Finite Group Databasebeta

← #6495020

Subgroups — #6495020

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2868C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic
3#33316C212yesmaximal, complemented
4#33293C212yesmaximal, complemented
5#30309C214yesmaximal, complemented
6#29208C212yesmaximal, complemented
7#29206C212yesmaximal, complemented
8#28694C214yesmaximal, complemented
9#27950C212yesmaximal
10#27948C212yesmaximal
11#25816C214yesmaximal
12#21987C214yesmaximal
13C4×#432C211yesmaximal, characteristic
14C22×C4⋊Q8—21no—
15C22×C4⋊Q8—21no—
16C22×C4⋊Q8C2212yes—
17C2×#501C2218yes—
18C2×#432C2212yes—
19#2868—21no—
20#2868—21no—
21#2868—21no—
22#2868—21no—
23#2868C2212yes—
24#2868C2212yes—
25#2868C2212yes—
26#2868C2212yes—
27#2864C2212yes—
28#2863—21no—
29#2863—21no—
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2855—21no—
35#2855—21no—
36#2851C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×#125C2212yes—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8—21no—
42#2295—21no—
43#2295—21no—
44C2×#384—21no—
45C2×#384—21no—
46C2×#384C2212yes—
47C2×#384C2212yes—
48#2283C2212yes—
49C2×C43C2211yescharacteristic
50#2084C2218yes—
51#2083C2218yes—
52#2058C2218yes—
53#2057C2218yes—
54#1993C2218yes—
55#1990C2214yes—
56#1990C2214yes—
57#1948C2218yes—
58#1947C2218yes—
59#1853C2218yes—
60#1852C2218yes—
61#1847C2218yes—
62#1845C2214yes—
63#1844C2214yes—
64#1836C2218yes—
65#1767C2212yes—
66C4×C4⋊C8C2212yes—
67#1749C2212yes—
68C4×C22⋊C8C2212yes—
69#1745C2212yes—
70C4×#97C2212yes—
71C23×Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—48no—
83C2×C4⋊Q8—48no—
84C2×C4⋊Q8—48no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2314yes—
88C2×C4⋊Q8C2314yes—
89C2×#125C2312yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4C2314yes—
100C22×C42D411yescharacteristic
101C22×C42D411yescharacteristic
102C22×C42D411yescharacteristic
103C22×C42D411yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#501—416no—
108#432Q8116yescomplemented
109C2×#107C23116yes—
110C2×#106C23116yes—
111C2×C4⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×#97C2314yes—
114C2×#97C2314yes—
115C2×C4×C8C2314yes—
116#391—21no—
117#391—21no—
118#391—21no—
119#391—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#391—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389C2312yes—
133#389C2312yes—
134#389C2312yes—
135#389C2312yes—
136#387—22no—
137#387—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#385—22no—
145#385—22no—
146#384—22no—
147#384—22no—
148#384—22no—
149#384—22no—
150#384—22no—
151#384—22no—
152#384—22no—
153#384—22no—
154#384—22no—
155#384—22no—
156#384—22no—
157#384—22no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2314yes—
161#384C2314yes—
162#384C2314yes—
163#384C2314yes—
164#383—22no—
165#382C2312yes—
166#382C2312yes—
167#382C2312yes—
168#381—22no—
169#380—21no—
170#380—21no—
171#380C2312yes—
172#380C2312yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—22no—
180#376—22no—
181C43D411yescharacteristic
182C43D411yescharacteristic
183C43D411yescharacteristic
184C43D411yescharacteristic
185C43C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#340C23116yes—
190#340C23116yes—
191#336C2314yes—
192#336C2314yes—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—48no—
198C22×Q8—48no—
199C22×Q8—48no—
200C23×C4—24no—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2411yescharacteristic
208C22×C8C2418yes—
209C4⋊Q8—216no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—48no—
213C4⋊Q8—48no—
214C4⋊Q8—48no—
215C4⋊Q8—48no—
216C4⋊Q8—48no—
217C4⋊Q8—48no—
218C4⋊Q8—48no—
219C4⋊Q8—48no—
220C4⋊Q8—48no—
221C4⋊Q8—48no—
222C4⋊Q8—48no—
223#125—28no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C4×Q8—44no—
227C4×Q8—44no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4C2418yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×Q818yes—
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—416no—
302#106—416no—
303C4⋊C8C2×Q8116yescomplemented
304C22⋊C8C2×Q8116yes—
305#97—28no—
306#97C2×Q8116yescomplemented
307C4×C8—28no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322C24C22×D411yescharacteristic
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—48no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—832no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4C4⋊Q814yes—
361C22×C4C4⋊Q814yes—
362C22×C4AGL(1,ℤ/8)14yes—
363C22×C4#13714yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q814yes—
378C22×C4C2511yescharacteristic
379C2×C8—28no—
380C2×C8—28no—
381C2×C8C22×Q8116yes—
382C4⋊C4—24no—
383C4⋊C4—216no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—44no—
433C42—44no—
434C42C4⋊Q814yes—
435C42C4⋊Q814yes—
436C42AGL(1,ℤ/8)14yes—
437C42#13714yes—
438C42C22×Q814yes—
439C42C22×Q814yes—
440C42H(5,𝔽2)14yes—
441C422−1+414yes—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23C2×C4⋊Q812yes—
447C23C2×AGL(1,ℤ/8)12yes—
448C23C2×#13712yes—
449C23C23×D411yescharacteristic
450Q8—816no—
451Q8—816no—
452Q8—84no—
453Q8—84no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—816no—
497C2×C4C2×C4⋊Q812yes—
498C2×C4C2×C4⋊Q812yes—
499C2×C4C2×C4⋊Q812yes—
500C2×C4C2×C4⋊Q812yes—
501C2×C4C2×C4⋊Q812yes—
502C2×C4#53512yes—
503C2×C4#53512yes—
504C2×C4#53612yes—
505C2×C4#53612yes—
506C2×C4C2×AGL(1,ℤ/8)12yes—
507C2×C4C2×AGL(1,ℤ/8)12yes—
508C2×C4C2×AGL(1,ℤ/8)12yes—
509C2×C4C2×AGL(1,ℤ/8)12yes—
510C2×C4C2×AGL(1,ℤ/8)12yes—
511C2×C4C2×#13712yes—
512C2×C4C2×#13712yes—
513C2×C4C2×#13712yes—
514C2×C4C2×#13712yes—
515C2×C4C2×#13712yes—
516C2×C4C23×Q812yes—
517C2×C4C2×H(5,𝔽2)12yes—
518C2×C4C2×2−1+412yes—
519C8—216no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22#316514yes—
527C22C22×C4⋊Q811yescharacteristic
528C22C2×#53511yescharacteristic
529C22C2×#53611yescharacteristic
530C22C22×AGL(1,ℤ/8)11yescharacteristic
531C22C22×AGL(1,ℤ/8)11yescharacteristic
532C22C22×#13711yescharacteristic
533C22C22×#13711yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—82no—
545C4—88no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#316514yes—
550C4#323814yes—
551C4#323914yes—
552C4#350314yes—
553C2—24no—
554C2#3528111yescharacteristic
555C2C2×#316511yescharacteristic
556C2C2×#316511yescharacteristic
557C2#6079811yescharacteristic
558C2#6080011yescharacteristic
559C2C2×#323811yescharacteristic
560C2C2×#323911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33293C2semidirect
5. #30309C2semidirect
6. #29208C2semidirect
7. #29206C2semidirect
8. #28694C2semidirect
108. #432Q8semidirect
303. C4⋊C8C2×Q8semidirect
306. #97C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
#366352
#333162
#332932
#303092
#292082
#292062
#286942
#279502
#279482
#258162
#219872
C4×#4322