Finite Group Databasebeta

← #204397

Subgroups — #204397

Order 512.

1187 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C211yesmaximal, characteristic
2#21378C211yesmaximal, characteristic, complemented
3#21190C211yesmaximal, characteristic, complemented
4C2×#1574C211yesmaximal, characteristic, complemented
5C2×#1495C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9666C211yesmaximal, characteristic, complemented
9#9665C211yesmaximal, characteristic, complemented
10#9663C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9660C211yesmaximal, characteristic, complemented
13#9659C211yesmaximal, characteristic, complemented
14#9657C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3105—24no—
19#3105—24no—
20C2×#484C411yescharacteristic
21C2×#481C411yescharacteristic
22C2×#478C411yescharacteristic
23C2×#478C411yescharacteristic
24C2×#463C411yescharacteristic
25C2×#463C411yescharacteristic
26C2×#450C411yescharacteristic
27C2×#447C411yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×C4×SD16C2211yescharacteristic
30C2×C4×D8C2211yescharacteristic
31C2×C4×D8C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#423C2211yescharacteristic
36C2×#407—22no—
37C2×C4×M16C2211yescharacteristic
38#1984C2211yescharacteristic, complemented
39#1984C2211yescharacteristic, complemented
40#1955—22no—
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1954C2211yescharacteristic, complemented
46#1954C2211yescharacteristic, complemented
47#1870—22no—
48#1750—22no—
49#1574—24no—
50C2×#335C2211yescharacteristic, complemented
51C2×#335C2211yescharacteristic, complemented
52C2×#334C2211yescharacteristic, complemented
53C2×#334C2211yescharacteristic, complemented
54#1495—24no—
55#1478—24no—
56C2×Q8⋊C8C2211yescharacteristic, complemented
57C2×Q8⋊C8C2211yescharacteristic, complemented
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic, complemented
63C2×D4⋊C8C2211yescharacteristic, complemented
64#1453—24no—
65C2×#322C2211yescharacteristic, complemented
66C2×#322C2211yescharacteristic, complemented
67C2×#322C2211yescharacteristic, complemented
68C2×C82C2211yescharacteristic, complemented
69C23×D4C2×C411yescharacteristic
70C24×C4—21no—
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×D8C2×C411yescharacteristic
74C22×D8C2×C411yescharacteristic
75C22×M16—21no—
76C22×M16D412yes—
77#529—22no—
78#529—22no—
79#529—22no—
80#529—22no—
81C2×#126—21no—
82C2×#125C2×C411yescharacteristic
83C2×#124C2×C411yescharacteristic
84C2×#123—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—21no—
94C2×C4×D4—22no—
95C2×C4×D4C2×C411yescharacteristic
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×#117—21no—
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C22⋊C4—21no—
103C22×C42C2311yescharacteristic
104#484—24no—
105#481—24no—
106#478—24no—
107#478—24no—
108#463—24no—
109#463—24no—
110#450—24no—
111#447—24no—
112C4×SD16—22no—
113C4×SD16—22no—
114C4×SD16—22no—
115C4×SD16—22no—
116C4×D8—22no—
117C4×D8—22no—
118C4×D8—22no—
119C4×D8—22no—
120#431—22no—
121#431—22no—
122#431—22no—
123#431D414yes—
124#431D414yes—
125#431D414yes—
126C2×#107C2×C411yescharacteristic
127C2×#107C2×C411yescharacteristic
128C2×#106C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130#423—22no—
131#423—22no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8C2×C411yescharacteristic
135C2×C4⋊C8C2×C411yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#103C2×C411yescharacteristic
142C2×#102C2×C411yescharacteristic
143C2×#102C2×C411yescharacteristic
144C2×#102C2×C411yescharacteristic
145C2×#102C2×C411yescharacteristic
146C2×#102C2×C411yescharacteristic
147C2×#102C2×C411yescharacteristic
148#407—48no—
149C2×C22⋊C8—21no—
150C2×C22⋊C8—21no—
151C2×C22⋊C8—22no—
152C2×C22⋊C8—22no—
153C2×C22⋊C8—22no—
154C2×C22⋊C8—22no—
155C2×C22⋊C8—22no—
156C2×C22⋊C8—22no—
157C2×C22⋊C8C2×C411yescharacteristic
158C2×C22⋊C8C2×C411yescharacteristic
159C4×M16—22no—
160C4×M16—22no—
161C2×#97D412yes—
162C2×C4×C8—21no—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#336—22no—
170#336—22no—
171#335—22no—
172#335—22no—
173#334—22no—
174#334—22no—
175Q8⋊C8—22no—
176Q8⋊C8—22no—
177D4⋊C8—22no—
178D4⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184#322—22no—
185#322—22no—
186C82—22no—
187C25—21no—
188C22×Q8C22×C411yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—21no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4C22×C411yescharacteristic
198C22×D4C22×C411yescharacteristic
199C22×D4C22×C411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4#4111yescharacteristic
213C2×SD16—22no—
214C2×SD16—22no—
215C2×SD16—22no—
216C2×SD16—22no—
217C2×SD16—41no—
218C2×SD16—41no—
219C2×D8—22no—
220C2×D8—22no—
221C2×D8—22no—
222C2×D8—22no—
223C2×D8—41no—
224C2×D8—41no—
225C2×M16—24no—
226C2×M16—24no—
227C2×M16—42no—
228C2×M16—44no—
229C2×M16—42no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8C22×C411yescharacteristic
237C22×C8C22×C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242#126—42no—
243#126—42no—
244#125—22no—
245#125—22no—
246#124—22no—
247#124—22no—
248#123—42no—
249#123—42no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252#121—22no—
253#121—22no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—42no—
280#117—42no—
281#117—42no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C42—21no—
305C2×C42C22×C411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42#4112yes—
311C2×C42#4111yescharacteristic
312C2×C42C2411yescharacteristic
313#107—22no—
314#107—22no—
315#106—22no—
316#106—22no—
317C4⋊C8—24no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—42no—
327C4⋊C8C22⋊C414yes—
328C4⋊C8C22⋊C414yes—
329#103—22no—
330#103—22no—
331#102—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336#102—22no—
337C22⋊C8—24no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340C22⋊C8—44no—
341C22⋊C8—42no—
342C22⋊C8—44no—
343C22⋊C8—44no—
344C22⋊C8—44no—
345C22⋊C8C22⋊C414yes—
346C22⋊C8C22⋊C414yes—
347#97—24no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—42no—
355C4×C8C2×D412yes—
356C4×C8C2×D412yes—
357C4×C8C2×D412yes—
358C4×C8C2×D412yes—
359C4×C8C2×D412yes—
360C4×C8C2×D412yes—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—42no—
365C24—42no—
366C24—44no—
367C24—42no—
368C24—82no—
369C24C4×D411yescharacteristic
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—82no—
401C2×D4#13112yes—
402C2×D4#13112yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4C2×C22⋊C412yes—
437C22×C4C4×D412yes—
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4#12112yes—
442C22×C4#12112yes—
443C22×C4#12412yes—
444C22×C4#12712yes—
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4111yescharacteristic
448SD16—44no—
449SD16—44no—
450D8—44no—
451D8—44no—
452M16—48no—
453M16—84no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8C2×C22⋊C412yes—
480C2×C8C2×C22⋊C412yes—
481C2×C8C2×C22⋊C412yes—
482C2×C8C2×C22⋊C412yes—
483C2×C8AGL(1,ℤ/8)14yes—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C42—41no—
511C42—41no—
512C42C2×C22⋊C412yes—
513C42C4×D412yes—
514C42#12112yes—
515C42#12112yes—
516C42#12412yes—
517C42#12712yes—
518C42C22×D411yescharacteristic
519C42C22×D411yescharacteristic
520C42C22×D411yescharacteristic
521C42C2×#4111yescharacteristic
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—48no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—84no—
543C23—84no—
544C23—88no—
545C23—88no—
546C23#39012yes—
547C23C2×C4×D411yescharacteristic
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4—48no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—84no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4#39012yes—
604C2×C4#39012yes—
605C2×C4#39012yes—
606C2×C4C22×C22⋊C411yescharacteristic
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12411yescharacteristic
611C2×C4C2×#12711yescharacteristic
612C2×C4C2×#13111yescharacteristic
613C2×C4C2×#13111yescharacteristic
614C2×C4#56811yescharacteristic
615C2×C4#56811yescharacteristic
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4#57512yes—
618C8—22no—
619C8—22no—
620C8—22no—
621C8—22no—
622C8—24no—
623C8—42no—
624C8—42no—
625C8—42no—
626C8—81no—
627C8—81no—
628C8—81no—
629C8—81no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—816no—
640C22—82no—
641C22—82no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—82no—
647C22—82no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22C2×#39011yescharacteristic
652C22#293311yescharacteristic
653C22#293311yescharacteristic
654C22C4×AGL(1,ℤ/8)12yes—
655C22#295712yes—
656C4—22no—
657C4—22no—
658C4—21no—
659C4—21no—
660C4—22no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4#288011yescharacteristic
667C4#288011yescharacteristic
668C4#292911yescharacteristic
669C4#293011yescharacteristic
670C4#305512yes—
671C4#315111yescharacteristic
672C4#315311yescharacteristic
673C2—42no—
674C2—82no—
675C2—82no—
676C2—161no—
677C2#1800612yes—
678C2#1800812yes—
679C2#2119111yescharacteristic
680C2#2123611yescharacteristic
681C2#2124011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#447C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21378C2semidirect
3. #21190C2semidirect
4. C2×#1574C2semidirect
5. C2×#1495C2semidirect
6. C2×#1478C2semidirect
7. C2×#1453C2semidirect
8. #9666C2semidirect
9. #9665C2semidirect
10. #9663C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9659C2semidirect
14. #9657C2semidirect
15. #9655C2semidirect
38. #1984C22semidirect
39. #1984C22semidirect
45. #1954C22semidirect
46. #1954C22semidirect
50. C2×#335C22semidirect
51. C2×#335C22semidirect
52. C2×#334C22semidirect
53. C2×#334C22semidirect
56. C2×Q8⋊C8C22semidirect
57. C2×Q8⋊C8C22semidirect
62. C2×D4⋊C8C22semidirect
63. C2×D4⋊C8C22semidirect
65. C2×#322C22semidirect
66. C2×#322C22semidirect
67. C2×#322C22semidirect
68. C2×C82C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
#213782
#211902
C2×#15742
C2×#14952
C2×#14782
C2×#14532
#96662
#96652
#96632
#96612
#96602
#96592
#96572
#96552