Finite Group Databasebeta

← #475738

Subgroups — #475738

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2036C212yesmaximal, complemented
2C2×#2035C212yesmaximal, complemented
3C2×#2001C212yesmaximal, complemented
4C2×#1971C211yesmaximal, characteristic, complemented
5C2×#1955C211yesmaximal, characteristic
6#15177C212yesmaximal
7#11095C212yesmaximal, complemented
8#11077C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10C23×Q16C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×#97C2211yescharacteristic
24C2×#391C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27#2036—28no—
28#2035—28no—
29#2001—28no—
30#1971—24no—
31#1955—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35#1444—22no—
36#1439—22no—
37#1439—22no—
38#1438—22no—
39C4⋊#95C2212yes—
40C23×Q8C2312yes—
41C23×D4D411yescharacteristic
42C24×C4C2312yes—
43C22×Q16—22no—
44C22×Q16—22no—
45C22×Q16—41no—
46C22×Q16—41no—
47C22×Q16—41no—
48C22×Q16—41no—
49C22×Q16D414yescomplemented
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8—42no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4—41no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42C2311yescharacteristic
83C2×#107—41no—
84C2×#107—41no—
85C2×#107D414yescomplemented
86C2×#106—22no—
87C2×#106—41no—
88C2×#106—41no—
89C2×#106—41no—
90C2×#106—41no—
91C2×#106D414yescomplemented
92C2×C4⋊C8—22no—
93C2×C4⋊C8—41no—
94C2×C4⋊C8—41no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—24no—
110C2×#97—41no—
111C2×#97—41no—
112C2×#97D414yes—
113#391—24no—
114#391—24no—
115#380—24no—
116#380—24no—
117#380—42no—
118#380—42no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128C25C2×D412yes—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×D414yes—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—41no—
146C22×D4—41no—
147C22×D4—42no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×Q16—28no—
171C2×Q16—44no—
172C2×Q16—44no—
173C2×Q16—44no—
174C2×Q16—44no—
175C2×Q16—44no—
176C2×Q16—44no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C22×C8#4112yes—
198C22⋊Q8—48no—
199C22⋊Q8—48no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
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—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C2×D412yes—
257C2×C42#4112yes—
258#107—44no—
259#106—44no—
260#106—44no—
261C4⋊C8—44no—
262#103—48no—
263#103—48no—
264#103—48no—
265C22⋊C8—48no—
266C22⋊C8—48no—
267#97—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—82no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24#13714yes—
289C24C22×D411yescharacteristic
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×D4—48no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—48no—
322C2×D4—44no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
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—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
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—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
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#12112yes—
395C22×C4#12112yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C42−1+411yescharacteristic
410Q16—416no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8#12114yes—
431C2×C8#12714yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C2×#12111yescharacteristic
492C23#53411yescharacteristic
493C23#53411yescharacteristic
494C23D4211yescharacteristic
495C23D4211yescharacteristic
496C23D4211yescharacteristic
497C23#54711yescharacteristic
498C23C2×#13712yes—
499C23#57512yes—
500C23#57712yes—
501C23#57812yes—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509Q8—82no—
510D4—416no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12711yescharacteristic
566C2×C4#53411yescharacteristic
567C2×C4#53411yescharacteristic
568C2×C4#53411yescharacteristic
569C2×C4#53411yescharacteristic
570C2×C4#53411yescharacteristic
571C2×C4#53511yescharacteristic
572C2×C4#53511yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#53711yescharacteristic
575C2×C4#53711yescharacteristic
576C2×C4#53811yescharacteristic
577C2×C4#53811yescharacteristic
578C2×C4#53811yescharacteristic
579C2×C4#53811yescharacteristic
580C2×C4D4211yescharacteristic
581C2×C4D4211yescharacteristic
582C2×C4#54711yescharacteristic
583C2×C4#55011yescharacteristic
584C2×C4#55011yescharacteristic
585C2×C4#55011yescharacteristic
586C2×C4#55011yescharacteristic
587C2×C4#55511yescharacteristic
588C2×C4H(3,𝔽4)11yescharacteristic
589C2×C4H(3,𝔽4)11yescharacteristic
590C2×C4#56211yescharacteristic
591C2×C4#56211yescharacteristic
592C8—24no—
593C8—24no—
594C8—82no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22#253711yescharacteristic
611C22#254111yescharacteristic
612C22#261711yescharacteristic
613C22#267911yescharacteristic
614C22#267911yescharacteristic
615C22#268211yescharacteristic
616C22#268211yescharacteristic
617C22#305612yes—
618C22#306312yes—
619C22#319511yescharacteristic
620C22#319612yes—
621C22#319711yescharacteristic
622C22#320112yes—
623C22#320211yescharacteristic
624C22#320411yescharacteristic
625C22#326612yes—
626C22#326712yes—
627C22#326912yes—
628C22#327012yes—
629C22#327012yes—
630C22#327112yes—
631C22#334412yes—
632C22#334912yes—
633C4—42no—
634C4—42no—
635C4—84no—
636C4—84no—
637C4—82no—
638C4—84no—
639C4—162no—
640C4#254111yescharacteristic
641C4#261711yescharacteristic
642C4#265611yescharacteristic
643C4#268211yescharacteristic
644C4#268211yescharacteristic
645C4#273311yescharacteristic
646C4#273311yescharacteristic
647C4#281911yescharacteristic
648C2—84no—
649C2#1706811yescharacteristic
650C2#2823812yes—
651C2#2849212yes—
652C2#2974311yescharacteristic
653C2#2974611yescharacteristic
654C2#3106712yes—
655C2#3106811yescharacteristic
656C2#3106911yescharacteristic
657C2#3120112yes—
658C2#3120311yescharacteristic
659C2#3120511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×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
1. C2×#2036C2semidirect
2. C2×#2035C2semidirect
3. C2×#2001C2semidirect
4. C2×#1971C2semidirect
7. #11095C2semidirect
8. #11077C2semidirect
9. #10863C2semidirect
49. C22×Q16D4semidirect
85. C2×#107D4semidirect
91. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20362
C2×#20352
C2×#20012
C2×#19712
C2×#19552
#151772
#110952
#110772
#108632