Finite Group Databasebeta

← #369441

Subgroups — #369441

Order 512.

1804 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5#13905C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10722C212yesmaximal, complemented
10#10721C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#435—24no—
17C2×#435—24no—
18C2×#435—24no—
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2563—24no—
24#2563C418yescomplemented
25C2×#393—21no—
26C2×#393—21no—
27C2×#390—21no—
28C2×#382—21no—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1993—24no—
37#1988—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1445—21no—
47#1438—21no—
48#1437—21no—
49#1437—21no—
50C23×Q8—21no—
51C23×D4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—24no—
58C22×M16—42no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—24no—
63C23×C8—42no—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C414yes—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2×C414yes—
79C2×C4×D4D414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#435—232no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8D414yes—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97D414yes—
108#393—28no—
109#393—28no—
110#390—28no—
111#382—28no—
112#382—22no—
113#382—22no—
114#380—24no—
115#380—24no—
116#380—24no—
117#380—24no—
118#380—42no—
119#380—42no—
120#380C2×C418yes—
121#380C2×C418yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×C414yes—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—42no—
141#340—42no—
142C25C2×D411yescharacteristic
143C25C2×D411yescharacteristic
144C22×Q8—24no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—82no—
148C22×Q8—82no—
149C22×D4—24no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4C22⋊C414yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C2×M16—28no—
184C2×M16—48no—
185C2×M16—48no—
186C2×M16—44no—
187C2×M16—44no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—28no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—48no—
205C22×C8—48no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8#4114yes—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C22⋊C414yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4#4114yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4—82no—
267C2×C22⋊C4C22⋊C414yes—
268C2×C22⋊C4C22⋊C414yes—
269C2×C42—24no—
270C2×C42—44no—
271C2×C42—42no—
272C2×C42—82no—
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C4⋊C8—28no—
280C22⋊C8—28no—
281C22⋊C8—28no—
282#97—28no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—42no—
289#95—42no—
290#95—42no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24C2×C22⋊C412yes—
307C24C2×C22⋊C412yes—
308C24#13714yes—
309C24C22×D411yescharacteristic
310C2×Q8—48no—
311C2×Q8—88no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8—88no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—44no—
318C2×D4—48no—
319C2×D4—48no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
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—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414M16—416no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—416no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8#12414yes—
436C4⋊C4—28no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C42—48no—
461C42—84no—
462C42C4×D414yes—
463C42C4×D414yes—
464C42#12014yes—
465C42#12314yes—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23#41914yes—
508C23C22×C22⋊C411yescharacteristic
509C23C2×#12111yescharacteristic
510C23C2×#12111yescharacteristic
511C23#54612yes—
512C23#54712yes—
513C23C2×#13712yes—
514C23#57512yes—
515C23#57612yes—
516C23#57712yes—
517Q8—816no—
518D4—416no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
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—44no—
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×C4#38614yes—
578C2×C4#38614yes—
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C4×#4112yes—
582C2×C4#52012yes—
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C2×C4C2×#12411yescharacteristic
586C2×C4#52912yes—
587C2×C4#53211yescharacteristic
588C2×C4#53511yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#54212yes—
591C2×C4#54612yes—
592C2×C4#54612yes—
593C2×C4#54712yes—
594C2×C4#54812yes—
595C2×C4#55612yes—
596C2×C4#55912yes—
597C8—24no—
598C8—42no—
599C8—42no—
600C8—42no—
601C22—28no—
602C22—24no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#38611yescharacteristic
620C22#233912yes—
621C22#242712yes—
622C22#254411yescharacteristic
623C22#254711yescharacteristic
624C22C2×#41911yescharacteristic
625C22#289611yescharacteristic
626C22#289712yes—
627C22#305612yes—
628C22#305612yes—
629C22#305711yescharacteristic
630C22#305811yescharacteristic
631C22#306212yes—
632C22#331412yes—
633C22#331612yes—
634C22#331912yes—
635C22#332112yes—
636C22#334412yes—
637C22#334612yes—
638C22#334712yes—
639C22#334812yes—
640C4—24no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—84no—
645C4—84no—
646C4—84no—
647C4—162no—
648C4C4×#12411yescharacteristic
649C4#232611yescharacteristic
650C4#236712yes—
651C4#243211yescharacteristic
652C4#243511yescharacteristic
653C4#244712yes—
654C2—44no—
655C2—82no—
656C2#1402711yescharacteristic
657C2#2097211yescharacteristic
658C2#2097511yescharacteristic
659C2#2249212yes—
660C2#2249712yes—
661C2#2535012yes—
662C2#2535612yes—
663C2#2861212yes—
664C2#2871512yes—
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
3. C2×#1993C2semidirect
4. C2×#1988C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10850C2semidirect
9. #10722C2semidirect
10. #10721C2semidirect
11. #10703C2semidirect
24. #2563C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#25632
C2×#19932
C2×#19882
#139052
#112002
#111992
#108502
#107222
#107212
#107032