Finite Group Databasebeta

← #360168

Subgroups — #360168

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C8C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3C2×#1893C212yesmaximal
4C2×#1802C212yesmaximal, complemented
5#14686C211yesmaximal, characteristic, complemented
6#11227C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10721C212yesmaximal, complemented
10C24×C8C2212yes—
11C22×#121C2211yescharacteristic
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C4⋊C8—24no—
16C22×C4⋊C8—28no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2212yes—
19C22×#102C2214yes—
20C4×#121—24no—
21C4×#121C418yescomplemented
22C2×#390C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#386—21no—
26C2×#384C2212yes—
27C2×#380—21no—
28C2×#380—21no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1893—28no—
34#1802—28no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—24no—
48C23×C8—28no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—44no—
52C23×C8C2314yes—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121C2×C414yes—
57C2×#121D414yes—
58C2×#121Q814yes—
59C2×C4×D4—28no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—44no—
64C2×C4×D4—44no—
65C2×C4×D4C2×C414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42—21no—
82C22×C42C2×C414yes—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D414yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—232no—
89C2×C4⋊C8—28no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8D418yescomplemented
94C2×#102—28no—
95C2×#102—28no—
96C2×#102—44no—
97#390—22no—
98#390C2×C414yes—
99#389—22no—
100#389C2×C414yes—
101#388—28no—
102#386—28no—
103#384—24no—
104#384—24no—
105#380—22no—
106#380—24no—
107#380—24no—
108#380—44no—
109#380—44no—
110#380C2×C414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C414yes—
113C4×C22⋊C4—28no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—28no—
121#340—28no—
122#340—44no—
123C25—21no—
124C25C2×D411yescharacteristic
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C22×C412yes—
140C22×D4C2×D412yes—
141C22×D4C2×Q812yes—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4C22⋊C414yes—
162C23×C4C22⋊C414yes—
163C23×C4D814yes—
164C23×C4SD1614yes—
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×Q812yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C22×C8—216no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—416no—
185C22×C8—416no—
186C22×C8—48no—
187C22×C8—48no—
188C22×C8C2×D418yes—
189#121—28no—
190#121—28no—
191#121—28no—
192#121C4⋊C4116yescomplemented
193C4×D4—216no—
194C4×D4—432no—
195C4×D4—44no—
196C4×D4—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—216no—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22⋊C414yes—
253C2×C42C22⋊C414yes—
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D414yes—
258C2×C42C2×D412yes—
259C2×C42#4114yes—
260C4⋊C8—216no—
261C4⋊C8—216no—
262#102—416no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—44no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—42no—
277C24—42no—
278C24—41no—
279C24—41no—
280C24—42no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24C2×C22⋊C412yes—
285C24#12112yes—
286C24C22⋊Q812yes—
287C24C2×D812yes—
288C24C2×SD1612yes—
289C24AGL(1,ℤ/8)12yes—
290C24#13712yes—
291C24C22×D411yescharacteristic
292C2×D4—216no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4C2×C4⋊C414yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—88no—
379C22×C4—88no—
380C22×C4#102116yescomplemented
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C4⋊C414yes—
387C22×C4#11714yes—
388C22×C4#12114yes—
389C22×C4#12112yes—
390C22×C4#12114yes—
391C22×C4#12114yes—
392C22×C4C22⋊Q814yes—
393C22×C4C22⋊Q812yes—
394C22×C4C2×D812yes—
395C22×C4C2×D812yes—
396C22×C4C2×SD1612yes—
397C22×C4C2×SD1612yes—
398C22×C4#13514yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409C2×C8—28no—
410C2×C8—28no—
411C2×C8—28no—
412C2×C8—432no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C4⋊C4C2×C4⋊C414yes—
434C22⋊C4—216no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—88no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—28no—
444C42—28no—
445C42—48no—
446C42C4×D418yes—
447C42#12014yes—
448C42#12314yes—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23#38014yes—
489C23C2×#10214yes—
490C23#41714yes—
491C23#45914yes—
492C23#46114yes—
493C23#47414yes—
494C23#47614yes—
495C23C22×C22⋊C411yescharacteristic
496C23C2×#12111yescharacteristic
497C23C2×#12112yes—
498C23C2×C22⋊Q811yescharacteristic
499C23#53712yes—
500C23C22×D811yescharacteristic
501C23C22×SD1611yescharacteristic
502C23C2×#13512yes—
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×#13711yescharacteristic
505C23#57512yes—
506D4—28no—
507D4—432no—
508D4—44no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—232no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—816no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4#38014yes—
564C2×C4#38614yes—
565C2×C4#38614yes—
566C2×C4C2×#10214yes—
567C2×C4C2×#10214yes—
568C2×C4#41614yes—
569C2×C4#46518yes—
570C2×C4#46618yes—
571C2×C4C22×C4⋊C411yescharacteristic
572C2×C4C2×#11711yescharacteristic
573C2×C4C2×C4×D412yes—
574C2×C4#52012yes—
575C2×C4C2×#12011yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4C2×#12311yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#53611yescharacteristic
581C2×C4D4212yes—
582C2×C4#54612yes—
583C2×C4#54712yes—
584C2×C4#54812yes—
585C2×C4D4×Q812yes—
586C2×C4#55212yes—
587C2×C4#55312yes—
588C2×C4#55412yes—
589C8—48no—
590C22—28no—
591C22—28no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—24no—
599C22—24no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—84no—
604C22—84no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22#180218yescomplemented
612C22C2×#38011yescharacteristic
613C22C2×#38611yescharacteristic
614C22#231111yescharacteristic
615C22#241612yes—
616C22#241712yes—
617C22C22×#10211yescharacteristic
618C22C2×#41611yescharacteristic
619C22C2×#41711yescharacteristic
620C22#289611yescharacteristic
621C22C2×#45911yescharacteristic
622C22C2×#46111yescharacteristic
623C22#303512yes—
624C22C2×#46512yes—
625C22C2×#46612yes—
626C22#305512yes—
627C22#305612yes—
628C22C2×#47411yescharacteristic
629C22C2×#47611yescharacteristic
630C22#307612yes—
631C22#324512yes—
632C22#324612yes—
633C22#324712yes—
634C22#324812yes—
635C4—24no—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—164no—
643C4#189318yescomplemented
644C4D4×C4⋊C412yes—
645C4#235212yes—
646C4#240811yescharacteristic
647C4#240911yescharacteristic
648C4#248611yescharacteristic
649C4#248711yescharacteristic
650C2—28no—
651C2—84no—
652C2#1387211yescharacteristic
653C2C2×#180211yescharacteristic
654C2#2064511yescharacteristic
655C2C2×#189311yescharacteristic
656C2#2097111yescharacteristic
657C2#2188411yescharacteristic
658C2#2188711yescharacteristic
659C2#2475012yes—
660C2#2475112yes—
661C2#2483512yes—
662C2#2483612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊C8C2semidirect
4. C2×#1802C2semidirect
5. #14686C2semidirect
6. #11227C2semidirect
7. #11076C2semidirect
8. #10849C2semidirect
9. #10721C2semidirect
21. C4×#121C4semidirect
93. C2×C4⋊C8D4semidirect
192. #121C4⋊C4semidirect
380. C22×C4#102semidirect
611. C22#1802semidirect
643. C4#1893semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C82
C2×C4×#1212
C2×#18932
C2×#18022
#146862
#112272
#110762
#108492
#107212