Finite Group Databasebeta

← #115492

Subgroups — #115492

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic
4C22×Q8⋊C8C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#475—22no—
11C2×#475—22no—
12C2×#475—22no—
13C2×#475—22no—
14C2×#475—22no—
15C2×#475C414yescomplemented
16C22×#106—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—21no—
20C22×#103—22no—
21C22×C4×C8C2211yescharacteristic
22C2×#391—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990—22no—
26#1990—22no—
27#1981—22no—
28#1981—22no—
29C2×#340—22no—
30C2×Q8⋊C8—22no—
31C2×Q8⋊C8—22no—
32C2×Q8⋊C8—22no—
33#1441—22no—
34C4⋊#95—22no—
35C4⋊#95—21no—
36#1296—22no—
37#1279—22no—
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40C23×C8—21no—
41C23×C8—21no—
42C23×C8—21no—
43C2×C4⋊Q8—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⋊Q8C2×C412yes—
53C2×C4⋊Q8D412yes—
54C2×C4⋊Q8D412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8D412yes—
67C22×C4⋊C4—21no—
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×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#475—28no—
79#475—28no—
80#475—28no—
81#475—28no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×C4⋊C8—21no—
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⋊C8C2×C412yes—
94C2×C4⋊C8D412yes—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8D412yes—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—28no—
101C2×#103—44no—
102C2×#103—44no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8D412yes—
109#391—42no—
110#391—42no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C2×#95—24no—
120C2×#95—21no—
121C2×#95—24no—
122#340—28no—
123Q8⋊C8—28no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8C22⋊C412yes—
146C22×Q8SD1612yes—
147C22×Q8Q1612yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—42no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—42no—
172C22×C8—42no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8C22⋊C414yes—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4SD1614yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272#106—216no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8C22⋊C414yes—
281#103—216no—
282#103—416no—
283C4×C8—22no—
284C4×C8—22no—
285C4×C8—41no—
286C4×C8—41no—
287#95—48no—
288#95—41no—
289#95—41no—
290#95—84no—
291C24#12011yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8#10314yescomplemented
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4#12011yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4#12011yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12311yescharacteristic
385C22×C4#12711yescharacteristic
386C22×C4#12711yescharacteristic
387C22×C4C2×SD1611yescharacteristic
388C22×C4C2×SD1612yes—
389C22×C4C2×Q1611yescharacteristic
390C22×C4#13511yescharacteristic
391C22×C4#13511yescharacteristic
392C22×C4#13512yes—
393C22×C4AGL(1,ℤ/8)11yescharacteristic
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4#13711yescharacteristic
396C22×C4#13711yescharacteristic
397C22×C4#13711yescharacteristic
398C22×C4#13712yes—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—44no—
402C2×C8—42no—
403C2×C8—44no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—44no—
413C2×C8—82no—
414C2×C8—81no—
415C2×C8—81no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—82no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—81no—
450C42—81no—
451C42—81no—
452C42—81no—
453C42C2×C22⋊C411yescharacteristic
454C42C2×C22⋊C411yescharacteristic
455C42C4×D411yescharacteristic
456C42C4×D411yescharacteristic
457C42#12011yescharacteristic
458C42#12011yescharacteristic
459C42#12011yescharacteristic
460C42#12111yescharacteristic
461C42#12111yescharacteristic
462C42#12111yescharacteristic
463C42#12111yescharacteristic
464C42#12311yescharacteristic
465C42#12311yescharacteristic
466C42#12311yescharacteristic
467C42#12411yescharacteristic
468C42#12411yescharacteristic
469C23#38611yescharacteristic
470C23#39211yescharacteristic
471C23#39211yescharacteristic
472C23#44812yes—
473C23#44912yes—
474C23#45011yescharacteristic
475C23#45111yescharacteristic
476C23#45211yescharacteristic
477C23#45211yescharacteristic
478C23#45512yes—
479C23#45612yes—
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—48no—
489Q8—84no—
490Q8—82no—
491Q8—82no—
492Q8—82no—
493Q8—82no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—84no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4#37911yescharacteristic
532C2×C4#38611yescharacteristic
533C2×C4#38611yescharacteristic
534C2×C4#38611yescharacteristic
535C2×C4#39011yescharacteristic
536C2×C4#39011yescharacteristic
537C2×C4#39211yescharacteristic
538C2×C4#39211yescharacteristic
539C2×C4#39411yescharacteristic
540C2×C4#39411yescharacteristic
541C2×C4#39411yescharacteristic
542C2×C4#39411yescharacteristic
543C2×C4C2×#10311yescharacteristic
544C2×C4#41611yescharacteristic
545C2×C4#41711yescharacteristic
546C2×C4#41911yescharacteristic
547C2×C4C4×SD1612yes—
548C2×C4#44012yes—
549C2×C4#44811yescharacteristic
550C2×C4#44811yescharacteristic
551C2×C4#44911yescharacteristic
552C2×C4#44911yescharacteristic
553C2×C4#45012yes—
554C2×C4#45111yescharacteristic
555C2×C4#45111yescharacteristic
556C2×C4#45212yes—
557C2×C4#45211yescharacteristic
558C2×C4#45211yescharacteristic
559C2×C4#46012yes—
560C2×C4#46112yes—
561C2×C4#46312yes—
562C2×C4#46412yes—
563C2×C4#46512yes—
564C2×C4#46511yescharacteristic
565C2×C4#46511yescharacteristic
566C2×C4#46711yescharacteristic
567C2×C4#46711yescharacteristic
568C2×C4#46812yes—
569C2×C4#46811yescharacteristic
570C2×C4#46811yescharacteristic
571C2×C4#47011yescharacteristic
572C2×C4#47011yescharacteristic
573C2×C4#48311yescharacteristic
574C2×C4#48411yescharacteristic
575C2×C4#48511yescharacteristic
576C2×C4#48511yescharacteristic
577C2×C4#49211yescharacteristic
578C2×C4#49511yescharacteristic
579C2×C4#49611yescharacteristic
580C2×C4#49711yescharacteristic
581C8—44no—
582C8—44no—
583C8—82no—
584C22#143811yescharacteristic
585C22#164012yes—
586C22#164212yes—
587C22#164312yes—
588C22#164512yes—
589C22#164812yes—
590C22#165012yes—
591C22#165112yes—
592C22#165312yes—
593C22#187211yescharacteristic
594C22#187311yescharacteristic
595C22#188012yes—
596C22#188212yes—
597C22#200012yes—
598C22#200011yescharacteristic
599C22#200011yescharacteristic
600C22#200111yescharacteristic
601C22#200111yescharacteristic
602C22#200411yescharacteristic
603C22#200511yescharacteristic
604C22#200712yes—
605C22#201211yescharacteristic
606C22#201311yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—44no—
612C4—82no—
613C4—84no—
614C4—84no—
615C4—162no—
616C4#178711yescharacteristic
617C4#178811yescharacteristic
618C4#187712yes—
619C4#189211yescharacteristic
620C4#189411yescharacteristic
621C4#196811yescharacteristic
622C4#197311yescharacteristic
623C4#201411yescharacteristic
624C4#201511yescharacteristic
625C4#203511yescharacteristic
626C4#203511yescharacteristic
627C4#203611yescharacteristic
628C4#203611yescharacteristic
629C4#205512yes—
630C2#898212yes—
631C2#898312yes—
632C2#1007111yescharacteristic
633C2#1007311yescharacteristic
634C2#1007511yescharacteristic
635C2#1007711yescharacteristic
636C2#1023212yes—
637C2#1023412yes—
638C2#1097011yescharacteristic
639C2#1101711yescharacteristic
640C2#1102311yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
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
15. C2×#475C4semidirect
337. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#19902
C2×#19812
C22×Q8⋊C82
#139262
C2×#12962
C2×#12792