Finite Group Databasebeta

← #362007

Subgroups — #362007

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×#1977C212yesmaximal
3C2×#1848C211yesmaximal, characteristic, complemented
4#20647C211yesmaximal, characteristic, complemented
5#20646C211yesmaximal, characteristic
6C2×#1801C211yesmaximal, characteristic, complemented
7C2×#1800C211yesmaximal, characteristic
8#20437C211yesmaximal, characteristic, complemented
9#11128C212yesmaximal, complemented
10#11127C212yesmaximal
11#10869C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×SD16—24no—
16C2×C4×SD16—21no—
17C2×C4×SD16—21no—
18C2×C4×SD16—21no—
19C2×C4×SD16—21no—
20C2×C4×SD16C418yescomplemented
21C22×#108C2212yes—
22C22×#108C2212yes—
23C22×#106C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#419C2212yes—
28C2×#418C2212yes—
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#101C2212yes—
32C22×#100C2212yes—
33C2×#405C2212yes—
34C22×C4×C8C2211yescharacteristic
35#2285C2211yescharacteristic
36#1977—28no—
37#1848—24no—
38#1805—22no—
39#1804—22no—
40#1803—22no—
41#1802—22no—
42#1801—24no—
43#1800—24no—
44#1770—21no—
45#1770—21no—
46C8×C4⋊C4—21no—
47C8×C4⋊C4—21no—
48C2×#343C2212yes—
49C2×#343C2212yes—
50#1736C2212yes—
51#1736C2212yes—
52#1729C2212yes—
53C23×Q8C2311yescharacteristic
54C23×D4C2311yescharacteristic
55C24×C4—21no—
56C22×SD16—21no—
57C22×SD16—21no—
58C22×SD16—22no—
59C22×SD16—22no—
60C22×SD16—22no—
61C22×SD16C2×C414yes—
62C22×SD16D414yes—
63C22×SD16Q814yes—
64C22×M16C2312yes—
65C22×M16C2312yes—
66C22×M16C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×C4×Q8—22no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C414yes—
77C2×C4×D4—22no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4C2×C414yes—
85C2×#117C2312yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92C4×SD16—28no—
93C4×SD16—28no—
94C4×SD16—28no—
95C4×SD16—28no—
96C2×#108—24no—
97C2×#108—24no—
98C2×#108—24no—
99C2×#108—24no—
100C2×#108—42no—
101C2×#108—42no—
102C2×#106—21no—
103C2×#106—21no—
104C2×#106—22no—
105C2×#106C2×C414yes—
106#423—22no—
107#423—22no—
108#423—22no—
109#423—22no—
110#423—22no—
111#423D414yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118#419—24no—
119#419—42no—
120#418—24no—
121#418—42no—
122C2×#103—21no—
123C2×#103—21no—
124C2×#103—22no—
125C2×#103—24no—
126C2×#103C2×C414yes—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—22no—
130C2×#102—24no—
131C2×#102C2×C414yes—
132C2×#101—24no—
133C2×#101—24no—
134C2×#101—42no—
135C2×#100—24no—
136C2×#100—24no—
137C2×#100—42no—
138#405—24no—
139#405—24no—
140C2×#97—22no—
141C2×C4×C8—22no—
142C2×C4×C8C2×C414yes—
143C2×C4×C8D414yes—
144C2×C4×C8D411yescharacteristic
145C2×C4×C8D411yescharacteristic
146C2×C4×C8D411yescharacteristic
147C2×C4×C8D411yescharacteristic
148#384D412yes—
149#382D412yes—
150#380—21no—
151#380—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154#343—24no—
155#343—24no—
156#340—24no—
157#340—24no—
158#336—22no—
159#336—22no—
160C25—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8C22×C412yes—
173C22×Q8C2×D412yes—
174C22×Q8C2×Q812yes—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4C22×C412yes—
187C22×D4C2×D412yes—
188C22×D4C2×Q812yes—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—24no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—42no—
196C23×C4—84no—
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4C2411yescharacteristic
200C2×SD16—24no—
201C2×SD16—24no—
202C2×SD16—28no—
203C2×SD16—24no—
204C2×SD16—24no—
205C2×SD16—28no—
206C2×SD16—24no—
207C2×SD16—24no—
208C2×SD16—28no—
209C2×SD16C4⋊C4116yescomplemented
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16C2×D414yes—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D412yes—
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×Q812yes—
245C22×C8#4112yes—
246C22×C8#4112yes—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C22×C412yes—
285C2×C4⋊C4C2×D414yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—42no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—41no—
294C2×C42—41no—
295C2×C42—42no—
296C2×C42C22×C412yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D412yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42#4112yes—
304#108—48no—
305#108—48no—
306#106—24no—
307#106—24no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310C4⋊C8—42no—
311C4⋊C8—42no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314#103—24no—
315#103—24no—
316#103—48no—
317#102—24no—
318#102—24no—
319#102—48no—
320#101—48no—
321#100—48no—
322#97—44no—
323C4×C8—44no—
324C4×C8C22⋊C414yes—
325C4×C8C22⋊C414yes—
326C4×C8C22⋊C414yes—
327C4×C8C22⋊C414yes—
328#95—22no—
329#95—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—41no—
334C24—41no—
335C24—41no—
336C24—41no—
337C24—82no—
338C24—82no—
339C24C2×#4111yescharacteristic
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8C2×C4⋊C414yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4C2×C4⋊C414yes—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—84no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4C4×D412yes—
421C22×C4C4×D412yes—
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12412yes—
427C22×C4#12412yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×Q811yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C42−1+411yescharacteristic
441C22×C42−1+411yescharacteristic
442SD16—216no—
443SD16—216no—
444M16—24no—
445M16—24no—
446M16—42no—
447M16—42no—
448M16—42no—
449M16—42no—
450M16—42no—
451M16—42no—
452M16—42no—
453M16—42no—
454M16—42no—
455M16—42no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—24no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8C2×C22⋊C412yes—
474C2×C8C2×C22⋊C412yes—
475C2×C8C2×C22⋊C412yes—
476C2×C8C2×C22⋊C412yes—
477C2×C8C2×C4⋊C414yes—
478C2×C8#11714yes—
479C2×C8#12112yes—
480C2×C8#12112yes—
481C2×C8#12114yes—
482C2×C8C22⋊Q812yes—
483C2×C8C22⋊Q812yes—
484C2×C8C22⋊Q814yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—44no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—44no—
507C42C2×C22⋊C412yes—
508C42C2×C22⋊C412yes—
509C42C4×D412yes—
510C42C4×D412yes—
511C42#12112yes—
512C42#12112yes—
513C42#12712yes—
514C42#12712yes—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—82no—
525C23—82no—
526C23—84no—
527C23—84no—
528C23—82no—
529C23—82no—
530C23—84no—
531C23—84no—
532C23—82no—
533C23—82no—
534C23C2×C4×D411yescharacteristic
535C23C2×#12111yescharacteristic
536C23C2×#12411yescharacteristic
537C23#54711yescharacteristic
538C23#55411yescharacteristic
539C23#55511yescharacteristic
540C23#55611yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549D4—24no—
550D4—24no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—88no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—88no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4#39012yes—
585C2×C4#39012yes—
586C2×C4#39012yes—
587C2×C4#39012yes—
588C2×C4#47114yes—
589C2×C4C22×C22⋊C411yescharacteristic
590C2×C4C22×C4⋊C411yescharacteristic
591C2×C4C2×#11711yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4#52011yescharacteristic
594C2×C4#52011yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4C2×#12711yescharacteristic
601C2×C4#53511yescharacteristic
602C2×C4#53611yescharacteristic
603C2×C4#53711yescharacteristic
604C2×C4#53711yescharacteristic
605C2×C4D4211yescharacteristic
606C2×C4D4211yescharacteristic
607C2×C4#54711yescharacteristic
608C2×C4#54711yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4D4×Q811yescharacteristic
611C2×C4D4×Q811yescharacteristic
612C2×C4#55011yescharacteristic
613C2×C4#55011yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55511yescharacteristic
616C2×C4#55611yescharacteristic
617C8—22no—
618C8—22no—
619C8—44no—
620C8—82no—
621C8—82no—
622C8#38014yes—
623C8#38014yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22C2×#39011yescharacteristic
642C22D4×C4⋊C411yescharacteristic
643C22#235411yescharacteristic
644C22#253711yescharacteristic
645C22#253811yescharacteristic
646C22#254711yescharacteristic
647C22#256311yescharacteristic
648C22C2×#47112yes—
649C22#306712yes—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4#197714yescomplemented
658C4C2×#38011yescharacteristic
659C4#231111yescharacteristic
660C4D4×C4⋊C411yescharacteristic
661C4#235411yescharacteristic
662C4#253711yescharacteristic
663C4#253811yescharacteristic
664C4#254111yescharacteristic
665C4#254211yescharacteristic
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—84no—
671C2#1390611yescharacteristic
672C2C2×#197711yescharacteristic
673C2#2124811yescharacteristic
674C2#2823912yes—
675C2#2829612yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1848C2semidirect
4. #20647C2semidirect
6. C2×#1801C2semidirect
8. #20437C2semidirect
9. #11128C2semidirect
11. #10869C2semidirect
20. C2×C4×SD16C4semidirect
209. C2×SD16C4⋊C4semidirect
657. C4#1977semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#19772
C2×#18482
#206472
#206462
C2×#18012
C2×#18002
#204372
#111282
#111272
#108692