Finite Group Databasebeta

← #6986972

Subgroups — #6986972

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic, complemented
3#32775C212yesmaximal, complemented
4#32033C214yesmaximal, complemented
5#32008C214yesmaximal, complemented
6#30188C212yesmaximal, complemented
7#30175C212yesmaximal, complemented
8#29922C211yesmaximal, characteristic, complemented
9#29159C212yesmaximal, complemented
10#28598C212yesmaximal, complemented
11#26277C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25646C212yesmaximal, complemented
14#25339C212yesmaximal, complemented
15#24299C211yesmaximal, characteristic, complemented
16#23754C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#487C2212yes—
22C2×#476C2212yes—
23C2×C8×D4C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2850C2212yes—
28#2628—22no—
29#2568—21no—
30#2567C2212yes—
31#2563C2212yes—
32#2542—21no—
33#2542C2211yescharacteristic
34#2540C2212yes—
35#2537C2211yescharacteristic
36#2536C2212yes—
37#2487C2214yes—
38#2479—22no—
39#2467—22no—
40#2423—21no—
41#2409—22no—
42#2407—21no—
43#2363—21no—
44#2355C2211yescharacteristic
45#2354—21no—
46#2317—21no—
47#2311—21no—
48C4×C22⋊Q8C2212yes—
49C4×#121C2211yescharacteristic
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2096C2212yes—
54#2093C2211yescharacteristic
55#2085C2214yes—
56#2083C2211yescharacteristic
57#2077C2214yes—
58#2075C2212yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2058C2212yes—
62#2042C2214yes—
63#2042C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2034C2214yes—
68#2033C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1987C2212yes—
74#1984C2212yes—
75#1981C2212yes—
76#1948C2212yes—
77#1919C2214yes—
78#1918C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1892C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1869C2214yes—
85#1865C2214yes—
86#1865C2214yes—
87#1853C2212yes—
88#1817C2212yes—
89#1807C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#556—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#126—22no—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C2×#121C2311yescharacteristic
115C2×#121C2312yes—
116C4×#41—44no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117D411yescharacteristic
129C2×#117D412yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#487—44no—
135#476—44no—
136C8×D4—44no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#399—22no—
155#399—24no—
156#399—22no—
157#398—22no—
158#395—24no—
159#394—22no—
160#394—22no—
161#393—24no—
162#392—22no—
163#391—22no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#390—22no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390C2311yescharacteristic
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389C2314yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—21no—
185#387—22no—
186#387—22no—
187#386—24no—
188#386—22no—
189#386—22no—
190#386—21no—
191#385—22no—
192#385—24no—
193#384—22no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—22no—
199#382—21no—
200#382C2312yes—
201#380—22no—
202#380—22no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380C2314yes—
208#380C2314yes—
209#380C2312yes—
210#380C2311yescharacteristic
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2412yes—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4C2×D412yes—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C22×C8—44no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269#126—48no—
270#124—44no—
271C22⋊Q8—24no—
272C22⋊Q8—42no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—84no—
277C22⋊Q8—84no—
278#121—22no—
279#121—41no—
280#121—44no—
281#121—44no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291#117—44no—
292#117—42no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—42no—
359#106—44no—
360#106—44no—
361C4⋊C8—24no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—42no—
367#102—42no—
368#102—44no—
369C22⋊C8—42no—
370C22⋊C8—42no—
371C22⋊C8—44no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—22no—
388C24—21no—
389C24C22×D411yescharacteristic
390C24C22×D412yes—
391#41—416no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—816no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—81no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4#13714yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4#13514yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—88no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C42−1+412yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—82no—
536C42H(5,𝔽2)14yes—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—22no—
548C23—21no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×#13512yes—
557C23C2×#13714yes—
558C23C23×D411yescharacteristic
559Q8—44no—
560Q8—44no—
561Q8—44no—
562Q8—88no—
563Q8—88no—
564D4—28no—
565D4—48no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4—164no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—84no—
622C2×C4#53412yes—
623C2×C4#53412yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13712yes—
627C2×C4#57712yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+411yescharacteristic
630C2×C4#58512yes—
631C2×C4#58511yescharacteristic
632C8—42no—
633C8—44no—
634C8—42no—
635C22—22no—
636C22—24no—
637C22—22no—
638C22—24no—
639C22—22no—
640C22—24no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22C2×#53411yescharacteristic
657C22#348911yescharacteristic
658C22#350011yescharacteristic
659C22C22×#13511yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57711yescharacteristic
662C22#358511yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—81no—
672C4—81no—
673C4—82no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#319014yes—
679C4#355412yes—
680C4#356212yes—
681C2—42no—
682C2—44no—
683C2—161no—
684C2#4291311yescharacteristic
685C2C2×#319011yescharacteristic
686C2#6075711yescharacteristic
687C2#6209811yescharacteristic
688C2#6213111yescharacteristic
689C2#6253211yescharacteristic
690C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36504C2
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
2. #35279C2semidirect
3. #32775C2semidirect
4. #32033C2semidirect
5. #32008C2semidirect
6. #30188C2semidirect
7. #30175C2semidirect
8. #29922C2semidirect
9. #29159C2semidirect
10. #28598C2semidirect
11. #26277C2semidirect
12. #25902C2semidirect
13. #25646C2semidirect
14. #25339C2semidirect
15. #24299C2semidirect
16. #23754C2semidirect
17. #22469C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#352792
#327752
#320332
#320082
#301882
#301752
#299222
#291592
#285982
#262772
#259022
#256462
#253392
#242992
#237542
#224692