Finite Group Databasebeta

← #369263

Subgroups — #369263

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#486C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic
5#14713C211yesmaximal, characteristic
6#11066C214yesmaximal, complemented
7#10931C214yesmaximal, complemented
8#10841C211yesmaximal, characteristic, complemented
9C4×#336C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×C4×Q8—21no—
13C2×#486—24no—
14C2×#486—24no—
15C2×#486C418yescomplemented
16C22×#102C2214yes—
17C22×#102C2214yes—
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C4×C4⋊Q8—24no—
21C4×C4⋊Q8—24no—
22C2×#391—21no—
23C2×#390—21no—
24C2×#390—21no—
25C2×#390—21no—
26C2×#390—21no—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389—21no—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2212yes—
35C2×C43C2211yescharacteristic
36#1990—24no—
37#1957—24no—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C23×Q8—21no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8C2×C414yes—
52C2×C4⋊Q8D414yes—
53C2×C4⋊Q8D414yes—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127C2×C414yes—
60C2×C4×Q8—28no—
61C2×C4×Q8—44no—
62C2×C4×Q8—44no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—22no—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#486—232no—
85C2×#102—28no—
86C2×#102—28no—
87C2×#102—28no—
88C2×#102—28no—
89C2×#102—44no—
90C2×#102C2×C4116yes—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8C2×C414yes—
97#391—28no—
98#390—28no—
99#390—44no—
100#390—44no—
101#390—44no—
102#390—44no—
103#390—44no—
104#389—28no—
105#389—24no—
106#389—24no—
107#384—22no—
108#384—22no—
109#384—22no—
110#384D414yes—
111C4×C4⋊C4—28no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—24no—
114C2×#95—22no—
115C2×#95—22no—
116C43—22no—
117C43—22no—
118#340—28no—
119#340—44no—
120#336—24no—
121#336—24no—
122C25—22no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—24no—
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×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4C22×C414yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
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—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C22×C414yes—
177C22×C8#4114yes—
178C4⋊Q8—28no—
179C4⋊Q8—28no—
180C4⋊Q8—28no—
181C4⋊Q8C22⋊C4116yescomplemented
182#127—28no—
183#127—28no—
184#127—44no—
185#127—44no—
186C4×Q8—432no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
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×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—28no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42D814yes—
252C2×C42SD1614yes—
253C2×C42C22×C412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42#4114yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260#102—232no—
261#102—416no—
262C4×C8—28no—
263C4×C8—44no—
264#95—28no—
265#95—28no—
266C24—24no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24C22×D411yescharacteristic
282C2×Q8—216no—
283C2×Q8—48no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—28no—
292C2×D4—28no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—24no—
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—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
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—42no—
361C22×C4—42no—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12412yes—
369C22×C4#12412yes—
370C22×C4C2×D812yes—
371C22×C4C2×D812yes—
372C22×C4C2×D812yes—
373C22×C4C2×SD1612yes—
374C22×C4C2×SD1612yes—
375C22×C4C2×SD1612yes—
376C22×C4#13512yes—
377C22×C4#13512yes—
378C22×C4AGL(1,ℤ/8)12yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4#13712yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—216no—
410C4⋊C4—28no—
411C4⋊C4—416no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4C2×C22⋊C418yes—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C42—24no—
425C42—24no—
426C42—28no—
427C42—28no—
428C42—416no—
429C42—42no—
430C42—42no—
431C42—44no—
432C42—44no—
433C42C2×C22⋊C414yes—
434C42C4×D414yes—
435C42#12114yes—
436C42#12314yes—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
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—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23C2×C4×D411yescharacteristic
473C23C2×#12011yescharacteristic
474C23C2×#12111yescharacteristic
475C23C2×#12411yescharacteristic
476C23#53211yescharacteristic
477C23#53511yescharacteristic
478C23#53611yescharacteristic
479C23C22×D811yescharacteristic
480C23C22×SD1611yescharacteristic
481C23C2×#13511yescharacteristic
482C23C2×#13511yescharacteristic
483C23C2×AGL(1,ℤ/8)11yescharacteristic
484C23C2×AGL(1,ℤ/8)11yescharacteristic
485C23C2×AGL(1,ℤ/8)11yescharacteristic
486C23C2×#13711yescharacteristic
487Q8—432no—
488D4—48no—
489D4—48no—
490D4—816no—
491D4—816no—
492D4—84no—
493D4—84no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#38614yes—
543C2×C4#38614yes—
544C2×C4C4×D814yes—
545C2×C4C4×SD1614yes—
546C2×C4#44014yes—
547C2×C4#44214yes—
548C2×C4#44714yes—
549C2×C4#44814yes—
550C2×C4#44914yes—
551C2×C4#44914yes—
552C2×C4#45914yes—
553C2×C4#46014yes—
554C2×C4#46314yes—
555C2×C4#46414yes—
556C2×C4#48614yescomplemented
557C2×C4C22×C22⋊C411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C4×#4112yes—
560C2×C4#52012yes—
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4#52912yes—
564C2×C4#54212yes—
565C2×C4#54612yes—
566C2×C4#54812yes—
567C2×C4#54812yes—
568C2×C4#54812yes—
569C2×C4#55012yes—
570C2×C4#55012yes—
571C2×C4#55612yes—
572C2×C4#55912yes—
573C8—44no—
574C8—44no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—162no—
583C22—162no—
584C22—162no—
585C22—162no—
586C22—162no—
587C22—162no—
588C22—162no—
589C22—162no—
590C22C2×#38611yescharacteristic
591C22C4×#12411yescharacteristic
592C22#232611yescharacteristic
593C22#243111yescharacteristic
594C22#243611yescharacteristic
595C22#254511yescharacteristic
596C22#254811yescharacteristic
597C22C2×C4×D811yescharacteristic
598C22C2×C4×SD1611yescharacteristic
599C22C2×#44011yescharacteristic
600C22C2×#44211yescharacteristic
601C22C2×#44711yescharacteristic
602C22C2×#44811yescharacteristic
603C22C2×#44911yescharacteristic
604C22C2×#44911yescharacteristic
605C22C2×#45911yescharacteristic
606C22C2×#46311yescharacteristic
607C22C2×#46011yescharacteristic
608C22C2×#46411yescharacteristic
609C22C2×#48611yescharacteristic
610C22#313111yescharacteristic
611C22#313811yescharacteristic
612C22#314111yescharacteristic
613C22#316711yescharacteristic
614C22#316911yescharacteristic
615C22#318211yescharacteristic
616C22#318411yescharacteristic
617C22#320511yescharacteristic
618C22#320711yescharacteristic
619C22#320811yescharacteristic
620C22#321011yescharacteristic
621C22#322611yescharacteristic
622C22#322711yescharacteristic
623C22#322811yescharacteristic
624C22#322811yescharacteristic
625C4—24no—
626C4—42no—
627C4—48no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—162no—
632C4#187918yes—
633C4#188018yes—
634C4#234112yes—
635C4#236712yes—
636C4#242612yes—
637C4#244812yes—
638C2—164no—
639C2#1402611yescharacteristic
640C2C2×#187911yescharacteristic
641C2C2×#188011yescharacteristic
642C2C4×#48611yescharacteristic
643C2#2164611yescharacteristic
644C2#2221911yescharacteristic
645C2#2222011yescharacteristic
646C2#2559811yescharacteristic
647C2#2559911yescharacteristic
648C2#2590311yescharacteristic
649C2#2590511yescharacteristic
650C2#2865411yescharacteristic
651C2#2865511yescharacteristic
652C2#2874911yescharacteristic
653C2#2875011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×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×C4×C4⋊Q8C2semidirect
3. C2×#1990C2semidirect
6. #11066C2semidirect
7. #10931C2semidirect
8. #10841C2semidirect
9. C4×#336C2semidirect
15. C2×#486C4semidirect
181. C4⋊Q8C22⋊C4semidirect
556. C2×C4#486semidirect

Maximal subgroups

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

subgroupindex
C22×#4862
C2×C4×C4⋊Q82
C2×#19902
C2×#19572
#147132
#110662
#109312
#108412
C4×#3362