Finite Group Databasebeta

← #6484989

Subgroups — #6484989

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#33095C212yesmaximal, complemented
5#29912C212yesmaximal, complemented
6#29906C212yesmaximal, complemented
7#26332C214yesmaximal, complemented
8#26292C214yesmaximal, complemented
9#25072C214yesmaximal, complemented
10#25006C214yesmaximal, complemented
11#21986C211yesmaximal, characteristic, complemented
12#21865C211yesmaximal, characteristic
13#21822C211yesmaximal, characteristic, complemented
14#21395C211yesmaximal, characteristic
15#21365C211yesmaximal, characteristic, complemented
16C2×#536—21no—
17C2×#532C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21#2854C2212yes—
22#2854C2212yes—
23#2618—21no—
24#2618—21no—
25#2548C2212yes—
26#2546C2212yes—
27#2545—22no—
28#2538C2212yes—
29#2538C2212yes—
30#2536—22no—
31#2355—22no—
32#2355—22no—
33#2355C2214yes—
34#2355C2214yes—
35#2311—21no—
36C2×#391—21no—
37C2×#387—21no—
38C2×#387—21no—
39#2285C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2101C2214yes—
48#2099C2214yes—
49#2093C2214yes—
50#2092C2214yes—
51#2085C2214yes—
52#2083C2214yes—
53#2073C2214yes—
54#2071C2214yes—
55#1993C2212yes—
56#1987C2212yes—
57#1985C2214yes—
58#1985C2214yes—
59#1982C2212yes—
60#1961C2212yes—
61#1949C2214yes—
62#1948C2212yes—
63#1947C2212yes—
64#1926C2214yes—
65#1926C2214yes—
66#1926C2214yes—
67#1925C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1919C2214yes—
71#1919C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1874C2212yes—
75#1874C2212yes—
76#1853C2212yes—
77#1853C2212yes—
78#1852C2212yes—
79#1852C2212yes—
80#1847C2212yes—
81#1836C2212yes—
82#1817C2212yes—
83#1817C2212yes—
84#1804C2214yes—
85#1776C2214yes—
86#1762C2214yes—
87C22×#41—21no—
88C23×Q8D411yescharacteristic
89#536—48no—
90#532—24no—
91#532—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#124—22no—
96C2×#124—21no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8D412yes—
106C2×C4×Q8D412yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×#117—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#432—24no—
128#432—24no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2312yes—
142C2×#97C2312yes—
143#398—22no—
144#398—22no—
145#396—22no—
146#396—22no—
147#396—22no—
148#396—22no—
149#394—22no—
150#394—22no—
151#391—24no—
152#391—24no—
153#391—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—24no—
168#387—24no—
169#387—24no—
170#387—24no—
171#386—22no—
172#386—22no—
173#384—22no—
174#384C2314yes—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382—22no—
178#382—21no—
179#382—21no—
180#382—21no—
181#382C2312yes—
182#382C2314yes—
183#382C2312yes—
184#380—21no—
185#380—22no—
186#380—22no—
187#380C2312yes—
188C4×C4⋊C4—24no—
189C4×C4⋊C4—24no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219C2×#41—44no—
220C2×#41—44no—
221C2×#41—44no—
222C22×Q8—24no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—42no—
226C22×Q8C2×D412yes—
227C22×Q8C2×D412yes—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—48no—
249#127—22no—
250#127—22no—
251#124—48no—
252#124—44no—
253#123—48no—
254#123—48no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257#121—24no—
258#121—24no—
259#121—44no—
260C4×Q8—28no—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—44no—
269#117—44no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—44no—
328C2×C42—82no—
329C2×C42—82no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42#4114yes—
335C2×C42#4114yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—44no—
341#107—44no—
342#106—28no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346#102—28no—
347#102—44no—
348#102—44no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352#97—24no—
353#97—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24—22no—
371C24C22×D411yescharacteristic
372#41—816no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—416no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—81no—
395C2×D4—81no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4#12314yes—
436C22×C4#12314yes—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—84no—
486C4⋊C42−1+414yes—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—48no—
505C42—44no—
506C42—82no—
507C42—82no—
508C42—84no—
509C42#12314yes—
510C42#12314yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C2×#12312yes—
527C23C2×AGL(1,ℤ/8)12yes—
528C23#57512yes—
529C23C23×D411yescharacteristic
530Q8—48no—
531Q8—48no—
532Q8—84no—
533D4—44no—
534D4—44no—
535D4—84no—
536D4—84no—
537D4—88no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4C2×#12312yes—
597C2×C4C2×#12312yes—
598C2×C4C2×AGL(1,ℤ/8)14yes—
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×2−1+412yes—
602C2×C4#58512yes—
603C8—44no—
604C8—44no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22#309414yes—
619C22C22×#12311yescharacteristic
620C22#348611yescharacteristic
621C22#348711yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C2×#57511yescharacteristic
624C22#358511yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—22no—
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#346112yes—
643C4#350712yes—
644C2—24no—
645C2—162no—
646C2#3518211yescharacteristic
647C2C2×#309411yescharacteristic
648C2#6051711yescharacteristic
649C2#6182811yescharacteristic
650C2#6183111yescharacteristic
651C2#6189111yescharacteristic
652C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #33096C2semidirect
4. #33095C2semidirect
5. #29912C2semidirect
6. #29906C2semidirect
7. #26332C2semidirect
8. #26292C2semidirect
9. #25072C2semidirect
10. #25006C2semidirect
11. #21986C2semidirect
13. #21822C2semidirect
15. #21365C2semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#23552
#330962
#330952
#299122
#299062
#263322
#262922
#250722
#250062
#219862
#218652
#218222
#213952
#213652