Finite Group Databasebeta

← #6856254

Subgroups — #6856254

Order 512.

1667 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#36587C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#32442C214yesmaximal, complemented
5#31650C214yesmaximal, complemented
6#29719C212yesmaximal, complemented
7#28418C212yesmaximal, complemented
8#27978C212yesmaximal, complemented
9#27738C212yesmaximal, complemented
10#27677C212yesmaximal, complemented
11#25903C211yesmaximal, characteristic
12#25322C212yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22047C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#486C2212yes—
21C2×#477C2212yes—
22C2×#460C2212yes—
23C2×#451C2211yescharacteristic
24C2×#448C2212yes—
25C2×C4×Q16C2212yes—
26#2856C2211yescharacteristic
27#2850C2212yes—
28#2597—22no—
29#2571C2211yescharacteristic
30#2567—22no—
31#2567—21no—
32#2567—21no—
33#2567C2212yes—
34#2567C2212yes—
35#2567C2214yes—
36#2548C2211yescharacteristic
37#2513—21no—
38#2513C2212yes—
39#2504—21no—
40#2447C2212yes—
41#2444—22no—
42C2×#395—22no—
43#2366C2211yescharacteristic
44#2361—21no—
45#2357—21no—
46#2355C2211yescharacteristic
47C4×C22⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389—21no—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2214yes—
55#2093C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2071C2214yes—
59#2057C2212yes—
60#2056C2214yes—
61#2055C2214yes—
62#2024C2212yes—
63#2023C2214yes—
64#2015C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1973C2214yes—
68#1959C2214yes—
69#1947C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1928C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1878C2212yes—
76#1877C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1852C2211yescharacteristic
84#1852C2212yes—
85#1836C2214yes—
86#1803C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#124C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—44no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#486—44no—
131#477—44no—
132#460—44no—
133#451—24no—
134#448—28no—
135C4×Q16—44no—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#397—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#395—24no—
160#395—24no—
161#395—24no—
162#395—24no—
163#394—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#391C2312yes—
169#391C2311yescharacteristic
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#389—22no—
179#389—22no—
180#389—24no—
181#389—22no—
182#389—24no—
183#389—22no—
184#389—21no—
185#389C2311yescharacteristic
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—21no—
193#387—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#385—21no—
199#385—22no—
200#384—22no—
201#384—22no—
202#384C2314yes—
203#384C2314yes—
204#384C2311yescharacteristic
205#382—21no—
206#382—21no—
207#382C2312yes—
208#381—22no—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×Q16—24no—
262C2×Q16—42no—
263C2×Q16—42no—
264C2×SD16—28no—
265C2×SD16—44no—
266C2×SD16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#127—24no—
276#124—44no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279#121—24no—
280#121—24no—
281#121—44no—
282#121—82no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—84no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357#107—42no—
358#107—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#103—28no—
362#103—24no—
363#103—42no—
364#103—42no—
365#103—42no—
366#103—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370C22⋊C8—24no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373#97—44no—
374C4×C8—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24C22×D411yescharacteristic
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—816no—
417C2×Q8C2×#4114yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—81no—
430C2×D4—81no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4#13714yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477Q16—48no—
478SD16—88no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—81no—
519C4⋊C4#13714yes—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—24no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—44no—
541C42—42no—
542C42—42no—
543C42—82no—
544C42H(5,𝔽2)14yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23#54812yes—
557C23C2×#13712yes—
558C23#57612yes—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—24no—
562Q8—48no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—88no—
568Q8—88no—
569D4—44no—
570D4—44no—
571D4—88no—
572D4—84no—
573D4—162no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—84no—
626C2×C4—88no—
627C2×C4—164no—
628C2×C4#53512yes—
629C2×C4#53512yes—
630C2×C4#54812yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13714yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58512yes—
639C8—44no—
640C8—42no—
641C8—42no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22#337514yes—
656C22C2×#53511yescharacteristic
657C22C2×#54811yescharacteristic
658C22#348511yescharacteristic
659C22C22×#13711yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57611yescharacteristic
662C22#358511yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—41no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4—82no—
678C4—84no—
679C4#321814yes—
680C4#350512yes—
681C4#354112yes—
682C2—24no—
683C2—162no—
684C2#3972211yescharacteristic
685C2C2×#321811yescharacteristic
686C2#6081111yescharacteristic
687C2C2×#337511yescharacteristic
688C2#6131911yescharacteristic
689C2#6171911yescharacteristic
690C2#6172611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#2567C2semidirect
4. #32442C2semidirect
5. #31650C2semidirect
6. #29719C2semidirect
7. #28418C2semidirect
8. #27978C2semidirect
9. #27738C2semidirect
10. #27677C2semidirect
12. #25322C2semidirect
13. #24682C2semidirect
14. #24302C2semidirect
15. #22629C2semidirect
16. #22047C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25672
#329032
#324422
#316502
#297192
#284182
#279782
#277382
#276772
#259032
#253222
#246822
#243022
#226292
#220472
C4×#4512