Finite Group Databasebeta

← #460497

Subgroups — #460497

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2100C212yesmaximal, complemented
3#14686C211yesmaximal, characteristic
4#11385C212yesmaximal, complemented
5#11268C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#11199C214yesmaximal
8#10712C211yesmaximal, characteristic
9C22×#125C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2851—28no—
13C2×#389C2212yes—
14C2×#387—22no—
15C2×#385—22no—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#380—22no—
20C2×#380C2212yes—
21C2×#380C2212yes—
22#2100—28no—
23C2×#340C2214yes—
24C2×#340C2214yes—
25C2×#340C2214yes—
26C2×#340C2214yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C4⋊#95—22no—
30C4⋊#95C2212yes—
31C24×C4D412yes—
32C24×C4C2311yescharacteristic
33C23×C8C2314yes—
34C2×#125—24no—
35C2×#125—24no—
36C2×#125—42no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—22no—
47C22×C42—24no—
48C22×C42—24no—
49C22×C42—42no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C2×C22⋊C8—24no—
54C2×C22⋊C8—42no—
55C2×C22⋊C8—42no—
56#389—24no—
57#389D418yes—
58#387—216no—
59#385—216no—
60#384—24no—
61#384—42no—
62#382—24no—
63#380—24no—
64#380—24no—
65#380—28no—
66#380—48no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—24no—
72#340—28no—
73#340—28no—
74#340—44no—
75#340—44no—
76#336—24no—
77#336—24no—
78C25C2×D411yescharacteristic
79C23×C4—24no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4#4114yes—
101C23×C4C2411yescharacteristic
102C22×C8—24no—
103C22×C8—24no—
104C22×C8—24no—
105C22×C8—24no—
106C22×C8—24no—
107C22×C8—24no—
108C22×C8—24no—
109#125—416no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—48no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—416no—
151C2×C42—42no—
152C2×C42C2×D414yes—
153C2×C42#4114yes—
154C22⋊C8—48no—
155#95—28no—
156#95—28no—
157#95—28no—
158#95—48no—
159#95—44no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24#12412yes—
170C24AGL(1,ℤ/8)14yes—
171C24C22×D411yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—216no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4#12014yes—
230C22×C4#12314yes—
231C22×C4#12412yes—
232C22×C4#13514yes—
233C22×C4#13714yes—
234C22×C4C22×D412yes—
235C22×C4C2×#4112yes—
236C22×C4C2×#4112yes—
237C22×C4H(5,𝔽2)11yescharacteristic
238C22×C4H(5,𝔽2)12yes—
239C22×C42−1+411yescharacteristic
240C22×C42−1+412yes—
241C22×C42−1+412yes—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—28no—
252C4⋊C4—416no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—416no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—88no—
261C4⋊C4—88no—
262C22⋊C4—416no—
263C22⋊C4—88no—
264C42—28no—
265C42—28no—
266C42—48no—
267C23—24no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—22no—
279C23—41no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23#49014yes—
290C23C2×#12011yescharacteristic
291C23C2×#12311yescharacteristic
292C23C2×#12411yescharacteristic
293C23#53411yescharacteristic
294C23#53611yescharacteristic
295C23#53611yescharacteristic
296C23#53711yescharacteristic
297C23C2×#13512yes—
298C23C2×AGL(1,ℤ/8)12yes—
299C23C2×#13712yes—
300C23#57512yes—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—432no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4#45218yes—
349C2×C4#53612yes—
350C2×C4#53712yes—
351C2×C4#53812yes—
352C2×C4#53912yes—
353C2×C4#54112yes—
354C2×C4#54212yes—
355C2×C4#54614yes—
356C2×C4#54814yes—
357C2×C4#55214yes—
358C2×C4#56314yes—
359C8—84no—
360C22—24no—
361C22—24no—
362C22—24no—
363C22—44no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22—42no—
368C22—44no—
369C22—42no—
370C22—44no—
371C22#240511yescharacteristic
372C22#243611yescharacteristic
373C22#250311yescharacteristic
374C22#261611yescharacteristic
375C22#262511yescharacteristic
376C22#263911yescharacteristic
377C22#281711yescharacteristic
378C22C2×#45212yes—
379C22#300612yes—
380C22#309112yes—
381C22#309212yes—
382C22#313111yescharacteristic
383C22C2×#49011yescharacteristic
384C22#313911yescharacteristic
385C22#314211yescharacteristic
386C22#318812yes—
387C22#319612yes—
388C22#323112yes—
389C22#323312yes—
390C22#323412yes—
391C22#323812yes—
392C22#324712yes—
393C22#325212yes—
394C4—28no—
395C4—44no—
396C4—42no—
397C4—84no—
398C4—84no—
399C4—162no—
400C4—162no—
401C4#269014yes—
402C4#270214yes—
403C2—44no—
404C2#1636611yescharacteristic
405C2#2400712yes—
406C2#2590411yescharacteristic
407C2#2591311yescharacteristic
408C2#2796311yescharacteristic
409C2#2796611yescharacteristic
410C2#2969212yes—
411C2#2994412yes—
412C2#3018911yescharacteristic
413C2#3020211yescharacteristic
414C2#3278712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
4. #11385C2semidirect
5. #11268C2semidirect
6. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#21002
#146862
#113852
#112682
#112072
#111992
#107122