Finite Group Databasebeta

← #198397

Subgroups — #198397

Order 512.

1447 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic, complemented
3#21759C211yesmaximal, characteristic, complemented
4C2×#1964C211yesmaximal, characteristic
5C2×#1519C211yesmaximal, characteristic
6#9262C214yesmaximal, complemented
7#9261C214yesmaximal, complemented
8C2×#1289C211yesmaximal, characteristic, complemented
9C2×#1282C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12#3187—22no—
13#3187—22no—
14#3187—22no—
15#3187—22no—
16C2×#481C2212yes—
17C2×#483C412yescomplemented
18C2×#465C414yescomplemented
19C2×#448C2212yes—
20C2×#450C412yescomplemented
21C22×#99—21no—
22C22×#99—21no—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2368—22no—
26#2360—22no—
27#2345—22no—
28#2345C2212yescomplemented
29Q8×C22⋊C4C2212yescomplemented
30#2329—22no—
31#2315—21no—
32C4×C22⋊Q8—22no—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35#2273—21no—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1964—22no—
38#1964C2214yes—
39#1892—22no—
40#1872—22no—
41#1824C2212yescomplemented
42#1787C2212yescomplemented
43#1519—22no—
44#1519C2214yes—
45C2×#327—22no—
46#1289C2214yescomplemented
47#1282C2214yescomplemented
48C23×Q8C2311yescharacteristic
49C23×D4—21no—
50C23×D4C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×SD16C2×C414yescomplemented
53C23×C8—21no—
54C23×C8C2×C412yes—
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⋊Q8C2×C412yes—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8D412yes—
64C2×C22⋊Q8D412yes—
65C2×C22⋊Q8D412yes—
66C2×C22⋊Q8D412yes—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121C2×C412yes—
78C2×#121C2312yes—
79C2×C4×Q8—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—21no—
92#483—24no—
93#483—24no—
94#481—24no—
95#481—24no—
96#465—28no—
97#465—28no—
98#450—24no—
99#450—24no—
100#448—24no—
101#448—24no—
102C2×#106C2×C414yescomplemented
103C2×#103—24no—
104C2×#103C2×C414yescomplemented
105C2×#102C2×C414yescomplemented
106C2×#99—24no—
107C2×#99—24no—
108C2×#99—44no—
109C2×#99—44no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
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⋊C8C2×C412yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122#391—22no—
123#391—24no—
124#389—24no—
125#389—22no—
126#387—24no—
127#387—24no—
128#387—24no—
129#385—24no—
130#385—24no—
131#380—21no—
132#380—21no—
133#380—22no—
134#380—22no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380D412yes—
139#380C2312yes—
140#380C2312yes—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—22no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4D412yes—
150C4×C22⋊C4C2312yes—
151C2×#95—21no—
152C2×#95—21no—
153C2×#95D412yes—
154#340—24no—
155#327—28no—
156C25—21no—
157C25C2×D411yescharacteristic
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—82no—
162C22×Q8C22×C412yes—
163C22×Q8C2×D412yes—
164C22×Q8C2×D412yes—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—82no—
181C22×D4C22×C412yes—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—41no—
189C23×C4—82no—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4#4111yescharacteristic
200C23×C4#4111yescharacteristic
201C23×C4#4111yescharacteristic
202C23×C4#4111yescharacteristic
203C23×C4C2411yescharacteristic
204C2×SD16—28no—
205C2×SD16—84no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—41no—
210C22×C8—41no—
211C22×C8—82no—
212C22×C8—82no—
213C22×C8C22×C414yes—
214C22⋊Q8—22no—
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—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8C22⋊C414yes—
227C22⋊Q8C22⋊C414yes—
228C22⋊Q8C22⋊C414yes—
229C22⋊Q8C22⋊C414yes—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—41no—
239#121—41no—
240#121—41no—
241#121—41no—
242#121—41no—
243#121—41no—
244#121—41no—
245#121—41no—
246C4×Q8—84no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
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—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
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—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4SD1614yescomplemented
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—82no—
306#106—28no—
307#103—28no—
308#103—84no—
309#102—28no—
310#99—48no—
311#99—48no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—41no—
319C22⋊C8—41no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—41no—
329#95C2×D414yes—
330C24—22no—
331C24—41no—
332C24—41no—
333C24—42no—
334C24—42no—
335C24—81no—
336C24—81no—
337C24—82no—
338C24—82no—
339C24C4×D411yescharacteristic
340C24C4×D411yescharacteristic
341C24#12011yescharacteristic
342C24#12011yescharacteristic
343C24#12111yescharacteristic
344C24#12111yescharacteristic
345C24C2×SD1612yes—
346C24#13512yes—
347C24AGL(1,ℤ/8)12yes—
348C24#13712yes—
349C24C22×D411yescharacteristic
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—82no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×Q8C2×C22⋊C414yes—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4—82no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—82no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4#12011yescharacteristic
442C22×C4#12011yescharacteristic
443C22×C4#12111yescharacteristic
444C22×C4#12111yescharacteristic
445C22×C4#12111yescharacteristic
446C22×C4#12111yescharacteristic
447C22×C4#12111yescharacteristic
448C22×C4#12111yescharacteristic
449C22×C4#12311yescharacteristic
450C22×C4#12311yescharacteristic
451C22×C4#12311yescharacteristic
452C22×C4#12311yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459SD16—88no—
460C2×C8—24no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8—42no—
464C2×C8—82no—
465C2×C8—82no—
466C2×C8—82no—
467C2×C8—82no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—82no—
485C4⋊C4C2×C22⋊C414yes—
486C22⋊C4—22no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—82no—
505C22⋊C4C2×SD1614yes—
506C42—44no—
507C42—84no—
508C42—82no—
509C42—84no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—164no—
540C23—164no—
541C23#38611yescharacteristic
542C23#38611yescharacteristic
543C23#38611yescharacteristic
544C23#38611yescharacteristic
545C23C4×SD1612yes—
546C23C4×SD1612yes—
547C23#44012yes—
548C23#44012yes—
549C23#44812yes—
550C23#44812yes—
551C23#44912yes—
552C23#44912yes—
553C23#46012yes—
554C23#46012yes—
555C23#46412yes—
556C23#46412yes—
557C23C2×C4×D411yescharacteristic
558C23C2×#12011yescharacteristic
559C23C2×#12111yescharacteristic
560C23C22×SD1611yescharacteristic
561C23C2×#13511yescharacteristic
562C23C2×AGL(1,ℤ/8)11yescharacteristic
563C23C2×#13711yescharacteristic
564Q8—28no—
565Q8—44no—
566Q8—82no—
567Q8—82no—
568D4—44no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—84no—
573D4—84no—
574D4—82no—
575D4—82no—
576D4—82no—
577D4—82no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
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—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4#38611yescharacteristic
636C2×C4#38611yescharacteristic
637C2×C4#38611yescharacteristic
638C2×C4#38611yescharacteristic
639C2×C4C22×C22⋊C411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12311yescharacteristic
643C8—44no—
644C8—82no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—162no—
679C22—162no—
680C22—164no—
681C22—164no—
682C22#188012yes—
683C22#188012yes—
684C22#188012yes—
685C22#188012yes—
686C22C2×#38611yescharacteristic
687C22C2×C4×SD1611yescharacteristic
688C22C2×#44011yescharacteristic
689C22C2×#44811yescharacteristic
690C22C2×#44911yescharacteristic
691C22C2×#46011yescharacteristic
692C22C2×#46411yescharacteristic
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4—162no—
707C4#288312yes—
708C2—21no—
709C2—21no—
710C2—21no—
711C2—22no—
712C2—22no—
713C2—21no—
714C2—21no—
715C2—22no—
716C2—22no—
717C2—21no—
718C2—162no—
719C2C2×#188011yescharacteristic
720C2#2098411yescharacteristic
721C2#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
2. #34212C2semidirect
3. #21759C2semidirect
6. #9262C2semidirect
7. #9261C2semidirect
8. C2×#1289C2semidirect
9. C2×#1282C2semidirect
17. C2×#483C4semidirect
18. C2×#465C4semidirect
20. C2×#450C4semidirect
28. #2345C22semidirect
29. Q8×C22⋊C4C22semidirect
41. #1824C22semidirect
42. #1787C22semidirect
46. #1289C22semidirect
47. #1282C22semidirect
52. C22×SD16C2×C4semidirect
102. C2×#106C2×C4semidirect
104. C2×#103C2×C4semidirect
105. C2×#102C2×C4semidirect
298. 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
#342122
#217592
C2×#19642
C2×#15192
#92622
#92612
C2×#12892
C2×#12822