Finite Group Databasebeta

← #416512

Subgroups — #416512

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2858C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic
5C2×#1957C211yesmaximal, characteristic
6#14712C211yesmaximal, characteristic, complemented
7#11189C214yesmaximal, complemented
8#11066C214yesmaximal, complemented
9C4×#336C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×#124—21no—
13C22×#124—21no—
14C22×#102C2214yes—
15C22×#102C2214yes—
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2858—24no—
19#2858—24no—
20C2×#390—21no—
21C2×#390—21no—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1990—24no—
31#1990—24no—
32#1957—24no—
33#1957—24no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×D4—21no—
43C23×D4—21no—
44C23×D4C2312yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D414yes—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127D414yes—
59C2×#124—28no—
60C2×#124—28no—
61C2×#124—44no—
62C2×#124—44no—
63C2×#124—44no—
64C2×#124—44no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#102—28no—
86C2×#102—28no—
87C2×#102—28no—
88C2×#102—28no—
89C2×#102—44no—
90C2×#102—44no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97#390—28no—
98#390—28no—
99#390—24no—
100#390—24no—
101#390—24no—
102#390—24no—
103#389—44no—
104#389—44no—
105#389—44no—
106#384—22no—
107#384—22no—
108#384—22no—
109#384D414yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C43—22no—
114C43—22no—
115#340—28no—
116#340—28no—
117#336—24no—
118#336—24no—
119C25—22no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4C2×D414yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8#4114yes—
174C22×C8#4114yes—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—44no—
178C4⋊Q8—44no—
179#127—28no—
180#127—28no—
181#127—44no—
182#127—44no—
183#124—432no—
184#124—432no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42D814yes—
248C2×C42SD1614yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#102—416no—
256#102—416no—
257C4×C8—44no—
258C4×C8—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262C24—24no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24C22×D411yescharacteristic
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4#12412yes—
365C22×C4#12412yes—
366C22×C4C2×D812yes—
367C22×C4C2×D812yes—
368C22×C4C2×D812yes—
369C22×C4C2×SD1612yes—
370C22×C4C2×SD1612yes—
371C22×C4C2×SD1612yes—
372C22×C4#13512yes—
373C22×C4#13512yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C42−1+412yes—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—88no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—416no—
417C22⋊C4—416no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C42—28no—
421C42—28no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—44no—
428C42#12114yes—
429C42#12114yes—
430C42#12314yes—
431C42#12314yes—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23C2×#12011yescharacteristic
468C23C2×#12111yescharacteristic
469C23C2×#12411yescharacteristic
470C23#53211yescharacteristic
471C23#53211yescharacteristic
472C23#53511yescharacteristic
473C23#53611yescharacteristic
474C23C22×D811yescharacteristic
475C23C22×SD1611yescharacteristic
476C23C2×#13511yescharacteristic
477C23C2×#13511yescharacteristic
478C23C2×AGL(1,ℤ/8)11yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×AGL(1,ℤ/8)11yescharacteristic
481C23C2×#13711yescharacteristic
482Q8—816no—
483D4—48no—
484D4—48no—
485D4—84no—
486D4—84no—
487D4—816no—
488D4—816no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#39414yes—
539C2×C4#39414yes—
540C2×C4#44714yes—
541C2×C4#44914yes—
542C2×C4#45014yes—
543C2×C4#45214yes—
544C2×C4#45914yes—
545C2×C4#46014yes—
546C2×C4#46314yes—
547C2×C4#46414yes—
548C2×C4#48614yes—
549C2×C4#48814yes—
550C2×C4#48914yes—
551C2×C4#49014yes—
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4#52912yes—
557C2×C4#52912yes—
558C2×C4#54212yes—
559C2×C4#54212yes—
560C2×C4#54612yes—
561C2×C4#54612yes—
562C2×C4#54712yes—
563C2×C4#54812yes—
564C2×C4#54812yes—
565C2×C4#55012yes—
566C2×C4#55612yes—
567C2×C4#55912yes—
568C8—44no—
569C8—44no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—162no—
578C22—162no—
579C22—162no—
580C22—162no—
581C22—162no—
582C22—162no—
583C22—162no—
584C22—162no—
585C22C2×#39411yescharacteristic
586C22#243111yescharacteristic
587C22#243211yescharacteristic
588C22#254411yescharacteristic
589C22#254511yescharacteristic
590C22#264011yescharacteristic
591C22#286111yescharacteristic
592C22C2×#44711yescharacteristic
593C22C2×#45011yescharacteristic
594C22C2×#44911yescharacteristic
595C22C2×#45211yescharacteristic
596C22C2×#45911yescharacteristic
597C22C2×#46311yescharacteristic
598C22C2×#46011yescharacteristic
599C22C2×#46411yescharacteristic
600C22C2×#48611yescharacteristic
601C22C2×#48811yescharacteristic
602C22C2×#48911yescharacteristic
603C22C2×#49011yescharacteristic
604C22#316711yescharacteristic
605C22#316711yescharacteristic
606C22#316911yescharacteristic
607C22#316911yescharacteristic
608C22#318211yescharacteristic
609C22#318211yescharacteristic
610C22#318411yescharacteristic
611C22#318411yescharacteristic
612C22#320511yescharacteristic
613C22#320611yescharacteristic
614C22#320711yescharacteristic
615C22#320711yescharacteristic
616C22#322611yescharacteristic
617C22#322711yescharacteristic
618C22#322811yescharacteristic
619C22#322811yescharacteristic
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—84no—
625C4—84no—
626C4—162no—
627C4#204118yes—
628C4#204218yes—
629C4#242612yes—
630C4#242712yes—
631C4#244712yes—
632C4#244812yes—
633C2—164no—
634C2#1522611yescharacteristic
635C2C2×#204111yescharacteristic
636C2C2×#204211yescharacteristic
637C2#2559711yescharacteristic
638C2#2559911yescharacteristic
639C2#2567111yescharacteristic
640C2#2567311yescharacteristic
641C2#2861411yescharacteristic
642C2#2861511yescharacteristic
643C2#2865411yescharacteristic
644C2#2865511yescharacteristic
645C2#3020311yescharacteristic
646C2#3020411yescharacteristic
647C2#3327711yescharacteristic
648C2#3328611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1990C2semidirect
6. #14712C2semidirect
7. #11189C2semidirect
8. #11066C2semidirect
9. C4×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
C2×#19902
C2×#19572
#147122
#111892
#110662
C4×#3362