Finite Group Databasebeta

← #418666

Subgroups — #418666

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5C2×#1862C212yesmaximal
6#11380C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11066C212yesmaximal
10#10931C211yesmaximal, characteristic
11#10884C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C22×#107C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20#2546—28no—
21#2538—24no—
22#2538—24no—
23C2×#398—21no—
24C2×#398—21no—
25C2×#396—21no—
26C2×#396—21no—
27C2×#391—21no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#384C2211yescharacteristic
35C2×#380—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#2057—24no—
38#2023—24no—
39#1862—28no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2211yescharacteristic
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—28no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—44no—
61C2×C4⋊Q8—44no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—44no—
64C2×C22⋊Q8—44no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—42no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—42no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—42no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—42no—
105C2×#102Q818yescomplemented
106#398—28no—
107#398—28no—
108#396—28no—
109#396—28no—
110#391—28no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390D414yes—
115#389—22no—
116#389—22no—
117#389—24no—
118#389—24no—
119#389—22no—
120#389—24no—
121#389D414yes—
122#389D418yes—
123#384—22no—
124#384—22no—
125#380—28no—
126C4×C4⋊C4—24no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—22no—
138#336—24no—
139#336—22no—
140#336—42no—
141C25—21no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8D814yes—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4C2×D412yes—
163C22×D4C2×Q814yes—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C2×Q814yes—
197C4⋊Q8—48no—
198C4⋊Q8—48no—
199C4⋊Q8—432no—
200C22⋊Q8—432no—
201C4×Q8—48no—
202C4×Q8—48no—
203C4×D4—48no—
204C4×D4—48no—
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—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
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—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C4⋊C4#4114yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×D414yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286#107—48no—
287#102—216no—
288#102—48no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—28no—
294C24—22no—
295C24—24no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24C22×D411yescharacteristic
306C2×Q8—28no—
307C2×Q8—28no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4C4⋊Q814yescomplemented
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4C2×D812yes—
406C22×C4C2×D812yes—
407C22×C4C2×D812yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×Q812yes—
414C22×C4C2×#4112yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+412yes—
417C22×C42−1+412yes—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—84no—
457C4⋊C4C22⋊Q818yescomplemented
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—48no—
468C42—42no—
469C42—42no—
470C23—24no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—81no—
488C23—81no—
489C23—82no—
490C23—81no—
491C23—81no—
492C23—82no—
493C23C2×#12011yescharacteristic
494C23C2×#12011yescharacteristic
495C23#53411yescharacteristic
496C23#54812yes—
497C23D4×Q812yes—
498C23C22×D811yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×AGL(1,ℤ/8)11yescharacteristic
501C23#57511yescharacteristic
502C23#57712yes—
503C23#57812yes—
504Q8—416no—
505Q8—816no—
506D4—28no—
507D4—44no—
508D4—44no—
509D4—44no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
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—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—816no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4#44714yes—
577C2×C4#44714yes—
578C2×C4C2×C22⋊Q812yes—
579C2×C4C2×C4⋊Q811yescharacteristic
580C2×C4#53211yescharacteristic
581C2×C4#53611yescharacteristic
582C2×C4#53612yes—
583C2×C4#53611yescharacteristic
584C2×C4#54212yes—
585C2×C4#54412yes—
586C2×C4D4212yes—
587C2×C4D4212yes—
588C2×C4#54712yes—
589C2×C4#54812yes—
590C2×C4D4×Q812yes—
591C2×C4D4×Q812yes—
592C2×C4#55412yes—
593C2×C4#56312yes—
594C8—44no—
595C8—82no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—84no—
600C22—84no—
601C22—82no—
602C22—84no—
603C22—82no—
604C22—84no—
605C22—82no—
606C22—82no—
607C22#240311yescharacteristic
608C22#240912yes—
609C22#243211yescharacteristic
610C22#243411yescharacteristic
611C22#272912yes—
612C22C2×#44711yescharacteristic
613C22C2×#44711yescharacteristic
614C22#300611yescharacteristic
615C22#300611yescharacteristic
616C22#300911yescharacteristic
617C22#301111yescharacteristic
618C22#301312yes—
619C22#318611yescharacteristic
620C22#319211yescharacteristic
621C22#320112yes—
622C22#335812yes—
623C22#336712yes—
624C22#336812yes—
625C22#337712yes—
626C22Q8×D812yes—
627C22#338412yes—
628C22#338612yes—
629C22#338812yes—
630C4—24no—
631C4—24no—
632C4—48no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4—81no—
641C4#244012yes—
642C4#245212yes—
643C4#267412yes—
644C4#267712yes—
645C2—84no—
646C2#1526511yescharacteristic
647C2#2391811yescharacteristic
648C2#2392711yescharacteristic
649C2#2432612yes—
650C2#2434612yes—
651C2#2564511yescharacteristic
652C2#2566111yescharacteristic
653C2#2583811yescharacteristic
654C2#2585111yescharacteristic
655C2#3256412yes—
656C2#3258912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2546C2semidirect
3. C2×#2057C2semidirect
6. #11380C2semidirect
7. #11328C2semidirect
8. #11207C2semidirect
11. #10884C2semidirect
105. C2×#102Q8semidirect
335. C2×D4C4⋊Q8semidirect
457. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25382
C2×#20572
C2×#20232
C2×#18622
#113802
#113282
#112072
#110662
#109312
#108842