Finite Group Databasebeta

← #204415

Subgroups — #204415

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3106C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic, complemented
3#21789C211yesmaximal, characteristic, complemented
4C2×#1573C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic, complemented
6C2×#1488C211yesmaximal, characteristic, complemented
7C2×#1454C211yesmaximal, 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×#532C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18#3106—24no—
19#3106—24no—
20C2×#482C411yescharacteristic
21C2×#482C411yescharacteristic
22C2×#477C411yescharacteristic
23C2×#476C411yescharacteristic
24C2×#460C411yescharacteristic
25C2×#459C411yescharacteristic
26C2×#449C411yescharacteristic
27C2×#449C411yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×C4×SD16C2211yescharacteristic
30C2×C4×D8C2211yescharacteristic
31C2×C4×D8C2211yescharacteristic
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37#1984C2211yescharacteristic
38#1984C2211yescharacteristic
39#1955—22no—
40#1954C2211yescharacteristic
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1954C2211yescharacteristic
46#1869—22no—
47#1797—22no—
48#1780—22no—
49#1573—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1488—24no—
55#1480—24no—
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1454—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C22×#41—21no—
70C22×#41C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×D8C2×C411yescharacteristic
74C22×D8C2×C411yescharacteristic
75C23×C8—21no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#127C2×C411yescharacteristic
82C2×#124—21no—
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C4×#41—22no—
87C4×#41—22no—
88C4×#41—22no—
89C4×#41—22no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2×C411yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C2×#117C2×C411yescharacteristic
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103#482—24no—
104#482—24no—
105#477—24no—
106#476—24no—
107#460—24no—
108#459—24no—
109#449—24no—
110#449—24no—
111C4×SD16—22no—
112C4×SD16—22no—
113C4×SD16—22no—
114C4×SD16—22no—
115C4×D8—22no—
116C4×D8—22no—
117C4×D8—22no—
118C4×D8—22no—
119#432—22no—
120#432—22no—
121#431—22no—
122#431—22no—
123#431—22no—
124#431D414yes—
125#431D414yes—
126#431D414yes—
127C2×#107C2×C411yescharacteristic
128C2×#107C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130C2×#106C2×C411yescharacteristic
131C2×C4⋊C8—21no—
132C2×C4⋊C8—22no—
133C2×C4⋊C8—22no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8C2×C411yescharacteristic
136C2×C4⋊C8C2×C411yescharacteristic
137C2×C4⋊C8D412yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#102C2×C411yescharacteristic
145C2×#102C2×C411yescharacteristic
146C2×#102C2×C411yescharacteristic
147C2×#102C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×C22⋊C8—21no—
151C2×C22⋊C8—21no—
152C2×C22⋊C8—22no—
153C2×C22⋊C8—22no—
154C2×C22⋊C8—22no—
155C2×C22⋊C8—22no—
156C2×C22⋊C8—22no—
157C2×C22⋊C8—22no—
158C2×C22⋊C8C2×C411yescharacteristic
159C2×C22⋊C8C2×C411yescharacteristic
160C2×C22⋊C8D412yes—
161C2×#97—21no—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168#336—22no—
169#336—22no—
170#335—22no—
171#335—22no—
172#334—22no—
173#334—22no—
174Q8⋊C8—22no—
175Q8⋊C8—22no—
176D4⋊C8—22no—
177D4⋊C8—22no—
178D4⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182#322—22no—
183#322—22no—
184#322—22no—
185C82—22no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—41no—
191C2×#41—41no—
192C2×#41—42no—
193C2×#41—42no—
194C2×#41—82no—
195C2×#41—82no—
196C22×Q8—21no—
197C22×Q8C22×C411yescharacteristic
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C22×C411yescharacteristic
204C22×D4C22×C411yescharacteristic
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4#4111yescharacteristic
214C2×SD16—22no—
215C2×SD16—22no—
216C2×SD16—22no—
217C2×SD16—22no—
218C2×SD16—41no—
219C2×SD16—41no—
220C2×D8—22no—
221C2×D8—22no—
222C2×D8—22no—
223C2×D8—22no—
224C2×D8—41no—
225C2×D8—41no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—42no—
233C22×C8—82no—
234C22×C8C22×C411yescharacteristic
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C22×C411yescharacteristic
238C22×C8C2×D412yes—
239C22×C8C2×D412yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242#127—22no—
243#127—22no—
244#124—42no—
245#124—42no—
246#121—22no—
247#121—22no—
248#121—22no—
249#121—22no—
250#121—42no—
251#121—42no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—21no—
261C4×D4—21no—
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—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278#117—22no—
279#117—22no—
280#117—42no—
281#117—42no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D411yescharacteristic
306C2×C42#4112yes—
307C2×C42#4111yescharacteristic
308C2×C42C2411yescharacteristic
309#107—22no—
310#107—22no—
311#106—22no—
312#106—22no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—42no—
324C4⋊C8C22⋊C414yes—
325C4⋊C8C22⋊C414yes—
326#103—22no—
327#103—22no—
328#102—22no—
329#102—22no—
330#102—22no—
331#102—22no—
332#102—22no—
333#102—22no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—42no—
337C22⋊C8—82no—
338C22⋊C8—82no—
339C22⋊C8—82no—
340C22⋊C8—82no—
341C22⋊C8—82no—
342C22⋊C8—82no—
343C22⋊C8C22⋊C414yes—
344C22⋊C8C22⋊C414yes—
345#97—42no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8C2×D412yes—
353C4×C8C2×D412yes—
354C4×C8C2×D412yes—
355C4×C8C2×D412yes—
356C4×C8C2×D412yes—
357C4×C8C2×D412yes—
358C24—21no—
359C24—21no—
360C24—21no—
361C24C4×D411yescharacteristic
362#41—44no—
363#41—44no—
364#41—44no—
365#41—44no—
366#41—84no—
367#41—84no—
368#41—88no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—42no—
375C2×Q8—82no—
376C2×Q8#13112yes—
377C2×D4—24no—
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—21no—
394C2×D4—21no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—82no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4#13112yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4C2×C22⋊C412yes—
436C22×C4C4×D412yes—
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4#12112yes—
441C22×C4#12112yes—
442C22×C4#12412yes—
443C22×C4#12712yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447SD16—44no—
448SD16—44no—
449D8—44no—
450D8—44no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—84no—
478C2×C8C2×C22⋊C412yes—
479C2×C8C2×C22⋊C412yes—
480C2×C8C2×C22⋊C412yes—
481C2×C8C2×C22⋊C412yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C42—41no—
505C42—41no—
506C42—42no—
507C42—42no—
508C42C2×C22⋊C412yes—
509C42C4×D412yes—
510C42#12112yes—
511C42#12112yes—
512C42#12412yes—
513C42#12712yes—
514C42C22×D411yescharacteristic
515C42C22×D411yescharacteristic
516C42C22×D411yescharacteristic
517C42C2×#4111yescharacteristic
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23#39012yes—
538C23C2×C4×D411yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—84no—
544D4—22no—
545D4—22no—
546D4—48no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—84no—
558D4—84no—
559D4—84no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#39012yes—
598C2×C4#39012yes—
599C2×C4#39012yes—
600C2×C4C22×C22⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12411yescharacteristic
605C2×C4C2×#12711yescharacteristic
606C2×C4C2×#13111yescharacteristic
607C2×C4C2×#13111yescharacteristic
608C2×C4#56811yescharacteristic
609C2×C4#56811yescharacteristic
610C2×C4#57612yes—
611C2×C4#57712yes—
612C8—22no—
613C8—22no—
614C8—22no—
615C8—22no—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C8—81no—
621C8—81no—
622C8—81no—
623C8—81no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C2×#39011yescharacteristic
645C22#292911yescharacteristic
646C22#293011yescharacteristic
647C22#294612yes—
648C22#295712yes—
649C4—22no—
650C4—22no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—42no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4#288011yescharacteristic
660C4#288011yescharacteristic
661C4#293311yescharacteristic
662C4#293311yescharacteristic
663C4#306212yes—
664C4#315411yescharacteristic
665C4#315511yescharacteristic
666C2—42no—
667C2—82no—
668C2—161no—
669C2—161no—
670C2#1792012yes—
671C2#1794812yes—
672C2#2119111yescharacteristic
673C2#2123511yescharacteristic
674C2#2124011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#3106C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21847C2semidirect
3. #21789C2semidirect
4. C2×#1573C2semidirect
5. C2×#1480C2semidirect
6. C2×#1488C2semidirect
7. C2×#1454C2semidirect
8. #9666C2semidirect
9. #9665C2semidirect
10. #9663C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9659C2semidirect
14. #9657C2semidirect
15. #9655C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31062
#218472
#217892
C2×#15732
C2×#14802
C2×#14882
C2×#14542
#96662
#96652
#96632
#96612
#96602
#96592
#96572
#96552