Finite Group Databasebeta

← #323894

Subgroups — #323894

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24659C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4#10650C211yesmaximal, characteristic, complemented
5#10616C211yesmaximal, characteristic, complemented
6#10545C211yesmaximal, characteristic, complemented
7#10264C211yesmaximal, characteristic, complemented
8#9566C212yesmaximal, complemented
9#9217C212yesmaximal, complemented
10#9140C212yesmaximal, complemented
11#7875C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#461C2212yes—
14C2×C8×D4—21no—
15C2×C8×D4C2211yescharacteristic
16C2×#432C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2036—22no—
21#2000—22no—
22#1990C2212yes—
23#1981C2212yes—
24#1973—22no—
25#1954C2211yescharacteristic
26#1947C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2211yescharacteristic
30#1935—22no—
31#1923—21no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×SD16—22no—
48C23×C8—21no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51C2×C4⋊Q8C2312yes—
52C2×C22⋊Q8—22no—
53C2×#121D411yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#500—42no—
58#461—24no—
59#461—42no—
60C8×D4—42no—
61C8×D4—44no—
62#432—24no—
63#432—42no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#102—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×#97—22no—
84C2×C4×C8—21no—
85C2×C4×C8—42no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—22no—
91#390D411yescharacteristic
92#389—22no—
93#389—22no—
94#389D411yescharacteristic
95#388—21no—
96#386—21no—
97#384C2312yes—
98#380—22no—
99#380—22no—
100#380D411yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C4×C22⋊C4—21no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#332—24no—
111#332—42no—
112C22×Q8—22no—
113C22×D4—21no—
114C22×D4C2×D411yescharacteristic
115C23×C4—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C2×SD16—44no—
120C2×SD16—44no—
121C2×SD16—82no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C22⋊Q8—82no—
144C22⋊Q8—82no—
145#121—22no—
146#121—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—42no—
150C4×D4—42no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—42no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42C2411yescharacteristic
174#107—42no—
175#107—42no—
176#107—42no—
177#107—42no—
178#107—82no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—42no—
185C4⋊C8#4114yes—
186#103—44no—
187#103—44no—
188#103—82no—
189#102—82no—
190C22⋊C8—44no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195#97—44no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—41no—
199C4×C8—41no—
200C4×C8—42no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—42no—
205#95—42no—
206C24—21no—
207C24#12111yescharacteristic
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—22no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4#13512yes—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251SD16—88no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—24no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—84no—
276C2×C8AGL(1,ℤ/8)12yes—
277C2×C8#13712yes—
278C2×C8#13712yes—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4#13512yes—
294C22⋊C4—24no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—84no—
300C42—21no—
301C42—21no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C42C2×#4112yes—
307C42H(5,𝔽2)12yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#47312yes—
322C23C2×#12111yescharacteristic
323Q8—82no—
324Q8—82no—
325Q8—82no—
326Q8—82no—
327D4—24no—
328D4—48no—
329D4—42no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#53211yescharacteristic
368C2×C4#53411yescharacteristic
369C2×C4#54612yes—
370C2×C4#55312yes—
371C2×C4C2×#13511yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C2×C4C2×#13711yescharacteristic
376C2×C4#57511yescharacteristic
377C2×C4#57612yes—
378C8—22no—
379C8—22no—
380C8—22no—
381C8—41no—
382C8—44no—
383C8—81no—
384C8—81no—
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22#254311yescharacteristic
401C22#303611yescharacteristic
402C22#303911yescharacteristic
403C22#305011yescharacteristic
404C22#305811yescharacteristic
405C22C2×#47311yescharacteristic
406C22#306711yescharacteristic
407C4—22no—
408C4—42no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—162no—
415C4—162no—
416C4#317211yescharacteristic
417C4#318211yescharacteristic
418C4#319311yescharacteristic
419C4#319611yescharacteristic
420C4#329712yes—
421C4#330512yes—
422C2—42no—
423C2—82no—
424C2#1925911yescharacteristic
425C2#1927311yescharacteristic
426C2#1933511yescharacteristic
427C2#1935811yescharacteristic
428C2#2855211yescharacteristic
429C2#2857711yescharacteristic
430C2#2859111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33209C2semidirect
4. #10650C2semidirect
5. #10616C2semidirect
6. #10545C2semidirect
7. #10264C2semidirect
8. #9566C2semidirect
9. #9217C2semidirect
10. #9140C2semidirect
11. #7875C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#246592
C8×#1212
#106502
#106162
#105452
#102642
#95662
#92172
#91402
#78752