Finite Group Databasebeta

← #314622

Subgroups — #314622

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#10665C212yesmaximal, complemented
4#10627C212yesmaximal, complemented
5#9582C212yesmaximal, complemented
6#9581C212yesmaximal, complemented
7#8863C212yesmaximal, complemented
8#8733C211yesmaximal, characteristic, complemented
9#8409C212yesmaximal
10#3449C2211yescharacteristic
11#2921C2211yescharacteristic
12#2298C2211yescharacteristic
13#1980—22no—
14#1963C2212yes—
15#1962C2212yes—
16#1924—22no—
17#1917—22no—
18#1843C2211yescharacteristic, complemented
19#1832C2211yescharacteristic, complemented
20#1798C2212yes—
21#1789—22no—
22#1767C2212yescomplemented
23#1571C2212yescomplemented
24#1570C2212yescomplemented
25#1540C2212yes—
26#1540C2212yes—
27#1480C2212yes—
28#1480C2212yes—
29#1458C2212yes—
30#1301C2212yescomplemented
31#1301C2212yescomplemented
32#1276C2211yescharacteristic, complemented
33#1276C2211yescharacteristic, complemented
34#1274C2212yescomplemented
35C22×#41—21no—
36#537—22no—
37#536—22no—
38#535—22no—
39#534—22no—
40#532—21no—
41#532C2312yes—
42#525—41no—
43#525—41no—
44C2×#117C2311yescharacteristic
45#433—41no—
46#433—41no—
47#431C2312yes—
48#431C2312yes—
49#423C2312yes—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#104—22no—
54C2×#104—22no—
55C2×#104—22no—
56C2×#104—22no—
57#407—21no—
58C2×C22⋊C8—22no—
59C4×M16—21no—
60C2×#97—22no—
61#388—22no—
62#387—22no—
63C4×C4⋊C4D412yes—
64C4×C22⋊C4D412yes—
65#376C2311yescharacteristic
66C43C2311yescharacteristic
67#336—22no—
68#336—22no—
69#336—22no—
70#335—42no—
71#334—42no—
72#332—22no—
73#332—22no—
74#331—22no—
75#331—22no—
76D4⋊C8—42no—
77D4⋊C8—42no—
78#324—22no—
79#323—22no—
80C2×#41—22no—
81C2×#41—22no—
82C2×#41—41no—
83C2×#41—41no—
84C2×#41—82no—
85C2×#41—82no—
86C22×Q8—21no—
87C22×D4—21no—
88C22×D4—41no—
89C23×C4—41no—
90C23×C4C2×D411yescharacteristic
91C2×M16—22no—
92C22×C8—22no—
93C22×C8—22no—
94C4⋊Q8—21no—
95C4⋊Q8#4112yes—
96#127—21no—
97#127#4112yes—
98#124—22no—
99#124—22no—
100#123—42no—
101#123—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104#121—22no—
105#121—22no—
106#121—42no—
107#121—42no—
108#121—42no—
109#120—42no—
110#120—42no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—42no—
114#117—22no—
115#117—21no—
116#117—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4#4112yes—
119C2×C22⋊C4—22no—
120C2×C22⋊C4#4112yes—
121C2×C42—22no—
122C2×C42C2×D412yes—
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2411yescharacteristic
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—82no—
135#104—82no—
136#104—82no—
137#104—82no—
138#104—82no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—42no—
142C22⋊C8—82no—
143#97—41no—
144#97—82no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—41no—
148#95—22no—
149#95—22no—
150#95—22no—
151C24—41no—
152C24#12311yescharacteristic
153#41—42no—
154#41—42no—
155#41—42no—
156#41—42no—
157#41—84no—
158#41—82no—
159#41—82no—
160#41—84no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—84no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—84no—
185C22×C4#12311yescharacteristic
186C22×C4#12312yes—
187C22×C4C22×D411yescharacteristic
188C22×C4C2×#4112yes—
189M16—42no—
190M16—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—82no—
213C42—21no—
214C42—21no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—82no—
220C42C2×#4112yes—
221C42H(5,𝔽2)11yescharacteristic
222C42H(5,𝔽2)11yescharacteristic
223C23—21no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—82no—
228C23—162no—
229C23C2×#12311yescharacteristic
230Q8—42no—
231Q8—42no—
232Q8—82no—
233D4—42no—
234D4—42no—
235D4—82no—
236D4—82no—
237D4—82no—
238D4—84no—
239D4—82no—
240D4—82no—
241D4—162no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—84no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4#52511yescharacteristic
262C2×C4#53511yescharacteristic
263C2×C4#55112yes—
264C2×C4#55112yes—
265C2×C4#57612yes—
266C2×C4#57712yes—
267C8—82no—
268C8—82no—
269C8—82no—
270C8—82no—
271C8—82no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—164no—
280C22#248211yescharacteristic
281C22#309612yes—
282C4—22no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—82no—
287C4—82no—
288C4—82no—
289C4#312211yescharacteristic
290C4#312311yescharacteristic
291C4#322412yes—
292C2—22no—
293C2—81no—
294C2—162no—
295C2#2759111yescharacteristic
296C2#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. #10665C2semidirect
4. #10627C2semidirect
5. #9582C2semidirect
6. #9581C2semidirect
7. #8863C2semidirect
8. #8733C2semidirect
18. #1843C22semidirect
19. #1832C22semidirect
22. #1767C22semidirect
23. #1571C22semidirect
24. #1570C22semidirect
30. #1301C22semidirect
31. #1301C22semidirect
32. #1276C22semidirect
33. #1276C22semidirect
34. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#219272
#218172
#106652
#106272
#95822
#95812
#88632
#87332
#84092