Finite Group Databasebeta

← #364752

Subgroups — #364752

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#10885C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10730C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20#2487—24no—
21#2487C418yescomplemented
22C2×#400—21no—
23C2×#399—21no—
24D4×C4⋊C4—24no—
25D4×C4⋊C4—24no—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#386—21no—
29C2×#385—21no—
30C2×#385—21no—
31C2×#385—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1869—24no—
39#1824—24no—
40C2×#340—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—42no—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—28no—
68C2×C4×D4—44no—
69C2×C4×D4—44no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2×C414yescomplemented
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4D414yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4D411yescharacteristic
104C22×C22⋊C4D411yescharacteristic
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107C2×C22⋊C8—22no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—42no—
113#400—28no—
114#399—28no—
115#389—28no—
116#389—24no—
117#389C2×C418yescomplemented
118#386—28no—
119#385—28no—
120#385—28no—
121#385—28no—
122#384—22no—
123#384—22no—
124#380—22no—
125#380—24no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380C2×C418yescomplemented
130#380C2×C414yescomplemented
131#380D414yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340—42no—
147#340—42no—
148#336—24no—
149C25C2×D411yescharacteristic
150C25C2×D411yescharacteristic
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4C22⋊C414yes—
156C22×D4SD1614yes—
157C22×D4Q1614yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—82no—
188C23×C4—82no—
189C23×C4—82no—
190C23×C4—82no—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4#4112yes—
202C23×C4#4112yes—
203C23×C4C2411yescharacteristic
204C22×C8—28no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—48no—
219C22×C8—48no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8#4114yes—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—432no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4C22⋊C414yes—
277C2×C4⋊C4C22×C414yes—
278C2×C4⋊C4C22×C414yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4#4114yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4—82no—
300C2×C22⋊C4C22⋊C414yes—
301C2×C22⋊C4C22⋊C414yes—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C22×C412yes—
310C2×C42C2×D412yes—
311C22⋊C8—48no—
312C22⋊C8—44no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24C2×C22⋊C412yes—
332C24C2×C22⋊C412yes—
333C24#12012yes—
334C24#12312yes—
335C24C2×SD1612yes—
336C24C2×SD1612yes—
337C24C2×Q1612yes—
338C24C2×Q1612yes—
339C24AGL(1,ℤ/8)12yes—
340C24#13712yes—
341C24C22×D411yescharacteristic
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4#103116yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C4×D414yes—
449C22×C4C4×D414yes—
450C22×C4#12012yes—
451C22×C4#12114yes—
452C22×C4#12114yes—
453C22×C4#12312yes—
454C22×C4C2×SD1612yes—
455C22×C4C2×Q1612yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—416no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C4⋊C4—84no—
507C4⋊C4#12314yes—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—88no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—48no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23#38614yes—
558C23C2×#10314yes—
559C23C2×#10314yes—
560C23#41714yes—
561C23#45014yes—
562C23#45114yes—
563C23#48314yes—
564C23#48414yes—
565C23C22×C22⋊C411yescharacteristic
566C23C2×#12011yescharacteristic
567C23C2×#12311yescharacteristic
568C23#54612yes—
569C23#54712yes—
570C23C22×SD1611yescharacteristic
571C23C22×Q1611yescharacteristic
572C23C2×AGL(1,ℤ/8)11yescharacteristic
573C23C2×#13711yescharacteristic
574C23#57612yes—
575C23#57712yes—
576D4—232no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—816no—
630C2×C4—816no—
631C2×C4—816no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4#38614yes—
647C2×C4C2×#10314yes—
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C4×#4112yes—
651C2×C4#52012yes—
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×#12311yescharacteristic
655C2×C4#52511yescharacteristic
656C2×C4#52912yes—
657C2×C4#53411yescharacteristic
658C2×C4#53611yescharacteristic
659C2×C4#53912yes—
660C2×C4#54612yes—
661C2×C4#54612yes—
662C2×C4#54712yes—
663C2×C4#54712yes—
664C2×C4#55112yes—
665C2×C4#55212yes—
666C8—44no—
667C8—82no—
668C22—24no—
669C22—24no—
670C22—28no—
671C22—28no—
672C22—24no—
673C22—24no—
674C22—28no—
675C22—28no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—28no—
680C22—42no—
681C22—42no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22#187218yes—
690C22C2×#38611yescharacteristic
691C22#233912yes—
692C22#240511yescharacteristic
693C22#240611yescharacteristic
694C22#241912yes—
695C22C22×#10311yescharacteristic
696C22C2×#41711yescharacteristic
697C22#289712yes—
698C22C2×#45011yescharacteristic
699C22C2×#45111yescharacteristic
700C22#300811yescharacteristic
701C22#301011yescharacteristic
702C22C2×#48311yescharacteristic
703C22C2×#48411yescharacteristic
704C22#309612yes—
705C22#329512yes—
706C22#329912yes—
707C22#331812yes—
708C22#332012yes—
709C22#333512yes—
710C22#333812yes—
711C22#334512yes—
712C22#334612yes—
713C4—24no—
714C4—24no—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—82no—
719C4—84no—
720C4—84no—
721C4—82no—
722C4—82no—
723C4C4×#12311yescharacteristic
724C4#232211yescharacteristic
725C4#235912yes—
726C4#242512yes—
727C4#248611yescharacteristic
728C4#253411yescharacteristic
729C2—28no—
730C2—44no—
731C2#1396811yescharacteristic
732C2C2×#187211yescharacteristic
733C2#2091411yescharacteristic
734C2#2248712yes—
735C2#2249312yes—
736C2#2403311yescharacteristic
737C2#2404211yescharacteristic
738C2#2408911yescharacteristic
739C2#2409811yescharacteristic
740C2#2493512yes—
741C2#2493912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1824C2semidirect
5. C22×#340C2semidirect
6. #11207C2semidirect
7. #11200C2semidirect
8. #10885C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10721C2semidirect
21. #2487C4semidirect
90. C22×C4⋊C4C2×C4semidirect
117. #389C2×C4semidirect
129. #380C2×C4semidirect
130. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×D4×C4⋊C42
C2×#18692
C2×#18242
C22×#3402
#112072
#112002
#108852
#107302
#107212