Finite Group Databasebeta

← #7569825

Subgroups — #7569825

Order 512.

1939 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#32034C214yesmaximal, complemented
4#28748C211yesmaximal, characteristic, complemented
5#28748C211yesmaximal, characteristic, complemented
6#28600C212yesmaximal, complemented
7#28598C212yesmaximal, complemented
8#26233C212yesmaximal, complemented
9#26231C212yesmaximal, complemented
10#25886C212yesmaximal, complemented
11#25882C212yesmaximal, complemented
12#24043C214yesmaximal, complemented
13#23173C211yesmaximal, characteristic, complemented
14#22417C212yesmaximal, complemented
15#22417C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#547C2212yes—
19C2×#547C2212yes—
20C2×#547C2212yes—
21C2×#487C2212yes—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#476C2211yescharacteristic
25C2×#474C2211yescharacteristic
26C2×C8×D4C2212yes—
27C2×C8×D4C2212yes—
28C2×#416C2214yes—
29#2868C2211yescharacteristic
30#2856—21no—
31#2808—22no—
32#2592—22no—
33#2563C2212yes—
34#2548—21no—
35#2547C2212yes—
36#2542C2212yes—
37#2542C2212yes—
38#2542C2212yes—
39#2513—22no—
40#2501—22no—
41#2501—22no—
42#2501—22no—
43#2485C2214yes—
44#2355—21no—
45#2354C2212yes—
46#2354C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C2×#384—21no—
51C42×D4C2211yescharacteristic
52#2043C2214yes—
53#2043C2214yes—
54#2036C2214yes—
55#2036C2214yes—
56#2036C2214yes—
57#2035C2214yes—
58#2035C2214yes—
59#2035C2214yes—
60#1988C2211yescharacteristic
61#1987C2211yescharacteristic
62#1981C2212yes—
63#1981C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1956C2212yes—
67#1956C2212yes—
68#1954C2212yes—
69#1954C2212yes—
70#1948C2211yescharacteristic
71#1947C2211yescharacteristic
72#1920C2212yes—
73#1918C2212yes—
74#1892C2214yes—
75#1892C2214yes—
76#1892C2214yes—
77#1873C2214yes—
78#1873C2214yes—
79#1869C2214yes—
80#1864C2212yes—
81#1862C2212yes—
82#1802C2211yescharacteristic
83#1802C2211yescharacteristic
84#1788C2214yes—
85#1788C2214yes—
86#1788C2214yes—
87C4×#107C2212yes—
88C4×#106C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×#41C2314yes—
94C23×C8C2314yes—
95#547—28no—
96#547—28no—
97#547—44no—
98#547—44no—
99#547—44no—
100#547—44no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127D411yescharacteristic
107C2×#124—22no—
108C2×#123—24no—
109C2×#123—24no—
110C2×#123—24no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117C2314yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C42—21no—
130#487—44no—
131#487—44no—
132#485—48no—
133#476—24no—
134#474—24no—
135C8×D4—44no—
136C8×D4—44no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145#416—48no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163#399—24no—
164#399—24no—
165#399—24no—
166#397—24no—
167#397—24no—
168#394—24no—
169#393—24no—
170#391—22no—
171#391—21no—
172#391—21no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390D412yes—
180#390D412yes—
181#390D412yes—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—24no—
190#384—22no—
191#384—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384D411yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#380C2314yes—
202#380C2314yes—
203#380C2314yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#336C2312yes—
226#336C2312yes—
227C2×#41—28no—
228C2×#41—28no—
229C2×#41—28no—
230C2×#41—44no—
231C2×#41—44no—
232C2×#41—44no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8C2414yes—
236C22×D4—24no—
237C22×D4C2×D412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4C2414yes—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263#127—24no—
264#124—84no—
265#123—416no—
266#123—416no—
267#123—416no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—48no—
275#121—48no—
276#121—48no—
277C4×Q8—44no—
278C4×Q8—82no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4#4114yes—
290C4×D4#4114yes—
291#117—44no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C22⋊C4C2×D414yes—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—24no—
352#106—24no—
353#106—24no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—24no—
368#102—24no—
369#102—24no—
370C22⋊C8—44no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373C4×C8—24no—
374C4×C8—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385C24—21no—
386C24C22×D414yes—
387#41—416no—
388#41—88no—
389#41—88no—
390#41—88no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4C2×#4112yes—
419C2×D4C2×#4112yes—
420C2×D4H(5,𝔽2)12yes—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—88no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4C2×#4112yes—
500C4⋊C4C2×#4112yes—
501C4⋊C4H(5,𝔽2)12yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—44no—
524C42—44no—
525C42—84no—
526C42#13514yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C42H(5,𝔽2)12yes—
530C42H(5,𝔽2)12yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C23×D411yescharacteristic
546Q8—44no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—84no—
551Q8—84no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4—48no—
556D4—48no—
557D4—84no—
558D4—816no—
559D4#55914yes—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—48no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—816no—
604C2×C4#54612yes—
605C2×C4#54612yes—
606C2×C4#54612yes—
607C2×C4#54612yes—
608C2×C4C2×#13512yes—
609C2×C4C2×#13512yes—
610C2×C4C2×#13512yes—
611C2×C4#57712yes—
612C2×C4#57812yes—
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C8—42no—
620C8—42no—
621C8—42no—
622C8—42no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22C2×#54611yescharacteristic
638C22C2×#54611yescharacteristic
639C22#348411yescharacteristic
640C22C22×#13511yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57811yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—41no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#330214yes—
658C4#330314yes—
659C4C2×#55911yescharacteristic
660C4#350811yescharacteristic
661C4#350811yescharacteristic
662C4#354311yescharacteristic
663C2—44no—
664C2—84no—
665C2#5914311yescharacteristic
666C2C2×#330211yescharacteristic
667C2C2×#330311yescharacteristic
668C2#6120911yescharacteristic
669C2#6121011yescharacteristic
670C2#6156511yescharacteristic
671C2#6166711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59863C2semidirect
3. #32034C2semidirect
4. #28748C2semidirect
5. #28748C2semidirect
6. #28600C2semidirect
7. #28598C2semidirect
8. #26233C2semidirect
9. #26231C2semidirect
10. #25886C2semidirect
11. #25882C2semidirect
12. #24043C2semidirect
13. #23173C2semidirect
14. #22417C2semidirect
15. #22417C2semidirect
16. C4×#476C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#361792
#320342
#287482
#286002
#285982
#262332
#262312
#258862
#258822
#240432
#231732
#224172
C4×#4762
C4×#4742