Finite Group Databasebeta

← #6489658

Subgroups — #6489658

Order 512.

3366 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#3234C211yesmaximal, characteristic, complemented
3C2×#3109C211yesmaximal, characteristic
4#35178C211yesmaximal, characteristic, complemented
5#33004C212yesmaximal, complemented
6#33002C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#25946C214yesmaximal, complemented
10#25851C214yesmaximal, complemented
11#24347C218yesmaximal, complemented
12#21838C211yesmaximal, characteristic, complemented
13#21822C211yesmaximal, characteristic, complemented
14#20738C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C2×#536—21no—
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#3234—28no—
35C2×#501C2212yescomplemented
36C2×#490C2212yescomplemented
37#3109—28no—
38C2×#481C2212yes—
39C2×#481C2212yes—
40C2×#480C2212yes—
41C2×#478C2214yescomplemented
42C2×#476C2212yes—
43C2×#459C2212yes—
44C2×#447C2212yes—
45C2×#442C2214yes—
46C2×#432C2211yescharacteristic
47C2×#432C2211yescharacteristic
48#2854C2212yes—
49#2854C2212yes—
50#2618—21no—
51#2618—21no—
52#2548C2212yescomplemented
53#2546C2212yescomplemented
54#2545—22no—
55#2538C2214yescomplemented
56#2536—22no—
57#2311—21no—
58#2285C2211yescharacteristic
59#2059C2218yescomplemented
60#2058C2218yescomplemented
61#2025C2218yescomplemented
62#2023C2218yescomplemented
63#1992C2214yescomplemented
64#1989C2214yescomplemented
65#1985C2214yescomplemented
66#1985C2214yescomplemented
67#1879C2218yes—
68#1879C2218yes—
69#1864C2218yes—
70#1862C2218yes—
71#1824C2212yes—
72#1824C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1802C2214yes—
77#1786C2212yes—
78#1762C2214yes—
79C22×#41—21no—
80C23×Q8—21no—
81C23×Q8D411yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4—21no—
86C22×D8C2314yes—
87#554—232no—
88#554—232no—
89#554—216no—
90#554—216no—
91#554—216no—
92#554—216no—
93#536—48no—
94#532—24no—
95#532—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#125—24no—
104C2×#125—24no—
105C2×#125—22no—
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—24no—
109C2×#125D418yes—
110C2×#125C2314yes—
111C2×#125C2312yes—
112C2×#125C2312yes—
113C2×#124—22no—
114C2×#124—21no—
115C2×#123—22no—
116C2×#123—22no—
117C2×C22⋊Q8—216no—
118C2×C22⋊Q8—216no—
119C2×C22⋊Q8—216no—
120C2×C22⋊Q8—24no—
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×#121C2312yes—
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×D4D418yes—
146C2×C4×D4C2312yes—
147C2×C4×D4C2314yes—
148C2×C4×D4C2314yes—
149C2×C4×D4C2312yes—
150C2×C4×D4C2312yes—
151C2×#117—21no—
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#501—28no—
173#490—28no—
174#481—28no—
175#481—28no—
176#480—28no—
177#478—216no—
178#476—28no—
179#459—28no—
180#447—28no—
181#442—216no—
182#432—24no—
183#432—24no—
184C2×#107C2314yes—
185C2×#106C2314yes—
186C2×#106C2314yes—
187C2×C4⋊C8C2312yes—
188C2×C4⋊C8C2312yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×#102C2318yes—
192C2×#102C2314yes—
193C2×#102C2314yes—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×#97C2312yes—
197C2×#97C2312yes—
198#398—22no—
199#398—22no—
200#396—24no—
201#396—24no—
202#394—22no—
203#394—22no—
204#391—22no—
205#390—22no—
206#390C2314yes—
207#390C2314yes—
208#389C2314yes—
209#389C2314yes—
210#388—24no—
211#388—24no—
212#386—22no—
213#386—22no—
214#384C2312yes—
215#382—21no—
216#380—21no—
217#380—24no—
218#380C2312yes—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2312yes—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225#336C2318yes—
226C25—21no—
227C2×#41—44no—
228C2×#41—44no—
229C2×#41—44no—
230C22×Q8—28no—
231C22×Q8—24no—
232C22×Q8—24no—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—42no—
236C22×Q8—84no—
237C22×Q8C2×D412yes—
238C22×Q8C2×D412yes—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D414yes—
246C22×D4C2412yes—
247C22×D4C2412yes—
248C23×C4—28no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—24no—
254C23×C4—28no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—84no—
262C23×C4—84no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×D8—28no—
271C2×D8—216no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—216no—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—48no—
280#127—22no—
281#127—22no—
282#125—216no—
283#125—28no—
284#125—28no—
285#125—28no—
286#125—28no—
287#125—24no—
288#125—24no—
289#125C2×D418yes—
290#125C2×D418yes—
291#124—48no—
292#124—44no—
293#123—48no—
294#123—48no—
295C22⋊Q8—232no—
296C22⋊Q8—232no—
297C22⋊Q8—232no—
298C22⋊Q8—232no—
299C22⋊Q8—232no—
300C22⋊Q8—232no—
301C22⋊Q8—416no—
302#121—24no—
303#121—24no—
304#121—44no—
305C4×Q8—28no—
306C4×Q8—28no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×D4—216no—
312C4×D4—216no—
313C4×D4—28no—
314C4×D4—28no—
315C4×D4—28no—
316C4×D4—28no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—44no—
324C4×D4C2×D418yes—
325C4×D4C2×D418yes—
326#117—44no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—232no—
330C2×C4⋊C4—232no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42C2×D412yes—
389C2×C42C2×D412yes—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D414yes—
393C2×C42C2×D414yes—
394C2×C42C2412yes—
395C2×C42C2412yes—
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—28no—
399#106—28no—
400#106—28no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403#102—28no—
404#102—216no—
405#102—28no—
406#102—28no—
407#102—28no—
408C22⋊C8—24no—
409C22⋊C8—24no—
410#97—24no—
411#97—24no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416C24—28no—
417C24—24no—
418C24—22no—
419C24—22no—
420C24—44no—
421C24—42no—
422C24—42no—
423C24C22×D411yescharacteristic
424#41—816no—
425C2×Q8—216no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—416no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—88no—
437C2×Q8—816no—
438C2×D4—28no—
439C2×D4—216no—
440C2×D4—216no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—28no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—82no—
455C2×D4C22×D414yes—
456C2×D42−1+418yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—216no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—216no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—432no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—88no—
495C22×C4—816no—
496C22×C4—816no—
497C22×C4#12014yes—
498C22×C4#12014yes—
499C22×C4AGL(1,ℤ/8)14yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C2511yescharacteristic
512D8—416no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—28no—
516C2×C8—28no—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—28no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—48no—
543C4⋊C4—82no—
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#12014yes—
566C42#12014yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C42C22×D412yes—
570C42C22×D412yes—
571C23—22no—
572C23—22no—
573C23—216no—
574C23—24no—
575C23—28no—
576C23—48no—
577C23—416no—
578C23—416no—
579C23—44no—
580C23—48no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23—42no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23C2×#12012yes—
589C23C2×AGL(1,ℤ/8)12yes—
590C23#57512yes—
591C23C23×D411yescharacteristic
592Q8—48no—
593Q8—48no—
594Q8—84no—
595Q8—816no—
596D4—216no—
597D4—24no—
598D4—28no—
599D4—44no—
600D4—44no—
601D4—84no—
602D4—84no—
603D4—88no—
604D4#53618yescomplemented
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—44no—
629C2×C4—432no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4—816no—
650C2×C4C2×#12012yes—
651C2×C4C2×#12012yes—
652C2×C4C2×#12012yes—
653C2×C4C2×#12012yes—
654C2×C4C2×#12012yes—
655C2×C4C2×AGL(1,ℤ/8)14yes—
656C2×C4C23×D411yescharacteristic
657C2×C4C23×D411yescharacteristic
658C2×C4C2×2−1+414yes—
659C8—44no—
660C8—44no—
661C22—24no—
662C22—24no—
663C22—22no—
664C22—22no—
665C22—28no—
666C22—432no—
667C22—48no—
668C22—48no—
669C22—44no—
670C22—44no—
671C22—84no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#300614yes—
677C22C22×#12011yescharacteristic
678C22#348612yes—
679C22C22×AGL(1,ℤ/8)11yescharacteristic
680C22C2×#57511yescharacteristic
681C22#358512yes—
682C4—22no—
683C4—24no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—84no—
695C4—82no—
696C4C2×#53612yes—
697C4#348612yes—
698C2—24no—
699C2—48no—
700C2—162no—
701C2#3524211yescharacteristic
702C2C2×#300611yescharacteristic
703C2#6036711yescharacteristic
704C2#6182512yes—
705C2#6183112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
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×#3234C2semidirect
4. #35178C2semidirect
5. #33004C2semidirect
6. #33002C2semidirect
7. #29640C2semidirect
8. #29639C2semidirect
9. #25946C2semidirect
10. #25851C2semidirect
11. #24347C2semidirect
12. #21838C2semidirect
13. #21822C2semidirect
14. #20738C2semidirect
35. C2×#501C22semidirect
36. C2×#490C22semidirect
41. C2×#478C22semidirect
52. #2548C22semidirect
53. #2546C22semidirect
55. #2538C22semidirect
59. #2059C22semidirect
60. #2058C22semidirect
61. #2025C22semidirect
62. #2023C22semidirect
63. #1992C22semidirect
64. #1989C22semidirect
65. #1985C22semidirect
66. #1985C22semidirect
604. D4#536semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32342
C2×#31092
#351782
#330042
#330022
#296402
#296392
#259462
#258512
#243472
#218382
#218222
#207382