Finite Group Databasebeta

← #244308

Subgroups — #244308

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22268C212yesmaximal
2#22246C212yesmaximal, complemented
3C4×#498C211yesmaximal, characteristic, complemented
4C4×#492C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10681C212yesmaximal, complemented
7#10677C211yesmaximal, characteristic
8#10316C211yesmaximal, characteristic, complemented
9#10315C212yesmaximal
10#10315C211yesmaximal, characteristic
11#8133C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#498C2212yes—
14C2×#492C2211yescharacteristic
15C2×#492C2212yes—
16C2×C4×SD16—22no—
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19C4×#127C2211yescharacteristic
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1968—22no—
24#1957C2211yescharacteristic
25#1957C2212yes—
26#1956C2211yescharacteristic
27#1956C2212yes—
28#1941—22no—
29#1920—24no—
30#1919—24no—
31#1880—24no—
32#1877—24no—
33#1844C2211yescharacteristic
34#1844C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37C4×#106—22no—
38#1744C2212yes—
39C42×C8C2211yescharacteristic
40#1273C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C22×SD16—24no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×#127C2311yescharacteristic
48C2×C4×Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×D4—21no—
51C2×C4×D4—22no—
52#498—24no—
53#498—24no—
54#498—24no—
55#492—22no—
56#492—24no—
57#492—22no—
58#492—24no—
59C4×SD16—48no—
60C2×#106—24no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#103—22no—
65C2×#103—24no—
66C2×#102—22no—
67C2×#102—24no—
68C2×C4×C8D412yes—
69C2×C4×C8D412yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#391—21no—
76#390—22no—
77#390—21no—
78#389—22no—
79#389—22no—
80#389—22no—
81#384C2311yescharacteristic
82#384C2312yes—
83#384C2311yescharacteristic
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C43C2311yescharacteristic
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#336D414yes—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×D4—21no—
98C22×D4—22no—
99C23×C4—41no—
100C2×SD16—28no—
101C2×SD16—216no—
102C2×SD16—44no—
103C2×SD16—44no—
104C2×SD16—84no—
105C2×SD16—84no—
106C22×C8C2×D414yes—
107C22×C8C2×D414yes—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—21no—
115C4⋊Q8#4112yes—
116C4⋊Q8#4114yes—
117#127—21no—
118#127—22no—
119#127#4112yes—
120C4×Q8—44no—
121C4×Q8—84no—
122C4×D4—44no—
123C4×D4—84no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—41no—
138C2×C42—42no—
139C2×C42—41no—
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2411yescharacteristic
145#106—28no—
146#106—216no—
147#106—44no—
148#106—44no—
149#103—44no—
150#103—84no—
151#102—44no—
152#102—84no—
153C4×C8—21no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—42no—
161C4×C8SD1618yescomplemented
162#95—42no—
163#95—42no—
164#95—42no—
165C24—41no—
166C2×Q8—22no—
167C2×Q8—24no—
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×D4—22no—
175C2×D4—24no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—44no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—84no—
188C22×C4—82no—
189C22×C4#12712yes—
190C22×C4#12711yescharacteristic
191C22×C4#12711yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194SD16—416no—
195SD16—816no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8C2×SD1614yes—
207C2×C8C2×SD1614yes—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—84no—
222C4⋊C4—82no—
223C22⋊C4—84no—
224C22⋊C4—82no—
225C42—22no—
226C42—24no—
227C42—24no—
228C42—84no—
229C42—82no—
230C42C2×SD1614yes—
231C42C2×#4111yescharacteristic
232C42C2×#4112yes—
233C42C2×#4111yescharacteristic
234C23—42no—
235C23—81no—
236C23—81no—
237C23—82no—
238C23—82no—
239C23C2×#12711yescharacteristic
240Q8—44no—
241Q8—48no—
242Q8—82no—
243Q8—84no—
244D4—44no—
245D4—48no—
246D4—82no—
247D4—84no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—164no—
267C2×C4#49212yes—
268C2×C4#49212yes—
269C2×C4#52511yescharacteristic
270C2×C4#52511yescharacteristic
271C2×C4#52512yes—
272C2×C4#52512yes—
273C2×C4C22×SD1612yes—
274C2×C4#57512yes—
275C2×C4#57714yes—
276C2×C4#57714yes—
277C8—24no—
278C8—24no—
279C8—44no—
280C8—44no—
281C22—84no—
282C22—82no—
283C22—161no—
284C22—161no—
285C22—162no—
286C22#239711yescharacteristic
287C22C2×#49211yescharacteristic
288C22#315011yescharacteristic
289C22#315412yes—
290C22#315412yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—84no—
295C4—82no—
296C4—161no—
297C4—161no—
298C4—162no—
299C4#310112yes—
300C4#310312yes—
301C4#312314yes—
302C4#312614yes—
303C2—162no—
304C2#1965111yescharacteristic
305C2#1965411yescharacteristic
306C2#1970912yes—
307C2#2379911yescharacteristic
308C2#2381112yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22246C2semidirect
3. C4×#498C2semidirect
5. #10686C2semidirect
6. #10681C2semidirect
8. #10316C2semidirect
11. #8133C2semidirect
161. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#222682
#222462
C4×#4982
C4×#4922
#106862
#106812
#106772
#103162
#103152
#81332