Finite Group Databasebeta

← #503286

Subgroups — #503286

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic
2C2×#2255C212yesmaximal, complemented
3#33208C211yesmaximal, characteristic
4#13452C212yesmaximal, complemented
5#13451C212yesmaximal, complemented
6#13283C212yesmaximal, complemented
7#12920C212yesmaximal
8#12919C212yesmaximal
9#12599C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C2×#500—28no—
12C2×#500—21no—
13C2×#500—21no—
14C2×#500C2212yes—
15C2×#500C2212yes—
16C2×#499C2211yescharacteristic
17C22×#107—21no—
18C22×#107C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2851—21no—
21#2255—28no—
22C2×#367C2214yes—
23C2×#366C2214yes—
24C2×C22⋊C16C2212yes—
25C2×#346C2212yes—
26#1991—21no—
27#1942C2211yescharacteristic
28#1942C2211yescharacteristic
29#1941—21no—
30#1846C2211yescharacteristic
31#1808C2211yescharacteristic
32#1808C2211yescharacteristic
33#1807—21no—
34#1380C2214yes—
35#1380C2214yes—
36#1374C2212yes—
37C23×Q8—41no—
38C23×C8D411yescharacteristic
39C23×C8C2311yescharacteristic
40C2×C4⋊Q8—28no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—44no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×#125—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—41no—
49C22×C42D411yescharacteristic
50C22×C16C2314yes—
51#500—28no—
52#500—28no—
53#500—28no—
54#500—24no—
55#500D418yes—
56#499—24no—
57C2×#107—28no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—28no—
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#107C2314yes—
67C2×#106—21no—
68C2×#106—21no—
69C2×C4×C8—24no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#389—21no—
75#389—21no—
76#387—41no—
77#387—41no—
78#385—41no—
79#385—41no—
80#382—21no—
81#380—21no—
82#380—21no—
83#367—28no—
84#366—28no—
85C22⋊C16Q818yescomplemented
86#346—24no—
87#340—21no—
88#340—21no—
89#340—21no—
90#340—21no—
91C22×Q8—48no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—84no—
95C22×Q8—84no—
96C23×C4—42no—
97C23×C4D811yescharacteristic
98C23×C4D811yescharacteristic
99C23×C4C2×D411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8Q1614yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×Q814yes—
109C22×C8C2411yescharacteristic
110C4⋊Q8—28no—
111C4⋊Q8—28no—
112C4⋊Q8—44no—
113C4⋊Q8—48no—
114C4⋊Q8—48no—
115C4⋊Q8—44no—
116#125—42no—
117#125—42no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42D811yescharacteristic
138C2×C42D811yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C16—24no—
142C2×C16—24no—
143C2×C16C2×Q818yescomplemented
144#107—28no—
145#107—28no—
146#107—28no—
147#107—24no—
148#107—22no—
149#107—22no—
150#107—22no—
151#107—22no—
152#107C2×D418yes—
153#106—42no—
154#106—42no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8C2×D414yes—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24C2×D811yescharacteristic
166C2×Q8—48no—
167C2×Q8—48no—
168C2×Q8—832no—
169C2×Q8—84no—
170C2×Q8—84no—
171C2×Q8—84no—
172C2×Q8—84no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—48no—
184C22×C4—44no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4C4⋊Q812yes—
192C22×C4C2×D811yescharacteristic
193C22×C4C2×D811yescharacteristic
194C22×C4C2×D811yescharacteristic
195C22×C4C2×D811yescharacteristic
196C22×C4C2×D811yescharacteristic
197C22×C4C2×Q1612yes—
198C22×C4C22×D411yescharacteristic
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—28no—
204C2×C8—28no—
205C2×C8#12012yes—
206C2×C8C4⋊Q812yes—
207C2×C8C2×Q1612yes—
208C2×C8C2×Q1612yes—
209C2×C8#13712yes—
210C2×C8C22×D412yes—
211C2×C8C22×Q812yes—
212C2×C82−1+412yes—
213C4⋊C4—28no—
214C4⋊C4—48no—
215C4⋊C4—48no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—22no—
236C42#12012yes—
237C42AGL(1,ℤ/8)12yes—
238C16—28no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23#50012yes—
244C23#50912yes—
245C23#51012yes—
246C23C22×D811yescharacteristic
247Q8—816no—
248Q8—816no—
249C2×C4—28no—
250C2×C4—28no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—48no—
266C2×C4—816no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4#44712yes—
282C2×C4#44712yes—
283C2×C4#50012yes—
284C2×C4#50912yes—
285C2×C4#51012yes—
286C2×C4C2×#12011yescharacteristic
287C2×C4C2×C4⋊Q811yescharacteristic
288C2×C4#53611yescharacteristic
289C2×C4C22×Q1611yescharacteristic
290C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
291C2×C4C2×#13711yescharacteristic
292C8—22no—
293C8—22no—
294C8#45114yes—
295C8D4×Q814yes—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22#225514yescomplemented
303C22C2×#44711yescharacteristic
304C22C2×#50011yescharacteristic
305C22#322611yescharacteristic
306C22C2×#50911yescharacteristic
307C22C2×#50911yescharacteristic
308C22C2×#51011yescharacteristic
309C22C2×#51011yescharacteristic
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—88no—
316C4—161no—
317C4—161no—
318C4—161no—
319C4—161no—
320C4#243411yescharacteristic
321C4C2×#45111yescharacteristic
322C4#316511yescharacteristic
323C4#323011yescharacteristic
324C2—24no—
325C2#2581311yescharacteristic
326C2#3351711yescharacteristic
327C2#3351811yescharacteristic
328C2C2×#225511yescharacteristic
329C2#3373111yescharacteristic
330C2#3380411yescharacteristic
331C2#3380511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2255C2semidirect
4. #13452C2semidirect
5. #13451C2semidirect
6. #13283C2semidirect
85. C22⋊C16Q8semidirect
143. C2×C16C2×Q8semidirect
302. C22#2255semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
C2×#22552
#332082
#134522
#134512
#132832
#129202
#129192
#125992