Finite Group Databasebeta

← #393495

Subgroups — #393495

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2624C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3C2×#1875C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5C2×#1818C211yesmaximal, characteristic, complemented
6#10931C212yesmaximal, complemented
7#10931C212yesmaximal, complemented
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10730C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2624—24no—
22#2624C418yescomplemented
23C2×#399—21no—
24C2×#397—21no—
25C2×#394—22no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#380C2211yescharacteristic
29#2274—24no—
30#2274—24no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4—22no—
36#1875—24no—
37#1874—24no—
38#1818—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C414yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—24no—
67C2×#117—24no—
68C2×#117C2×C414yes—
69C2×#117D414yes—
70C2×#117D414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D414yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4D412yes—
90C22×C42—21no—
91C22×C42—22no—
92C22×C42—22no—
93C22×C42C2312yes—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107#399—28no—
108#397—28no—
109#394—216no—
110#390—24no—
111#390C2×C418yescomplemented
112#389—24no—
113#389C2×C418yescomplemented
114#380—22no—
115#380C2×C414yes—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C22⋊C4—216no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336—24no—
136C25—21no—
137C25C2×D411yescharacteristic
138C22×D4—24no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—82no—
151C22×D4—82no—
152C22×D4C22×C412yes—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—81no—
165C23×C4—82no—
166C23×C4—81no—
167C23×C4C22×C412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194#121—28no—
195#121—28no—
196#121—44no—
197#121—44no—
198#117—216no—
199#117—28no—
200#117—28no—
201#117—28no—
202#117—28no—
203#117—28no—
204#117—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22⋊C414yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4#4114yes—
242C2×C4⋊C4#4114yes—
243C2×C4⋊C4#4114yes—
244C2×C4⋊C4#4114yes—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—82no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—81no—
260C2×C22⋊C4—81no—
261C2×C22⋊C4C22⋊C418yes—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—82no—
271C2×C42—82no—
272C2×C42—82no—
273C2×C42—82no—
274C2×C42C22×C414yes—
275C2×C42C2×D414yes—
276#102—48no—
277#102—44no—
278#102—44no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—28no—
285C24—22no—
286C24—22no—
287C24—21no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—21no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—42no—
297C24—42no—
298C24—41no—
299C24—42no—
300C24—41no—
301C24—42no—
302C24—41no—
303C24—41no—
304C24C2×C22⋊C412yes—
305C24#12012yes—
306C24#12312yes—
307C24C22×D411yescharacteristic
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—48no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—84no—
330C2×D4—84no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—216no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4C2×C22⋊C414yes—
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C4×D414yes—
397C22×C4C4×D414yes—
398C22×C4#12012yes—
399C22×C4#12114yes—
400C22×C4#12114yes—
401C22×C4#12312yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—216no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—88no—
448C4⋊C4#12318yes—
449C4⋊C4#12314yes—
450C4⋊C4#12314yes—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C22⋊C4—84no—
457C42—216no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—88no—
463C42—88no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—82no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23#38614yes—
510C23C22×C22⋊C411yescharacteristic
511C23C2×#12011yescharacteristic
512C23C2×#12311yescharacteristic
513C23#52512yes—
514C23#52512yes—
515C23#57612yes—
516C23#57611yescharacteristic
517C23#57712yes—
518C23#57711yescharacteristic
519C23#57711yescharacteristic
520C23#57811yescharacteristic
521D4—44no—
522D4—42no—
523D4—42no—
524D4—816no—
525D4—82no—
526D4—81no—
527D4—81no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—816no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4#38614yes—
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C4×#4112yes—
586C2×C4C4×#4112yes—
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4C2×#12312yes—
592C2×C4#52512yes—
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52912yes—
596C2×C4#52912yes—
597C2×C4#52912yes—
598C2×C4#52912yes—
599C2×C4#53212yes—
600C2×C4#53212yes—
601C8—82no—
602C8—81no—
603C8—81no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22—162no—
621C22—162no—
622C22—161no—
623C22—161no—
624C22—161no—
625C22—161no—
626C22C2×#38611yescharacteristic
627C22#229712yes—
628C22#284911yescharacteristic
629C22#284911yescharacteristic
630C22#285312yes—
631C22#289712yes—
632C22#289711yescharacteristic
633C22#289811yescharacteristic
634C22#300811yescharacteristic
635C22#301011yescharacteristic
636C22#301211yescharacteristic
637C22#301311yescharacteristic
638C22#309612yes—
639C22#309811yescharacteristic
640C22#309911yescharacteristic
641C22#311812yes—
642C22#311912yes—
643C22#311912yes—
644C22#312112yes—
645C22#312212yes—
646C22#312412yes—
647C22#312512yes—
648C22#312612yes—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—48no—
655C4—84no—
656C4—82no—
657C4C4×#12312yes—
658C4C4×#12311yescharacteristic
659C4C4×#12311yescharacteristic
660C4#285412yes—
661C4#285411yescharacteristic
662C4#285411yescharacteristic
663C2—44no—
664C2—162no—
665C2#1470611yescharacteristic
666C2#2091411yescharacteristic
667C2#2091511yescharacteristic
668C2#2139712yes—
669C2#2139912yes—
670C2#3301211yescharacteristic
671C2#3301311yescharacteristic
672C2#3301411yescharacteristic
673C2#3301511yescharacteristic
674C2#3310212yes—
675C2#3310312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
5. C2×#1818C2semidirect
6. #10931C2semidirect
7. #10931C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
10. #10731C2semidirect
11. #10730C2semidirect
22. #2624C4semidirect
111. #390C2×C4semidirect
113. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26242
C2×#22742
C2×#18752
C2×#18742
C2×#18182
#109312
#108852
#108842
#107312
#107302