Finite Group Databasebeta

← #361689

Subgroups — #361689

Order 512.

4138 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic, complemented
3C2×#1961C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic, complemented
5C2×#1805C211yesmaximal, characteristic, complemented
6C2×#1788C214yesmaximal, complemented
7C2×#1763C211yesmaximal, characteristic, complemented
8#13905C211yesmaximal, characteristic, complemented
9#10969C214yesmaximal, complemented
10C23×#41C2212yes—
11C2×#550—24no—
12C2×#550—24no—
13C2×#550—24no—
14C2×#550—24no—
15C2×#550—24no—
16C2×#550C418yescomplemented
17C22×#127—21no—
18C22×#127C2211yescharacteristic
19C22×#121—22no—
20C22×#121C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#102C2214yes—
28C22×#102C2214yes—
29C22×C22⋊C8C2214yes—
30C22×#97C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1982—24no—
35#1961—24no—
36#1955—24no—
37#1805—24no—
38#1788—216no—
39#1763—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1445—21no—
43#1438—21no—
44#1437—22no—
45C22×#41—24no—
46C22×#41—24no—
47C22×#41—24no—
48C22×#41—24no—
49C22×#41—24no—
50C22×#41—28no—
51C22×#41—28no—
52C22×#41—44no—
53C22×#41—44no—
54C22×#41C2×C418yes—
55C23×Q8C2311yescharacteristic
56C23×D4—22no—
57C23×D4—22no—
58C23×D4—21no—
59C23×D4—21no—
60C23×D4D411yescharacteristic
61C23×D4C2312yes—
62C24×C4—22no—
63C24×C4C2312yes—
64C23×C8C2312yes—
65C23×C8C2312yes—
66#550—232no—
67#550—232no—
68#550—232no—
69#550—232no—
70C2×#127—28no—
71C2×#127—28no—
72C2×#127—28no—
73C2×#127—22no—
74C2×#127—22no—
75C2×#127—22no—
76C2×#127C2×C414yes—
77C2×#127D414yes—
78C2×#127D414yes—
79C2×#121—216no—
80C2×#121—216no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—216no—
87C2×#121C2×C418yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8C2×C414yes—
92C2×C4×Q8D414yes—
93C2×C4×Q8D414yes—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4C2×C414yes—
102C2×C4×D4D414yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—22no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4D412yes—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—28no—
121C2×#103D418yescomplemented
122C2×#102—28no—
123C2×#102—28no—
124C2×#102—28no—
125C2×#102—28no—
126C2×#102—44no—
127C2×#102—44no—
128C2×C22⋊C8—28no—
129C2×C22⋊C8—28no—
130C2×C22⋊C8—44no—
131C2×#97—22no—
132C2×#97—22no—
133C2×#97—22no—
134#382—22no—
135#382—22no—
136#380—28no—
137#380—44no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C2×#95—21no—
141C2×#95—24no—
142C2×#95—21no—
143C2×#95—21no—
144#340—24no—
145#340—24no—
146C25—22no—
147C25C2×D412yes—
148C2×#41—216no—
149C2×#41—216no—
150C2×#41—216no—
151C2×#41—216no—
152C2×#41—232no—
153C2×#41—416no—
154C2×#41—416no—
155C2×#41—416no—
156C2×#41—416no—
157C2×#41—416no—
158C2×#41—416no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—48no—
162C2×#41—48no—
163C2×#41—416no—
164C2×#41—48no—
165C2×#41—48no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8C22×C412yes—
172C22×Q8C2×D412yes—
173C22×Q8C2×D412yes—
174C22×Q8C2×D414yes—
175C22×Q8#4114yes—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—44no—
198C22×D4—48no—
199C22×D4—48no—
200C22×D4—48no—
201C22×D4—82no—
202C22×D4—82no—
203C22×D4—84no—
204C22×D4—84no—
205C22×D4C22⋊C414yes—
206C22×D4C22×C414yes—
207C22×D4C2×D414yes—
208C22×D4C2×D414yes—
209C23×C4—21no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—21no—
213C23×C4—28no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—48no—
225C23×C4—48no—
226C23×C4C22×C414yes—
227C23×C4C2×D412yes—
228C23×C4C2×D414yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2411yescharacteristic
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—24no—
241C22×C8—22no—
242C22×C8—24no—
243C22×C8C2×D414yes—
244#127—264no—
245#127—28no—
246#127—28no—
247#127—28no—
248#127C22⋊C4116yescomplemented
249#121—216no—
250#121—216no—
251#121—2128no—
252#121—216no—
253#121—216no—
254C4×Q8—28no—
255C4×Q8—28no—
256C4×Q8—28no—
257C4×Q8C22⋊C4116yescomplemented
258C4×D4—264no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4C22⋊C414yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—84no—
293C2×C22⋊C4C22⋊C418yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—82no—
299C2×C42C22×C412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C4⋊C8—44no—
306#103—216no—
307#103—48no—
308#102—416no—
309#102—416no—
310C22⋊C8—416no—
311#97—44no—
312#95—48no—
313#95—42no—
314#95—42no—
315#95—42no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—44no—
328C24—44no—
329C24—48no—
330C24—44no—
331C24—44no—
332C24C2×C22⋊C414yes—
333C24C22×D411yescharacteristic
334#41—464no—
335#41—432no—
336#41—464no—
337#41—432no—
338#41—432no—
339#41—432no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8C2×C22⋊C414yes—
354C2×Q8C4×D418yes—
355C2×Q8#12118yes—
356C2×Q8#12414yes—
357C2×Q8#12714yes—
358C2×D4—232no—
359C2×D4—232no—
360C2×D4—24no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—48no—
373C2×D4—432no—
374C2×D4—416no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—432no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—88no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—88no—
394C2×D4—816no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—816no—
398C2×D4C2×C22⋊C418yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—232no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—432no—
425C22×C4—432no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4C2×C22⋊C414yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4#12012yes—
453C22×C4#12012yes—
454C22×C4#12012yes—
455C22×C4#12012yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—44no—
477C2×C8—44no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4C2×C22⋊C414yes—
491C22⋊C4—232no—
492C22⋊C4—216no—
493C22⋊C4—816no—
494C42—216no—
495C42—84no—
496C42C2×C22⋊C414yes—
497C42C2×C22⋊C414yes—
498C42#12014yes—
499C42#12014yes—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—88no—
534C23—88no—
535C23—88no—
536C23—88no—
537C23C22×C22⋊C411yescharacteristic
538C23C2×#12011yescharacteristic
539C23C2×#12011yescharacteristic
540C23D4212yes—
541C23#54612yes—
542C23#57514yes—
543C23#57612yes—
544C23#57712yes—
545Q8—24no—
546Q8—28no—
547Q8—28no—
548Q8—24no—
549Q8—24no—
550Q8#39018yes—
551D4—216no—
552D4—416no—
553D4—432no—
554D4—464no—
555D4—44no—
556D4—48no—
557D4—44no—
558D4—44no—
559D4—44no—
560D4—816no—
561D4—832no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—464no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4#37914yes—
602C2×C4#37914yes—
603C2×C4C22×C22⋊C411yescharacteristic
604C2×C4C22×C22⋊C411yescharacteristic
605C2×C4C2×C4×D412yes—
606C2×C4#51812yes—
607C2×C4C2×#12011yescharacteristic
608C2×C4C2×#12011yescharacteristic
609C2×C4C2×#12112yes—
610C2×C4C2×#12411yescharacteristic
611C2×C4C2×#12711yescharacteristic
612C2×C4#53512yes—
613C2×C4#53511yescharacteristic
614C2×C4#53511yescharacteristic
615C2×C4D4212yes—
616C2×C4D4212yes—
617C2×C4D4212yes—
618C2×C4#54612yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C8—44no—
622C8—82no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—84no—
632C22—88no—
633C22—816no—
634C22—84no—
635C22—88no—
636C22—84no—
637C22—88no—
638C22—84no—
639C22—88no—
640C22C2×#37911yescharacteristic
641C22D4×C22⋊C412yes—
642C22#240812yes—
643C22#243311yescharacteristic
644C22#243511yescharacteristic
645C22#289612yes—
646C22#289712yes—
647C22#300512yes—
648C22#300512yes—
649C22#300811yescharacteristic
650C22#300911yescharacteristic
651C22#301212yes—
652C22#326814yes—
653C22#327114yes—
654C22#330914yes—
655C22#331714yes—
656C4—28no—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—44no—
661C4—84no—
662C4—162no—
663C4C2×#39011yescharacteristic
664C4#230911yescharacteristic
665C4D4×C22⋊C412yes—
666C4#240812yes—
667C4#243311yescharacteristic
668C4#243511yescharacteristic
669C2—84no—
670C2—88no—
671C2#1390211yescharacteristic
672C2#2060011yescharacteristic
673C2#2060211yescharacteristic
674C2#2242214yes—
675C2#2425012yes—
676C2#2425412yes—
677C2#2576512yes—
678C2#2589312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1982C2semidirect
3. C2×#1961C2semidirect
4. C2×#1955C2semidirect
5. C2×#1805C2semidirect
6. C2×#1788C2semidirect
7. C2×#1763C2semidirect
8. #13905C2semidirect
9. #10969C2semidirect
16. C2×#550C4semidirect
121. C2×#103D4semidirect
248. #127C22⋊C4semidirect
257. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5502
C2×#19822
C2×#19612
C2×#19552
C2×#18052
C2×#17882
C2×#17632
#139052
#109692