Finite Group Databasebeta

← #401365

Subgroups — #401365

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4C2×C4×#106C211yesmaximal, characteristic
5#14053C211yesmaximal, characteristic
6#14006C211yesmaximal, characteristic
7#11328C211yesmaximal, characteristic
8#11325C211yesmaximal, characteristic
9#11220C211yesmaximal, characteristic
10#11220C211yesmaximal, characteristic
11#11219C211yesmaximal, characteristic
12#10908C211yesmaximal, characteristic
13#10907C211yesmaximal, characteristic
14#10862C211yesmaximal, characteristic
15#10797C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#391—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2085—24no—
33#2058—24no—
34#2057—24no—
35C4×#106—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51#1446—21no—
52#1442—21no—
53#1441—21no—
54#1441—21no—
55#1436—21no—
56#1436—21no—
57C4⋊#95—21no—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C23×Q8—21no—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—41no—
73C2×C4⋊Q8—41no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—41no—
100C2×#107—41no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106—22no—
106C2×#106—22no—
107C2×#106—41no—
108C2×#106D414yes—
109C2×#106Q814yescomplemented
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8Q814yescomplemented
113#391—44no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—41no—
120#389—41no—
121#389—41no—
122#382—22no—
123#382—41no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—41no—
127C4×C4⋊C4—41no—
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—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95D411yescharacteristic
145C2×#95D411yescharacteristic
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—41no—
157#340D414yes—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162#336Q814yescomplemented
163C22×Q8—24no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—81no—
169C22×Q8—81no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
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×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×Q812yes—
208C22×C8C2×Q812yes—
209C22×C8C2×Q812yes—
210C22×C8C2×Q812yes—
211C22×C8C2×Q812yes—
212C22×C8#4112yes—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
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—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—81no—
266C2×C4⋊C4—81no—
267C2×C4⋊C4—81no—
268C2×C4⋊C4—81no—
269C2×C4⋊C4—81no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×Q812yes—
275C2×C4⋊C4C2×Q812yes—
276C2×C4⋊C4#4112yes—
277C2×C4⋊C4#4112yes—
278C2×C4⋊C4#4112yes—
279C2×C4⋊C4#4112yes—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—81no—
301C2×C42—81no—
302C2×C42C2×Q812yes—
303C2×C42C2×Q812yes—
304#107—44no—
305#107—44no—
306#107—44no—
307#106—28no—
308#106—28no—
309#106—44no—
310C4×C8—28no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—81no—
329#95—81no—
330#95—81no—
331C24C22×D411yescharacteristic
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—84no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
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×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4C4⋊Q812yes—
432C22×C4C4⋊Q812yes—
433C22×C4#13512yes—
434C22×C4#13512yes—
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4#13712yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q811yescharacteristic
440C22×C4C22×Q811yescharacteristic
441C22×C4C22×Q811yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C42−1+411yescharacteristic
450C22×C42−1+411yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8C22⋊Q814yes—
478C2×C8C4⋊Q814yes—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4#12114yes—
515C4⋊C4C22⋊Q814yes—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—84no—
529C42—84no—
530C23#52511yescharacteristic
531C23#52511yescharacteristic
532C23C2×C4⋊Q811yescharacteristic
533C23D4211yescharacteristic
534C23D4211yescharacteristic
535C23#54711yescharacteristic
536C23D4×Q811yescharacteristic
537C23C2×#13511yescharacteristic
538C23C2×#13511yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23C2×#13711yescharacteristic
541C23#57611yescharacteristic
542C23#57711yescharacteristic
543C23#57711yescharacteristic
544C23#57811yescharacteristic
545Q8—816no—
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—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
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—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4#50114yes—
644C2×C4C2×#12111yescharacteristic
645C2×C4C2×C22⋊Q811yescharacteristic
646C2×C4C2×C22⋊Q811yescharacteristic
647C2×C4C2×C4⋊Q811yescharacteristic
648C2×C4#53211yescharacteristic
649C2×C4#53311yescharacteristic
650C2×C4#53311yescharacteristic
651C2×C4#53311yescharacteristic
652C2×C4#53311yescharacteristic
653C2×C4#53311yescharacteristic
654C2×C4#53811yescharacteristic
655C2×C4#53911yescharacteristic
656C2×C4#54111yescharacteristic
657C2×C4#54211yescharacteristic
658C2×C4#54411yescharacteristic
659C2×C4#54411yescharacteristic
660C2×C4#54611yescharacteristic
661C2×C4#54611yescharacteristic
662C2×C4#54711yescharacteristic
663C2×C4#54811yescharacteristic
664C2×C4D4×Q811yescharacteristic
665C2×C4D4×Q811yescharacteristic
666C2×C4#55111yescharacteristic
667C2×C4#55211yescharacteristic
668C2×C4#55411yescharacteristic
669C2×C4#55411yescharacteristic
670C2×C4#55711yescharacteristic
671C2×C4Q8211yescharacteristic
672C8—24no—
673C8—24no—
674C8—42no—
675C8—42no—
676C22#239911yescharacteristic
677C22#242411yescharacteristic
678C22#244011yescharacteristic
679C22#244911yescharacteristic
680C22#246111yescharacteristic
681C22#246211yescharacteristic
682C22#255111yescharacteristic
683C22#310511yescharacteristic
684C22#310611yescharacteristic
685C22#310711yescharacteristic
686C22#310811yescharacteristic
687C22#311811yescharacteristic
688C22#311811yescharacteristic
689C22#312111yescharacteristic
690C22#312111yescharacteristic
691C22#316011yescharacteristic
692C22C2×#50111yescharacteristic
693C22#316311yescharacteristic
694C22#316411yescharacteristic
695C22#326511yescharacteristic
696C22#326611yescharacteristic
697C22#327411yescharacteristic
698C22#327611yescharacteristic
699C22#328011yescharacteristic
700C22#328211yescharacteristic
701C22#328211yescharacteristic
702C22#328511yescharacteristic
703C22#334111yescharacteristic
704C22#334211yescharacteristic
705C22#334411yescharacteristic
706C22#334511yescharacteristic
707C22#338111yescharacteristic
708C22#338111yescharacteristic
709C22#338511yescharacteristic
710C22#338611yescharacteristic
711C4—24no—
712C4—44no—
713C4—42no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4—81no—
722C4—82no—
723C4—161no—
724C4—161no—
725C4#242911yescharacteristic
726C4#243811yescharacteristic
727C4#245111yescharacteristic
728C4#255011yescharacteristic
729C4#255611yescharacteristic
730C4#258611yescharacteristic
731C4#266611yescharacteristic
732C4#285711yescharacteristic
733C2#1481611yescharacteristic
734C2#2383811yescharacteristic
735C2#2384211yescharacteristic
736C2#2515211yescharacteristic
737C2#2515711yescharacteristic
738C2#2606411yescharacteristic
739C2#2606811yescharacteristic
740C2#2637211yescharacteristic
741C2#2637711yescharacteristic
742C2#2679511yescharacteristic
743C2#2679711yescharacteristic
744C2#2685911yescharacteristic
745C2#2686111yescharacteristic
746C2#2881011yescharacteristic
747C2#2881611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
109. C2×#106Q8semidirect
112. C2×C4×C8Q8semidirect
162. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20852
C2×#20582
C2×#20572
C2×C4×#1062
#140532
#140062
#113282
#113252
#112202
#112192
#109082
#109072
#108622
#107972