Finite Group Databasebeta

← #329434

Subgroups — #329434

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic, complemented
2#32956C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic
4C2×#1954C211yesmaximal, characteristic
5#10621C212yesmaximal, complemented
6#10484C212yesmaximal, complemented
7#10480C212yesmaximal, complemented
8#10275C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic, complemented
10#9127C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#487C2211yescharacteristic, complemented
14C2×#476C2212yes—
15C2×#474C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8—21no—
18C22×C22⋊C8—21no—
19C22×C4×C8C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#1990C2211yescharacteristic, complemented
23#1956C2211yescharacteristic, complemented
24#1954—21no—
25#1954—21no—
26#1954C2212yes—
27#1954C2212yes—
28#1933—21no—
29#1933—21no—
30#1892—21no—
31#1892—21no—
32#1864—22no—
33#1862—22no—
34#1853—21no—
35#1852—21no—
36#1834C2211yescharacteristic, complemented
37#1802C2212yes—
38#1802C2212yes—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1786—21no—
42#1786—21no—
43C2×#334C2212yescomplemented
44C2×#333C2212yescomplemented
45C2×#332C2212yescomplemented
46#1296C2212yescomplemented
47#1296C2212yescomplemented
48#1277C2212yescomplemented
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4D412yes—
63C2×C4×D4C2312yes—
64C22×C4⋊C4D411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C42C2311yescharacteristic
67#487—42no—
68#476—42no—
69#476—42no—
70#474—42no—
71#474—42no—
72C2×#107—22no—
73C2×#106—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—21no—
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—44no—
94C2×C22⋊C8—44no—
95C2×C4×C8—42no—
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#340—21no—
113#340—21no—
114#340—21no—
115#340—21no—
116#340—22no—
117#340—22no—
118#336—21no—
119#336—21no—
120#334—42no—
121#333—42no—
122#332—42no—
123C25—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×D4—24no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4C2×D412yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—82no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42C2411yescharacteristic
202#107—44no—
203#106—44no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—42no—
211#103—42no—
212#103—42no—
213#103—42no—
214#103—42no—
215#102—44no—
216#102—44no—
217#102—44no—
218#102—44no—
219C22⋊C8—44no—
220C22⋊C8—44no—
221C22⋊C8—44no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C22⋊C8—44no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—41no—
228C4×C8—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233C24—24no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24#12111yescharacteristic
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4#13514yes—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#13512yes—
296C22×C4AGL(1,ℤ/8)12yes—
297C22×C4#13712yes—
298C22×C4#13712yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—84no—
325C2×C8AGL(1,ℤ/8)12yes—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—22no—
351C42—21no—
352C42—22no—
353C42H(5,𝔽2)12yes—
354C42H(5,𝔽2)12yes—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—24no—
360C23—48no—
361C23—48no—
362C23—48no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—84no—
370C23—84no—
371C23—84no—
372C23—84no—
373C23#46312yes—
374C23#47012yes—
375C23#47112yes—
376C23C2×#12111yescharacteristic
377Q8—81no—
378Q8—81no—
379Q8—81no—
380Q8—81no—
381Q8—162no—
382D4—24no—
383D4—24no—
384D4—48no—
385D4—42no—
386D4—42no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—88no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4#46312yes—
425C2×C4#47012yes—
426C2×C4#53411yescharacteristic
427C2×C4#53412yes—
428C2×C4#53511yescharacteristic
429C2×C4#53812yes—
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×#13512yes—
432C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4#57511yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8—41no—
441C8—82no—
442C8—82no—
443C8—82no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—816no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22#170314yes—
459C22#261711yescharacteristic
460C22C2×#46311yescharacteristic
461C22#303611yescharacteristic
462C22#305011yescharacteristic
463C22C2×#47011yescharacteristic
464C22C2×#47111yescharacteristic
465C22#306711yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—41no—
470C4—82no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4#318811yescharacteristic
476C4#319012yes—
477C4#319012yes—
478C4#319411yescharacteristic
479C4#321111yescharacteristic
480C4#321711yescharacteristic
481C2—24no—
482C2—84no—
483C2#1911011yescharacteristic
484C2C2×#170311yescharacteristic
485C2#1925711yescharacteristic
486C2#1928111yescharacteristic
487C2#2971811yescharacteristic
488C2#2973311yescharacteristic
489C2#2975011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #33043C2semidirect
5. #10621C2semidirect
6. #10484C2semidirect
7. #10480C2semidirect
8. #10275C2semidirect
9. #9131C2semidirect
10. #9127C2semidirect
11. #8595C2semidirect
13. C2×#487C22semidirect
22. #1990C22semidirect
23. #1956C22semidirect
36. #1834C22semidirect
43. C2×#334C22semidirect
44. C2×#333C22semidirect
45. C2×#332C22semidirect
46. #1296C22semidirect
47. #1296C22semidirect
48. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#330432
#329562
#329532
C2×#19542
#106212
#104842
#104802
#102752
#91312
#91272
#85952