Finite Group Databasebeta

← #198388

Subgroups — #198388

Order 512.

1569 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#34205C211yesmaximal, characteristic
3#21758C211yesmaximal, characteristic
4C2×#1996C211yesmaximal, characteristic, complemented
5C2×#1520C211yesmaximal, characteristic, complemented
6#9256C214yesmaximal, complemented
7#9255C214yesmaximal, complemented
8C2×#1289C211yesmaximal, characteristic, complemented
9C2×#1282C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#518—21no—
13#3187—22no—
14#3187—22no—
15#3187—22no—
16#3187—22no—
17C2×#481C412yescomplemented
18C2×#483C2212yes—
19C2×#465C414yescomplemented
20C2×#448C412yescomplemented
21C2×#450C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2364—22no—
25#2361—22no—
26#2344C2212yes—
27#2342—22no—
28D4×C22⋊C4C2212yes—
29#2325—22no—
30#2308—21no—
31C4×#121—22no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1996—22no—
36#1996C2214yes—
37#1893—22no—
38#1874—22no—
39#1823C2212yes—
40#1786C2212yes—
41C2×#342—21no—
42C2×#342—21no—
43#1520—22no—
44#1520C2214yes—
45C2×#328—22no—
46#1289C2214yes—
47#1282C2214yes—
48C23×Q8C2×C411yescharacteristic, complemented
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×SD16C2×C414yescomplemented
53C23×C8—21no—
54C23×C8C2×C412yescomplemented
55C2×C22⋊Q8—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×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yescomplemented
66C2×C22⋊Q8C2312yes—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121C2×C412yes—
74C2×#121C2×C412yes—
75C2×#121D412yes—
76C2×#121D412yes—
77C2×#121D412yes—
78C2×#121D412yes—
79#518—44no—
80#518—44no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×#117—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C411yescharacteristic, complemented
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98#483—24no—
99#483—24no—
100#481—24no—
101#481—24no—
102#465—28no—
103#465—28no—
104#450—24no—
105#450—24no—
106#448—24no—
107#448—24no—
108C2×#106C2×C414yescomplemented
109C2×#103C2×C414yescomplemented
110C2×#102—24no—
111C2×#102C2×C414yescomplemented
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8C2×C412yescomplemented
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2312yes—
124#390—22no—
125#389—22no—
126#388—24no—
127#388—24no—
128#387—24no—
129#386—24no—
130#386—24no—
131#386—24no—
132#385—24no—
133#381—22no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380D412yes—
140#380C2312yes—
141#380C2312yes—
142#379—22no—
143C4×C4⋊C4—22no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4D412yes—
148C4×C22⋊C4C2312yes—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95D412yes—
152#342—24no—
153#342—24no—
154#340—24no—
155#328—28no—
156C25—21no—
157C25C2×D411yescharacteristic
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—82no—
164C22×Q8C22×C412yes—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—82no—
179C22×D4C22×C412yes—
180C22×D4C2×D412yes—
181C22×D4C2×D412yes—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—41no—
190C23×C4—82no—
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4#4111yescharacteristic
201C23×C4#4111yescharacteristic
202C23×C4#4111yescharacteristic
203C23×C4#4111yescharacteristic
204C23×C4C2411yescharacteristic
205C2×SD16—28no—
206C2×SD16—84no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—41no—
211C22×C8—41no—
212C22×C8—82no—
213C22×C8—82no—
214C22×C8C22×C414yes—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C22⋊Q8—41no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121—22no—
241#121—22no—
242#121—22no—
243#121C22⋊C414yes—
244#121C22⋊C414yes—
245#121C22⋊C414yes—
246#121C22⋊C414yes—
247C4×D4—416no—
248C4×D4—416no—
249C4×D4—84no—
250#117—44no—
251#117—44no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4—82no—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4SD1614yescomplemented
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—82no—
314#106—28no—
315#103—28no—
316#102—28no—
317#102—84no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323C22⋊C8—22no—
324C22⋊C8—41no—
325C22⋊C8—41no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—41no—
333#95C2×D414yes—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—81no—
342C24—81no—
343C24—82no—
344C24—82no—
345C24C4×D411yescharacteristic
346C24C4×D411yescharacteristic
347C24#12011yescharacteristic
348C24#12011yescharacteristic
349C24#12111yescharacteristic
350C24#12111yescharacteristic
351C24C2×SD1612yes—
352C24#13512yes—
353C24AGL(1,ℤ/8)12yes—
354C24#13712yes—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—44no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4C2×C22⋊C414yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
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—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C4×D411yescharacteristic
448C22×C4C4×D411yescharacteristic
449C22×C4C4×D411yescharacteristic
450C22×C4#12011yescharacteristic
451C22×C4#12011yescharacteristic
452C22×C4#12111yescharacteristic
453C22×C4#12111yescharacteristic
454C22×C4#12111yescharacteristic
455C22×C4#12111yescharacteristic
456C22×C4#12111yescharacteristic
457C22×C4#12111yescharacteristic
458C22×C4#12311yescharacteristic
459C22×C4#12311yescharacteristic
460C22×C4#12311yescharacteristic
461C22×C4#12311yescharacteristic
462C22×C4C23×C411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4111yescharacteristic
468SD16—88no—
469C2×C8—24no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—42no—
473C2×C8—82no—
474C2×C8—82no—
475C2×C8—82no—
476C2×C8—82no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4C2×C22⋊C414yes—
494C22⋊C4—22no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—82no—
516C22⋊C4C2×SD1614yes—
517C42—44no—
518C42—48no—
519C42—82no—
520C23—24no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—162no—
552C23—162no—
553C23—162no—
554C23—162no—
555C23#38611yescharacteristic
556C23#38611yescharacteristic
557C23#38611yescharacteristic
558C23#38611yescharacteristic
559C23C4×SD1612yes—
560C23C4×SD1612yes—
561C23#44012yes—
562C23#44012yes—
563C23#45012yes—
564C23#45012yes—
565C23#45212yes—
566C23#45212yes—
567C23#46112yes—
568C23#46112yes—
569C23#46312yes—
570C23#46312yes—
571C23C2×C4×D411yescharacteristic
572C23C2×#12011yescharacteristic
573C23C2×#12111yescharacteristic
574C23C22×SD1611yescharacteristic
575C23C2×#13511yescharacteristic
576C23C2×AGL(1,ℤ/8)11yescharacteristic
577C23C2×#13711yescharacteristic
578Q8—44no—
579Q8—82no—
580Q8—82no—
581Q8—82no—
582D4—28no—
583D4—44no—
584D4—48no—
585D4—48no—
586D4—48no—
587D4—48no—
588D4—82no—
589D4—82no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
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—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—84no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—84no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—168no—
648C2×C4—164no—
649C2×C4—164no—
650C2×C4#38611yescharacteristic
651C2×C4#38611yescharacteristic
652C2×C4#38611yescharacteristic
653C2×C4#38611yescharacteristic
654C2×C4C22×C22⋊C411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×#12111yescharacteristic
657C2×C4C2×#12311yescharacteristic
658C8—44no—
659C8—82no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—84no—
694C22—84no—
695C22—162no—
696C22—162no—
697C22—162no—
698C22—162no—
699C22#187712yes—
700C22#187712yes—
701C22#187712yes—
702C22#187712yes—
703C22C2×#38611yescharacteristic
704C22C2×C4×SD1611yescharacteristic
705C22C2×#44011yescharacteristic
706C22C2×#45011yescharacteristic
707C22C2×#45211yescharacteristic
708C22C2×#46111yescharacteristic
709C22C2×#46311yescharacteristic
710C4—21no—
711C4—21no—
712C4—21no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—21no—
717C4—44no—
718C4—44no—
719C4—82no—
720C4—84no—
721C4—82no—
722C4—82no—
723C4—162no—
724C4#288312yes—
725C2—21no—
726C2—21no—
727C2—21no—
728C2—21no—
729C2—21no—
730C2—21no—
731C2—22no—
732C2—22no—
733C2—22no—
734C2—22no—
735C2—84no—
736C2C2×#187711yescharacteristic
737C2#2098411yescharacteristic
738C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1996C2semidirect
5. C2×#1520C2semidirect
6. #9256C2semidirect
7. #9255C2semidirect
8. C2×#1289C2semidirect
9. C2×#1282C2semidirect
17. C2×#481C4semidirect
19. C2×#465C4semidirect
20. C2×#448C4semidirect
48. C23×Q8C2×C4semidirect
52. C22×SD16C2×C4semidirect
54. C23×C8C2×C4semidirect
65. C2×C22⋊Q8C2×C4semidirect
88. C22×C4⋊C4C2×C4semidirect
108. C2×#106C2×C4semidirect
109. C2×#103C2×C4semidirect
111. C2×#102C2×C4semidirect
121. C2×C22⋊C8C2×C4semidirect
307. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#342052
#217582
C2×#19962
C2×#15202
#92562
#92552
C2×#12892
C2×#12822