Finite Group Databasebeta

← #387419

Subgroups — #387419

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3#13872C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11199C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10806C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9#10721C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#124—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2547—24no—
18#2547C418yescomplemented
19C2×#397—21no—
20C2×#397—21no—
21C2×#391—21no—
22C2×#386—21no—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1869—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1441—21no—
39#1439—21no—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C23×Q8—21no—
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—41no—
50C23×C8—41no—
51C23×C8—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—41no—
56C2×C4⋊Q8—41no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8—41no—
59C2×C4⋊Q8C2×C414yescomplemented
60C2×#124—28no—
61C2×#124—44no—
62C2×#124—44no—
63C2×C4×D4—41no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4C2×C414yes—
68C2×C4×D4D414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—41no—
73C22×C4⋊C4—41no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94#397—28no—
95#397—28no—
96#391—28no—
97#386—44no—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—42no—
102#380—41no—
103#380—41no—
104#380C2×C418yes—
105#380C2×C418yescomplemented
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C414yescomplemented
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#340—42no—
122#340—42no—
123#336—24no—
124C25C2×D411yescharacteristic
125C25C2×D411yescharacteristic
126C22×Q8—24no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×D4—41no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4C22⋊C414yes—
138C22×D4D814yes—
139C22×D4SD1614yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—82no—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—44no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8#4114yes—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C4⋊Q8—44no—
197#124—432no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—44no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4C22⋊C414yes—
254C2×C22⋊C4C22⋊C414yes—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C22×C412yes—
263C2×C42C2×D412yes—
264C22⋊C8—48no—
265C22⋊C8—44no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—44no—
270#95—42no—
271#95—44no—
272#95—82no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24C2×C22⋊C412yes—
290C24C2×C22⋊C412yes—
291C24C2×D812yes—
292C24C2×D812yes—
293C24C2×SD1612yes—
294C24C2×SD1612yes—
295C24C22×D411yescharacteristic
296C2×Q8—48no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4#102116yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
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—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×D812yes—
400C22×C4C2×SD1612yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+412yes—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—416no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—88no—
450C22⋊C4—84no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—48no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C2×#10214yes—
501C23C2×#10214yes—
502C23C22×C22⋊C411yescharacteristic
503C23#53411yescharacteristic
504C23#53411yescharacteristic
505C23#54612yes—
506C23#54712yes—
507C23C22×D811yescharacteristic
508C23C22×SD1611yescharacteristic
509C23#57511yescharacteristic
510C23#57511yescharacteristic
511C23#57712yes—
512C23#57812yes—
513Q8—816no—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—816no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4C2×#10214yes—
578C2×C4C4×#4112yes—
579C2×C4#51811yescharacteristic
580C2×C4#51811yescharacteristic
581C2×C4#52012yes—
582C2×C4#53411yescharacteristic
583C2×C4#53411yescharacteristic
584C2×C4#53812yes—
585C2×C4#53812yes—
586C2×C4#54212yes—
587C2×C4#54212yes—
588C2×C4#54612yes—
589C2×C4#54712yes—
590C2×C4#55111yescharacteristic
591C2×C4#55111yescharacteristic
592C2×C4#55512yes—
593C2×C4H(3,𝔽4)11yescharacteristic
594C2×C4#56212yes—
595C2×C4#56311yescharacteristic
596C8—44no—
597C8—82no—
598C22—28no—
599C22—28no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22#230811yescharacteristic
621C22#233912yes—
622C22#268411yescharacteristic
623C22#268612yes—
624C22#271711yescharacteristic
625C22C22×#10211yescharacteristic
626C22#289611yescharacteristic
627C22#289812yes—
628C22#318611yescharacteristic
629C22#318611yescharacteristic
630C22#318711yescharacteristic
631C22#318711yescharacteristic
632C22#320011yescharacteristic
633C22#320112yes—
634C22#320211yescharacteristic
635C22#329412yes—
636C22#329812yes—
637C22#331112yes—
638C22#331312yes—
639C22#333412yes—
640C22#333512yes—
641C22#334912yes—
642C22#335012yes—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—84no—
648C4—84no—
649C4—84no—
650C4—162no—
651C4#233712yes—
652C4#237211yescharacteristic
653C4#237911yescharacteristic
654C4#268411yescharacteristic
655C4#268612yes—
656C4#271711yescharacteristic
657C2—44no—
658C2—44no—
659C2#1447311yescharacteristic
660C2#2175811yescharacteristic
661C2#2177111yescharacteristic
662C2#2248512yes—
663C2#2249412yes—
664C2#3126011yescharacteristic
665C2#3126211yescharacteristic
666C2#3131812yes—
667C2#3132012yes—
668C2#3225811yescharacteristic
669C2#3226011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
4. #11384C2semidirect
5. #11199C2semidirect
6. #11199C2semidirect
7. #10806C2semidirect
8. #10730C2semidirect
9. #10721C2semidirect
18. #2547C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
105. #380C2×C4semidirect
107. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#18692
#138722
#113842
#111992
#108062
#107302
#107212