Finite Group Databasebeta

← #365010

Subgroups — #365010

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#470C211yesmaximal, characteristic
2C22×#441C211yesmaximal, characteristic, complemented
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5C2×#1753C211yesmaximal, characteristic
6#13969C212yesmaximal
7#11190C212yesmaximal, complemented
8#10970C212yesmaximal
9#10863C211yesmaximal, characteristic, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×Q8C2212yes—
16C2×#470—24no—
17C2×#470—24no—
18C2×#470C418yescomplemented
19C2×#441—24no—
20C2×#441—24no—
21C2×#441—24no—
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#103C2212yes—
28C22×C22⋊C8C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#391C2212yes—
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1969—24no—
35#1809—24no—
36#1753—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—22no—
43#1439—22no—
44#1437—22no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16C2×C414yes—
55C22×Q16D414yes—
56C22×Q16D414yes—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—22no—
63C22×M16C2×C414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—42no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8C2×C418yes—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—24no—
77C2×C4×Q8—24no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D412yes—
88C22×C42—22no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#470—232no—
92#441—232no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106C2×C414yes—
101C2×#106D414yes—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—42no—
110C2×#103C2×C418yes—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97D414yes—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120#391—24no—
121#391—42no—
122#380—24no—
123#380—42no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D412yes—
134#340—24no—
135#340—42no—
136#340—42no—
137#336—22no—
138#336—22no—
139C25C2×D411yescharacteristic
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C22×C414yes—
152C22×Q8C2×D414yes—
153C22×Q8C2×D414yes—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C22×C412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×Q16—28no—
177C2×Q16—28no—
178C2×Q16—28no—
179C2×Q16—28no—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16C22⋊C4116yescomplemented
183C2×M16—28no—
184C2×M16—28no—
185C2×M16—28no—
186C2×M16—44no—
187C2×M16—44no—
188C2×M16—44no—
189C2×M16—44no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C22×C412yes—
209C22×C8C22×C412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C22×C8#4112yes—
218C22⋊Q8—216no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C4×Q8—216no—
223C4×Q8—48no—
224C4×Q8—48no—
225C4×Q8—48no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—82no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42C2×D412yes—
273C2×C42#4112yes—
274#107—44no—
275#106—28no—
276#106—28no—
277#103—216no—
278#103—216no—
279#103—48no—
280C22⋊C8—28no—
281#97—28no—
282C4×C8—44no—
283#95—24no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—82no—
290#95—82no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24C4×D412yes—
302C24C22×D411yescharacteristic
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8C2×C22⋊C418yes—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C4×D412yes—
406C22×C4#12114yes—
407C22×C4#13714yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C42−1+411yescharacteristic
422Q16—232no—
423M16—416no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8C2×C22⋊C414yes—
446C2×C8C4×D414yes—
447C2×C8C4×D414yes—
448C2×C8#12014yes—
449C2×C8#12114yes—
450C2×C8#12114yes—
451C2×C8#12314yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—88no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—416no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23C2×C4×D411yescharacteristic
502C23C2×#12111yescharacteristic
503C23#53211yescharacteristic
504C23D4211yescharacteristic
505C23D4211yescharacteristic
506C23D4211yescharacteristic
507C23#54711yescharacteristic
508C23C2×#13712yes—
509C23#57512yes—
510C23#57712yes—
511C23#57812yes—
512Q8—28no—
513Q8—28no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518Q8—44no—
519Q8—44no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4C22×C22⋊C411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4#51811yescharacteristic
582C2×C4#52011yescharacteristic
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4#52911yescharacteristic
589C2×C4#53511yescharacteristic
590C2×C4#53711yescharacteristic
591C2×C4#54011yescharacteristic
592C2×C4#54211yescharacteristic
593C2×C4#54611yescharacteristic
594C2×C4#54611yescharacteristic
595C2×C4#54611yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4#54811yescharacteristic
601C2×C4#55011yescharacteristic
602C2×C4#55311yescharacteristic
603C2×C4#55311yescharacteristic
604C2×C4#55511yescharacteristic
605C8—42no—
606C8—42no—
607C8—42no—
608C8#38618yes—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22—81no—
620C22C4×#12111yescharacteristic
621C22D4×C22⋊C411yescharacteristic
622C22#236111yescharacteristic
623C22#240811yescharacteristic
624C22#242211yescharacteristic
625C22#242811yescharacteristic
626C22#253711yescharacteristic
627C22C4×#13712yes—
628C22#294712yes—
629C22#305612yes—
630C22#306312yes—
631C22#317311yescharacteristic
632C22#317711yescharacteristic
633C22#318311yescharacteristic
634C22#318511yescharacteristic
635C22#327012yes—
636C22#327112yes—
637C22#327512yes—
638C22#327612yes—
639C22#327712yes—
640C22#327812yes—
641C22#334412yes—
642C22#334912yes—
643C4—44no—
644C4—42no—
645C4—48no—
646C4—44no—
647C4—84no—
648C4—84no—
649C4—162no—
650C4C2×#38611yescharacteristic
651C4#232411yescharacteristic
652C4#233911yescharacteristic
653C4#236311yescharacteristic
654C4#240711yescharacteristic
655C4#242011yescharacteristic
656C4#242711yescharacteristic
657C4#253511yescharacteristic
658C2—82no—
659C2#1397011yescharacteristic
660C2#2153911yescharacteristic
661C2#2154211yescharacteristic
662C2#2244512yes—
663C2#2303812yes—
664C2#2427612yes—
665C2#2504711yescharacteristic
666C2#2504911yescharacteristic
667C2#2542411yescharacteristic
668C2#2542711yescharacteristic
669C2#2823812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#441C2semidirect
3. C2×#1969C2semidirect
7. #11190C2semidirect
9. #10863C2semidirect
10. #10861C2semidirect
11. #10789C2semidirect
18. C2×#470C4semidirect
182. C2×Q16C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4702
C22×#4412
C2×#19692
C2×#18092
C2×#17532
#139692
#111902
#109702
#108632
#108612
#107892