Finite Group Databasebeta

← #360246

Subgroups — #360246

Order 512.

2216 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#1928C212yesmaximal, complemented
3C2×#1808C211yesmaximal, characteristic
4C2×#1787C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#10932C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#103—28no—
17C22×#103C418yescomplemented
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1928—28no—
33#1808—24no—
34#1787—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C23×Q8—21no—
47C23×Q8—22no—
48C23×Q8—24no—
49C23×Q8—24no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—42no—
55C23×Q8—42no—
56C23×Q8C2×C414yes—
57C23×Q8C2311yescharacteristic
58C23×Q8C2311yescharacteristic
59C24×C4—21no—
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—24no—
65C23×C8—42no—
66C23×C8C2×C414yes—
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4C2×C414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—22no—
84C22×C42C2312yes—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—232no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103C2×C414yes—
99C2×#103C2×C414yes—
100C2×#103C2×C414yes—
101C2×#103C2×C414yes—
102C2×#103D418yes—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106#391—24no—
107#391—24no—
108#391—42no—
109#391—42no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380D414yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—42no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—44no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95D412yes—
129#340—24no—
130#340—24no—
131#340—42no—
132#336—24no—
133#336—42no—
134C25C2×D411yescharacteristic
135C22×Q8—24no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—216no—
141C22×Q8—216no—
142C22×Q8—28no—
143C22×Q8—28no—
144C22×Q8—28no—
145C22×Q8—28no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8—48no—
163C22×Q8—48no—
164C22×Q8—48no—
165C22×Q8—48no—
166C22×Q8—48no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×Q8C22×C412yes—
175C22×Q8C22×C412yes—
176C22×Q8C2×D414yes—
177C22×Q8#4114yes—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—82no—
198C23×C4C22×C412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D412yes—
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4#4112yes—
206C23×C4#4112yes—
207C23×C4C2411yescharacteristic
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—28no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—28no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—44no—
225C22×C8—48no—
226C22×C8—48no—
227C22×C8C22×C412yes—
228C22×C8C22×C412yes—
229C22×C8C2×D414yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4Q1614yes—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—82no—
276#107—48no—
277#103—28no—
278#103—28no—
279#103—28no—
280#103—48no—
281#103—48no—
282#103C22⋊C4116yescomplemented
283C22⋊C8—44no—
284#95—24no—
285#95—24no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—82no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C4×D412yes—
299C24#12112yes—
300C24#12312yes—
301C24C2×Q1612yes—
302C24#13512yes—
303C24AGL(1,ℤ/8)12yes—
304C24#13712yes—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—232no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—416no—
330C2×Q8—416no—
331C2×Q8—416no—
332C2×Q8—416no—
333C2×Q8—416no—
334C2×Q8—432no—
335C2×Q8—432no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—48no—
351C2×Q8C2×C22⋊C414yes—
352C2×Q8#11714yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4#11714yes—
427C22×C4C4×D414yes—
428C22×C4C4×D412yes—
429C22×C4#12112yes—
430C22×C4#12312yes—
431C22×C4#12414yes—
432C22×C4#12614yes—
433C22×C4C2×Q1614yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—416no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8C2×C22⋊C414yes—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4C2×C22⋊C414yes—
477C22⋊C4—28no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—84no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—88no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23#38814yes—
503C23C4×Q1614yes—
504C23#44014yes—
505C23#46214yes—
506C23#46314yes—
507C23#48414yes—
508C23#48514yes—
509C23C2×C4×D411yescharacteristic
510C23C2×#12111yescharacteristic
511C23C2×#12311yescharacteristic
512C23D4212yes—
513C23#54612yes—
514C23C22×Q1611yescharacteristic
515C23C2×#13511yescharacteristic
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23C2×#13711yescharacteristic
518C23#57712yes—
519C23#57812yes—
520Q8—24no—
521Q8—24no—
522Q8—24no—
523Q8—24no—
524Q8—432no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—44no—
530Q8—44no—
531Q8—44no—
532Q8—44no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—432no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—816no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4#38814yes—
594C2×C4C22×C22⋊C411yescharacteristic
595C2×C4C2×#11711yescharacteristic
596C2×C4C2×#11711yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4#51811yescharacteristic
599C2×C4#51811yescharacteristic
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4C2×#12411yescharacteristic
603C2×C4C2×#12611yescharacteristic
604C2×C4#53812yes—
605C2×C4#53812yes—
606C2×C4D4212yes—
607C2×C4#54612yes—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4#55112yes—
611C2×C4#55112yes—
612C8—24no—
613C8—42no—
614C8—82no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—28no—
625C22#192818yescomplemented
626C22C2×#38811yescharacteristic
627C22D4×C22⋊C411yescharacteristic
628C22#234211yescharacteristic
629C22#241812yes—
630C22#242412yes—
631C22C2×C4×Q1611yescharacteristic
632C22C2×#44011yescharacteristic
633C22#294911yescharacteristic
634C22#295011yescharacteristic
635C22C2×#46311yescharacteristic
636C22C2×#46211yescharacteristic
637C22#304612yes—
638C22C2×#48511yescharacteristic
639C22C2×#48411yescharacteristic
640C22#309712yes—
641C22#326512yes—
642C22#326612yes—
643C22#328512yes—
644C22D4×Q1612yes—
645C22#329912yes—
646C22#330112yes—
647C22#330812yes—
648C22#331212yes—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4—162no—
658C4#228011yescharacteristic
659C4#231211yescharacteristic
660C4D4×C22⋊C411yescharacteristic
661C4#234211yescharacteristic
662C4#247212yes—
663C4#248012yes—
664C2—28no—
665C2#1387311yescharacteristic
666C2C2×#192811yescharacteristic
667C2#2106611yescharacteristic
668C2#2243711yescharacteristic
669C2C22⋊C4×Q1611yescharacteristic
670C2#2256311yescharacteristic
671C2#2257511yescharacteristic
672C2#2486712yes—
673C2#2487512yes—
674C2#2508212yes—
675C2#2509012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
2. C2×#1928C2semidirect
7. #11190C2semidirect
8. #11124C2semidirect
9. #10932C2semidirect
17. C22×#103C4semidirect
282. #103C22⋊C4semidirect
625. C22#1928semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#19282
C2×#18082
C2×#17872
#139222
C22⋊C4×C4⋊C42
#111902
#111242
#109322
#107302
#107142