Finite Group Databasebeta

← #360436

Subgroups — #360436

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1926C212yesmaximal
2C2×#1844C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6C4⋊C42C211yesmaximal, characteristic, complemented
7#11257C212yesmaximal, complemented
8#11121C212yesmaximal
9#10908C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#390C2212yes—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#380—21no—
24C2×#380—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1926—28no—
30#1844—24no—
31#1802—24no—
32C4×#102C418yescomplemented
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1439—22no—
41#1436—22no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102C2×C414yes—
89C2×#102C2×C414yes—
90C2×#102D414yes—
91C2×#102Q814yes—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8C2×C414yes—
95#390—24no—
96#390—42no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384D414yes—
101#380—44no—
102#380—44no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—42no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4C2×C414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—42no—
120#340C2×C414yes—
121#340C2×C414yes—
122#336—24no—
123#336—24no—
124C25—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C22×D4C2×D412yes—
138C22×D4C2×Q812yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C22×C412yes—
172C22×C8C22×C412yes—
173C22×C8C2×D412yes—
174C22×C8C2×Q812yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
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—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4SD1614yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—82no—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256#106—48no—
257#106—48no—
258C4⋊C8—44no—
259#102—28no—
260#102—48no—
261#102C4⋊C4116yescomplemented
262C4×C8—28no—
263#95—24no—
264#95—24no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—82no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24C22×D411yescharacteristic
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4C2×C4⋊C414yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12312yes—
378C22×C4#12312yes—
379C22×C4C2×SD1614yes—
380C22×C4C2×SD1612yes—
381C22×C4#13512yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8C2×C4⋊C414yes—
418C2×C8#11714yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—28no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4C2×C4⋊C414yes—
445C22⋊C4—48no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—48no—
453C42—88no—
454C42#11714yes—
455C42C4×D414yes—
456C42#12414yes—
457C42#12614yes—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
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—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23C2×C4×D411yescharacteristic
494C23C2×#12111yescharacteristic
495C23C2×#12311yescharacteristic
496C23D4211yescharacteristic
497C23#54711yescharacteristic
498C23D4×Q811yescharacteristic
499C23#55011yescharacteristic
500C23C22×SD1611yescharacteristic
501C23C2×#13511yescharacteristic
502C23C2×AGL(1,ℤ/8)11yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23#57711yescharacteristic
505C23#57711yescharacteristic
506C23#57711yescharacteristic
507C23#57711yescharacteristic
508D4—24no—
509D4—24no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—816no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4#38814yes—
573C2×C4#38814yes—
574C2×C4C4×SD1614yes—
575C2×C4#44214yes—
576C2×C4#46014yes—
577C2×C4#46314yes—
578C2×C4#48114yes—
579C2×C4#48214yes—
580C2×C4C22×C4⋊C411yescharacteristic
581C2×C4C2×#11711yescharacteristic
582C2×C4C2×#11711yescharacteristic
583C2×C4C2×C4×D411yescharacteristic
584C2×C4#52011yescharacteristic
585C2×C4#52011yescharacteristic
586C2×C4C2×#12411yescharacteristic
587C2×C4C2×#12611yescharacteristic
588C2×C4#53211yescharacteristic
589C2×C4#53211yescharacteristic
590C2×C4#53311yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#54011yescharacteristic
593C2×C4#54111yescharacteristic
594C2×C4#54211yescharacteristic
595C2×C4#54211yescharacteristic
596C2×C4D4211yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4#54811yescharacteristic
601C2×C4D4×Q811yescharacteristic
602C2×C4#55211yescharacteristic
603C2×C4#55311yescharacteristic
604C2×C4#55411yescharacteristic
605C2×C4#55411yescharacteristic
606C2×C4#55511yescharacteristic
607C2×C4#55611yescharacteristic
608C8—24no—
609C8—24no—
610C8—82no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×#38811yescharacteristic
627C22D4×C4⋊C411yescharacteristic
628C22#235411yescharacteristic
629C22#242211yescharacteristic
630C22#242311yescharacteristic
631C22#242811yescharacteristic
632C22#242911yescharacteristic
633C22C2×C4×SD1611yescharacteristic
634C22C2×#44211yescharacteristic
635C22#295211yescharacteristic
636C22#295211yescharacteristic
637C22C2×#46311yescharacteristic
638C22C2×#46011yescharacteristic
639C22#304211yescharacteristic
640C22#304311yescharacteristic
641C22C2×#48211yescharacteristic
642C22C2×#48111yescharacteristic
643C22#309911yescharacteristic
644C22#309911yescharacteristic
645C22#327211yescharacteristic
646C22#327211yescharacteristic
647C22D4×SD1611yescharacteristic
648C22#328411yescharacteristic
649C22#333511yescharacteristic
650C22#334011yescharacteristic
651C22#335411yescharacteristic
652C22#335411yescharacteristic
653C22Q8×SD1611yescharacteristic
654C22#338111yescharacteristic
655C22#338411yescharacteristic
656C22#338411yescharacteristic
657C22#339211yescharacteristic
658C22#339411yescharacteristic
659C22#339711yescharacteristic
660C22#339711yescharacteristic
661C4—24no—
662C4—44no—
663C4—44no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—162no—
669C4#192618yescomplemented
670C4#228511yescharacteristic
671C4#231411yescharacteristic
672C4D4×C4⋊C411yescharacteristic
673C4#235211yescharacteristic
674C4#249111yescharacteristic
675C4#249211yescharacteristic
676C4#249611yescharacteristic
677C4#249711yescharacteristic
678C2—84no—
679C2#1387611yescharacteristic
680C2C2×#192611yescharacteristic
681C2#2106811yescharacteristic
682C2#2267011yescharacteristic
683C2C4⋊C4×SD1611yescharacteristic
684C2#2280511yescharacteristic
685C2#2281311yescharacteristic
686C2#2501211yescharacteristic
687C2#2501311yescharacteristic
688C2#2505811yescharacteristic
689C2#2505911yescharacteristic
690C2#2536611yescharacteristic
691C2#2536711yescharacteristic
692C2#2545111yescharacteristic
693C2#2545211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1844C2semidirect
6. C4⋊C42C2semidirect
7. #11257C2semidirect
9. #10908C2semidirect
10. #10849C2semidirect
11. #10834C2semidirect
32. C4×#102C4semidirect
261. #102C4⋊C4semidirect
669. C4#1926semidirect

Maximal subgroups

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

subgroupindex
C2×#19262
C2×#18442
C2×#18022
C2×C4×#1022
#139062
C4⋊C422
#112572
#111212
#109082
#108492
#108342