C2semidirect
3. #10667C2semidirect
4. #10666C2semidirect
5. #10629C2semidirect
6. #10629C2semidirect
7. #9582C2semidirect
8. #9581C2semidirect
9. #8863C2semidirect
10. #8732C2semidirect
22. #1843C22semidirect
23. #1832C22semidirect
26. #1767C22semidirect
27. #1571C22semidirect
28. #1570C22semidirect
34. #1301C22semidirect
35. #1301C22semidirect
36. #1276C22semidirect
37. #1274C22semidirect
38. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#219272
#218172
#106672
#106662
#106292
#95822
#95812
#88632
#87322
#84092