Finite Group Databasebeta

← #6631348

Subgroups — #6631348

Order 512.

1645 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic, complemented
2#35255C211yesmaximal, characteristic
3#32752C214yesmaximal, complemented
4#32751C214yesmaximal, complemented
5#32745C212yesmaximal, complemented
6#32742C212yesmaximal, complemented
7#32740C212yesmaximal, complemented
8#32739C212yesmaximal, complemented
9#23668C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23643C212yesmaximal, complemented
12#23627C211yesmaximal, characteristic, complemented
13#23616C212yesmaximal, complemented
14C2×#2058C211yesmaximal, characteristic, complemented
15#23489C212yesmaximal
16#23452C211yesmaximal, characteristic, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C22×#106C2212yes—
23C2×#424C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26C2×#408C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2855C2212yes—
30#2854C2212yes—
31#2628C2214yes—
32#2625C2212yes—
33#2624C2212yes—
34#2568—21no—
35#2567—21no—
36#2563—21no—
37#2542—21no—
38#2487—22no—
39#2444—22no—
40#2443—22no—
41#2423—21no—
42#2422—21no—
43#2421—21no—
44#2420—21no—
45#2409—22no—
46#2317C2212yes—
47#2317C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2212yes—
51#2297C2211yescharacteristic
52C2×#389C2211yescharacteristic
53#2285C2211yescharacteristic
54#2088C2214yes—
55#2088C2214yes—
56#2088C2214yes—
57#2088C2214yes—
58#2085C2214yes—
59#2084C2212yes—
60#2083C2212yes—
61#2077C2214yes—
62#2077C2214yes—
63#2076C2214yes—
64#2075C2214yes—
65#2074C2212yes—
66#2073C2212yes—
67#2072C2211yescharacteristic
68#2072C2211yescharacteristic
69#2071C2211yescharacteristic
70#2071C2211yescharacteristic
71#2068C2214yes—
72#2067C2214yes—
73#2066C2214yes—
74#2065C2214yes—
75#2058C2212yes—
76#2058C2212yes—
77#2058C2214yes—
78#2035C2214yes—
79#2035C2214yes—
80#2034C2214yes—
81#2034C2214yes—
82#2024C2212yes—
83#2024C2212yes—
84#2023C2212yes—
85#2023C2212yes—
86#1798C2214yes—
87#1797C2212yes—
88#1797C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8—21no—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#537—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121C2312yes—
114C4×#41—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#428—44no—
133C2×#107C2314yes—
134C2×#106—44no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138#424—44no—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141#417—44no—
142#416—44no—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149#408—44no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152#400—24no—
153#400—22no—
154#399—22no—
155#399—24no—
156#398—22no—
157#397—22no—
158#397—22no—
159#397—22no—
160#395—24no—
161#394—22no—
162#394—22no—
163#393—24no—
164#391—22no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391C2312yes—
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#390—22no—
172#390—21no—
173#390C2312yes—
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#386—24no—
191#386—22no—
192#386—22no—
193#386—21no—
194#386—21no—
195#386—22no—
196#385—22no—
197#385—24no—
198#385—22no—
199#384—22no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—22no—
204#382—21no—
205#382C2312yes—
206#381—22no—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216#380C2314yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—82no—
249C22×Q8—82no—
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2411yescharacteristic
252C22×D4—21no—
253C22×D4—22no—
254C22×D4C2×D412yes—
255C22×D4C2×D411yescharacteristic
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#123—84no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—84no—
284C22⋊Q8—84no—
285#121—44no—
286#121—48no—
287#121—44no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117—44no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#107—44no—
362#106—44no—
363#106—44no—
364#106—44no—
365C4⋊C8—44no—
366C4⋊C8—44no—
367#103—42no—
368#103—42no—
369#103—44no—
370#102—42no—
371#102—42no—
372#102—44no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393C24—21no—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396#41—48no—
397#41—48no—
398#41—48no—
399#41—48no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—816no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—82no—
429C2×D4—82no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4H(5,𝔽2)14yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+414yes—
477C22×C42−1+412yes—
478C22×C4C2511yescharacteristic
479M16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—48no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—82no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—82no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—44no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23#53412yes—
566C23#53612yes—
567C23#53712yes—
568C23C23×D411yescharacteristic
569Q8—44no—
570Q8—42no—
571Q8—42no—
572Q8—88no—
573Q8—88no—
574D4—48no—
575D4—42no—
576D4—42no—
577D4—44no—
578D4—84no—
579D4—164no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—162no—
633C2×C4#53412yes—
634C2×C4#53612yes—
635C2×C4#53712yes—
636C2×C4#56218yes—
637C2×C4#56318yes—
638C2×C4#57712yes—
639C2×C4#57712yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×2−1+412yes—
643C2×C4C2×2−1+411yescharacteristic
644C8—42no—
645C8—42no—
646C8—44no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22#281414yes—
668C22C2×#53411yescharacteristic
669C22C2×#53611yescharacteristic
670C22C2×#53711yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358511yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—22no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—84no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4#281414yes—
691C4C2×#56212yes—
692C4C2×#56312yes—
693C2—24no—
694C2—82no—
695C2—161no—
696C2C2×#281411yescharacteristic
697C2#6075711yescharacteristic
698C2#6075811yescharacteristic
699C2#6088111yescharacteristic
700C2#6088711yescharacteristic
701C2#6092911yescharacteristic
702C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35279C2semidirect
3. #32752C2semidirect
4. #32751C2semidirect
5. #32745C2semidirect
6. #32742C2semidirect
7. #32740C2semidirect
8. #32739C2semidirect
9. #23668C2semidirect
10. #23657C2semidirect
11. #23643C2semidirect
12. #23627C2semidirect
13. #23616C2semidirect
14. C2×#2058C2semidirect
16. #23452C2semidirect
17. #20630C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#352552
#327522
#327512
#327452
#327422
#327402
#327392
#236682
#236572
#236432
#236272
#236162
C2×#20582
#234892
#234522
#206302