Finite Group Databasebeta

← #309512

Subgroups — #309512

Order 512.

658 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 158 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#25012C212yesmaximal
2#22963C212yesmaximal, complemented
3#10685C211yesmaximal, characteristic, complemented
4#10480C212yesmaximal, complemented
5#10235C211yesmaximal, characteristic
6#9612C212yesmaximal, complemented
7#9542C212yesmaximal, complemented
8#8928C211yesmaximal, characteristic, complemented
9#8886C212yesmaximal, complemented
10C2×#481—22no—
11C2×#476C2212yes—
12#2537C2211yescharacteristic
13#2369C2211yescharacteristic
14D4×C4⋊C4C2212yes—
15#2077—22no—
16#2034—22no—
17#1990C2212yes—
18#1956C2211yescharacteristic
19#1954C2212yes—
20#1948C2212yes—
21#1948C2212yes—
22#1941—22no—
23#1919—22no—
24#1864—22no—
25#1853—22no—
26#1844C2212yes—
27#1844C2211yescharacteristic
28#1802C2212yes—
29C8×C4⋊C4C2212yes—
30C4×#102—22no—
31C4×#102C2212yes—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×D4⋊C8C2212yes—
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2311yescharacteristic
40C2×#121—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2312yes—
43C22×C4⋊C4—22no—
44#481—48no—
45#476—24no—
46#476—24no—
47C2×#106—22no—
48C2×#106—22no—
49C2×#106—22no—
50C2×#106—22no—
51C2×#106—22no—
52C2×#106—41no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×#103—41no—
56C2×#102—22no—
57C2×#102—22no—
58C2×#102D412yes—
59C2×#102D412yes—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C4×C8—22no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2312yes—
65#394—41no—
66#391—41no—
67#389—22no—
68#389—22no—
69#385—42no—
70#384C2312yes—
71#384C2311yescharacteristic
72#380—22no—
73#380—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2312yes—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340D412yes—
82#340D412yes—
83#336—22no—
84#334—24no—
85#333—22no—
86D4⋊C8—24no—
87C22×Q8—21no—
88C22×D4—41no—
89C22×D4C2×D412yes—
90C23×C4—22no—
91C23×C4—42no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4⋊Q8—22no—
98C4⋊Q8—21no—
99C4⋊Q8#4112yes—
100#121—44no—
101#121—82no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—44no—
107C4×D4—82no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C42—22no—
125C2×C42—41no—
126C2×C42C2411yescharacteristic
127#106—28no—
128#106—44no—
129#106—44no—
130#106—82no—
131#106—82no—
132#106—82no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8#4114yes—
137#103—82no—
138#102—28no—
139#102—42no—
140#102—44no—
141#102SD1618yes—
142C22⋊C8—44no—
143C22⋊C8—44no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—82no—
148#95—22no—
149#95—42no—
150#95—41no—
151#95—42no—
152C24—22no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—84no—
163C2×D4—161no—
164C2×D4C2×SD1614yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4—162no—
180C22×C4C22×D412yes—
181C22×C4H(5,𝔽2)11yescharacteristic
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8C2×SD1614yes—
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—84no—
211C4⋊C4—81no—
212C4⋊C4—82no—
213C4⋊C4—161no—
214C4⋊C4C2×SD1614yes—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—84no—
218C22⋊C4—84no—
219C42—42no—
220C42—42no—
221C42—161no—
222C42C2×#4111yescharacteristic
223C42C2×#4112yes—
224C422−1+411yescharacteristic
225C23—24no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23D4211yescharacteristic
233Q8—82no—
234Q8—161no—
235D4—24no—
236D4—24no—
237D4—42no—
238D4—42no—
239D4—164no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—168no—
264C2×C4#52512yes—
265C2×C4#53911yescharacteristic
266C2×C4#54712yes—
267C2×C4#55911yescharacteristic
268C2×C4C22×SD1612yes—
269C2×C4#57512yes—
270C2×C4#57712yes—
271C2×C4#57712yes—
272C8—24no—
273C8—42no—
274C8—42no—
275C8—44no—
276C8—82no—
277C22—44no—
278C22—44no—
279C22—44no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22#245511yescharacteristic
285C22#326712yes—
286C22D4×SD1612yes—
287C22#328811yescharacteristic
288C22#329111yescharacteristic
289C4—21no—
290C4—21no—
291C4—44no—
292C4—82no—
293C4—82no—
294C4—82no—
295C4—161no—
296C4—162no—
297C4#310312yes—
298C4#312512yes—
299C4#333512yes—
300C4#335112yes—
301C2—84no—
302C2#1980612yes—
303C2SD16211yescharacteristic
304C2#2014311yescharacteristic
305C2#2655112yes—
306C2#2657911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#2537C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22963C2semidirect
3. #10685C2semidirect
4. #10480C2semidirect
6. #9612C2semidirect
7. #9542C2semidirect
8. #8928C2semidirect
9. #8886C2semidirect

Maximal subgroups

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

subgroupindex
#250122
#229632
#106852
#104802
#102352
#96122
#95422
#89282
#88862