Finite Group Databasebeta

← #243998

Subgroups — #243998

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic, complemented
2#21805C211yesmaximal, characteristic
3#21410C211yesmaximal, characteristic
4#21401C211yesmaximal, characteristic
5#20824C211yesmaximal, characteristic, complemented
6#20760C211yesmaximal, characteristic, complemented
7#10686C211yesmaximal, characteristic, complemented
8#10681C211yesmaximal, characteristic, complemented
9#10679C211yesmaximal, characteristic, complemented
10#10678C211yesmaximal, characteristic, complemented
11#10678C211yesmaximal, characteristic, complemented
12#10676C211yesmaximal, characteristic, complemented
13#10672C211yesmaximal, characteristic, complemented
14#10672C211yesmaximal, characteristic, complemented
15#7848C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1983—22no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1949—22no—
37#1895—22no—
38#1893—22no—
39#1854—22no—
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1835—22no—
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1809—22no—
54#1788—22no—
55#1776—22no—
56#1762—22no—
57C4×#102—22no—
58#1735—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×#41—21no—
64C22×M16D412yes—
65C23×C8D412yes—
66#532—22no—
67#532—22no—
68#532—22no—
69#532Q814yes—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×#124—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×C4⋊C8D412yes—
88C2×#103—22no—
89C2×#103—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×C22⋊C8D412yes—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97D412yes—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8D412yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117#382—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142C2×#41—24no—
143C2×#41—42no—
144C2×#41—82no—
145C2×#41—82no—
146C22×Q8—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—41no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—24no—
156C2×M16—24no—
157C2×M16—42no—
158C22×C8—24no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C2×Q812yes—
171#127—21no—
172#127—21no—
173#127—21no—
174#127C2×Q812yes—
175#124—24no—
176#124—42no—
177#121—24no—
178#121—24no—
179#121—42no—
180#121—42no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—42no—
184C4×D4—42no—
185#117—44no—
186#117—44no—
187#117—82no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42C2411yescharacteristic
224#107—44no—
225#107—82no—
226#106—44no—
227#106—82no—
228C4⋊C8—24no—
229#103—82no—
230#103—82no—
231#102—44no—
232#102—44no—
233#102—44no—
234#102—44no—
235#102—82no—
236#102—82no—
237C22⋊C8—24no—
238#97—24no—
239C4×C8—24no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258C24—41no—
259C24#12711yescharacteristic
260#41—88no—
261#41—88no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—44no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—82no—
280C2×D4—82no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C22⋊Q812yes—
298C22×C4C22⋊Q812yes—
299C22×C4C22⋊Q812yes—
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4#12711yescharacteristic
304C22×C4#12711yescharacteristic
305C22×C4#12711yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—48no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—48no—
328C2×C8#13514yes—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—82no—
357C42—82no—
358C42—82no—
359C42C22⋊Q812yes—
360C42C22⋊Q812yes—
361C42C22⋊Q812yes—
362C42#12312yes—
363C42#12312yes—
364C42#12312yes—
365C42C22×Q811yescharacteristic
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—42no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23#39912yes—
383C23C2×#12711yescharacteristic
384Q8—44no—
385Q8—82no—
386D4—44no—
387D4—44no—
388D4—44no—
389D4—88no—
390D4—82no—
391D4—82no—
392D4—82no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—88no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
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—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#39912yes—
428C2×C4#39912yes—
429C2×C4#39912yes—
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×#13512yes—
437C2×C4#57512yes—
438C2×C4#57512yes—
439C2×C4#57512yes—
440C2×C4#57612yes—
441C2×C4#57712yes—
442C8—24no—
443C8—42no—
444C8—42no—
445C8—42no—
446C8—42no—
447C8—42no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22C2×#39911yescharacteristic
463C22#314711yescharacteristic
464C22#315011yescharacteristic
465C22#315311yescharacteristic
466C22#315311yescharacteristic
467C22#315411yescharacteristic
468C22#315511yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—82no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4#307612yes—
480C4#308012yes—
481C4#308112yes—
482C4#309112yes—
483C4#309312yes—
484C4#309612yes—
485C2—42no—
486C2—162no—
487C2#1964411yescharacteristic
488C2#1964911yescharacteristic
489C2#1969211yescharacteristic
490C2#1969511yescharacteristic
491C2#2373011yescharacteristic
492C2#2373311yescharacteristic
493C2#2373511yescharacteristic
subgroupG/H
center, socleC23C2×#127
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. #21984C2semidirect
5. #20824C2semidirect
6. #20760C2semidirect
7. #10686C2semidirect
8. #10681C2semidirect
9. #10679C2semidirect
10. #10678C2semidirect
11. #10678C2semidirect
12. #10676C2semidirect
13. #10672C2semidirect
14. #10672C2semidirect
15. #7848C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#218052
#214102
#214012
#208242
#207602
#106862
#106812
#106792
#106782
#106762
#106722
#78482