Finite Group Databasebeta

← #6447692

Subgroups — #6447692

Order 512.

3474 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#3142C211yesmaximal, characteristic, complemented
3C2×#3036C211yesmaximal, characteristic
4#34955C211yesmaximal, characteristic, complemented
5#25946C214yesmaximal, complemented
6#25945C214yesmaximal, complemented
7#25938C218yesmaximal, complemented
8#23453C212yesmaximal, complemented
9#23452C212yesmaximal, complemented
10#21314C211yesmaximal, characteristic, complemented
11#21303C211yesmaximal, characteristic, complemented
12#20925C212yesmaximal, complemented
13#20925C212yesmaximal, complemented
14C4×#418C211yesmaximal, characteristic, complemented
15C22×AGL(1,ℤ/8)C2212yes—
16C2×#554—28no—
17C2×#554—22no—
18C2×#554—22no—
19C2×#554—28no—
20C2×#554C2214yes—
21C2×#554C2214yes—
22C2×#532C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#125C2211yescharacteristic
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8—21no—
28C22×C22⋊Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C22×C4×D4C2211yescharacteristic
33C23×C4⋊C4—21no—
34#3142—28no—
35C2×#491C2212yescomplemented
36C2×#490C2212yescomplemented
37C2×#488C2214yescomplemented
38#3036—28no—
39C2×#463C2214yes—
40C2×#461C2212yes—
41C2×#459C2212yes—
42C2×#423C2211yescharacteristic
43C2×#417C2212yes—
44C2×#417C2212yes—
45C2×#417C2212yes—
46C2×#418C2212yes—
47C2×C4×M16C2211yescharacteristic
48#2592C2214yescomplemented
49#2591—21no—
50#2591—21no—
51#2568C2212yescomplemented
52#2548C2212yescomplemented
53#2504—21no—
54#2436—21no—
55#2398—22no—
56C4×#124—22no—
57#2297C2212yes—
58#2297C2212yes—
59C4×#117C2211yescharacteristic
60#2026C2218yescomplemented
61#2025C2218yescomplemented
62#2024C2218yescomplemented
63#2023C2218yescomplemented
64#1989C2214yescomplemented
65#1989C2214yescomplemented
66#1985C2214yescomplemented
67#1984C2214yescomplemented
68#1879C2218yes—
69#1879C2218yes—
70#1877C2218yes—
71#1877C2218yes—
72#1762C2214yes—
73C4×#102C2214yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1728C2212yes—
78#1728C2212yes—
79C22×#41C2312yes—
80C23×Q8—21no—
81C23×Q8D411yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4—21no—
86C2×AGL(1,ℤ/8)—28no—
87C2×AGL(1,ℤ/8)—216no—
88C22×SD16C2314yes—
89C22×D8C2314yes—
90C22×M16C2312yes—
91C22×M16C2312yes—
92#554—232no—
93#554—232no—
94#554—216no—
95#554—216no—
96#554—216no—
97#554—216no—
98#532—24no—
99#532—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×#127C2311yescharacteristic
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—22no—
109C2×#125—21no—
110C2×#125—21no—
111C2×#125—24no—
112C2×#125D418yes—
113C2×#125C2314yes—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#124—22no—
117C2×#124C2312yes—
118C2×C22⋊Q8—216no—
119C2×C22⋊Q8—216no—
120C2×C22⋊Q8—216no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—24no—
127C2×#121—21no—
128C2×#121—21no—
129C2×C4×Q8—24no—
130C2×C4×Q8—24no—
131C2×C4×Q8D412yes—
132C2×C4×Q8D412yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×D4—24no—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4—24no—
139C2×C4×D4—22no—
140C2×C4×D4—22no—
141C2×C4×D4—21no—
142C2×C4×D4—21no—
143C2×C4×D4—24no—
144C2×C4×D4—21no—
145C2×C4×D4—21no—
146C2×C4×D4D418yes—
147C2×C4×D4C2314yes—
148C2×C4×D4C2314yes—
149C2×C4×D4C2312yes—
150C2×C4×D4C2312yes—
151C2×#117C2312yes—
152C22×C4⋊C4—216no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4—24no—
159C22×C4⋊C4—24no—
160C22×C4⋊C4D411yescharacteristic
161C22×C4⋊C4D411yescharacteristic
162C22×C4⋊C4D411yescharacteristic
163C22×C4⋊C4C2311yescharacteristic
164C22×C4⋊C4C2311yescharacteristic
165C22×C4⋊C4C2311yescharacteristic
166C22×C22⋊C4—21no—
167C22×C22⋊C4—21no—
168C22×C22⋊C4—21no—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171C22×C42C2311yescharacteristic
172#491—28no—
173#490—28no—
174#488—216no—
175#463—216no—
176#461—28no—
177#459—28no—
178#423—24no—
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2312yes—
181#418—28no—
182#417—28no—
183#417—28no—
184#417—28no—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2314yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×#102C2318yes—
191C2×#102C2314yes—
192C4×M16—24no—
193C2×#97C2312yes—
194C2×C4×C8C2312yes—
195#400—22no—
196#400—22no—
197#397—24no—
198#394—22no—
199#394—22no—
200#393—24no—
201#391—22no—
202#391C2314yes—
203#391C2314yes—
204#390—22no—
205#390C2314yes—
206#390C2314yes—
207#388—24no—
208#388—24no—
209#386—22no—
210#386—22no—
211#386—22no—
212#386—22no—
213#382—22no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—24no—
220#376—21no—
221C43—21no—
222#340C2318yes—
223#340C2318yes—
224#340C2318yes—
225#336C2318yes—
226#336C2318yes—
227C25—21no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—28no—
231C2×#41—28no—
232C22×Q8—28no—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—44no—
236C22×Q8—84no—
237C22×Q8C2×D412yes—
238C22×Q8C2×D412yes—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—24no—
246C22×D4C2×D414yes—
247C22×D4#4118yes—
248C22×D4C2412yes—
249C22×D4C2412yes—
250C23×C4—28no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—28no—
255C23×C4—24no—
256C23×C4—28no—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—48no—
262C23×C4—84no—
263C23×C4—84no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271AGL(1,ℤ/8)—432no—
272C2×SD16—28no—
273C2×SD16—216no—
274C2×D8—28no—
275C2×D8—216no—
276C2×M16—24no—
277C2×M16—24no—
278C2×M16—28no—
279C2×M16—24no—
280C2×M16—24no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—216no—
284C4⋊Q8—28no—
285C4⋊Q8—28no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288#127—22no—
289#127—22no—
290#125—216no—
291#125—28no—
292#125—28no—
293#125—28no—
294#125—28no—
295#125—24no—
296#125—24no—
297#125C2×D418yes—
298#125C2×D418yes—
299#124—24no—
300#124—24no—
301#124—48no—
302C22⋊Q8—232no—
303C22⋊Q8—232no—
304C22⋊Q8—232no—
305C22⋊Q8—232no—
306C22⋊Q8—232no—
307C22⋊Q8—232no—
308#121—44no—
309#121—44no—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×D4—216no—
317C4×D4—216no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—28no—
322C4×D4—28no—
323C4×D4—28no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4C2×D418yes—
329C4×D4C2×D418yes—
330#117—24no—
331#117—24no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—232no—
335C2×C4⋊C4—232no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D414yes—
396C2×C42C2×D414yes—
397C2×C42C2412yes—
398C2×C42C2412yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C4⋊C8—24no—
402C4⋊C8—24no—
403#103—28no—
404#103—28no—
405#103—28no—
406#102—216no—
407#102—28no—
408#102—28no—
409#102—28no—
410#97—24no—
411C4×C8—24no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418C24—24no—
419C24—28no—
420C24—22no—
421C24—22no—
422C24—44no—
423C24—44no—
424C24C22×D411yescharacteristic
425#41—48no—
426#41—48no—
427#41—416no—
428C2×Q8—216no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—416no—
440C2×Q8—88no—
441C2×Q8—816no—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—216no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—48no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4#124116yescomplemented
459C2×D4C22×D414yes—
460C2×D4C2×#4118yes—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—216no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—216no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—432no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4—816no—
495C22×C4—816no—
496C22×C4#12014yes—
497C22×C4#12014yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4114yes—
510C22×C42−1+414yes—
511C22×C4C2511yescharacteristic
512SD16—416no—
513D8—416no—
514M16—48no—
515M16—48no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—28no—
520C2×C8—24no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—216no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4C22×D414yes—
545C4⋊C4C22×D414yes—
546C4⋊C4C22×D414yes—
547C22⋊C4—216no—
548C22⋊C4—216no—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—48no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—44no—
565C42—44no—
566C42—82no—
567C42#12014yes—
568C42#12014yes—
569C42C22×D412yes—
570C42C22×D412yes—
571C42C22×D412yes—
572C42C22×D412yes—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—216no—
577C23—28no—
578C23—48no—
579C23—48no—
580C23—432no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—44no—
586C23—44no—
587C23C2×#12012yes—
588C23#54814yes—
589C23C23×D411yescharacteristic
590Q8—48no—
591Q8—48no—
592Q8—42no—
593Q8—42no—
594Q8—44no—
595Q8—816no—
596D4—24no—
597D4—216no—
598D4—28no—
599D4—42no—
600D4—42no—
601D4—44no—
602D4—84no—
603D4—84no—
604D4—88no—
605D4C2×#12418yes—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—432no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4—816no—
643C2×C4—88no—
644C2×C4C2×#12012yes—
645C2×C4C2×#12012yes—
646C2×C4C2×#12012yes—
647C2×C4C2×#12012yes—
648C2×C4C2×#12012yes—
649C2×C4C2×#12414yes—
650C2×C4#53614yes—
651C2×C4#54814yes—
652C2×C4#54818yes—
653C2×C4#57514yes—
654C2×C4C23×D411yescharacteristic
655C2×C4C23×D411yescharacteristic
656C2×C4C22×#4112yes—
657C2×C4C2×2−1+412yes—
658C8—44no—
659C8—44no—
660C22—28no—
661C22—22no—
662C22—22no—
663C22—28no—
664C22—432no—
665C22—416no—
666C22—44no—
667C22—44no—
668C22—88no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#243614yes—
673C22C22×#12011yescharacteristic
674C22C2×#54812yes—
675C22C2×#57512yes—
676C22#358512yes—
677C4—24no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—44no—
688C4—84no—
689C4—82no—
690C4#243614yes—
691C4C22×#12411yescharacteristic
692C4C2×#53611yescharacteristic
693C4C2×#54812yes—
694C2—24no—
695C2—48no—
696C2—162no—
697C2C2×#243611yescharacteristic
698C2#6035911yescharacteristic
699C2#6036711yescharacteristic
700C2#6131814yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3036C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#554C2semidirect
2. C2×#3142C2semidirect
4. #34955C2semidirect
5. #25946C2semidirect
6. #25945C2semidirect
7. #25938C2semidirect
8. #23453C2semidirect
9. #23452C2semidirect
10. #21314C2semidirect
11. #21303C2semidirect
12. #20925C2semidirect
13. #20925C2semidirect
14. C4×#418C2semidirect
35. C2×#491C22semidirect
36. C2×#490C22semidirect
37. C2×#488C22semidirect
48. #2592C22semidirect
51. #2568C22semidirect
52. #2548C22semidirect
60. #2026C22semidirect
61. #2025C22semidirect
62. #2024C22semidirect
63. #2023C22semidirect
64. #1989C22semidirect
65. #1989C22semidirect
66. #1985C22semidirect
67. #1984C22semidirect
458. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#31422
C2×#30362
#349552
#259462
#259452
#259382
#234532
#234522
#213142
#213032
#209252
C4×#4182