Finite Group Databasebeta

← #198724

Subgroups — #198724

Order 512.

1435 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic, complemented
3C2×#1964C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic, complemented
5C2×#1519C211yesmaximal, characteristic
6#9278C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic, complemented
8C2×#1282C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11#3187—22no—
12#3187—22no—
13#3187C414yescomplemented
14#3187C2214yes—
15C2×#481—21no—
16C2×#483C412yescomplemented
17C2×#465—22no—
18C2×#448—21no—
19C2×#450C412yescomplemented
20C22×#99—21no—
21C22×#99—21no—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2368C2214yescomplemented
25#2360—22no—
26#2345—22no—
27#2345—21no—
28Q8×C22⋊C4—21no—
29#2329—22no—
30#2315—21no—
31C4×C22⋊Q8—22no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34#2273—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1964—22no—
37#1964C2214yes—
38#1520—22no—
39#1520C2214yescomplemented
40#1519—22no—
41#1519—22no—
42C2×#328—22no—
43C2×#327C2214yes—
44#1294C2214yescomplemented
45#1282C2214yescomplemented
46C23×Q8D411yescharacteristic
47C23×D4—21no—
48C23×D4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×SD16—22no—
51C23×C8—21no—
52C23×C8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×C22⋊Q8D412yes—
62C2×C22⋊Q8D412yes—
63C2×C22⋊Q8D412yes—
64C2×C22⋊Q8C2312yes—
65C2×#121—21no—
66C2×#121—21no—
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×#121C2×C412yes—
76C2×#121C2312yes—
77C2×C4×Q8—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42—21no—
90#483—24no—
91#483C2×C418yescomplemented
92#481—24no—
93#481—24no—
94#465—28no—
95#465—28no—
96#450—24no—
97#450C2×C418yescomplemented
98#448—24no—
99#448—24no—
100C2×#106—22no—
101C2×#103—22no—
102C2×#102—22no—
103C2×#99—24no—
104C2×#99—24no—
105C2×#99—44no—
106C2×#99—44no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8C2×C412yes—
116C2×C22⋊C8C2×C412yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119#391—22no—
120#391—24no—
121#389—24no—
122#389—22no—
123#387—24no—
124#387—24no—
125#387—24no—
126#385—24no—
127#385—24no—
128#380—22no—
129#380—22no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135#380D412yes—
136#380D412yes—
137#380C2312yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—24no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4D412yes—
147C4×C22⋊C4C2312yes—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95C2312yes—
151#328—28no—
152#327—28no—
153C25—21no—
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—82no—
159C22×Q8C22⋊C412yes—
160C22×Q8C22⋊C412yes—
161C22×Q8C2×D412yes—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—82no—
178C22×D4C22×C412yes—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—41no—
186C23×C4—82no—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4#4111yescharacteristic
200C23×C4C2411yescharacteristic
201C2×SD16—28no—
202C2×SD16—84no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—41no—
208C22×C8—41no—
209C22×C8—82no—
210C22×C8—82no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
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⋊Q8C22⋊C414yes—
224C22⋊Q8C22⋊C414yes—
225C22⋊Q8C22×C414yes—
226C22⋊Q8C2×D414yes—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—41no—
236#121—41no—
237#121—41no—
238#121—41no—
239#121—41no—
240#121—41no—
241#121—41no—
242#121—41no—
243C4×Q8—84no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
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—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4C22⋊C412yes—
267C2×C4⋊C4C22⋊C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4SD1614yescomplemented
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—82no—
303#106—28no—
304#103—28no—
305#102—28no—
306#99—48no—
307#99—48no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8C22×C414yes—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—41no—
325#95C2×D414yes—
326C24—22no—
327C24—41no—
328C24—41no—
329C24—42no—
330C24—42no—
331C24—81no—
332C24—81no—
333C24—82no—
334C24—82no—
335C24C4×D411yescharacteristic
336C24C4×D411yescharacteristic
337C24#12011yescharacteristic
338C24#12011yescharacteristic
339C24#12111yescharacteristic
340C24#12111yescharacteristic
341C24C2×SD1612yes—
342C24#13512yes—
343C24AGL(1,ℤ/8)12yes—
344C24#13712yes—
345C24C22×D411yescharacteristic
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—82no—
352C2×Q8—82no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×Q8C2×C22⋊C414yes—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
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—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—82no—
421C22×C4—81no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4#12011yescharacteristic
439C22×C4#12011yescharacteristic
440C22×C4#12111yescharacteristic
441C22×C4#12111yescharacteristic
442C22×C4#12111yescharacteristic
443C22×C4#12111yescharacteristic
444C22×C4#12111yescharacteristic
445C22×C4#12111yescharacteristic
446C22×C4#12311yescharacteristic
447C22×C4#12311yescharacteristic
448C22×C4#12311yescharacteristic
449C22×C4#12311yescharacteristic
450C22×C4C23×C411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456SD16—88no—
457C2×C8—24no—
458C2×C8—44no—
459C2×C8—42no—
460C2×C8—82no—
461C2×C8—82no—
462C2×C8—82no—
463C2×C8—82no—
464C2×C8—82no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—82no—
478C4⋊C4—84no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4C2×C22⋊C414yes—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—82no—
501C22⋊C4C2×SD1614yescomplemented
502C42—44no—
503C42—84no—
504C42—82no—
505C42—84no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—164no—
536C23—164no—
537C23#38611yescharacteristic
538C23#38611yescharacteristic
539C23#38611yescharacteristic
540C23#38611yescharacteristic
541C23C4×SD1612yes—
542C23C4×SD1612yes—
543C23#44012yes—
544C23#44012yes—
545C23#44812yes—
546C23#44812yes—
547C23#44912yes—
548C23#44912yes—
549C23#46012yes—
550C23#46012yes—
551C23#46412yes—
552C23#46412yes—
553C23C2×C4×D411yescharacteristic
554C23C2×#12011yescharacteristic
555C23C2×#12111yescharacteristic
556C23C22×SD1611yescharacteristic
557C23C2×#13511yescharacteristic
558C23C2×AGL(1,ℤ/8)11yescharacteristic
559C23C2×#13711yescharacteristic
560Q8—44no—
561Q8—44no—
562Q8—44no—
563Q8—82no—
564D4—44no—
565D4—82no—
566D4—82no—
567D4—84no—
568D4—84no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—82no—
573D4—82no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
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—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—162no—
630C2×C4—162no—
631C2×C4—164no—
632C2×C4#38611yescharacteristic
633C2×C4#38611yescharacteristic
634C2×C4#38611yescharacteristic
635C2×C4#38611yescharacteristic
636C2×C4C22×C22⋊C411yescharacteristic
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12311yescharacteristic
640C8—44no—
641C8—82no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—41no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—162no—
676C22—162no—
677C22—164no—
678C22—164no—
679C22#188012yes—
680C22#188012yes—
681C22#188012yes—
682C22#188012yes—
683C22C2×#38611yescharacteristic
684C22C2×C4×SD1611yescharacteristic
685C22C2×#44011yescharacteristic
686C22C2×#44811yescharacteristic
687C22C2×#44911yescharacteristic
688C22C2×#46011yescharacteristic
689C22C2×#46411yescharacteristic
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—82no—
701C4—84no—
702C4—82no—
703C4—162no—
704C4#191312yes—
705C2—21no—
706C2—21no—
707C2—21no—
708C2—22no—
709C2—22no—
710C2—21no—
711C2—21no—
712C2—22no—
713C2—22no—
714C2—21no—
715C2—162no—
716C2C2×#188011yescharacteristic
717C2#2100011yescharacteristic
718C2C2×#191311yescharacteristic
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
4. C2×#1520C2semidirect
6. #9278C2semidirect
7. #8023C2semidirect
8. C2×#1282C2semidirect
13. #3187C4semidirect
16. C2×#483C4semidirect
19. C2×#450C4semidirect
24. #2368C22semidirect
39. #1520C22semidirect
44. #1294C22semidirect
45. #1282C22semidirect
91. #483C2×C4semidirect
97. #450C2×C4semidirect
295. C2×C22⋊C4SD16semidirect
501. C22⋊C4C2×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#342122
C2×#19642
C2×#15202
C2×#15192
#92782
#80232
C2×#12822