Finite Group Databasebeta

← #393115

Subgroups — #393115

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C211yesmaximal, characteristic
2C2×#1921C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#14707C211yesmaximal, characteristic
5C42×C4⋊C4C211yesmaximal, characteristic
6#10908C211yesmaximal, characteristic
7#10907C211yesmaximal, characteristic
8#10885C211yesmaximal, characteristic
9#10884C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11#10850C211yesmaximal, characteristic
12#10850C211yesmaximal, characteristic
13#10832C211yesmaximal, characteristic
14#10806C211yesmaximal, characteristic
15#10806C211yesmaximal, characteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C43—21no—
31C2×C43C2211yescharacteristic
32#1949C418yescomplemented
33#1921—24no—
34#1919—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1436—21no—
54#1436—21no—
55#1436—21no—
56#1436—21no—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4×#95—21no—
60C4×#95—21no—
61C4×#95—21no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#107—22no—
90C2×#107—22no—
91C2×#107C2×C414yes—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106C2×C414yes—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8C2×C414yes—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#382—22no—
106#382—22no—
107#382—22no—
108#382C2×C414yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C414yes—
118C4×C4⋊C4D414yes—
119C4×C4⋊C4Q814yes—
120C4×C4⋊C4Q814yes—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C43—22no—
138C43—22no—
139C43—44no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340C2×C414yes—
153#340C2×C414yes—
154#336—22no—
155#336—22no—
156#336—22no—
157#336—22no—
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×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
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—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C22×C412yes—
200C22×C8C22×C412yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×Q812yes—
248C2×C4⋊C4C2×Q812yes—
249C2×C4⋊C4C2×Q812yes—
250C2×C4⋊C4C2×Q812yes—
251C2×C4⋊C4#4112yes—
252C2×C4⋊C4#4112yes—
253C2×C4⋊C4#4112yes—
254C2×C4⋊C4#4112yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C412yes—
294C2×C42C2×D412yes—
295C2×C42C2×Q812yes—
296C2×C42C2×Q812yes—
297C2×C42#4112yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300C2×C42#4112yes—
301C2×C42#4112yes—
302C2×C42#4112yes—
303#107—28no—
304#106—28no—
305C4⋊C8—28no—
306C4⋊C8—44no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323C24C22×D411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4#12112yes—
411C22×C4#12112yes—
412C22×C4C22⋊Q812yes—
413C22×C4C22⋊Q812yes—
414C22×C4C22⋊Q812yes—
415C22×C4C22⋊Q812yes—
416C22×C4#12312yes—
417C22×C4#12312yes—
418C22×C4#13512yes—
419C22×C4#13512yes—
420C22×C4AGL(1,ℤ/8)12yes—
421C22×C4#13712yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4C4×D414yes—
495C4⋊C4C4×Q814yes—
496C4⋊C4#12514yes—
497C4⋊C4C4⋊Q814yes—
498C42—28no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—44no—
508C42—44no—
509C42—48no—
510C42—44no—
511C42—48no—
512C42—48no—
513C42—48no—
514C42—48no—
515C42—48no—
516C42#11714yes—
517C42C4×Q814yes—
518C42#12514yes—
519C42#12614yes—
520C23C2×C4×D411yescharacteristic
521C23C2×#12111yescharacteristic
522C23C2×C22⋊Q811yescharacteristic
523C23C2×C22⋊Q811yescharacteristic
524C23C2×#12311yescharacteristic
525C23#52511yescharacteristic
526C23#52511yescharacteristic
527C23C2×#13511yescharacteristic
528C23C2×#13511yescharacteristic
529C23C2×AGL(1,ℤ/8)11yescharacteristic
530C23C2×#13711yescharacteristic
531C23#57511yescharacteristic
532C23#57511yescharacteristic
533C23#57511yescharacteristic
534C23#57511yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4#38714yes—
630C2×C4#38714yes—
631C2×C4#46414yes—
632C2×C4#47814yes—
633C2×C4C2×#11711yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×Q811yescharacteristic
636C2×C4C2×C4×Q811yescharacteristic
637C2×C4C4×#4111yescharacteristic
638C2×C4C4×#4111yescharacteristic
639C2×C4C2×#12511yescharacteristic
640C2×C4C2×#12511yescharacteristic
641C2×C4C2×#12611yescharacteristic
642C2×C4#52911yescharacteristic
643C2×C4#52911yescharacteristic
644C2×C4#52911yescharacteristic
645C2×C4#52911yescharacteristic
646C2×C4#52911yescharacteristic
647C2×C4#52911yescharacteristic
648C2×C4#52911yescharacteristic
649C2×C4#52911yescharacteristic
650C2×C4C2×C4⋊Q811yescharacteristic
651C2×C4#53211yescharacteristic
652C2×C4#53211yescharacteristic
653C2×C4#53211yescharacteristic
654C2×C4#53211yescharacteristic
655C2×C4#53311yescharacteristic
656C2×C4#53311yescharacteristic
657C2×C4#53311yescharacteristic
658C2×C4#53311yescharacteristic
659C2×C4#53311yescharacteristic
660C2×C4#53311yescharacteristic
661C8—42no—
662C8—42no—
663C8—81no—
664C8—81no—
665C22C2×#38711yescharacteristic
666C22C4×#12111yescharacteristic
667C22C4×C22⋊Q811yescharacteristic
668C22#285111yescharacteristic
669C22#285411yescharacteristic
670C22#285511yescharacteristic
671C22#285711yescharacteristic
672C22C4×#13511yescharacteristic
673C22#294311yescharacteristic
674C22C4×AGL(1,ℤ/8)11yescharacteristic
675C22C4×#13711yescharacteristic
676C22C2×#46411yescharacteristic
677C22#303511yescharacteristic
678C22#303611yescharacteristic
679C22#304011yescharacteristic
680C22C2×#47811yescharacteristic
681C22#307611yescharacteristic
682C22#307611yescharacteristic
683C22#307611yescharacteristic
684C22#307711yescharacteristic
685C22#307811yescharacteristic
686C22#307811yescharacteristic
687C22#308011yescharacteristic
688C22#309111yescharacteristic
689C22#309111yescharacteristic
690C22#309211yescharacteristic
691C22#309311yescharacteristic
692C22#310211yescharacteristic
693C22#310211yescharacteristic
694C22#310511yescharacteristic
695C22#310811yescharacteristic
696C22#311111yescharacteristic
697C22#311111yescharacteristic
698C22#311411yescharacteristic
699C22#311711yescharacteristic
700C4—24no—
701C4—24no—
702C4—24no—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—82no—
711C4—82no—
712C4—161no—
713C4—161no—
714C4#229311yescharacteristic
715C4#229511yescharacteristic
716C4C4×#12511yescharacteristic
717C4C4×C4⋊Q811yescharacteristic
718C4#285911yescharacteristic
719C4#286011yescharacteristic
720C4#286311yescharacteristic
721C4#286411yescharacteristic
722C2#1469911yescharacteristic
723C2#2103211yescharacteristic
724C2#2103311yescharacteristic
725C2C4×#46411yescharacteristic
726C2#2150711yescharacteristic
727C2C4×#47811yescharacteristic
728C2#2161811yescharacteristic
729C2#3307011yescharacteristic
730C2#3307111yescharacteristic
731C2#3311011yescharacteristic
732C2#3311211yescharacteristic
733C2#3315111yescharacteristic
734C2#3315211yescharacteristic
735C2#3319911yescharacteristic
736C2#3320211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #1949C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#19212
C2×#19192
#147072
C42×C4⋊C42
#109082
#109072
#108852
#108842
#108502
#108322
#108062