Finite Group Databasebeta

← #410861

Subgroups — #410861

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#20846C212yesmaximal
3#13881C211yesmaximal, characteristic
4#11342C214yesmaximal, complemented
5#11211C214yesmaximal, 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×#382C2212yes—
18C2×#380—22no—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21#1854—24no—
22#1853—22no—
23#1852—22no—
24C2×#343C2214yes—
25#1736C2214yes—
26C2×#339C2218yes—
27#1442—22no—
28C4⋊#95C2212yes—
29C22⋊#95—22no—
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⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—22no—
53C22×C42C2312yes—
54#428—48no—
55C2×#107—24no—
56C2×#106—24no—
57#407—24no—
58#407Q818yes—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61#389—28no—
62#389—28no—
63#389—28no—
64#389—28no—
65#382—28no—
66#380—28no—
67#380—28no—
68#380—22no—
69#380—22no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4Q814yes—
72C2×#95—24no—
73C2×#95—21no—
74C2×#95—24no—
75C2×#95—21no—
76C2×#95—24no—
77#343—28no—
78#340—24no—
79#340—24no—
80#339—48no—
81C25C2×D411yescharacteristic
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—28no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—44no—
93C23×C4—82no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D414yes—
98C23×C4C2×Q814yes—
99C23×C4C2×Q812yes—
100C23×C4#4112yes—
101C23×C4C2411yescharacteristic
102C2×M16—24no—
103C2×M16—28no—
104C2×M16—28no—
105C2×M16C2×Q818yes—
106C22×C8—24no—
107#117—28no—
108#117—28no—
109#117—44no—
110#117—44no—
111#117—48no—
112#117—48no—
113C2×C4⋊C4—216no—
114C2×C4⋊C4—216no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4#4118yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—48no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—48no—
149C2×C42C2×Q814yes—
150#107—48no—
151#106—48no—
152C22⋊C8—28no—
153C22⋊C8—28no—
154#95—216no—
155#95—24no—
156#95—42no—
157#95—48no—
158#95—84no—
159C24—22no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—41no—
164C24—41no—
165C24#12011yescharacteristic
166C24#12011yescharacteristic
167C24C22⋊Q812yes—
168C24C4⋊Q812yes—
169C24C22×D411yescharacteristic
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—216no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—28no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—816no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4#12012yes—
214C22×C4C22⋊Q814yes—
215C22×C4C22⋊Q814yes—
216C22×C4C22⋊Q812yes—
217C22×C4#12318yes—
218C22×C4C4⋊Q814yes—
219C22×C4C4⋊Q814yes—
220C22×C4C4⋊Q812yes—
221C22×C4C22×D412yes—
222C22×C4C22×Q812yes—
223C22×C4C22×Q811yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C22×C4C2×#4114yes—
226C22×C4H(5,𝔽2)14yes—
227M16—216no—
228M16—48no—
229C2×C8—28no—
230C2×C8—44no—
231C2×C8—44no—
232C4⋊C4—216no—
233C4⋊C4—216no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—88no—
243C4⋊C4—88no—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C22⋊C4—48no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C42—28no—
250C42—48no—
251C42—48no—
252C42—44no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—24no—
262C23—24no—
263C23—21no—
264C23—21no—
265C23—24no—
266C23—24no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23—44no—
276C23—44no—
277C23#39512yes—
278C23#39512yes—
279C23#45412yes—
280C23#45512yes—
281C23C2×#12011yescharacteristic
282C23C2×C22⋊Q811yescharacteristic
283C23C2×#12312yes—
284C23C2×C4⋊Q811yescharacteristic
285C23#53512yes—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—416no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—84no—
309C2×C4—88no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4#39514yes—
325C2×C4C2×C22⋊Q812yes—
326C2×C4#52514yes—
327C2×C4C2×C4⋊Q812yes—
328C2×C4#53414yes—
329C2×C4#55314yes—
330C2×C4#55418yes—
331C2×C4#55414yes—
332C8—48no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—48no—
351C22—48no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22#206114yes—
357C22C2×#39511yescharacteristic
358C22#248712yes—
359C22#250212yes—
360C22#261311yescharacteristic
361C22#284911yescharacteristic
362C22C2×#45511yescharacteristic
363C22C2×#45411yescharacteristic
364C22#301911yescharacteristic
365C22#302011yescharacteristic
366C4—24no—
367C4—48no—
368C4—84no—
369C4—84no—
370C4—84no—
371C4—161no—
372C4—161no—
373C4#257914yes—
374C4#258314yes—
375C2—22no—
376C2—24no—
377C2—22no—
378C2—42no—
379C2#1509211yescharacteristic
380C2C2×#206111yescharacteristic
381C2#2356111yescharacteristic
382C2#2961711yescharacteristic
383C2#2961811yescharacteristic
384C2#3299111yescharacteristic
385C2#3299311yescharacteristic
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. #11211C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#208462
#138812
#113422
#112112
#107332
#107272