Finite Group Databasebeta

← #6449928

Subgroups — #6449928

Order 512.

2136 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic, complemented
2C2×#2925C211yesmaximal, characteristic, complemented
3#35334C211yesmaximal, characteristic, complemented
4C4×#535C211yesmaximal, characteristic
5#24716C214yesmaximal, complemented
6#24712C214yesmaximal, complemented
7#24706C214yesmaximal, complemented
8#24704C214yesmaximal, complemented
9#23643C212yesmaximal, complemented
10#23483C212yesmaximal
11#21311C211yesmaximal, characteristic, complemented
12#21172C211yesmaximal, characteristic, complemented
13#21160C212yesmaximal, complemented
14#21089C212yesmaximal, complemented
15C2×#1804C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3077—28no—
21C2×#478C2214yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24#2925—28no—
25C2×#433C2214yescomplemented
26C2×#432C2212yescomplemented
27C2×#431C2212yescomplemented
28C2×#428C2212yes—
29C2×#428C2212yes—
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C2×#418C2212yes—
33C22×#102C2212yes—
34C2×#408C2212yes—
35C2×#405C2212yes—
36#2851C2214yescomplemented
37#2628C2212yescomplemented
38#2628C2212yescomplemented
39#2581—21no—
40#2581—21no—
41#2580—21no—
42#2580—21no—
43#2564—22no—
44#2564—22no—
45#2328—21no—
46#2326—21no—
47#2324C2214yes—
48#2321—22no—
49#2317C2211yescharacteristic
50#2316C2212yes—
51#2311C2212yes—
52#2311C2211yescharacteristic
53#2309C2211yescharacteristic
54C4×#127—21no—
55C4×#124—21no—
56C4×#121C2214yes—
57C4×#120—22no—
58C2×#382C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2077C2214yescomplemented
61#2077C2214yescomplemented
62#2076C2214yescomplemented
63#2075C2214yescomplemented
64#2034C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#2033C2214yes—
68#1989C2212yescomplemented
69#1989C2212yescomplemented
70#1988C2212yescomplemented
71#1987C2212yescomplemented
72#1949C2214yescomplemented
73#1949C2214yescomplemented
74#1944C2214yescomplemented
75#1943C2214yescomplemented
76#1942C2214yescomplemented
77#1941C2214yescomplemented
78#1936C2214yes—
79#1936C2214yes—
80#1934C2214yes—
81#1934C2214yes—
82#1804C2212yes—
83#1804C2212yes—
84#1804C2214yes—
85#1736C2212yes—
86#1735C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89C22×#41C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C22×M16C2312yes—
93C23×C8C2312yes—
94#535—24no—
95#535—24no—
96#533—24no—
97#533—24no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127—21no—
101C2×#125C2312yes—
102C2×#124—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×#121C2314yes—
106C2×#120—22no—
107C4×#41—24no—
108C4×#41—24no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×D4—24no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—24no—
117C2×C4×D4—24no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4D412yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#478—216no—
135#476—28no—
136#474—28no—
137#433—216no—
138#432—28no—
139#431—28no—
140#428—28no—
141#428—28no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146#423D418yescomplemented
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153#418—28no—
154C2×#102—28no—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159#408—28no—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162#405—28no—
163C2×#97C2314yes—
164C2×C4×C8C2314yes—
165#400—22no—
166#400—22no—
167#399—22no—
168#399—22no—
169#395—22no—
170#395—22no—
171#393—22no—
172#393—22no—
173#391—22no—
174#391—21no—
175#390—21no—
176#390—22no—
177#390—21no—
178#390—21no—
179#390—24no—
180#390C2314yes—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#387—24no—
189#387—24no—
190#386—24no—
191#386—22no—
192#386—24no—
193#386—24no—
194#385—24no—
195#385—24no—
196#385—24no—
197#385—22no—
198#384—21no—
199#382—21no—
200#382—21no—
201#382—21no—
202#382C2312yes—
203#382C2312yes—
204#380—24no—
205#380C2314yes—
206#380C2314yes—
207#380C2314yes—
208#379—22no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—24no—
214#376—21no—
215C2×#95—21no—
216C2×#95—21no—
217C43—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230C25—21no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×D4—24no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4C2×D412yes—
248C22×D4C2×D411yescharacteristic
249C22×D4#4114yes—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—84no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16C2×D418yes—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276#127—28no—
277#125—24no—
278#125—24no—
279#124—28no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282#121—28no—
283#121—28no—
284#120—216no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—88no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—22no—
308#117—22no—
309#117#4114yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D414yes—
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—28no—
365#107—28no—
366#106—28no—
367#106—28no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—24no—
371C4⋊C8C2×D418yes—
372C4⋊C8C2×D418yes—
373#102—28no—
374#102—28no—
375#102—28no—
376#102—28no—
377C22⋊C8—28no—
378C22⋊C8—28no—
379#97—28no—
380C4×C8—28no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393C24—22no—
394C24—24no—
395C24—42no—
396C24—42no—
397C24—82no—
398C24—82no—
399C24C22×D412yes—
400C24C22×D411yescharacteristic
401#41—44no—
402#41—44no—
403#41—44no—
404#41—44no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—216no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4C2×#4114yes—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—84no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—816no—
462C22×C4#12114yes—
463C22×C4#12114yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C22×C4C2511yescharacteristic
479M16—216no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—48no—
489C2×C8C22×D418yes—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C2×#4114yes—
515C22⋊C4—216no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—24no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—82no—
538C42#12114yes—
539C42#12114yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—48no—
556C23—42no—
557C23—88no—
558C23—84no—
559C23—84no—
560C23—88no—
561C23C2×#12112yes—
562C23#52512yes—
563C23#53612yes—
564C23C23×D411yescharacteristic
565Q8—44no—
566Q8—84no—
567Q8—84no—
568D4—24no—
569D4—24no—
570D4—48no—
571D4—48no—
572D4—44no—
573D4—42no—
574D4—42no—
575D4—816no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—164no—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4#52512yes—
622C2×C4#53612yes—
623C2×C4#54714yes—
624C2×C4#54814yes—
625C2×C4#55214yes—
626C2×C4#55614yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C8—28no—
634C8—42no—
635C8—42no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—816no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22#241514yes—
652C22C22×#12111yescharacteristic
653C22C2×#52511yescharacteristic
654C22C2×#53611yescharacteristic
655C22#358412yes—
656C22#358512yes—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4#241514yes—
671C4C2×#54711yescharacteristic
672C4C2×#54811yescharacteristic
673C4C2×#55211yescharacteristic
674C4C2×#55611yescharacteristic
675C2—24no—
676C2—82no—
677C2—84no—
678C2C2×#241511yescharacteristic
679C2#6045311yescharacteristic
680C2#6045411yescharacteristic
681C2#6059212yes—
682C2#6089012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3077C2semidirect
2. C2×#2925C2semidirect
3. #35334C2semidirect
5. #24716C2semidirect
6. #24712C2semidirect
7. #24706C2semidirect
8. #24704C2semidirect
9. #23643C2semidirect
11. #21311C2semidirect
12. #21172C2semidirect
13. #21160C2semidirect
14. #21089C2semidirect
15. C2×#1804C2semidirect
25. C2×#433C22semidirect
26. C2×#432C22semidirect
27. C2×#431C22semidirect
36. #2851C22semidirect
37. #2628C22semidirect
38. #2628C22semidirect
60. #2077C22semidirect
61. #2077C22semidirect
62. #2076C22semidirect
63. #2075C22semidirect
68. #1989C22semidirect
69. #1989C22semidirect
70. #1988C22semidirect
71. #1987C22semidirect
72. #1949C22semidirect
73. #1949C22semidirect
74. #1944C22semidirect
75. #1943C22semidirect
76. #1942C22semidirect
77. #1941C22semidirect
146. #423D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C2×#29252
#353342
C4×#5352
#247162
#247122
#247062
#247042
#236432
#234832
#213112
#211722
#211602
#210892
C2×#18042