Finite Group Databasebeta

← #6631343

Subgroups — #6631343

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32756C214yesmaximal, complemented
4#32754C214yesmaximal, complemented
5#32743C212yesmaximal, complemented
6#32742C212yesmaximal, complemented
7#32739C212yesmaximal, complemented
8#32738C212yesmaximal, complemented
9#23669C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23643C212yesmaximal, complemented
12C2×#2071C211yesmaximal, characteristic, complemented
13#23614C212yesmaximal, complemented
14#23551C211yesmaximal, characteristic, complemented
15#23486C212yesmaximal
16#23454C211yesmaximal, characteristic, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#424C2212yes—
24C2×#418C2212yes—
25C22×#102C2212yes—
26C2×#408C2212yes—
27#2857C2212yescomplemented
28#2857C2212yescomplemented
29#2854C2212yes—
30#2854C2212yes—
31#2629C2214yescomplemented
32#2624C2212yes—
33#2624C2212yes—
34#2582—21no—
35#2582—21no—
36#2538—21no—
37#2538—21no—
38#2448—21no—
39#2448—21no—
40#2445—21no—
41#2445—21no—
42#2418—22no—
43#2418—22no—
44#2417—22no—
45#2417—22no—
46#2316C2212yes—
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2212yes—
50#2309C2211yescharacteristic
51#2296C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2088C2214yescomplemented
55#2088C2214yescomplemented
56#2087C2214yescomplemented
57#2086C2214yescomplemented
58#2084C2212yescomplemented
59#2084C2212yescomplemented
60#2083C2212yescomplemented
61#2083C2212yescomplemented
62#2077C2214yescomplemented
63#2077C2214yescomplemented
64#2076C2214yescomplemented
65#2075C2214yescomplemented
66#2071C2212yes—
67#2071C2212yes—
68#2071C2214yes—
69#2067C2214yes—
70#2067C2214yes—
71#2065C2214yes—
72#2065C2214yes—
73#2059C2212yescomplemented
74#2059C2212yescomplemented
75#2058C2211yescharacteristic, complemented
76#2058C2211yescharacteristic, complemented
77#2057C2211yescharacteristic, complemented
78#2057C2211yescharacteristic, complemented
79#2034C2214yes—
80#2034C2214yes—
81#2033C2214yes—
82#2033C2214yes—
83#2025C2214yes—
84#2023C2212yes—
85#2023C2212yes—
86#1798C2214yescomplemented
87#1797C2212yescomplemented
88#1797C2212yescomplemented
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#535—44no—
100#533—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127—21no—
104C2×#125C2312yes—
105C2×#124—21no—
106C2×#123—22no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×#120—22no—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4D412yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#428—44no—
138#427—44no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145#424—44no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#418—44no—
149C2×#102—44no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154#408—44no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157#399—22no—
158#399—22no—
159#398—24no—
160#398—24no—
161#397—22no—
162#397—22no—
163#396—24no—
164#396—24no—
165#394—22no—
166#394—22no—
167#394—22no—
168#394—22no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#390—21no—
174#390—22no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2311yescharacteristic
190#389C2311yescharacteristic
191#389C2311yescharacteristic
192#389C2311yescharacteristic
193#389C2314yes—
194#389C2314yes—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#386—24no—
202#386—22no—
203#386—22no—
204#386—22no—
205#385—21no—
206#385—21no—
207#384—21no—
208#384—21no—
209#384C2312yes—
210#384C2312yes—
211#380—22no—
212#380—22no—
213#380—24no—
214#380—22no—
215#380C2312yes—
216#380C2314yes—
217#380C2314yes—
218#380C2312yes—
219#379—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41—22no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8C2×D411yescharacteristic
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—24no—
249C22×D4—22no—
250C22×D4—24no—
251C22×D4—22no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274#127—44no—
275#125—42no—
276#125—42no—
277#124—44no—
278#123—84no—
279#123—84no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—84no—
283C22⋊Q8—84no—
284#121—44no—
285#121—44no—
286#120—48no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—22no—
300#117—22no—
301#117—44no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
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—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—82no—
355C2×C22⋊C4—82no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371#107—42no—
372#107—42no—
373#107—44no—
374#106—42no—
375#106—42no—
376#106—44no—
377C4⋊C8—44no—
378C4⋊C8—44no—
379#102—44no—
380#102—44no—
381#102—44no—
382#102—44no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397C24—22no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—81no—
402C24—81no—
403C24—81no—
404C24—81no—
405C24C22×D412yes—
406C24C22×D411yescharacteristic
407#41—48no—
408#41—48no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+414yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
482M16—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—48no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—42no—
554C42—42no—
555C42—44no—
556C42—42no—
557C42—42no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—162no—
580C23—162no—
581C23—162no—
582C23—162no—
583C23#53412yes—
584C23#53612yes—
585C23#53712yes—
586C23C23×D411yescharacteristic
587Q8—44no—
588Q8—84no—
589Q8—84no—
590D4—48no—
591D4—44no—
592D4—44no—
593D4—44no—
594D4—84no—
595D4—164no—
596D4—164no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—162no—
647C2×C4—164no—
648C2×C4#53412yes—
649C2×C4#53612yes—
650C2×C4#53712yes—
651C2×C4#56218yes—
652C2×C4#56318yes—
653C2×C4#57612yes—
654C2×C4#57712yes—
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×2−1+412yes—
658C2×C4C2×2−1+411yescharacteristic
659C8—42no—
660C8—42no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—168no—
683C22#281414yes—
684C22C2×#53411yescharacteristic
685C22C2×#53611yescharacteristic
686C22C2×#53711yescharacteristic
687C22C2×#57611yescharacteristic
688C22C2×#57711yescharacteristic
689C22#358511yescharacteristic
690C22#358611yescharacteristic
691C4—22no—
692C4—22no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—41no—
697C4—42no—
698C4—81no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4—82no—
703C4—81no—
704C4—84no—
705C4—81no—
706C4#281414yes—
707C4C2×#56212yes—
708C4C2×#56312yes—
709C2—24no—
710C2—82no—
711C2—162no—
712C2C2×#281411yescharacteristic
713C2#6076211yescharacteristic
714C2#6076511yescharacteristic
715C2#6087711yescharacteristic
716C2#6088511yescharacteristic
717C2#6092211yescharacteristic
718C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35280C2semidirect
3. #32756C2semidirect
4. #32754C2semidirect
5. #32743C2semidirect
6. #32742C2semidirect
7. #32739C2semidirect
8. #32738C2semidirect
9. #23669C2semidirect
10. #23658C2semidirect
11. #23643C2semidirect
12. C2×#2071C2semidirect
13. #23614C2semidirect
14. #23551C2semidirect
16. #23454C2semidirect
17. #20630C2semidirect
27. #2857C22semidirect
28. #2857C22semidirect
31. #2629C22semidirect
54. #2088C22semidirect
55. #2088C22semidirect
56. #2087C22semidirect
57. #2086C22semidirect
58. #2084C22semidirect
59. #2084C22semidirect
60. #2083C22semidirect
61. #2083C22semidirect
62. #2077C22semidirect
63. #2077C22semidirect
64. #2076C22semidirect
65. #2075C22semidirect
73. #2059C22semidirect
74. #2059C22semidirect
75. #2058C22semidirect
76. #2058C22semidirect
77. #2057C22semidirect
78. #2057C22semidirect
86. #1798C22semidirect
87. #1797C22semidirect
88. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#352802
#352542
#327562
#327542
#327432
#327422
#327392
#327382
#236692
#236582
#236432
C2×#20712
#236142
#235512
#234862
#234542
#206302