Finite Group Databasebeta

← #6489551

Subgroups — #6489551

Order 512.

3078 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×#544C211yesmaximal, characteristic, complemented
2C2×#3234C211yesmaximal, characteristic, complemented
3C2×#3226C211yesmaximal, characteristic, complemented
4#35301C211yesmaximal, characteristic
5#32980C212yesmaximal, complemented
6#32978C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#25915C214yesmaximal, complemented
10#25821C214yesmaximal, complemented
11#24308C218yesmaximal, complemented
12#21840C211yesmaximal, characteristic, complemented
13#21836C211yesmaximal, characteristic, complemented
14#20717C211yesmaximal, characteristic, complemented
15C2×#544—28no—
16C2×#544—28no—
17C2×#544—22no—
18C2×#544—22no—
19C2×#544C2214yes—
20C2×#544C2214yes—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8—21no—
29C22×C22⋊Q8—21no—
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3234—28no—
34#3226—28no—
35C2×#501C2212yescomplemented
36C2×#500C2212yescomplemented
37C2×#490C2212yescomplemented
38C2×#486C2212yescomplemented
39C2×#481C2212yes—
40C2×#480C2212yes—
41C2×#480C2212yes—
42C2×#480C2212yes—
43C2×#478C2214yescomplemented
44C2×#474C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2854C2212yes—
48#2847C2212yes—
49#2628C2212yes—
50#2619C2212yes—
51#2610—22no—
52#2598—22no—
53#2591—21no—
54#2539—21no—
55#2406—21no—
56#2404—21no—
57#2324C2214yes—
58#2283C2211yescharacteristic
59#2059C2218yescomplemented
60#2057C2218yescomplemented
61#2025C2218yescomplemented
62#2023C2218yescomplemented
63#1993C2214yescomplemented
64#1991C2214yescomplemented
65#1989C2214yescomplemented
66#1987C2214yescomplemented
67#1875C2218yes—
68#1874C2218yes—
69#1854C2218yes—
70#1852C2218yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1817C2212yes—
76#1798C2214yes—
77#1779C2212yes—
78#1750C2214yes—
79C22×#41—21no—
80C23×Q8—21no—
81C23×Q8—21no—
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84#544—232no—
85#544—232no—
86#544—216no—
87#544—216no—
88#544—216no—
89#544—216no—
90#532—24no—
91#532—24no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8D418yescomplemented
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—216no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—22no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—24no—
116C2×#125D418yescomplemented
117C2×#125C2314yes—
118C2×#125C2312yes—
119C2×#125C2312yes—
120C2×#124—21no—
121C2×#123—22no—
122C2×C22⋊Q8—216no—
123C2×C22⋊Q8—216no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—24no—
128C2×#121—21no—
129C2×#121C2312yes—
130C2×#121C2314yes—
131C2×#120—22no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—22no—
135C2×#117—24no—
136C2×#117—24no—
137C2×#117—24no—
138C2×#117—24no—
139C2×#117D412yes—
140C2×#117D412yes—
141C2×#117C2314yes—
142C2×#117C2312yes—
143C2×#117C2312yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—22no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2314yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C22⋊C4—21no—
162C22×C22⋊C4D411yescharacteristic
163C22×C22⋊C4D411yescharacteristic
164C22×C42—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167#501—28no—
168#500—28no—
169#490—28no—
170#486—28no—
171#481—28no—
172#480—28no—
173#480—28no—
174#480—28no—
175#478—216no—
176#474—216no—
177#432—24no—
178#431—24no—
179C2×#107C2314yes—
180C2×#107C2314yes—
181C2×#107C2314yes—
182C2×#106C2314yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185C2×#102C2314yes—
186C2×#102C2314yes—
187C2×#102C2314yes—
188C2×#102C2314yes—
189C2×C22⋊C8C2318yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#400—22no—
195#399—22no—
196#395—22no—
197#394—22no—
198#394—22no—
199#393—22no—
200#392—22no—
201#392—22no—
202#390—24no—
203#390C2314yes—
204#389C2314yes—
205#388—24no—
206#388—24no—
207#386—22no—
208#386—22no—
209#386—24no—
210#386—24no—
211#385—24no—
212#383—21no—
213#382C2312yes—
214#381—21no—
215#380C2314yes—
216#380C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219#376—21no—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C2×#41—44no—
227C2×#41—84no—
228C22×Q8—28no—
229C22×Q8—28no—
230C22×Q8—24no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8—84no—
235C22×Q8—84no—
236C22×D4—22no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—84no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2412yes—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—232no—
263C4⋊Q8—232no—
264C4⋊Q8—216no—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8C2×D418yes—
274C4⋊Q8C2×D418yes—
275#127—22no—
276#127—22no—
277#125—232no—
278#125—232no—
279#125—216no—
280#125—28no—
281#125—28no—
282#125—28no—
283#125—28no—
284#125—24no—
285#125—24no—
286#125C2×D418yes—
287#125C2×D418yes—
288#124—44no—
289#123—84no—
290C22⋊Q8—232no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—232no—
294#121—24no—
295#121—24no—
296#121—216no—
297#121—44no—
298#120—84no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—44no—
302#117—216no—
303#117—28no—
304#117—28no—
305#117—28no—
306#117—28no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—216no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—28no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C42—28no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—24no—
376C2×C42—22no—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D414yes—
381C2×C42C2×D414yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#107—28no—
388#106—28no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#102—28no—
392#102—28no—
393#102—28no—
394#102—28no—
395C22⋊C8—24no—
396C22⋊C8—24no—
397C22⋊C8—216no—
398#97—24no—
399C4×C8—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406C24—22no—
407C24—24no—
408C24—24no—
409C24—22no—
410C24—42no—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—816no—
414C2×Q8—216no—
415C2×Q8—28no—
416C2×Q8—28no—
417C2×Q8—42no—
418C2×Q8—816no—
419C2×Q8—84no—
420C2×Q8—84no—
421C2×Q8—88no—
422C2×Q8—82no—
423C2×Q8—816no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—28no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—84no—
436C2×D4—84no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—216no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—88no—
473C22×C4—816no—
474C22×C4#12014yes—
475C22×C4#12014yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C42−1+418yes—
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—28no—
491C2×C8—28no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—28no—
517C4⋊C4—42no—
518C4⋊C4—84no—
519C4⋊C4C22×D414yes—
520C4⋊C4C22×D414yes—
521C4⋊C4C22×D414yes—
522C4⋊C4C22×D414yes—
523C22⋊C4—216no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—216no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—82no—
540C42#12014yes—
541C42#12014yes—
542C42C22×D412yes—
543C42C22×D412yes—
544C42C22×D412yes—
545C42C22×D412yes—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—416no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23C2×#12012yes—
562C23#53614yes—
563C23#57614yes—
564C23C23×D411yescharacteristic
565Q8—88no—
566Q8—88no—
567Q8—84no—
568Q8—816no—
569D4—44no—
570D4—44no—
571D4—84no—
572D4—84no—
573D4—88no—
574D4—168no—
575C2×C4—216no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—816no—
616C2×C4C2×#12012yes—
617C2×C4C2×#12012yes—
618C2×C4C2×#12012yes—
619C2×C4C2×#12012yes—
620C2×C4C2×#12012yes—
621C2×C4#53614yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C23×D411yescharacteristic
624C2×C4C2×2−1+414yes—
625C8—44no—
626C8—44no—
627C22—22no—
628C22—28no—
629C22—22no—
630C22—24no—
631C22—28no—
632C22—416no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—82no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22#300814yes—
642C22C22×#12011yescharacteristic
643C22C2×#53612yes—
644C22C2×#57612yes—
645C22#358612yes—
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—84no—
659C4—82no—
660C4#348612yes—
661C4#348612yes—
662C2—24no—
663C2—44no—
664C2—162no—
665C2#3524211yescharacteristic
666C2C2×#300811yescharacteristic
667C2#6036811yescharacteristic
668C2#6087114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35301C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#544C2semidirect
2. C2×#3234C2semidirect
3. C2×#3226C2semidirect
5. #32980C2semidirect
6. #32978C2semidirect
7. #29640C2semidirect
8. #29638C2semidirect
9. #25915C2semidirect
10. #25821C2semidirect
11. #24308C2semidirect
12. #21840C2semidirect
13. #21836C2semidirect
14. #20717C2semidirect
35. C2×#501C22semidirect
36. C2×#500C22semidirect
37. C2×#490C22semidirect
38. C2×#486C22semidirect
43. C2×#478C22semidirect
44. C2×#474C22semidirect
59. #2059C22semidirect
60. #2057C22semidirect
61. #2025C22semidirect
62. #2023C22semidirect
63. #1993C22semidirect
64. #1991C22semidirect
65. #1989C22semidirect
66. #1987C22semidirect
101. C2×C4⋊Q8D4semidirect
116. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#32342
C2×#32262
#353012
#329802
#329782
#296402
#296382
#259152
#258212
#243082
#218402
#218362
#207172