Finite Group Databasebeta

← #115487

Subgroups — #115487

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4C22×D4⋊C8C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic, complemented
7C2×#1279C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#476—22no—
11C2×#476—22no—
12C2×#476—22no—
13C2×#476—22no—
14C2×#476—22no—
15C2×#476C414yescomplemented
16C22×#106—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—22no—
20C22×#102—21no—
21C22×C4×C8C2211yescharacteristic
22C2×#390—21no—
23C2×#384C2211yescharacteristic
24C2×#380—22no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1987—22no—
27#1987—22no—
28#1956—22no—
29#1956—22no—
30C2×#340—22no—
31C2×D4⋊C8—22no—
32C2×D4⋊C8—22no—
33C2×D4⋊C8—22no—
34#1439—22no—
35C4⋊#95—21no—
36#1296—22no—
37#1279—22no—
38C23×Q8—21no—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C23×C8—21no—
42C23×C8—21no—
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8D412yes—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4D412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—24no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#476—28no—
80#476—28no—
81#476—28no—
82#476—28no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C412yes—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8D412yes—
97C2×C4⋊C8D412yes—
98C2×#103—28no—
99C2×#103—48no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8D412yes—
109#390—42no—
110#390—42no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115#380—44no—
116#380—44no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C2×#95—24no—
122C2×#95—21no—
123#340—28no—
124D4⋊C8—28no—
125C25—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22⋊C412yes—
146C22×D4SD1612yes—
147C22×D4Q1612yes—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C22×C8—24no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—41no—
173C22×C8—41no—
174C22×C8—41no—
175C22×C8—41no—
176C22×C8—41no—
177C22×C8—41no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8—42no—
181C22×C8—42no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8C22⋊C414yescomplemented
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4SD1614yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—84no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278#106—216no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—41no—
285C4⋊C8—41no—
286C4⋊C8C22⋊C414yescomplemented
287#103—416no—
288#102—216no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—41no—
292C4×C8—41no—
293#95—48no—
294#95—41no—
295#95—41no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24#12011yescharacteristic
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—84no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×Q8—84no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4#10314yescomplemented
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12311yescharacteristic
406C22×C4#12711yescharacteristic
407C22×C4#12711yescharacteristic
408C22×C4C2×SD1611yescharacteristic
409C22×C4C2×SD1612yes—
410C22×C4C2×Q1611yescharacteristic
411C22×C4#13511yescharacteristic
412C22×C4#13511yescharacteristic
413C22×C4#13512yes—
414C22×C4AGL(1,ℤ/8)11yescharacteristic
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4#13711yescharacteristic
417C22×C4#13711yescharacteristic
418C22×C4#13711yescharacteristic
419C22×C4#13712yes—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—44no—
423C2×C8—42no—
424C2×C8—44no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—44no—
434C2×C8—82no—
435C2×C8—81no—
436C2×C8—81no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—82no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—816no—
467C42—81no—
468C42—81no—
469C42—81no—
470C42—81no—
471C42C2×C22⋊C411yescharacteristic
472C42C2×C22⋊C411yescharacteristic
473C42C4×D411yescharacteristic
474C42C4×D411yescharacteristic
475C42#12011yescharacteristic
476C42#12011yescharacteristic
477C42#12011yescharacteristic
478C42#12111yescharacteristic
479C42#12111yescharacteristic
480C42#12111yescharacteristic
481C42#12111yescharacteristic
482C42#12311yescharacteristic
483C42#12311yescharacteristic
484C42#12311yescharacteristic
485C42#12411yescharacteristic
486C42#12411yescharacteristic
487C23—24no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—81no—
506C23—81no—
507C23—82no—
508C23—81no—
509C23—81no—
510C23#38611yescharacteristic
511C23#39211yescharacteristic
512C23#39211yescharacteristic
513C23#45011yescharacteristic
514C23#45012yes—
515C23#45111yescharacteristic
516C23#45211yescharacteristic
517C23#45211yescharacteristic
518C23#45212yes—
519C23#45312yes—
520C23#45412yes—
521Q8—48no—
522Q8—84no—
523Q8—84no—
524Q8—84no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—42no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#37911yescharacteristic
570C2×C4#38611yescharacteristic
571C2×C4#38611yescharacteristic
572C2×C4#38611yescharacteristic
573C2×C4#39011yescharacteristic
574C2×C4#39011yescharacteristic
575C2×C4#39211yescharacteristic
576C2×C4#39211yescharacteristic
577C2×C4#39411yescharacteristic
578C2×C4#39411yescharacteristic
579C2×C4#39411yescharacteristic
580C2×C4#39411yescharacteristic
581C2×C4C2×#10311yescharacteristic
582C2×C4#41611yescharacteristic
583C2×C4#41711yescharacteristic
584C2×C4#41911yescharacteristic
585C2×C4C4×SD1612yes—
586C2×C4#44012yes—
587C2×C4#44812yes—
588C2×C4#44811yescharacteristic
589C2×C4#44811yescharacteristic
590C2×C4#44912yes—
591C2×C4#44911yescharacteristic
592C2×C4#44911yescharacteristic
593C2×C4#45111yescharacteristic
594C2×C4#45111yescharacteristic
595C2×C4#45211yescharacteristic
596C2×C4#45211yescharacteristic
597C2×C4#46012yes—
598C2×C4#46112yes—
599C2×C4#46312yes—
600C2×C4#46412yes—
601C2×C4#46512yes—
602C2×C4#46511yescharacteristic
603C2×C4#46511yescharacteristic
604C2×C4#46711yescharacteristic
605C2×C4#46711yescharacteristic
606C2×C4#46812yes—
607C2×C4#46811yescharacteristic
608C2×C4#46811yescharacteristic
609C2×C4#47011yescharacteristic
610C2×C4#47011yescharacteristic
611C2×C4#48311yescharacteristic
612C2×C4#48411yescharacteristic
613C2×C4#48511yescharacteristic
614C2×C4#48511yescharacteristic
615C2×C4#49211yescharacteristic
616C2×C4#49511yescharacteristic
617C2×C4#49611yescharacteristic
618C2×C4#49711yescharacteristic
619C8—44no—
620C8—44no—
621C8—82no—
622C22—48no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—84no—
633C22#143811yescharacteristic
634C22#162912yes—
635C22#163112yes—
636C22#163212yes—
637C22#163412yes—
638C22#163512yes—
639C22#163512yes—
640C22#163712yes—
641C22#163812yes—
642C22#187211yescharacteristic
643C22#187311yescharacteristic
644C22#187712yes—
645C22#188312yes—
646C22#200012yes—
647C22#200011yescharacteristic
648C22#200011yescharacteristic
649C22#200111yescharacteristic
650C22#200111yescharacteristic
651C22#200211yescharacteristic
652C22#200311yescharacteristic
653C22#200612yes—
654C22#201411yescharacteristic
655C22#201511yescharacteristic
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—84no—
661C4—82no—
662C4—84no—
663C4—162no—
664C4#178711yescharacteristic
665C4#178811yescharacteristic
666C4#188012yes—
667C4#189211yescharacteristic
668C4#189411yescharacteristic
669C4#196811yescharacteristic
670C4#197311yescharacteristic
671C4#201211yescharacteristic
672C4#201311yescharacteristic
673C4#203511yescharacteristic
674C4#203511yescharacteristic
675C4#203611yescharacteristic
676C4#203611yescharacteristic
677C4#205512yes—
678C2—84no—
679C2#897312yes—
680C2#897612yes—
681C2#1005411yescharacteristic
682C2#1005611yescharacteristic
683C2#1005811yescharacteristic
684C2#1006011yescharacteristic
685C2#1023512yes—
686C2#1023612yes—
687C2#1097011yescharacteristic
688C2#1101611yescharacteristic
689C2#1102311yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1987C2semidirect
3. C2×#1956C2semidirect
6. C2×#1296C2semidirect
7. C2×#1279C2semidirect
15. C2×#476C4semidirect
197. C4⋊Q8C22⋊C4semidirect
286. C4⋊C8C22⋊C4semidirect
347. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#19872
C2×#19562
C22×D4⋊C82
#139062
C2×#12962
C2×#12792