Finite Group Databasebeta

← #365100

Subgroups — #365100

Order 512.

2216 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#1928C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4C2×#1787C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#10932C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#103—28no—
18C22×#103—24no—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1928—28no—
32#1852—24no—
33#1787C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
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×Q8C2312yes—
58C24×C4—21no—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C23×C8—22no—
62C23×C8—24no—
63C23×C8—22no—
64C23×C8—22no—
65C23×C8—42no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4D414yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—22no—
84C22×C42C2312yes—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—24no—
88C2×#107—42no—
89C2×#103—232no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103C2×C418yescomplemented
100C2×#103C2×C418yescomplemented
101C2×#103D418yes—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8C2×C414yes—
105#391—24no—
106#391—24no—
107#391—42no—
108#391—42no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380C2×C414yes—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—42no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—44no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95D412yes—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—42no—
132#336—24no—
133#336—42no—
134C25C2×D411yescharacteristic
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—28no—
140C22×Q8—216no—
141C22×Q8—28no—
142C22×Q8—216no—
143C22×Q8—28no—
144C22×Q8—28no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8—48no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—48no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8C22×C414yes—
170C22×Q8C2×D414yes—
171C22×Q8#4114yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—82no—
192C23×C4C22×C412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4#4112yes—
201C23×C4C2411yescharacteristic
202C22×C8—28no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—48no—
218C22×C8—48no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8C22×C412yes—
222C22×C8C22×C412yes—
223C22×C8C2×D414yes—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4Q1614yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—82no—
272#107—44no—
273#107—44no—
274#103—216no—
275#103—216no—
276#103—216no—
277#103—48no—
278C22⋊C8—28no—
279#95—24no—
280#95—24no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—82no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24C4×D412yes—
294C24#12112yes—
295C24#12312yes—
296C24C2×Q1612yes—
297C24#13512yes—
298C24AGL(1,ℤ/8)12yes—
299C24#13712yes—
300C24C22×D411yescharacteristic
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—232no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—416no—
318C2×Q8—432no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—416no—
322C2×Q8—416no—
323C2×Q8—416no—
324C2×Q8—416no—
325C2×Q8—432no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—48no—
339C2×Q8C4×D418yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—416no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—416no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—84no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4#11714yes—
414C22×C4C4×D414yes—
415C22×C4C4×D412yes—
416C22×C4#12112yes—
417C22×C4#12312yes—
418C22×C4#12414yes—
419C22×C4#12614yes—
420C22×C4C2×Q1614yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—416no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4#12014yes—
466C22⋊C4—28no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—84no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—88no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23#38814yes—
492C23C4×Q1614yes—
493C23#44014yes—
494C23#46214yes—
495C23#46314yes—
496C23#48414yes—
497C23#48514yes—
498C23C2×C4×D411yescharacteristic
499C23C2×#12111yescharacteristic
500C23C2×#12311yescharacteristic
501C23D4212yes—
502C23#54612yes—
503C23C22×Q1611yescharacteristic
504C23C2×#13511yescharacteristic
505C23C2×AGL(1,ℤ/8)11yescharacteristic
506C23C2×#13711yescharacteristic
507C23#57712yes—
508C23#57812yes—
509Q8—28no—
510Q8—28no—
511Q8—432no—
512Q8—44no—
513Q8—44no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—432no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
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—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—816no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4#38814yes—
579C2×C4C2×#11711yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C2×C4×D412yes—
582C2×C4#51812yes—
583C2×C4C2×#12011yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4#52512yes—
586C2×C4C2×#12411yescharacteristic
587C2×C4C2×#12611yescharacteristic
588C2×C4#53411yescharacteristic
589C2×C4#53511yescharacteristic
590C2×C4#53812yes—
591C2×C4#54612yes—
592C2×C4#54612yes—
593C2×C4#54612yes—
594C2×C4#55112yes—
595C2×C4#55112yes—
596C2×C4#55112yes—
597C8—42no—
598C8—44no—
599C8—42no—
600C22—28no—
601C22—24no—
602C22—28no—
603C22—24no—
604C22—28no—
605C22—24no—
606C22—28no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22#192818yescomplemented
611C22C2×#38811yescharacteristic
612C22#235812yes—
613C22#241011yescharacteristic
614C22#241111yescharacteristic
615C22#241812yes—
616C22C2×C4×Q1611yescharacteristic
617C22C2×#44011yescharacteristic
618C22#294712yes—
619C22C2×#46311yescharacteristic
620C22C2×#46211yescharacteristic
621C22#304211yescharacteristic
622C22#304411yescharacteristic
623C22C2×#48511yescharacteristic
624C22C2×#48411yescharacteristic
625C22#309712yes—
626C22#326512yes—
627C22#327012yes—
628C22#328212yes—
629C22D4×Q1612yes—
630C22#329912yes—
631C22#330012yes—
632C22#330812yes—
633C22#332012yes—
634C4—24no—
635C4—44no—
636C4—44no—
637C4—48no—
638C4—82no—
639C4—82no—
640C4—84no—
641C4—161no—
642C4—161no—
643C4C4×#12011yescharacteristic
644C4#232111yescharacteristic
645C4#234212yes—
646C4#243511yescharacteristic
647C4#248012yes—
648C4#256111yescharacteristic
649C2—28no—
650C2#1397111yescharacteristic
651C2C2×#192811yescharacteristic
652C2#2106411yescharacteristic
653C2#2285512yes—
654C2#2287112yes—
655C2#2435911yescharacteristic
656C2#2436011yescharacteristic
657C2#2448111yescharacteristic
658C2#2448211yescharacteristic
659C2#2486712yes—
660C2#2487112yes—
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
33. #1787C4semidirect
99. C2×#103C2×C4semidirect
100. C2×#103C2×C4semidirect
610. 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×#18522
C2×#17872
#139222
C22⋊C4×C4⋊C42
#111902
#111242
#109322
#107212
#107142