Finite Group Databasebeta

← #6526686

Subgroups — #6526686

Order 512.

2188 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic
4#34213C211yesmaximal, characteristic, complemented
5#28120C214yesmaximal
6#27647C214yesmaximal, complemented
7#25451C214yesmaximal, complemented
8#25320C214yesmaximal, complemented
9#24813C214yesmaximal, complemented
10#24678C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21833C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C22×C22⋊Q8—21no—
18C22×C4×D4C2211yescharacteristic
19#3229—28no—
20C2×#498C2212yescomplemented
21C2×#487C2212yescomplemented
22#3101—28no—
23C2×#483C2212yes—
24C2×#483C2214yes—
25C2×#475C2214yescomplemented
26C2×#476C2212yes—
27C2×#461C2212yes—
28C2×#450C2212yes—
29C2×C4×SD16C2214yes—
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2542—22no—
33#2542—22no—
34#2542C2214yes—
35#2542C2214yes—
36C2×#399—22no—
37#2370—22no—
38#2369C2212yescomplemented
39#2365C2212yescomplemented
40#2355—22no—
41#2355C2212yescomplemented
42Q8×C4⋊C4C2212yescomplemented
43#2329—22no—
44#2317—21no—
45C4×C22⋊Q8C2214yes—
46C2×#390C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2212yes—
49#2274—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2084C2218yescomplemented
52#2072C2218yescomplemented
53#2065C2218yes—
54#2055C2218yes—
55#1990C2212yescomplemented
56#1990C2212yescomplemented
57#1988C2214yescomplemented
58#1948C2218yescomplemented
59#1939C2214yescomplemented
60#1939C2214yescomplemented
61#1926C2218yes—
62#1880C2218yes—
63#1823C2214yes—
64#1823C2212yes—
65#1803C2218yescomplemented
66#1802C2212yes—
67#1802C2212yes—
68#1797C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C23×Q8—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×SD16C2314yes—
75#533—24no—
76#533D418yes—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—48no—
92C2×C22⋊Q8—48no—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2314yes—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×#117—21no—
104C2×#117—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—22no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#498—28no—
118#487—28no—
119#483—28no—
120#483—216no—
121#476—28no—
122#475—216no—
123#461—28no—
124#450—28no—
125C4×SD16—216no—
126#431—24no—
127#431—24no—
128C2×#106C2318yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2318yes—
133C2×#103C2318yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#399—28no—
143#399—28no—
144#391—24no—
145#391C2314yes—
146#390—24no—
147#390C2318yes—
148#389—22no—
149#389—22no—
150#389—24no—
151#389C2314yes—
152#387—24no—
153#387—24no—
154#387—24no—
155#385—24no—
156#384—22no—
157#384—24no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2318yes—
161#382—22no—
162#382—21no—
163#380—21no—
164#380C2312yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—24no—
178C2×#95D412yes—
179#340C2318yes—
180#340C2314yes—
181#340C2314yes—
182#340C2318yes—
183#336C2318yes—
184C25—21no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8C2412yes—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4C2412yes—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×SD16—28no—
211C2×SD16—28no—
212C2×SD16—44no—
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—24no—
216C4⋊Q8—24no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223C4⋊Q8C2×D414yes—
224C4⋊Q8C2×D414yes—
225#125—28no—
226C22⋊Q8—28no—
227C22⋊Q8—24no—
228C22⋊Q8—416no—
229C22⋊Q8—416no—
230C22⋊Q8—416no—
231C22⋊Q8—416no—
232C22⋊Q8C2×D418yes—
233C4×Q8—28no—
234C4×Q8—24no—
235C4×Q8—48no—
236C4×Q8C2×D418yes—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—44no—
242C4×D4—44no—
243#117—28no—
244#117—44no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4SD1618yescomplemented
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—48no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D414yes—
298C2×C42#4114yes—
299C2×C42C2414yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#106—28no—
303#106—216no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306C4⋊C8—48no—
307#103—28no—
308#103—216no—
309#102—28no—
310#102—28no—
311#102—44no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C4×C8—24no—
315C4×C8—24no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—24no—
325C24—22no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24C22×D411yescharacteristic
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8—816no—
346C2×Q8C22×D414yes—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D42−1+414yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—816no—
386C22×C4#12114yes—
387C22×C4#12114yes—
388C22×C4C2×SD1618yes—
389C22×C4C2×SD1614yes—
390C22×C4AGL(1,ℤ/8)14yes—
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D414yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2511yescharacteristic
401SD16—416no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—416no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—416no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4C2×SD1618yes—
435C4⋊C4C2×SD1618yes—
436C4⋊C4C22×D414yes—
437C22⋊C4—28no—
438C22⋊C4—416no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—416no—
443C42—28no—
444C42—28no—
445C42—44no—
446C42—44no—
447C42—84no—
448C42—84no—
449C42#12114yes—
450C42#12114yes—
451C42C22×D412yes—
452C42C22×D412yes—
453C42C2×#4112yes—
454C42C2×#4112yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—48no—
461C23—48no—
462C23—48no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23C2×#12112yes—
474C23C22×SD1612yes—
475C23C2×AGL(1,ℤ/8)12yes—
476C23C23×D411yescharacteristic
477Q8—28no—
478Q8—44no—
479Q8—84no—
480Q8—88no—
481Q8—88no—
482D4—24no—
483D4—24no—
484D4—48no—
485D4—42no—
486D4—42no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4—88no—
523C2×C4—88no—
524C2×C4—88no—
525C2×C4#460116yes—
526C2×C4C2×#12112yes—
527C2×C4C2×#12112yes—
528C2×C4C2×#12112yes—
529C2×C4C2×#12112yes—
530C2×C4C2×#12112yes—
531C2×C4C22×SD1618yes—
532C2×C4C22×SD1612yes—
533C2×C4C22×SD1612yes—
534C2×C4C2×AGL(1,ℤ/8)12yes—
535C2×C4C2×AGL(1,ℤ/8)12yes—
536C2×C4C23×D411yescharacteristic
537C2×C4C22×#4111yescharacteristic
538C2×C4C2×2−1+412yes—
539C2×C4#58512yes—
540C8—44no—
541C8—44no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—816no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22C2×#46014yes—
557C22C22×#12111yescharacteristic
558C22#348511yescharacteristic
559C22#348611yescharacteristic
560C22C23×SD1611yescharacteristic
561C22C22×AGL(1,ℤ/8)11yescharacteristic
562C22#358611yescharacteristic
563C22#358611yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—28no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—84no—
575C4C2×#46014yes—
576C4C2×#46014yes—
577C4#348912yes—
578C4#349512yes—
579C2—24no—
580C2—84no—
581C2#3563311yescharacteristic
582C2C22×#46011yescharacteristic
583C2#6042711yescharacteristic
584C2#6169511yescharacteristic
585C2#6171711yescharacteristic
586C2#6181811yescharacteristic
587C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3229C2semidirect
4. #34213C2semidirect
6. #27647C2semidirect
7. #25451C2semidirect
8. #25320C2semidirect
9. #24813C2semidirect
10. #24678C2semidirect
11. #21951C2semidirect
12. #21833C2semidirect
20. C2×#498C22semidirect
21. C2×#487C22semidirect
25. C2×#475C22semidirect
38. #2369C22semidirect
39. #2365C22semidirect
41. #2355C22semidirect
42. Q8×C4⋊C4C22semidirect
51. #2084C22semidirect
52. #2072C22semidirect
55. #1990C22semidirect
56. #1990C22semidirect
57. #1988C22semidirect
58. #1948C22semidirect
59. #1939C22semidirect
60. #1939C22semidirect
65. #1803C22semidirect
270. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#31012
C2×#25422
#342132
#281202
#276472
#254512
#253202
#248132
#246782
#219512
#218332
#207322