Finite Group Databasebeta

← #6458874

Subgroups — #6458874

Order 512.

1843 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35255C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#27874C214yesmaximal, complemented
4#27870C214yesmaximal, complemented
5#27863C212yesmaximal, complemented
6#27861C212yesmaximal, complemented
7#27858C212yesmaximal, complemented
8#27856C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic, complemented
11C2×#2083C211yesmaximal, characteristic, complemented
12#23630C211yesmaximal, characteristic, complemented
13#21067C212yesmaximal, complemented
14#21039C212yesmaximal, complemented
15#20976C212yesmaximal
16#20663C212yesmaximal, complemented
17#20336C211yesmaximal, characteristic, complemented
18C22×#131C2212yes—
19C2×#537C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#427C2212yes—
22C22×#107C2212yes—
23C2×#417C2212yes—
24C2×#416C2212yes—
25#2855C2212yes—
26#2854C2212yes—
27#2625C2212yes—
28#2624C2212yes—
29#2444—22no—
30#2443—22no—
31#2423—21no—
32#2422—21no—
33#2421—21no—
34#2420—21no—
35#2355—21no—
36#2355—21no—
37#2354—21no—
38#2354—21no—
39#2352—22no—
40#2352—22no—
41#2317C2211yescharacteristic
42#2314C2214yes—
43#2311C2211yescharacteristic
44#2311C2212yes—
45#2311C2211yescharacteristic
46#2297C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53#2101C2214yes—
54#2100C2212yes—
55#2099C2212yes—
56#2097C2212yes—
57#2096C2211yescharacteristic
58#2095C2211yescharacteristic
59#2094C2211yescharacteristic
60#2093C2211yescharacteristic
61#2092C2212yes—
62#2083C2212yes—
63#2083C2212yes—
64#2083C2214yes—
65#2072C2212yes—
66#2072C2212yes—
67#2071C2212yes—
68#2071C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1919C2214yes—
76#1918C2214yes—
77#1895C2214yes—
78#1894C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1809C2214yes—
82#1809C2214yes—
83#1808C2214yes—
84#1808C2214yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C2×#131—28no—
98C2×#131—28no—
99C2×#131—44no—
100C22×M16C2312yes—
101C22×M16C2314yes—
102C23×C8C2314yes—
103C23×C8C2312yes—
104#537—28no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121—22no—
110C2×#121C2312yes—
111C4×#41—28no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×D4D412yes—
115C2×C4×D4D412yes—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×#117—22no—
119C2×#117—22no—
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#427—28no—
133C2×#107—44no—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140#417—44no—
141#416—28no—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#394—22no—
155#394—22no—
156#391—22no—
157#391—21no—
158#391C2312yes—
159#391C2311yescharacteristic
160#391C2311yescharacteristic
161#390—22no—
162#390—21no—
163#390C2312yes—
164#390C2311yescharacteristic
165#390C2311yescharacteristic
166#389—21no—
167#389—21no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#387—22no—
179#387—22no—
180#386—24no—
181#386—22no—
182#386—22no—
183#386—21no—
184#386—21no—
185#386—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—21no—
196#382—22no—
197#382—22no—
198#382C2312yes—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380—21no—
205#380—21no—
206#380C2314yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41#4118yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—22no—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262#131—432no—
263C2×M16—28no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—28no—
268C2×M16—44no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—28no—
272C22×C8—28no—
273C22×C8—44no—
274C22×C8#4118yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278#123—84no—
279C22⋊Q8—28no—
280C22⋊Q8—44no—
281#121—28no—
282#121—44no—
283#121—48no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290#117—28no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—84no—
296#117—84no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—28no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—24no—
363#107—24no—
364#107—44no—
365#107—44no—
366#106—24no—
367#106—24no—
368#103—24no—
369#103—24no—
370#103—44no—
371#102—24no—
372#102—24no—
373#102—44no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24—21no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—216no—
396#41—216no—
397#41—48no—
398#41—48no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8#12314yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4#12318yes—
421C2×D4#12314yes—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+412yes—
470C22×C4C2511yescharacteristic
471M16—416no—
472M16—48no—
473C2×C8—216no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—416no—
482C2×C8C2×#4114yes—
483C2×C8C2×#4114yes—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—82no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—22no—
539C42—42no—
540C42—44no—
541C42—84no—
542C42—84no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C2×#12314yes—
561C23C2×#12312yes—
562C23#53212yes—
563C23#53712yes—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—42no—
567Q8—42no—
568D4—216no—
569D4—28no—
570D4—42no—
571D4—42no—
572D4—84no—
573D4—164no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—162no—
625C2×C4—164no—
626C2×C4C2×#12314yes—
627C2×C4C2×#12312yes—
628C2×C4#53212yes—
629C2×C4#53712yes—
630C2×C4#55118yes—
631C2×C4#55218yes—
632C2×C4#57612yes—
633C2×C4#57812yes—
634C2×C4C22×#4112yes—
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—24no—
639C8—24no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#249814yes—
662C22C22×#12311yescharacteristic
663C22C2×#53211yescharacteristic
664C22C2×#53711yescharacteristic
665C22C2×#57611yescharacteristic
666C22C2×#57811yescharacteristic
667C22#358511yescharacteristic
668C22#358611yescharacteristic
669C4—22no—
670C4—22no—
671C4—24no—
672C4—41no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—82no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4#249814yes—
685C4C2×#55112yes—
686C4C2×#55212yes—
687C2—24no—
688C2—44no—
689C2—161no—
690C2C2×#249811yescharacteristic
691C2#6052111yescharacteristic
692C2#6052611yescharacteristic
693C2#6071811yescharacteristic
694C2#6072111yescharacteristic
695C2#6092211yescharacteristic
696C2#6092511yescharacteristic
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
2. #34188C2semidirect
3. #27874C2semidirect
4. #27870C2semidirect
5. #27863C2semidirect
6. #27861C2semidirect
7. #27858C2semidirect
8. #27856C2semidirect
9. #23747C2semidirect
10. #23730C2semidirect
11. C2×#2083C2semidirect
12. #23630C2semidirect
13. #21067C2semidirect
14. #21039C2semidirect
16. #20663C2semidirect
17. #20336C2semidirect

Maximal subgroups

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

subgroupindex
#352552
#341882
#278742
#278702
#278632
#278612
#278582
#278562
#237472
#237302
C2×#20832
#236302
#210672
#210392
#209762
#206632
#203362