Finite Group Databasebeta

← #6628092

Subgroups — #6628092

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33107C212yesmaximal, complemented
4#33107C211yesmaximal, characteristic, complemented
5#29328C212yesmaximal, complemented
6#29235C212yesmaximal, complemented
7#27130C214yesmaximal, complemented
8#26692C214yesmaximal, complemented
9#26508C212yesmaximal, complemented
10#25903C211yesmaximal, characteristic, complemented
11#25274C212yesmaximal, complemented
12#24299C211yesmaximal, characteristic, complemented
13#23215C212yesmaximal, complemented
14#23061C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#481C211yesmaximal, characteristic, complemented
17C4×#481C212yesmaximal, complemented
18C2×#553—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#486C2212yes—
21C2×#481C2212yes—
22C2×#481C2211yescharacteristic
23C2×#476C2212yes—
24#2856C2211yescharacteristic
25#2856C2212yes—
26#2554—22no—
27#2552C2211yescharacteristic
28#2543—22no—
29#2541C2211yescharacteristic
30#2512C2212yes—
31#2490—21no—
32#2445C2212yes—
33#2416—21no—
34#2369—21no—
35#2369—21no—
36#2369—22no—
37#2369C2212yes—
38#2369C2214yes—
39#2369C2212yes—
40#2366C2211yescharacteristic
41#2363—21no—
42#2354C2211yescharacteristic
43C4×#121C2212yes—
44C4×#121C2211yescharacteristic
45C2×#391—22no—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yes—
53#2088C2214yes—
54#2065C2214yes—
55#2058C2212yes—
56#2041C2214yes—
57#2023C2214yes—
58#2023C2212yes—
59#1990C2212yes—
60#1987C2212yes—
61#1982C2214yes—
62#1982C2212yes—
63#1948C2212yes—
64#1948C2214yes—
65#1939C2212yes—
66#1934C2214yes—
67#1934C2214yes—
68#1926C2214yes—
69#1926C2212yes—
70#1926C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1920C2214yes—
74#1919C2214yes—
75#1919C2212yes—
76#1916C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2211yescharacteristic
81#1874C2212yes—
82#1864C2214yes—
83#1853C2211yescharacteristic
84#1853C2212yes—
85#1824C2211yescharacteristic
86#1797C2212yes—
87#1797C2211yescharacteristic
88C4×#106C2214yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#102C2212yes—
92C4×#102C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96#553—48no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#127C2312yes—
104C2×#126—22no—
105C2×#125—21no—
106C2×#123—22no—
107C2×#121—21no—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×#121C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C42—21no—
129C22×C42—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#486—28no—
133#481—24no—
134#481—28no—
135#476—28no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#398—22no—
155#397—22no—
156#396—22no—
157#394—22no—
158#394—22no—
159#392—22no—
160#391—24no—
161#391—24no—
162#391—24no—
163#391—24no—
164#390—21no—
165#390—21no—
166#390—22no—
167#390—22no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#390C2311yescharacteristic
174#389—21no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2314yes—
182#389C2312yes—
183#389C2314yes—
184#388—22no—
185#388—21no—
186#388—22no—
187#388—22no—
188#386—22no—
189#386—21no—
190#386—22no—
191#386—22no—
192#384C2311yescharacteristic
193#384C2312yes—
194#380—21no—
195#380C2311yescharacteristic
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95—21no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—82no—
238C22×Q8—82no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8#4118yes—
265#127—24no—
266#127—44no—
267#126—48no—
268#125—44no—
269#123—48no—
270#121—24no—
271#121—22no—
272#121—24no—
273#121—22no—
274#121—44no—
275#121—48no—
276#121—44no—
277#121—44no—
278#121—82no—
279#121—84no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—44no—
285#117—44no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—84no—
305C2×C4⋊C4SD1618yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42—84no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#106—28no—
354#106—28no—
355#106—24no—
356#106—42no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#102—28no—
360#102—24no—
361#102—28no—
362#102—28no—
363#102—28no—
364#102—44no—
365#102—42no—
366C22⋊C8—22no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369C4×C8—24no—
370C4×C8—44no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24—22no—
388C24C22×D411yescharacteristic
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—816no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—81no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—81no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C2×SD1614yes—
457C22×C4C2×SD1618yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4112yes—
468C22×C42−1+412yes—
469C22×C4C2511yescharacteristic
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4C2×SD1614yes—
506C4⋊C4C2×SD1614yes—
507C4⋊C4C2×SD1618yes—
508C4⋊C4C2×#4114yes—
509C4⋊C4C2×#4118yes—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—81no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—44no—
536C42—84no—
537C42—84no—
538C42C2×#4114yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23#54812yes—
558C23C22×SD1612yes—
559C23#57612yes—
560C23C23×D411yescharacteristic
561Q8—88no—
562Q8—88no—
563D4—42no—
564D4—44no—
565D4—42no—
566D4—44no—
567D4—88no—
568D4—84no—
569D4—84no—
570D4—164no—
571D4—162no—
572D4—164no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—416no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—164no—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4#54812yes—
627C2×C4C22×SD1614yes—
628C2×C4C22×SD1612yes—
629C2×C4C22×SD1614yes—
630C2×C4C22×SD1612yes—
631C2×C4#57512yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4112yes—
634C2×C4C2×2−1+411yescharacteristic
635C2×C4#58512yes—
636C8—42no—
637C8—44no—
638C8—42no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—82no—
646C22—81no—
647C22—81no—
648C22—82no—
649C22—81no—
650C22—82no—
651C22—82no—
652C22—81no—
653C22—81no—
654C22—82no—
655C22—82no—
656C22—81no—
657C22—81no—
658C22—82no—
659C22#336214yes—
660C22C2×#52511yescharacteristic
661C22C2×#54811yescharacteristic
662C22#348511yescharacteristic
663C22C23×SD1611yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57611yescharacteristic
666C22#358511yescharacteristic
667C4—24no—
668C4—24no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—41no—
676C4—48no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4—81no—
682C4#310314yes—
683C4#348012yes—
684C4#351612yes—
685C2—24no—
686C2—162no—
687C2—161no—
688C2#3658211yescharacteristic
689C2C2×#310311yescharacteristic
690C2#6057811yescharacteristic
691C2C2×#336211yescharacteristic
692C2#6131811yescharacteristic
693C2#6169711yescharacteristic
694C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36596C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. #33107C2semidirect
4. #33107C2semidirect
5. #29328C2semidirect
6. #29235C2semidirect
7. #27130C2semidirect
8. #26692C2semidirect
9. #26508C2semidirect
10. #25903C2semidirect
11. #25274C2semidirect
12. #24299C2semidirect
13. #23215C2semidirect
14. #23061C2semidirect
15. #22625C2semidirect
16. C4×#481C2semidirect
17. C4×#481C2semidirect

Maximal subgroups

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

subgroupindex
#365962
C2×#23692
#331072
#293282
#292352
#271302
#266922
#265082
#259032
#252742
#242992
#232152
#230612
#226252
C4×#4812