Finite Group Databasebeta

← #6497261

Subgroups — #6497261

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33147C214yesmaximal, complemented
4#29937C214yesmaximal, complemented
5#27961C212yesmaximal, complemented
6#27948C212yesmaximal, complemented
7#27876C214yesmaximal, complemented
8#27613C214yesmaximal, complemented
9#27601C214yesmaximal
10#21981C212yesmaximal, complemented
11#21833C212yesmaximal, complemented
12#20714C211yesmaximal, characteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C22×#117—21no—
16C23×C42C2211yescharacteristic
17C2×#431C2212yes—
18#2857—21no—
19#2857—21no—
20#2857—21no—
21#2857—21no—
22#2857C2212yes—
23#2857C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26C2×#391—21no—
27C2×#389—21no—
28C2×#389C2212yes—
29#2285—21no—
30#2285—21no—
31#2285—21no—
32#2285—21no—
33#2285C2212yes—
34#2285C2212yes—
35#2285C2212yes—
36#2285C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#382—21no—
41C2×#380—21no—
42C2×#380—21no—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2212yes—
47#2100C2218yescomplemented
48#2084C2218yescomplemented
49#2077C22116yes—
50#1990C2214yescomplemented
51#1990C2214yescomplemented
52#1988C2218yescomplemented
53#1948C2218yescomplemented
54#1919C22116yescomplemented
55#1853C2218yes—
56#1853C2218yes—
57#1844C2214yescomplemented
58#1834C2214yescomplemented
59#1824C2218yes—
60#1823C2218yes—
61#1817C2212yes—
62#1797C2212yes—
63#1797C2212yes—
64#1779C2212yes—
65C4×C22⋊C8C2212yes—
66C4×C22⋊C8C2212yes—
67C23×Q8—21no—
68C24×C4D411yescharacteristic
69C24×C4D411yescharacteristic
70C24×C4C2311yescharacteristic
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8C2312yes—
75C2×C4⋊Q8C2312yes—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—48no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—48no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C42—28no—
101C22×C42—22no—
102C22×C42D411yescharacteristic
103C22×C42D411yescharacteristic
104C22×C42D411yescharacteristic
105C22×C42D411yescharacteristic
106C22×C42D411yescharacteristic
107C22×C42D411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#431—28no—
115C2×#106C23116yes—
116C2×C4⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C4×C8C2314yes—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389C2314yes—
132#389C2314yes—
133#384—21no—
134#384—21no—
135#384—21no—
136#384—21no—
137#384—21no—
138#384—21no—
139#384C2312yes—
140#384C2312yes—
141#384C2312yes—
142#384C2312yes—
143#384C2312yes—
144#384C2312yes—
145#382—22no—
146#382—22no—
147#382—22no—
148#382—22no—
149#380—22no—
150#380—22no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380—22no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380C2314yes—
162#380C2314yes—
163#380C2314yes—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C2×#95—22no—
174C2×#95—22no—
175C2×#95—22no—
176C2×#95—22no—
177#340C23116yes—
178#340C23116yes—
179#340C23116yes—
180#336C2314yes—
181#336C2314yes—
182C25C2×D411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—44no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4SD1618yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8C2418yes—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—44no—
223C4⋊Q8—44no—
224C22⋊Q8—48no—
225C22⋊Q8—48no—
226C22⋊Q8—48no—
227C22⋊Q8—48no—
228#117—48no—
229#117—48no—
230#117—48no—
231#117—48no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—48no—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4118yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#106—416no—
306C4⋊C8—28no—
307C22⋊C8—28no—
308C22⋊C8—28no—
309C22⋊C8—44no—
310C4×C8—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—41no—
324C24—41no—
325C24C2×SD1614yes—
326C24C22×D411yescharacteristic
327C24C22×D411yescharacteristic
328C24C22×D411yescharacteristic
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—816no—
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—232no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—416no—
384C22×C4—416no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4#12314yes—
398C22×C4#12314yes—
399C22×C4C2×SD1614yes—
400C22×C4C2×SD1614yes—
401C22×C4C2×SD1614yes—
402C22×C4C2×SD1614yes—
403C22×C4C2×SD1614yes—
404C22×C4C2×SD1614yes—
405C22×C4AGL(1,ℤ/8)14yes—
406C22×C4#13714yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4114yes—
419C22×C42−1+414yes—
420C22×C4C2511yescharacteristic
421C2×C8—28no—
422C2×C8—28no—
423C2×C8—28no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—48no—
465C42—48no—
466C42#12314yes—
467C42#12314yes—
468C42C2×#4114yes—
469C42C2×#4114yes—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—28no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—28no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23C2×#12312yes—
492C23#53612yes—
493C23C22×SD1612yes—
494C23C22×SD1612yes—
495C23C22×SD1612yes—
496C23C2×AGL(1,ℤ/8)12yes—
497C23C2×#13712yes—
498C23C23×D411yescharacteristic
499Q8—88no—
500Q8—88no—
501C2×C4—28no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4—88no—
552C2×C4#481116yes—
553C2×C4C2×#12312yes—
554C2×C4C2×#12312yes—
555C2×C4C2×#12312yes—
556C2×C4C2×#12312yes—
557C2×C4C2×#12312yes—
558C2×C4#53612yes—
559C2×C4C22×SD1612yes—
560C2×C4C22×SD1612yes—
561C2×C4C22×SD1612yes—
562C2×C4C22×SD1612yes—
563C2×C4C2×AGL(1,ℤ/8)12yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C2×C4C2×#13712yes—
566C2×C4C2×#13712yes—
567C2×C4#57512yes—
568C2×C4C22×#4112yes—
569C2×C4C2×2−1+412yes—
570C2×C4#58512yes—
571C8—48no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—416no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22C2×#48114yes—
593C22#322914yes—
594C22C22×#12311yescharacteristic
595C22C2×#53611yescharacteristic
596C22#344911yescharacteristic
597C22C23×SD1611yescharacteristic
598C22C22×AGL(1,ℤ/8)11yescharacteristic
599C22C22×#13711yescharacteristic
600C22C2×#57511yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—41no—
608C4—41no—
609C4—84no—
610C4—84no—
611C4—84no—
612C4—84no—
613C4C2×#48114yes—
614C4C2×#48114yes—
615C4#309514yes—
616C4#349514yes—
617C2—24no—
618C2—24no—
619C2—42no—
620C2#3531111yescharacteristic
621C2C22×#48111yescharacteristic
622C2C2×#309511yescharacteristic
623C2C2×#322911yescharacteristic
624C2#6086411yescharacteristic
625C2#6094111yescharacteristic
626C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33147C2semidirect
4. #29937C2semidirect
5. #27961C2semidirect
6. #27948C2semidirect
7. #27876C2semidirect
8. #27613C2semidirect
10. #21981C2semidirect
11. #21833C2semidirect
47. #2100C22semidirect
48. #2084C22semidirect
50. #1990C22semidirect
51. #1990C22semidirect
52. #1988C22semidirect
53. #1948C22semidirect
54. #1919C22semidirect
57. #1844C22semidirect
58. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#22852
#331472
#299372
#279612
#279482
#278762
#276132
#276012
#219812
#218332
#207142