Finite Group Databasebeta

← #410860

Subgroups — #410860

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#20846C212yesmaximal
3#13880C211yesmaximal, characteristic
4#11342C214yesmaximal, complemented
5#11210C214yesmaximal, complemented
6#10733C212yesmaximal
7#10727C211yesmaximal, characteristic
8C22×#117C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10C2×#428C2214yes—
11C2×#407C2212yes—
12#2316—24no—
13#2316—22no—
14#2316C2214yes—
15C2×#389—22no—
16C2×#389—22no—
17C2×#385—22no—
18C2×#384C2212yes—
19C2×#380—22no—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1854—24no—
23#1853—22no—
24#1852—22no—
25C2×#343C2214yes—
26#1736C2214yes—
27C2×#339C2218yes—
28#1444—22no—
29C4⋊#95C2212yes—
30C24×C4D412yes—
31C24×C4C2311yescharacteristic
32C22×M16C2314yes—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—48no—
36C2×#117D414yes—
37C2×#117C2314yes—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—28no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C42—22no—
54C22×C42C2312yes—
55#428—48no—
56C2×#107—24no—
57C2×#106—24no—
58#407—24no—
59#407Q818yes—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62#389—28no—
63#389—28no—
64#389—28no—
65#389—28no—
66#385—48no—
67#384—28no—
68#380—28no—
69#380—28no—
70#380—22no—
71#380—22no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4Q814yes—
74C2×#95—24no—
75C2×#95—21no—
76C2×#95—24no—
77C2×#95—21no—
78#343—28no—
79#340—24no—
80#340—24no—
81#339—48no—
82C25C2×D411yescharacteristic
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—28no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—44no—
94C23×C4—82no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D414yes—
99C23×C4C2×Q814yes—
100C23×C4C2×Q812yes—
101C23×C4#4112yes—
102C23×C4C2411yescharacteristic
103C2×M16—24no—
104C2×M16—28no—
105C2×M16—28no—
106C2×M16C2×Q818yes—
107C22×C8—24no—
108#117—28no—
109#117—28no—
110#117—44no—
111#117—44no—
112#117—48no—
113#117—48no—
114C2×C4⋊C4—216no—
115C2×C4⋊C4—216no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—48no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—84no—
133C2×C4⋊C4—84no—
134C2×C4⋊C4—84no—
135C2×C4⋊C4C2×D414yes—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4#4118yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42—48no—
153C2×C42C2×Q814yes—
154#107—48no—
155#106—48no—
156C22⋊C8—28no—
157C22⋊C8—28no—
158#95—216no—
159#95—24no—
160#95—48no—
161#95—42no—
162C24—22no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—41no—
167C24—41no—
168C24#12011yescharacteristic
169C24#12011yescharacteristic
170C24C22⋊Q812yes—
171C24C4⋊Q812yes—
172C24C22×D411yescharacteristic
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—216no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—28no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—816no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4#12012yes—
217C22×C4C22⋊Q814yes—
218C22×C4C22⋊Q814yes—
219C22×C4C22⋊Q812yes—
220C22×C4#12318yes—
221C22×C4C4⋊Q814yes—
222C22×C4C4⋊Q814yes—
223C22×C4C4⋊Q812yes—
224C22×C4C22×D412yes—
225C22×C4C22×Q812yes—
226C22×C4C22×Q811yescharacteristic
227C22×C4C2×#4111yescharacteristic
228C22×C4C2×#4114yes—
229C22×C4H(5,𝔽2)14yes—
230M16—216no—
231M16—48no—
232C2×C8—28no—
233C2×C8—44no—
234C2×C8—44no—
235C4⋊C4—216no—
236C4⋊C4—216no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—88no—
246C4⋊C4—88no—
247C4⋊C4—816no—
248C22⋊C4—216no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—28no—
254C42—48no—
255C42—48no—
256C42—44no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—24no—
266C23—24no—
267C23—21no—
268C23—21no—
269C23—24no—
270C23—24no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—44no—
279C23—44no—
280C23—44no—
281C23#39512yes—
282C23#39512yes—
283C23#45312yes—
284C23#45612yes—
285C23C2×#12011yescharacteristic
286C23C2×C22⋊Q811yescharacteristic
287C23C2×#12312yes—
288C23C2×C4⋊Q811yescharacteristic
289C23#53512yes—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—416no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—84no—
317C2×C4—88no—
318C2×C4—88no—
319C2×C4—88no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4#39514yes—
329C2×C4C2×C22⋊Q812yes—
330C2×C4#52514yes—
331C2×C4C2×C4⋊Q812yes—
332C2×C4#53414yes—
333C2×C4#55314yes—
334C2×C4#55418yes—
335C2×C4#55414yes—
336C8—48no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—48no—
355C22—48no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22#206014yes—
361C22C2×#39511yescharacteristic
362C22#248712yes—
363C22#250212yes—
364C22#261311yescharacteristic
365C22#284911yescharacteristic
366C22C2×#45311yescharacteristic
367C22C2×#45611yescharacteristic
368C22#301911yescharacteristic
369C22#302011yescharacteristic
370C4—24no—
371C4—48no—
372C4—48no—
373C4—84no—
374C4—84no—
375C4—161no—
376C4—161no—
377C4#257914yes—
378C4#258314yes—
379C2—22no—
380C2—24no—
381C2—22no—
382C2—42no—
383C2#1509211yescharacteristic
384C2C2×#206011yescharacteristic
385C2#2356111yescharacteristic
386C2#2961611yescharacteristic
387C2#2961911yescharacteristic
388C2#3299011yescharacteristic
389C2#3299411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11342C2semidirect
5. #11210C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#208462
#138802
#113422
#112102
#107332
#107272