Finite Group Databasebeta

← #6627613

Subgroups — #6627613

Order 512.

1631 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2C4×Q82C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#33031C212yesmaximal
5#29354C212yesmaximal
6#29273C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28280C211yesmaximal, characteristic
9#27567C214yesmaximal, complemented
10#27147C214yesmaximal, complemented
11#24990C212yesmaximal, complemented
12#24329C212yesmaximal, complemented
13#23067C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22759C212yesmaximal, complemented
16#21616C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#556—21no—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#460C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2858C2212yes—
26#2547—21no—
27#2545—22no—
28#2538—21no—
29#2538C2212yes—
30#2512—22no—
31#2512C2214yes—
32#2369—22no—
33#2369—21no—
34#2369C2212yes—
35#2369C2211yescharacteristic
36#2354—21no—
37#2354C2212yes—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C42×Q8—21no—
49C42×Q8C2212yes—
50C42×Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2100C2214yes—
53#2099C2214yes—
54#2071C2214yes—
55#2065C2214yes—
56#2058C2214yes—
57#2023C2214yes—
58#1990C2212yes—
59#1987C2212yes—
60#1987C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1982C2211yescharacteristic
65#1948C2211yescharacteristic
66#1948C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1936C2214yes—
70#1936C2214yes—
71#1926C2212yes—
72#1925C2214yes—
73#1925C2214yes—
74#1919C2214yes—
75#1919C2214yes—
76#1918C2214yes—
77#1917C2214yes—
78#1880C2214yes—
79#1880C2214yes—
80#1880C2212yes—
81#1865C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#102C2212yes—
95C22×SD16C2312yes—
96Q82—24no—
97Q82—24no—
98#556—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#126—24no—
108C2×#124—22no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128#477—28no—
129#476—28no—
130#460—24no—
131C4×SD16—28no—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#397—24no—
150#396—24no—
151#396—24no—
152#391—22no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390—22no—
162#390C2312yes—
163#390C2314yes—
164#390C2314yes—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—24no—
180#388—24no—
181#384—21no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#380—22no—
191#380—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C43—22no—
219C43—21no—
220C43C2311yescharacteristic
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C2×SD16—24no—
245C2×SD16—24no—
246C2×SD16—24no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—216no—
251C4⋊Q8—28no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8#4114yes—
260#127—22no—
261#127—22no—
262#126—416no—
263#124—48no—
264#124—84no—
265C22⋊Q8—48no—
266C4×Q8—28no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—41no—
279C4×Q8—48no—
280C4×Q8—44no—
281C4×Q8—82no—
282C4×Q8—84no—
283C4×Q8#4114yes—
284C4×Q8#4114yes—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—44no—
288C4×D4—44no—
289#117—48no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—28no—
358#106—24no—
359#106—28no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—42no—
365#103—24no—
366#103—28no—
367#102—24no—
368#102—24no—
369#102—28no—
370#102—42no—
371#102—44no—
372#97—44no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—22no—
389C2×Q8—28no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—81no—
400C2×Q8—82no—
401C2×Q8—81no—
402C2×Q8—82no—
403C2×Q8AGL(1,ℤ/8)14yes—
404C2×Q8C2×#4112yes—
405C2×Q8C2×#4112yes—
406C2×Q82−1+412yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2511yescharacteristic
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4C2×#4114yes—
491C4⋊C4C2×#4112yes—
492C4⋊C4C2×#4112yes—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C42—28no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—42no—
515C42—41no—
516C42—42no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—42no—
521C42—44no—
522C42—42no—
523C42—81no—
524C42—82no—
525C42#13714yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C422−1+412yes—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—22no—
540Q8—22no—
541Q8—22no—
542Q8—84no—
543Q8—88no—
544Q8—88no—
545Q8#55614yescomplemented
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—88no—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4#54812yes—
600C2×C4#54812yes—
601C2×C4C2×AGL(1,ℤ/8)14yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4#57712yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4#58511yescharacteristic
613C8—42no—
614C8—44no—
615C8—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#52511yescharacteristic
624C22C2×#54811yescharacteristic
625C22#348911yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358511yescharacteristic
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—82no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4#311214yes—
647C4#336914yes—
648C4C2×#55611yescharacteristic
649C4#348011yescharacteristic
650C4#350711yescharacteristic
651C4#352311yescharacteristic
652C2—162no—
653C2#3657911yescharacteristic
654C2C2×#311211yescharacteristic
655C2#6059711yescharacteristic
656C2C2×#336911yescharacteristic
657C2#6132111yescharacteristic
658C2#6211411yescharacteristic
659C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×Q82C2semidirect
3. #33198C2semidirect
6. #29273C2semidirect
7. #29218C2semidirect
9. #27567C2semidirect
10. #27147C2semidirect
11. #24990C2semidirect
12. #24329C2semidirect
13. #23067C2semidirect
14. #22825C2semidirect
15. #22759C2semidirect
16. #21616C2semidirect
545. Q8#556semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×Q822
#331982
#330312
#293542
#292732
#292182
#282802
#275672
#271472
#249902
#243292
#230672
#228252
#227592
#216162
C4×#4602