Finite Group Databasebeta

← #6464152

Subgroups — #6464152

Order 512.

1705 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29363C214yesmaximal, complemented
4#29361C214yesmaximal, complemented
5#29355C212yesmaximal, complemented
6#29352C212yesmaximal, complemented
7#29350C212yesmaximal, complemented
8#29347C212yesmaximal, complemented
9#23630C211yesmaximal, characteristic
10C2×#2072C211yesmaximal, characteristic, complemented
11#23455C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic
13#21315C212yesmaximal, complemented
14#21262C212yesmaximal
15#21090C212yesmaximal, complemented
16#21087C212yesmaximal
17#20430C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#536C2211yescharacteristic
20C2×#533C2211yescharacteristic
21C2×#424C2212yes—
22C2×#417C2212yes—
23C2×#419C2212yes—
24C2×#416C2212yes—
25C22×#103C2212yes—
26C2×#405C2212yes—
27#2862C2212yes—
28#2861C2212yes—
29#2642C2212yes—
30#2640C2212yes—
31#2492—21no—
32#2492—21no—
33#2489—21no—
34#2489—21no—
35#2488—22no—
36#2488—22no—
37#2360—22no—
38#2360—22no—
39#2355—21no—
40#2355—21no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4—21no—
43#2330C2214yes—
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2310C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2212yes—
49#2297C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53C4×#117C2211yescharacteristic
54#2072C2212yes—
55#2072C2212yes—
56#2072C2214yes—
57#2072C2212yes—
58#2072C2212yes—
59#2071C2212yes—
60#2071C2212yes—
61#2026C2212yes—
62#2026C2214yes—
63#2025C2212yes—
64#2024C2211yescharacteristic
65#2024C2211yescharacteristic
66#2024C2212yes—
67#2024C2212yes—
68#2023C2211yescharacteristic
69#2023C2211yescharacteristic
70#1989C2214yes—
71#1989C2214yes—
72#1988C2214yes—
73#1987C2214yes—
74#1983C2214yes—
75#1983C2214yes—
76#1981C2214yes—
77#1981C2214yes—
78#1936C2214yes—
79#1935C2214yes—
80#1935C2214yes—
81#1935C2214yes—
82#1934C2214yes—
83#1933C2214yes—
84#1933C2214yes—
85#1933C2214yes—
86#1768C2214yes—
87#1767C2212yes—
88C4×C4⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#536—28no—
101#533—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#125—21no—
108C2×#125C2312yes—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C42C2311yescharacteristic
130#424—28no—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133#419—28no—
134#417—44no—
135#416—28no—
136C2×#103—44no—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147#405—44no—
148C2×#97C2314yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#397—22no—
155#397—22no—
156#393—22no—
157#393—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2311yescharacteristic
171#391C2312yes—
172#391C2311yescharacteristic
173#390C2312yes—
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#389—22no—
177#389—22no—
178#389—24no—
179#389—24no—
180#389C2314yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—24no—
187#387—24no—
188#386—21no—
189#386—21no—
190#386—22no—
191#385—24no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382—22no—
200#381—22no—
201#380—21no—
202#380—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—24no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376C2312yes—
219C2×#95—21no—
220C2×#95—21no—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2312yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8C2412yes—
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—44no—
274#125—24no—
275#125—24no—
276#125—28no—
277#125—44no—
278#124—28no—
279#123—48no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—84no—
288C4×Q8—84no—
289#117—24no—
290#117—22no—
291#117—22no—
292#117—24no—
293#117#4118yes—
294#117#4118yes—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C4⋊C8—28no—
355C4⋊C8—28no—
356#103—24no—
357#103—24no—
358#103—28no—
359#103—28no—
360#103—44no—
361#103—44no—
362#103—44no—
363#102—24no—
364#102—24no—
365#102—44no—
366#97—44no—
367C4×C8—44no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C24C22×D411yescharacteristic
392#41—48no—
393#41—48no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463M16—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—48no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4#12414yes—
501C4⋊C4#12414yes—
502C4⋊C4C2×#4114yes—
503C4⋊C4C2×#4114yes—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C22⋊C4—81no—
515C22⋊C4#12418yes—
516C42—22no—
517C42—22no—
518C42—28no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—84no—
529C42—84no—
530C42C2×#4118yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23#52512yes—
543C23C2×#12412yes—
544C23#53712yes—
545C23C23×D411yescharacteristic
546Q8—48no—
547Q8—48no—
548Q8—44no—
549Q8—42no—
550Q8—44no—
551Q8—42no—
552Q8—84no—
553Q8—164no—
554D4—42no—
555D4—42no—
556D4—84no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—162no—
606C2×C4—164no—
607C2×C4#52512yes—
608C2×C4C2×#12414yes—
609C2×C4C2×#12414yes—
610C2×C4C2×#12412yes—
611C2×C4#53712yes—
612C2×C4#55618yes—
613C2×C4#55918yes—
614C2×C4#57712yes—
615C2×C4#57812yes—
616C2×C4C22×#4112yes—
617C2×C4C22×#4112yes—
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C8—42no—
621C8—42no—
622C8—44no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22#259214yes—
637C22C2×#52511yescharacteristic
638C22C22×#12411yescharacteristic
639C22C2×#53711yescharacteristic
640C22C2×#57711yescharacteristic
641C22C2×#57811yescharacteristic
642C22#358511yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—24no—
647C4—24no—
648C4—41no—
649C4—44no—
650C4—41no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—84no—
660C4#259214yes—
661C4C2×#55612yes—
662C4C2×#55912yes—
663C2—24no—
664C2—161no—
665C2C2×#259211yescharacteristic
666C2#6058511yescharacteristic
667C2#6058911yescharacteristic
668C2#6062311yescharacteristic
669C2#6062511yescharacteristic
670C2#6092411yescharacteristic
671C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #29363C2semidirect
4. #29361C2semidirect
5. #29355C2semidirect
6. #29352C2semidirect
7. #29350C2semidirect
8. #29347C2semidirect
10. C2×#2072C2semidirect
11. #23455C2semidirect
13. #21315C2semidirect
15. #21090C2semidirect
17. #20430C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#342142
#293632
#293612
#293552
#293522
#293502
#293472
#236302
C2×#20722
#234552
#234512
#213152
#212622
#210902
#210872
#204302