Finite Group Databasebeta

← #238127

Subgroups — #238127

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2#21859C211yesmaximal, characteristic
3#21386C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5#10356C214yesmaximal, complemented
6#10354C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×#125—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#417—21no—
16C2×#417—21no—
17#2552—22no—
18#2552—22no—
19#2552—22no—
20#2552C2214yes—
21C2×#398—22no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#384—21no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2212yes—
29#1988C2212yes—
30#1983—22no—
31#1982C2212yes—
32#1982C2212yes—
33#1981C2212yes—
34#1957C2212yes—
35#1874—22no—
36#1874—21no—
37#1874—21no—
38#1802—22no—
39C4×#102—22no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×#41—21no—
44C22×M16—21no—
45#532—22no—
46#532—22no—
47#532—22no—
48#532Q814yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—44no—
54C2×#125—44no—
55C2×#124—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431—22no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83#417—44no—
84#417—44no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—24no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#398—28no—
97#398—28no—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#384—24no—
103#384—24no—
104#384—21no—
105#384—21no—
106#384D412yes—
107#384C2312yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C4⋊C4C2312yes—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119#340—24no—
120#340—22no—
121#340—22no—
122C2×#41—24no—
123C2×#41—42no—
124C2×#41—42no—
125C2×#41—42no—
126C2×#41—42no—
127C2×#41—42no—
128C22×Q8—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—41no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—42no—
143C2×M16—42no—
144C2×M16—42no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8C2×Q812yes—
152#127—21no—
153#127—21no—
154#127—21no—
155#127C2×Q812yes—
156#125—48no—
157#125—416no—
158#125—48no—
159#124—24no—
160#124—42no—
161#121—24no—
162#121—24no—
163#121—42no—
164#121—42no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—42no—
168C4×D4—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4D814yes—
195C2×C4⋊C4SD1614yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224#103—44no—
225#103—44no—
226#102—44no—
227#102—44no—
228#102—44no—
229#102—44no—
230#102—84no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—42no—
236#95—28no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C24—41no—
242C24#12011yescharacteristic
243#41—48no—
244#41—48no—
245#41—84no—
246#41—84no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4C22⋊Q812yes—
304C22×C4#12412yes—
305C22×C4#12412yes—
306C22×C4#12412yes—
307C22×C4C2×D812yes—
308C22×C4C2×SD1612yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—84no—
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—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—416no—
339C4⋊C4—416no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—82no—
347C4⋊C4—88no—
348C4⋊C4C2×D814yes—
349C4⋊C4C2×SD1614yes—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C42—44no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—44no—
357C42C22⋊Q812yes—
358C42C22⋊Q812yes—
359C42C22⋊Q812yes—
360C42#12412yes—
361C42#12412yes—
362C42#12412yes—
363C42C22×Q811yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—42no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23#39312yes—
379C23#44712yes—
380C23#44812yes—
381C23C2×#12011yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—81no—
385Q8—81no—
386D4—44no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—82no—
391D4—88no—
392D4—81no—
393D4—81no—
394D4—82no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—88no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—88no—
429C2×C4#39312yes—
430C2×C4#39312yes—
431C2×C4#39312yes—
432C2×C4#44712yes—
433C2×C4#44812yes—
434C2×C4#47414yes—
435C2×C4#47614yes—
436C2×C4#48614yes—
437C2×C4#48914yes—
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C2×#12411yescharacteristic
443C2×C4C2×#12411yescharacteristic
444C2×C4C22×D811yescharacteristic
445C2×C4C22×SD1611yescharacteristic
446C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4#57512yes—
449C2×C4#57512yes—
450C8—82no—
451C8—82no—
452C8—82no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22#202314yes—
467C22C2×#39311yescharacteristic
468C22C2×#44711yescharacteristic
469C22C2×#44811yescharacteristic
470C22#300511yescharacteristic
471C22#300511yescharacteristic
472C22#301911yescharacteristic
473C22#302011yescharacteristic
474C4—22no—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—44no—
479C4—44no—
480C4—82no—
481C4—82no—
482C4—84no—
483C4—82no—
484C4#202314yes—
485C4C2×#47411yescharacteristic
486C4C2×#47611yescharacteristic
487C4#308012yes—
488C4C2×#48611yescharacteristic
489C4C2×#48911yescharacteristic
490C4#314212yes—
491C2—24no—
492C2—162no—
493C2#1874811yescharacteristic
494C2#1875611yescharacteristic
495C2#1876611yescharacteristic
496C2#1877811yescharacteristic
497C2C2×#202311yescharacteristic
498C2#2345311yescharacteristic
499C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. C2×#2552C2semidirect
5. #10356C2semidirect
6. #10354C2semidirect
7. #8043C2semidirect
8. #7901C2semidirect
9. #7892C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#218592
#213862
#212582
#103562
#103542
#80432
#79012
#78922