Finite Group Databasebeta

← #361685

Subgroups — #361685

Order 512.

3966 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
1C22×#547C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic, complemented
3C2×#1960C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic, complemented
6C2×#1788C214yesmaximal, complemented
7C2×#1762C211yesmaximal, characteristic, complemented
8#13905C211yesmaximal, characteristic, complemented
9#10970C214yesmaximal, complemented
10C23×#41C2212yes—
11C2×#547—24no—
12C2×#547—24no—
13C2×#547—24no—
14C2×#547—24no—
15C2×#547—24no—
16C2×#547C418yescomplemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#123—22no—
19C22×C22⋊Q8C2212yes—
20C22×#121—21no—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2214yes—
26C22×#103C2214yes—
27C22×#102C2212yes—
28C22×#102C2212yes—
29C22×C22⋊C8C2214yes—
30C22×#97C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1981—24no—
35#1960—24no—
36#1955—24no—
37#1804—24no—
38#1788—216no—
39#1762—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×Q8C2312yes—
56C23×D4—22no—
57C23×D4D411yescharacteristic
58C23×D4C2311yescharacteristic
59C24×C4—22no—
60C24×C4—21no—
61C24×C4C2312yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64#547—232no—
65#547—232no—
66#547—232no—
67#547—232no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C414yes—
72C2×C4⋊Q8D414yes—
73C2×C4⋊Q8D414yes—
74C2×#123—216no—
75C2×#123—216no—
76C2×#123—216no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8C2×C418yes—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—28no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4C2×C414yes—
94C2×C4×D4C2×C414yes—
95C2×C4×D4D414yes—
96C2×C4×D4D414yes—
97C2×C4×D4D414yes—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—22no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4D412yes—
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×#103—28no—
118C2×#103—28no—
119C2×#103—28no—
120C2×#103—28no—
121C2×#103—44no—
122C2×#103—44no—
123C2×#102—24no—
124C2×#102—24no—
125C2×#102—24no—
126C2×#102—28no—
127C2×#102D418yescomplemented
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—21no—
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—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8C22×C414yes—
174C22×Q8C2×D414yes—
175C22×Q8C2×D414yes—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4—48no—
189C22×D4C22⋊C414yes—
190C22×D4C22×C412yes—
191C22×D4C2×D412yes—
192C22×D4C2×D412yes—
193C22×D4C2×D414yes—
194C22×D4#4114yes—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—48no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4C22×C414yes—
217C23×C4C2×D412yes—
218C23×C4C2×D414yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—24no—
231C22×C8—22no—
232C22×C8—24no—
233C22×C8C2×D414yes—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8C22⋊C4116yescomplemented
238#123—2128no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—216no—
242C22⋊Q8—216no—
243#121—264no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—28no—
251C4×D4C22⋊C4116yescomplemented
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—264no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4C22⋊C414yes—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—84no—
292C2×C22⋊C4C22⋊C418yes—
293C2×C42—42no—
294C2×C42—82no—
295C2×C42C22×C412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C4⋊C8—44no—
302#103—416no—
303#103—416no—
304#102—216no—
305#102—48no—
306C22⋊C8—416no—
307#97—44no—
308#95—48no—
309#95—42no—
310#95—42no—
311#95—42no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—44no—
326C24—42no—
327C24—42no—
328C24C2×C22⋊C414yes—
329C24C22×D411yescharacteristic
330#41—464no—
331#41—432no—
332#41—464no—
333#41—432no—
334#41—432no—
335#41—432no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—44no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—44no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8C2×C22⋊C418yes—
354C2×D4—232no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—28no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—432no—
373C2×D4—432no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4C2×C22⋊C414yes—
377C2×D4C4×D418yes—
378C2×D4#12118yes—
379C2×D4#12414yes—
380C2×D4#12714yes—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—232no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—432no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—432no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C2×C22⋊C414yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4#12012yes—
438C22×C4#12012yes—
439C22×C4#12012yes—
440C22×C4#12012yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—44no—
462C2×C8—44no—
463C4⋊C4—232no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4C2×C22⋊C414yes—
478C22⋊C4—232no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—816no—
482C42—84no—
483C42C2×C22⋊C414yes—
484C42C2×C22⋊C414yes—
485C42#12014yes—
486C42#12014yes—
487C23—28no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—24no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C22×C22⋊C411yescharacteristic
525C23C2×#12011yescharacteristic
526C23C2×#12011yescharacteristic
527C23D4212yes—
528C23#54612yes—
529C23#57514yes—
530C23#57712yes—
531C23#57812yes—
532Q8—216no—
533Q8—44no—
534Q8—48no—
535Q8—44no—
536Q8—44no—
537Q8—44no—
538D4—24no—
539D4—28no—
540D4—28no—
541D4—24no—
542D4—24no—
543D4—416no—
544D4—464no—
545D4#39018yes—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—432no—
558C2×C4—464no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—84no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4#37914yes—
586C2×C4#37914yes—
587C2×C4C22×C22⋊C411yescharacteristic
588C2×C4C22×C22⋊C411yescharacteristic
589C2×C4C2×C4×D412yes—
590C2×C4#51812yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12112yes—
594C2×C4C2×#12411yescharacteristic
595C2×C4C2×#12711yescharacteristic
596C2×C4#53512yes—
597C2×C4#53511yescharacteristic
598C2×C4#53511yescharacteristic
599C2×C4D4212yes—
600C2×C4D4212yes—
601C2×C4D4212yes—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C8—44no—
606C8—82no—
607C22—28no—
608C22—48no—
609C22—44no—
610C22—44no—
611C22—416no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—48no—
619C22—48no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22C2×#37911yescharacteristic
625C22D4×C22⋊C412yes—
626C22#240812yes—
627C22#243311yescharacteristic
628C22#243511yescharacteristic
629C22#289612yes—
630C22#289812yes—
631C22#300512yes—
632C22#300512yes—
633C22#301011yescharacteristic
634C22#301111yescharacteristic
635C22#301312yes—
636C22#326714yes—
637C22#326914yes—
638C22#330714yes—
639C22#331514yes—
640C4—24no—
641C4—24no—
642C4—44no—
643C4—48no—
644C4—44no—
645C4—84no—
646C4—162no—
647C4C2×#39011yescharacteristic
648C4#230911yescharacteristic
649C4D4×C22⋊C412yes—
650C4#240812yes—
651C4#243311yescharacteristic
652C4#243511yescharacteristic
653C2—48no—
654C2—84no—
655C2#1390211yescharacteristic
656C2#2060011yescharacteristic
657C2#2060311yescharacteristic
658C2#2242014yes—
659C2#2424912yes—
660C2#2425312yes—
661C2#2576412yes—
662C2#2589212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#547C2
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×#1981C2semidirect
3. C2×#1960C2semidirect
4. C2×#1955C2semidirect
5. C2×#1804C2semidirect
6. C2×#1788C2semidirect
7. C2×#1762C2semidirect
8. #13905C2semidirect
9. #10970C2semidirect
16. C2×#547C4semidirect
127. C2×#102D4semidirect
237. C4⋊Q8C22⋊C4semidirect
251. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#19812
C2×#19602
C2×#19552
C2×#18042
C2×#17882
C2×#17622
#139052
#109702