Finite Group Databasebeta

← #368521

Subgroups — #368521

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1893C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic, complemented
4#14137C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#390C2212yes—
21C2×#386—21no—
22C2×#386—21no—
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1893C418yescomplemented
28#1874—24no—
29C2×#340—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1439—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C23×D4—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—42no—
52C23×C8—42no—
53C23×C8C2×C414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121C2×C414yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—22no—
85C22×C42C2312yes—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102C2×C418yescomplemented
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96#390—24no—
97#390—42no—
98#386—44no—
99#386—44no—
100#380—22no—
101#380C2×C414yes—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—42no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—44no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D412yes—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340C2×C418yescomplemented
121#336—42no—
122#336—42no—
123C25—21no—
124C25C2×D411yescharacteristic
125C22×D4—24no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4C22×C412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—28no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—48no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—48no—
192C22×C8C22×C414yes—
193#121—28no—
194#121—28no—
195#121—44no—
196#121—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4114yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4D814yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—82no—
260#102—48no—
261#102—44no—
262#102—44no—
263C22⋊C8—44no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—82no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24C4×D412yes—
287C24#12112yes—
288C24#12312yes—
289C24C2×D812yes—
290C24#13512yes—
291C24AGL(1,ℤ/8)12yes—
292C24#13712yes—
293C24C22×D411yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—84no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—48no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—48no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4#11714yes—
390C22×C4C4×D414yes—
391C22×C4C4×D412yes—
392C22×C4#12112yes—
393C22×C4#12312yes—
394C22×C4#12414yes—
395C22×C4#12614yes—
396C22×C4C2×D814yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)12yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—416no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4#12314yes—
445C22⋊C4—28no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—88no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—81no—
478C23—81no—
479C23—82no—
480C23—81no—
481C23—81no—
482C23—82no—
483C23—81no—
484C23—81no—
485C23—82no—
486C23—82no—
487C23—81no—
488C23—81no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23#38814yes—
496C23C4×D814yes—
497C23#44014yes—
498C23#45914yes—
499C23#46414yes—
500C23#48014yes—
501C23#48214yes—
502C23C2×C4×D411yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×#12311yescharacteristic
505C23#54612yes—
506C23#54612yes—
507C23C22×D811yescharacteristic
508C23C2×#13511yescharacteristic
509C23C2×AGL(1,ℤ/8)11yescharacteristic
510C23C2×#13711yescharacteristic
511C23#57712yes—
512C23#57712yes—
513D4—42no—
514D4—44no—
515D4—42no—
516D4—816no—
517D4—81no—
518D4—81no—
519D4—82no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—816no—
562C2×C4—816no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4#38814yes—
578C2×C4C2×#11711yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C4×#4112yes—
581C2×C4#51812yes—
582C2×C4C2×#12311yescharacteristic
583C2×C4#52511yescharacteristic
584C2×C4C2×#12411yescharacteristic
585C2×C4C2×#12611yescharacteristic
586C2×C4#52912yes—
587C2×C4#53411yescharacteristic
588C2×C4#53511yescharacteristic
589C2×C4#53812yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55312yes—
593C2×C4#55312yes—
594C2×C4#55312yes—
595C2×C4#55312yes—
596C8—44no—
597C8—81no—
598C8—81no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—82no—
614C22—162no—
615C22—162no—
616C22—161no—
617C22—161no—
618C22—161no—
619C22—161no—
620C22#192518yes—
621C22C2×#38811yescharacteristic
622C22#236412yes—
623C22#241111yescharacteristic
624C22#241211yescharacteristic
625C22#244412yes—
626C22C2×C4×D811yescharacteristic
627C22C2×#44011yescharacteristic
628C22#295212yes—
629C22C2×#45911yescharacteristic
630C22C2×#46411yescharacteristic
631C22#304211yescharacteristic
632C22#304311yescharacteristic
633C22C2×#48011yescharacteristic
634C22C2×#48211yescharacteristic
635C22#309912yes—
636C22#329412yes—
637C22#329412yes—
638C22#330112yes—
639C22#330112yes—
640C22#330612yes—
641C22#331012yes—
642C22#331612yes—
643C22#332012yes—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4C4×#12311yescharacteristic
653C4#232111yescharacteristic
654C4#235312yes—
655C4#250411yescharacteristic
656C4#251012yes—
657C4#259111yescharacteristic
658C2—28no—
659C2—162no—
660C2#1402011yescharacteristic
661C2C2×#192511yescharacteristic
662C2#2106811yescharacteristic
663C2#2312712yes—
664C2#2313412yes—
665C2#2447611yescharacteristic
666C2#2448311yescharacteristic
667C2#2452611yescharacteristic
668C2#2453311yescharacteristic
669C2#2619812yes—
670C2#2621212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#340C2semidirect
5. C22⋊C4×C4⋊C4C2semidirect
6. #11315C2semidirect
7. #11189C2semidirect
8. #10931C2semidirect
9. #10806C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
27. #1893C4semidirect
92. C2×#102C2×C4semidirect
120. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18932
C2×#18742
C22×#3402
#141372
C22⋊C4×C4⋊C42
#113152
#111892
#109312
#108062
#107302
#107142