Finite Group Databasebeta

← #6564140

Subgroups — #6564140

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29927C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic, complemented
5#27911C212yesmaximal, complemented
6#27905C211yesmaximal, characteristic, complemented
7#27685C212yesmaximal, complemented
8#27677C211yesmaximal, characteristic, complemented
9#26239C214yesmaximal, complemented
10#26233C212yesmaximal, complemented
11#25021C214yesmaximal, complemented
12#25013C212yesmaximal, complemented
13#23756C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#22725C212yesmaximal, complemented
16#22541C212yesmaximal, complemented
17#22530C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#482C2214yes—
22C2×#483C2214yes—
23C2×C8×Q8C2212yes—
24#2857C2211yescharacteristic
25#2850C2212yes—
26#2542C2211yescharacteristic
27#2540C2212yes—
28#2537C2211yescharacteristic
29#2536C2212yes—
30#2479—22no—
31#2467—22no—
32#2423—21no—
33#2407—21no—
34#2363—21no—
35#2355—21no—
36#2355C2212yes—
37#2355C2211yescharacteristic
38#2354—21no—
39#2354—21no—
40#2354C2212yes—
41#2352—22no—
42#2352C2214yes—
43#2314—22no—
44#2311—21no—
45C4×C22⋊Q8C2212yes—
46C4×#121C2211yescharacteristic
47#2285C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51#2274—21no—
52#2100C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2095C2212yes—
57#2094C2211yescharacteristic
58#2093C2212yes—
59#2084C2211yescharacteristic
60#2077C2214yes—
61#2077C2212yes—
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2036C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2034C2212yes—
68#2034C2214yes—
69#1983C2214yes—
70#1981C2212yes—
71#1958C2214yes—
72#1956C2212yes—
73#1954C2212yes—
74#1954C2214yes—
75#1949C2214yes—
76#1949C2214yes—
77#1947C2212yes—
78#1947C2212yes—
79#1874C2212yes—
80#1873C2214yes—
81#1854C2214yes—
82#1852C2212yes—
83#1846C2214yes—
84#1844C2212yes—
85#1805C2214yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89C4×#106C2212yes—
90C4×#106C2214yes—
91C4×#103C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96#556—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#126—22no—
99C2×#124C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121C2311yescharacteristic
106C2×#121C2312yes—
107C4×#41—28no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4D412yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—21no—
118C2×#117—22no—
119C2×#117—21no—
120C2×#117D412yes—
121C2×#117D411yescharacteristic
122C2×#117C2314yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#485—48no—
129#483—48no—
130#482—48no—
131C8×Q8—28no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#399—22no—
154#399—22no—
155#398—22no—
156#394—22no—
157#394—22no—
158#392—22no—
159#391—22no—
160#391—21no—
161#391C2312yes—
162#390—22no—
163#390—21no—
164#390—22no—
165#390—22no—
166#390C2311yescharacteristic
167#389—21no—
168#389—21no—
169#389C2311yescharacteristic
170#388—24no—
171#388—24no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—21no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#386—24no—
182#386—22no—
183#386—22no—
184#386—21no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#382—21no—
195#382—22no—
196#382C2314yes—
197#382C2312yes—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—21no—
205#380C2311yescharacteristic
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4—21no—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2×D412yes—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256#126—48no—
257#124—28no—
258C22⋊Q8—24no—
259C22⋊Q8—42no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263#121—22no—
264#121—41no—
265#121—44no—
266#121—44no—
267C4×Q8—24no—
268C4×Q8—28no—
269C4×Q8#4118yes—
270C4×D4—24no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—44no—
276#117—24no—
277#117—28no—
278#117—44no—
279#117—44no—
280#117—44no—
281#117—44no—
282#117—84no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—28no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348#107—44no—
349#107—44no—
350#106—28no—
351#106—24no—
352#106—24no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—28no—
356#103—28no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—24no—
361#102—42no—
362#102—44no—
363C22⋊C8—28no—
364C22⋊C8—24no—
365C22⋊C8—44no—
366C22⋊C8—42no—
367C4×C8—24no—
368C4×C8—28no—
369#95—21no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384C24—21no—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387#41—416no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8#12314yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—81no—
409C2×D4—82no—
410C2×D4—82no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4#13514yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8H(5,𝔽2)14yes—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4#12314yes—
494C4⋊C4#12318yes—
495C4⋊C42−1+412yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—44no—
520C42—42no—
521C42—44no—
522C42—82no—
523C42—84no—
524C42—84no—
525C42C2×#4114yes—
526C42C2×#4118yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×#13512yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—44no—
551Q8—44no—
552D4—48no—
553D4—44no—
554D4—42no—
555D4—42no—
556D4—84no—
557D4—164no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—164no—
608C2×C4C2×#12314yes—
609C2×C4C2×#12312yes—
610C2×C4C2×#12314yes—
611C2×C4C2×#12312yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4#57512yes—
615C2×C4#57712yes—
616C2×C4C22×#4112yes—
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58511yescharacteristic
620C8—24no—
621C8—44no—
622C8—42no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C22×#12311yescharacteristic
645C22#348911yescharacteristic
646C22#349811yescharacteristic
647C22C22×#13511yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57711yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—24no—
653C4—22no—
654C4—24no—
655C4—22no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4#309114yes—
667C4#351712yes—
668C4#352212yes—
669C2—42no—
670C2—82no—
671C2—161no—
672C2#3600511yescharacteristic
673C2C2×#309111yescharacteristic
674C2#6052211yescharacteristic
675C2#6209711yescharacteristic
676C2#6214511yescharacteristic
677C2#6241411yescharacteristic
678C2#6245611yescharacteristic
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. #34188C2semidirect
3. #29927C2semidirect
4. #29923C2semidirect
5. #27911C2semidirect
6. #27905C2semidirect
7. #27685C2semidirect
8. #27677C2semidirect
9. #26239C2semidirect
10. #26233C2semidirect
11. #25021C2semidirect
12. #25013C2semidirect
13. #23756C2semidirect
14. #23755C2semidirect
15. #22725C2semidirect
16. #22541C2semidirect
17. #22530C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#341882
#299272
#299232
#279112
#279052
#276852
#276772
#262392
#262332
#250212
#250132
#237562
#237552
#227252
#225412
#225302