Finite Group Databasebeta

← #503289

Subgroups — #503289

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic
2C2×#2255C212yesmaximal, complemented
3#33209C211yesmaximal, characteristic
4#13453C214yesmaximal, complemented
5#13284C212yesmaximal, complemented
6#12920C212yesmaximal
7#12919C212yesmaximal
8#12599C211yesmaximal, characteristic
9C22×#125—21no—
10C2×#500C2211yescharacteristic
11C2×#499—28no—
12C2×#499—21no—
13C2×#499—21no—
14C2×#499C2212yes—
15C2×#499C2212yes—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106—21no—
19C22×C4×C8C2211yescharacteristic
20#2857—21no—
21#2255—28no—
22C2×#367C2214yes—
23C2×#366C2214yes—
24C2×C22⋊C16C2212yes—
25C2×#346C2212yes—
26#1990—21no—
27#1942—21no—
28#1942C2212yes—
29#1845C2211yescharacteristic
30#1808—21no—
31#1808C2212yes—
32#1380C2214yes—
33#1380C2214yes—
34#1374C2212yes—
35C23×C8D411yescharacteristic
36C23×C8C2311yescharacteristic
37C2×C4⋊Q8—21no—
38C2×#125—28no—
39C2×#125—22no—
40C2×#125—22no—
41C2×#125—44no—
42C2×#125—41no—
43C2×#125—41no—
44C2×C22⋊Q8—41no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4—41no—
49C22×C42D411yescharacteristic
50C22×C16C2314yes—
51#500—24no—
52#499—28no—
53#499—28no—
54#499—24no—
55#499—24no—
56#499—24no—
57#499D418yes—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—24no—
61C2×#107—24no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107C2312yes—
65C2×#107C2312yes—
66C2×#107C2312yes—
67C2×#107C2312yes—
68C2×#106—28no—
69C2×#106—22no—
70C2×#106—22no—
71C2×C4×C8—24no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—41no—
77#389—22no—
78#389—41no—
79#384—21no—
80#380—22no—
81#380—41no—
82#367—28no—
83#366—28no—
84C22⋊C16Q818yescomplemented
85#346—24no—
86#340—22no—
87#340—22no—
88C22×Q8—41no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4D811yescharacteristic
92C23×C4D811yescharacteristic
93C23×C4C2×D411yescharacteristic
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8Q1614yes—
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×Q814yes—
103C22×C8C2411yescharacteristic
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106#125—28no—
107#125—28no—
108#125—44no—
109#125—48no—
110#125—48no—
111#125—44no—
112C22⋊Q8—84no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—42no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42D811yescharacteristic
137C2×C42D811yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C16—24no—
141C2×C16—24no—
142C2×C16C2×Q818yes—
143#107—24no—
144#107—24no—
145#107—22no—
146#107—22no—
147#107—24no—
148#107—24no—
149#107—42no—
150#107—42no—
151#107C2×D414yes—
152#107C2×D414yes—
153#106—28no—
154#106—28no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8C2×D414yes—
159#95—42no—
160#95—42no—
161C24C2×D811yescharacteristic
162C2×Q8—82no—
163C2×Q8—82no—
164C2×Q8—82no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4C4⋊Q812yes—
186C22×C4C2×D811yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4C2×D811yescharacteristic
189C22×C4C2×D811yescharacteristic
190C22×C4C2×D811yescharacteristic
191C22×C4C2×Q1612yes—
192C22×C4C22×D411yescharacteristic
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—28no—
198C2×C8—28no—
199C2×C8#12012yes—
200C2×C8C4⋊Q812yes—
201C2×C8C2×Q1612yes—
202C2×C8C2×Q1612yes—
203C2×C8#13712yes—
204C2×C8C22×D412yes—
205C2×C8C22×Q812yes—
206C2×C82−1+412yes—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—48no—
216C4⋊C4—48no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C4⋊C4—82no—
225C22⋊C4—84no—
226C42—21no—
227C42—21no—
228C42—22no—
229C42—22no—
230C42#12012yes—
231C42AGL(1,ℤ/8)12yes—
232C16—28no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23#50012yes—
238C23#50912yes—
239C23#51012yes—
240C23C22×D811yescharacteristic
241Q8—164no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—88no—
263C2×C4—88no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#44712yes—
275C2×C4#44712yes—
276C2×C4#50012yes—
277C2×C4C2×#12011yescharacteristic
278C2×C4C2×C4⋊Q811yescharacteristic
279C2×C4#53611yescharacteristic
280C2×C4C22×Q1611yescharacteristic
281C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
282C2×C4C2×#13711yescharacteristic
283C8—22no—
284C8—22no—
285C8#45114yes—
286C8D4×Q814yes—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22#225514yescomplemented
294C22C2×#44711yescharacteristic
295C22C2×#50011yescharacteristic
296C22#322611yescharacteristic
297C22C2×#50911yescharacteristic
298C22C2×#51011yescharacteristic
299C22#341411yescharacteristic
300C22#341411yescharacteristic
301C4—22no—
302C4—22no—
303C4—22no—
304C4—41no—
305C4—41no—
306C4—84no—
307C4—84no—
308C4—162no—
309C4—162no—
310C4#243411yescharacteristic
311C4C2×#45111yescharacteristic
312C4#316511yescharacteristic
313C4#323011yescharacteristic
314C2—24no—
315C2#2581311yescharacteristic
316C2#3352011yescharacteristic
317C2#3352111yescharacteristic
318C2C2×#225511yescharacteristic
319C2#3373211yescharacteristic
320C2#3380011yescharacteristic
321C2#3380211yescharacteristic
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. #13453C2semidirect
5. #13284C2semidirect
84. C22⋊C16Q8semidirect
293. C22#2255semidirect

Maximal subgroups

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

subgroupindex
C22×#4992
C2×#22552
#332092
#134532
#132842
#129202
#129192
#125992