Finite Group Databasebeta

← #115490

Subgroups — #115490

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C22×Q8⋊C8C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C23×Q16—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#462—22no—
12C2×#462—22no—
13C2×#462—22no—
14C2×#462—22no—
15C2×#462—22no—
16C2×#462C414yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—21no—
20C22×C4×C8C2211yescharacteristic
21C2×#391—21no—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990—22no—
25#1990—22no—
26#1988—22no—
27#1988—22no—
28C2×#340—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
40C22×Q16—24no—
41C22×Q16—24no—
42C22×Q16—24no—
43C22×Q16—42no—
44C22×Q16—42no—
45C22×Q16—42no—
46C22×Q16—42no—
47C23×C8—21no—
48C23×C8—21no—
49C23×C8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8D412yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C4⋊C4D412yes—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#462—28no—
86#462—28no—
87#462—28no—
88#462—28no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C412yes—
98C2×C4⋊C8D412yes—
99C2×C4⋊C8D412yes—
100C2×C4⋊C8D412yes—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8D412yes—
110#391—42no—
111#391—42no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C2×#95—24no—
121C2×#95—21no—
122C2×#95—24no—
123#340—28no—
124#340—28no—
125Q8⋊C8—28no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—41no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×Q8—41no—
144C22×Q8—41no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8C22⋊C412yes—
152C22×Q8D812yes—
153C22×Q8SD1612yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×Q16—216no—
163C2×Q16—48no—
164C2×Q16—48no—
165C2×Q16—48no—
166C2×Q16—48no—
167C2×Q16—48no—
168C2×Q16—48no—
169C22×C8—24no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—41no—
177C22×C8—41no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8—41no—
184C22×C8—42no—
185C22×C8—42no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C4⋊Q8C22⋊C414yes—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4Q1614yescomplemented
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42#4111yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—41no—
291C4⋊C8—41no—
292C4⋊C8C22⋊C414yes—
293#103—216no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—41no—
297C4×C8—41no—
298#95—48no—
299#95—41no—
300#95—41no—
301#95—84no—
302C24#12011yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—42no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8—81no—
358C2×Q8#10214yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12311yescharacteristic
408C22×C4#12711yescharacteristic
409C22×C4#12711yescharacteristic
410C22×C4C2×D811yescharacteristic
411C22×C4C2×SD1611yescharacteristic
412C22×C4C2×Q1612yes—
413C22×C4#13511yescharacteristic
414C22×C4#13511yescharacteristic
415C22×C4#13512yes—
416C22×C4AGL(1,ℤ/8)11yescharacteristic
417C22×C4AGL(1,ℤ/8)11yescharacteristic
418C22×C4AGL(1,ℤ/8)11yescharacteristic
419C22×C4#13711yescharacteristic
420C22×C4#13712yes—
421C22×C4#13712yes—
422Q16—432no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—44no—
426C2×C8—42no—
427C2×C8—44no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—44no—
437C2×C8—82no—
438C2×C8—81no—
439C2×C8—81no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—81no—
474C42—81no—
475C42—81no—
476C42—81no—
477C42C2×C22⋊C411yescharacteristic
478C42C2×C22⋊C411yescharacteristic
479C42C4×D411yescharacteristic
480C42C4×D411yescharacteristic
481C42#12011yescharacteristic
482C42#12011yescharacteristic
483C42#12011yescharacteristic
484C42#12111yescharacteristic
485C42#12111yescharacteristic
486C42#12111yescharacteristic
487C42#12111yescharacteristic
488C42#12311yescharacteristic
489C42#12311yescharacteristic
490C42#12311yescharacteristic
491C42#12411yescharacteristic
492C42#12411yescharacteristic
493C23#38611yescharacteristic
494C23#39211yescharacteristic
495C23#39211yescharacteristic
496C23#44711yescharacteristic
497C23#44811yescharacteristic
498C23#44911yescharacteristic
499C23#44911yescharacteristic
500C23#45112yes—
501C23#45212yes—
502C23#45512yes—
503C23#45612yes—
504Q8—24no—
505Q8—24no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—44no—
513Q8—44no—
514Q8—82no—
515Q8—82no—
516Q8—82no—
517Q8—82no—
518Q8—82no—
519Q8—82no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#37911yescharacteristic
560C2×C4#38611yescharacteristic
561C2×C4#38611yescharacteristic
562C2×C4#38611yescharacteristic
563C2×C4#39011yescharacteristic
564C2×C4#39011yescharacteristic
565C2×C4#39211yescharacteristic
566C2×C4#39211yescharacteristic
567C2×C4#39411yescharacteristic
568C2×C4#39411yescharacteristic
569C2×C4#39411yescharacteristic
570C2×C4#39411yescharacteristic
571C2×C4C2×#10211yescharacteristic
572C2×C4#41611yescharacteristic
573C2×C4#41711yescharacteristic
574C2×C4#41811yescharacteristic
575C2×C4C4×Q1612yes—
576C2×C4#44112yes—
577C2×C4#44711yescharacteristic
578C2×C4#44711yescharacteristic
579C2×C4#44911yescharacteristic
580C2×C4#44911yescharacteristic
581C2×C4#45011yescharacteristic
582C2×C4#45011yescharacteristic
583C2×C4#45112yes—
584C2×C4#45212yes—
585C2×C4#45211yescharacteristic
586C2×C4#45211yescharacteristic
587C2×C4#46212yes—
588C2×C4#46212yes—
589C2×C4#46412yes—
590C2×C4#46412yes—
591C2×C4#46511yescharacteristic
592C2×C4#46511yescharacteristic
593C2×C4#46611yescharacteristic
594C2×C4#46611yescharacteristic
595C2×C4#46712yes—
596C2×C4#46811yescharacteristic
597C2×C4#46811yescharacteristic
598C2×C4#46911yescharacteristic
599C2×C4#46911yescharacteristic
600C2×C4#47012yes—
601C2×C4#48011yescharacteristic
602C2×C4#48111yescharacteristic
603C2×C4#48211yescharacteristic
604C2×C4#48211yescharacteristic
605C2×C4#49411yescharacteristic
606C2×C4#49511yescharacteristic
607C2×C4#49711yescharacteristic
608C2×C4#49711yescharacteristic
609C8—44no—
610C8—44no—
611C8—82no—
612C22#143811yescharacteristic
613C22#164012yes—
614C22#164112yes—
615C22#164412yes—
616C22#164512yes—
617C22#164812yes—
618C22#164912yes—
619C22#165212yes—
620C22#165312yes—
621C22#187411yescharacteristic
622C22#187511yescharacteristic
623C22#187812yes—
624C22#188212yes—
625C22#199911yescharacteristic
626C22#199911yescharacteristic
627C22#200011yescharacteristic
628C22#200011yescharacteristic
629C22#200112yes—
630C22#200411yescharacteristic
631C22#200511yescharacteristic
632C22#200712yes—
633C22#201511yescharacteristic
634C22#201611yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—44no—
640C4—82no—
641C4—84no—
642C4—84no—
643C4—162no—
644C4#178611yescharacteristic
645C4#178811yescharacteristic
646C4#187812yes—
647C4#189311yescharacteristic
648C4#189511yescharacteristic
649C4#196911yescharacteristic
650C4#197111yescharacteristic
651C4#201511yescharacteristic
652C4#201611yescharacteristic
653C4#203311yescharacteristic
654C4#203311yescharacteristic
655C4#203411yescharacteristic
656C4#203411yescharacteristic
657C4#205612yes—
658C2#898612yes—
659C2#898712yes—
660C2#1006311yescharacteristic
661C2#1006511yescharacteristic
662C2#1006711yescharacteristic
663C2#1006911yescharacteristic
664C2#1023312yes—
665C2#1023412yes—
666C2#1096911yescharacteristic
667C2#1101711yescharacteristic
668C2#1102411yescharacteristic
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
16. C2×#462C4semidirect
260. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#19902
C2×#19882
C22×Q8⋊C82
#139262
C2×#12962
C2×#12792