Finite Group Databasebeta

← #364908

Subgroups — #364908

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2536C211yesmaximal, characteristic
3C2×#1892C212yesmaximal, complemented
4C2×#1805C211yesmaximal, characteristic, complemented
5C2×C4×#103C211yesmaximal, characteristic, complemented
6#13905C211yesmaximal, characteristic
7#11199C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal, complemented
12C24×C8C2212yes—
13C22×C22⋊Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×C8×D4—24no—
17C2×C8×D4—24no—
18C2×C8×D4—24no—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2536—24no—
26#2536C418yescomplemented
27C2×#398—21no—
28C2×#394—21no—
29C2×#388—22no—
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1892—28no—
37#1805—24no—
38C4×#103—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1445—21no—
47#1438—21no—
48#1437—22no—
49C23×Q8C2311yescharacteristic
50C23×D4D411yescharacteristic
51C24×C4C2312yes—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—42no—
58C23×C8—42no—
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—42no—
66C2×C22⋊Q8—42no—
67C2×C22⋊Q8C2×C418yes—
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⋊C4—22no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D412yes—
93C22×C42—21no—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96C8×D4—232no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8D414yes—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103D414yescomplemented
109C2×C22⋊C8—24no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8D414yes—
115#398—28no—
116#394—28no—
117#388—216no—
118#382—22no—
119#382—22no—
120#382—22no—
121#382C2×C414yes—
122#380—24no—
123#380—24no—
124#380—24no—
125#380—42no—
126#380—42no—
127#380C2×C418yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—21no—
138#340—24no—
139#340—24no—
140#340—22no—
141#340—22no—
142#340—24no—
143#340—42no—
144#340—42no—
145C25C2×D412yes—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8C22×C412yes—
158C22×Q8C2×D412yes—
159C22×Q8#4112yes—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4C22⋊C414yes—
165C23×C4—21no—
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×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4C22×C414yes—
184C23×C4C2×D412yes—
185C23×C4C2×D414yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—216no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—48no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8—48no—
210C22×C8—48no—
211C22×C8—48no—
212C22×C8C2×D414yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8#4112yes—
217C22⋊Q8—216no—
218C22⋊Q8—48no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C4×Q8—28no—
222C4×Q8—28no—
223C4×Q8—44no—
224C4×Q8—44no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4C22⋊C414yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4#4112yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4C22⋊C418yes—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—82no—
288C2×C42C22×C412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42#4112yes—
293C2×C42#4112yes—
294C4⋊C8—28no—
295#103—28no—
296#103—28no—
297#103—48no—
298C22⋊C8—216no—
299C4×C8—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—42no—
307#95—42no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24C2×C22⋊C414yes—
321C24#13514yes—
322C24#13514yes—
323C24C22×D411yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8C4×D414yes—
352C2×D4—216no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—216no—
356C2×D4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C2×C22⋊C414yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C412yes—
434C22×C4#12112yes—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C42−1+411yescharacteristic
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—432no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—41no—
469C2×C8—42no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8#12114yes—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—416no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C22⋊C4—216no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C42—48no—
503C42—84no—
504C42C4×D414yes—
505C42C4×D414yes—
506C42#12014yes—
507C42#12314yes—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23#41618yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#12111yescharacteristic
545C23C2×#12111yescharacteristic
546C23D4211yescharacteristic
547C23#54611yescharacteristic
548C23#54611yescharacteristic
549C23#54711yescharacteristic
550C23C2×#13512yes—
551C23C2×#13512yes—
552C23#57712yes—
553C23#57812yes—
554Q8—24no—
555Q8—24no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562D4—232no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4#38614yes—
623C2×C4#38614yes—
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C4×#4111yescharacteristic
628C2×C4#51811yescharacteristic
629C2×C4#52011yescharacteristic
630C2×C4C2×#12011yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×#12311yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4#52911yescharacteristic
635C2×C4#53211yescharacteristic
636C2×C4#53511yescharacteristic
637C2×C4#53711yescharacteristic
638C2×C4#54011yescharacteristic
639C2×C4#54211yescharacteristic
640C2×C4D4211yescharacteristic
641C2×C4D4211yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54711yescharacteristic
646C2×C4#54711yescharacteristic
647C2×C4#54811yescharacteristic
648C2×C4#55011yescharacteristic
649C2×C4#55311yescharacteristic
650C2×C4#55311yescharacteristic
651C2×C4#55511yescharacteristic
652C8—24no—
653C8—42no—
654C8—44no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22C2×#38611yescharacteristic
671C22D4×C22⋊C411yescharacteristic
672C22#233911yescharacteristic
673C22#242711yescharacteristic
674C22#242811yescharacteristic
675C22#253511yescharacteristic
676C22#253711yescharacteristic
677C22C2×#41612yes—
678C22#289812yes—
679C22#305012yes—
680C22#305011yescharacteristic
681C22#305011yescharacteristic
682C22#305711yescharacteristic
683C22#305911yescharacteristic
684C22#306312yes—
685C22#328212yes—
686C22#328512yes—
687C22#329612yes—
688C22#330012yes—
689C22#330012yes—
690C22#330112yes—
691C22#333612yes—
692C22#333712yes—
693C4—24no—
694C4—24no—
695C4—44no—
696C4—44no—
697C4—82no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4—162no—
702C4C4×#12111yescharacteristic
703C4#232411yescharacteristic
704C4#236111yescharacteristic
705C4#236311yescharacteristic
706C4#240711yescharacteristic
707C4#240811yescharacteristic
708C4#242011yescharacteristic
709C4#242211yescharacteristic
710C2—48no—
711C2#1397011yescharacteristic
712C2#2096711yescharacteristic
713C2#2097411yescharacteristic
714C2#2241612yes—
715C2#2248612yes—
716C2#2534312yes—
717C2#2538912yes—
718C2#2814411yescharacteristic
719C2#2814511yescharacteristic
720C2#2822411yescharacteristic
721C2#2822611yescharacteristic
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×#1892C2semidirect
4. C2×#1805C2semidirect
5. C2×C4×#103C2semidirect
7. #11199C2semidirect
8. #10970C2semidirect
9. #10850C2semidirect
10. #10842C2semidirect
11. #10721C2semidirect
26. #2536C4semidirect
108. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#25362
C2×#18922
C2×#18052
C2×C4×#1032
#139052
#111992
#109702
#108502
#108422
#107212