Finite Group Databasebeta

← #482637

Subgroups — #482637

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#11901C212yesmaximal, complemented
5#11878C212yesmaximal, complemented
6#11827C214yesmaximal, complemented
7#11755C214yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#99—21no—
11C2×#407—21no—
12#2848—21no—
13#2848—21no—
14#2633C412yescomplemented
15#2633C2212yes—
16#2547C411yescharacteristic, complemented
17#2547C2211yescharacteristic
18#2542C411yescharacteristic, complemented
19#2542C2211yescharacteristic
20#2536—21no—
21#2536—21no—
22#2502—21no—
23#2482C412yescomplemented
24#2482C2212yes—
25#2435—21no—
26#2326—21no—
27#2326—21no—
28#2283C411yescharacteristic, complemented
29C2×#380—21no—
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yes—
34C2×#342C2212yes—
35#1465—21no—
36#1465C2212yes—
37#1463—21no—
38#1462C2212yes—
39#1459—21no—
40#1459C2212yes—
41#1352C2218yescomplemented
42#1344C2218yescomplemented
43C23×D4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—41no—
47#533—41no—
48#533—41no—
49#533—41no—
50#533—41no—
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×C4⋊Q8C2×C411yescharacteristic, complemented
53C2×#125C2×C412yescomplemented
54C2×#124—21no—
55C2×#124—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C412yes—
77C22×C4⋊C4—41no—
78C22×C4⋊C4D411yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2311yescharacteristic
82C22×C42C2×C411yescharacteristic
83C2×#99—22no—
84C2×#99—22no—
85C2×#99—44no—
86#407—22no—
87#407—22no—
88#407—41no—
89#407—41no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—41no—
92#399C2×C412yes—
93#399C2×C412yescomplemented
94#399C2×C412yescomplemented
95#399C2312yes—
96#398—22no—
97#398—22no—
98#397C2×C412yes—
99#397C2×C412yescomplemented
100#397C2×C412yescomplemented
101#397C2312yes—
102#394—22no—
103#394—22no—
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#384—21no—
119#384—21no—
120#383C2×C412yescomplemented
121#382—21no—
122#381C2×C412yes—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—22no—
129#380—22no—
130#380—41no—
131#380—41no—
132#379—22no—
133#379—22no—
134C4×C4⋊C4C2×C411yescharacteristic, complemented
135C4×C4⋊C4C2×C411yescharacteristic, complemented
136C4×C22⋊C4—21no—
137#376—21no—
138C2×#95—41no—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C25C22⋊C411yescharacteristic
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—42no—
154C22×D4C22⋊C411yescharacteristic
155C22×D4C22⋊C411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—42no—
167C23×C4—82no—
168C23×C4—82no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—84no—
185C22×C8—42no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190#125—42no—
191#125—42no—
192#124—44no—
193#124—42no—
194#124—42no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—82no—
202C22⋊Q8—82no—
203C22⋊Q8—82no—
204C22⋊Q8—82no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225#117—42no—
226#117—42no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22×C411yescharacteristic
269C2×C22⋊C4C22×C411yescharacteristic
270C2×C22⋊C4C22×C411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2411yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285#99—44no—
286#99—44no—
287C22⋊C8—24no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—42no—
299#95—42no—
300#95C22×C412yes—
301#95C22×C412yes—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—81no—
314C24—81no—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C22×D411yescharacteristic
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4AGL(1,ℤ/8)12yes—
344C2×D4#13712yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4#9911yescharacteristic
400C22×C4#9911yescharacteristic
401C22×C4#9911yescharacteristic
402C22×C4#9911yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413M16—84no—
414M16—84no—
415C2×C8—44no—
416C2×C8—84no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—84no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—44no—
469C42—42no—
470C42—42no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—82no—
507C23—82no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23C2×#9911yescharacteristic
512C23C2×#9911yescharacteristic
513C23#41712yes—
514C23#42112yes—
515C23C22×C22⋊C411yescharacteristic
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23C2×#13711yescharacteristic
518Q8—82no—
519Q8—84no—
520Q8—82no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—42no—
526D4—42no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—168no—
590C2×C4C2×#9911yescharacteristic
591C2×C4C2×#9911yescharacteristic
592C2×C4C2×#9911yescharacteristic
593C2×C4C2×#9911yescharacteristic
594C2×C4#41712yes—
595C8—84no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—84no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22—84no—
633C22#212312yes—
634C22C22×#9911yescharacteristic
635C22C2×#41711yescharacteristic
636C22C2×#42111yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—164no—
653C4#212712yes—
654C2—21no—
655C2—21no—
656C2—22no—
657C2—41no—
658C2—41no—
659C2—41no—
660C2—42no—
661C2—42no—
662C2—81no—
663C2#1759211yescharacteristic
664C2C2×#212311yescharacteristic
665C2C2×#212711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11901C2semidirect
5. #11878C2semidirect
6. #11827C2semidirect
7. #11755C2semidirect
8. C2×#533C4semidirect
14. #2633C4semidirect
16. #2547C4semidirect
18. #2542C4semidirect
23. #2482C4semidirect
28. #2283C4semidirect
41. #1352C22semidirect
42. #1344C22semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
53. C2×#125C2×C4semidirect
93. #399C2×C4semidirect
94. #399C2×C4semidirect
99. #397C2×C4semidirect
100. #397C2×C4semidirect
120. #383C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×#17992
#173622
#119012
#118782
#118272
#117552