Finite Group Databasebeta

← #368435

Subgroups — #368435

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic, complemented
2C2×#1925C212yesmaximal, complemented
3C2×#1893C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5#14137C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic, complemented
7#11256C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107—28no—
17C22×#107—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#390C2212yes—
25C2×#386—21no—
26C2×#386—21no—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1925—28no—
32#1893C418yescomplemented
33#1874—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1439—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8C2×C414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—22no—
87C22×C42C2312yes—
88C2×#107—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107D418yescomplemented
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—24no—
100C2×#102—42no—
101C2×#102C2×C418yescomplemented
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105#390—24no—
106#390—42no—
107#386—44no—
108#386—44no—
109#380—22no—
110#380C2×C414yes—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—42no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—44no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D412yes—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—42no—
127#340C2×C418yescomplemented
128#336—24no—
129#336—24no—
130C25—21no—
131C25C2×D411yescharacteristic
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C22×C412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C22×C8—216no—
179C22×C8—28no—
180C22×C8—28no—
181C22×C8—28no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—216no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C22×C414yes—
198C22×C8C2×D414yes—
199C22×C8#4114yes—
200#121—28no—
201#121—28no—
202#121—44no—
203#121—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
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—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—82no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4D814yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—82no—
267#107—216no—
268#107—216no—
269#102—216no—
270#102—44no—
271#102—44no—
272C22⋊C8—44no—
273#95—24no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—82no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24—42no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—42no—
295C24C4×D412yes—
296C24#12112yes—
297C24#12312yes—
298C24C2×D812yes—
299C24#13512yes—
300C24AGL(1,ℤ/8)12yes—
301C24#13712yes—
302C24C22×D411yescharacteristic
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—84no—
327C2×D4—84no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—48no—
391C22×C4—84no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4#11714yes—
399C22×C4C4×D414yes—
400C22×C4C4×D412yes—
401C22×C4#12112yes—
402C22×C4#12312yes—
403C22×C4#12414yes—
404C22×C4#12614yes—
405C22×C4C2×D814yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4112yes—
413C22×C4C2×#4111yescharacteristic
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+412yes—
416C2×C8—232no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8C4×D418yes—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4#12014yes—
454C22⋊C4—28no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—88no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—82no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—81no—
494C23—81no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—81no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23#38814yes—
505C23C4×D814yes—
506C23#44014yes—
507C23#45914yes—
508C23#46414yes—
509C23#48014yes—
510C23#48214yes—
511C23C2×C4×D411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12311yescharacteristic
514C23D4212yes—
515C23#54712yes—
516C23C22×D811yescharacteristic
517C23C2×#13511yescharacteristic
518C23C2×AGL(1,ℤ/8)11yescharacteristic
519C23C2×#13711yescharacteristic
520C23#57612yes—
521C23#57812yes—
522D4—42no—
523D4—44no—
524D4—42no—
525D4—816no—
526D4—81no—
527D4—81no—
528D4—82no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—816no—
571C2×C4—816no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4#38814yes—
587C2×C4C2×#11711yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C4×D412yes—
590C2×C4#52012yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52512yes—
594C2×C4C2×#12411yescharacteristic
595C2×C4C2×#12611yescharacteristic
596C2×C4#53411yescharacteristic
597C2×C4#53611yescharacteristic
598C2×C4#53912yes—
599C2×C4#54612yes—
600C2×C4#54812yes—
601C2×C4#54812yes—
602C2×C4#55212yes—
603C2×C4#55312yes—
604C2×C4#55312yes—
605C8—28no—
606C8—81no—
607C8—81no—
608C22—24no—
609C22—24no—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—28no—
617C22—28no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—161no—
624C22—161no—
625C22—162no—
626C22—162no—
627C22—161no—
628C22—161no—
629C22#192518yescomplemented
630C22C2×#38811yescharacteristic
631C22#236112yes—
632C22#241011yescharacteristic
633C22#241311yescharacteristic
634C22#244312yes—
635C22C2×C4×D811yescharacteristic
636C22C2×#44011yescharacteristic
637C22#295112yes—
638C22C2×#45911yescharacteristic
639C22C2×#46411yescharacteristic
640C22#304111yescharacteristic
641C22#304411yescharacteristic
642C22C2×#48011yescharacteristic
643C22C2×#48211yescharacteristic
644C22#309812yes—
645C22#326412yes—
646C22#327012yes—
647C22D4×D812yes—
648C22#328512yes—
649C22#333412yes—
650C22#333912yes—
651C22#334312yes—
652C22#334612yes—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—84no—
661C4C4×#12011yescharacteristic
662C4#232211yescharacteristic
663C4#235212yes—
664C4#243611yescharacteristic
665C4#248112yes—
666C4#259111yescharacteristic
667C2—28no—
668C2—162no—
669C2#1401911yescharacteristic
670C2C2×#192511yescharacteristic
671C2#2106711yescharacteristic
672C2#2302512yes—
673C2#2303212yes—
674C2#2435411yescharacteristic
675C2#2436111yescharacteristic
676C2#2457611yescharacteristic
677C2#2458311yescharacteristic
678C2#2612612yes—
679C2#2614012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C23×#107C2semidirect
2. C2×#1925C2semidirect
6. C22⋊C4×C4⋊C4C2semidirect
7. #11256C2semidirect
8. #11123C2semidirect
9. #10907C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
32. #1893C4semidirect
94. C2×#107D4semidirect
101. C2×#102C2×C4semidirect
127. #340C2×C4semidirect
629. C22#1925semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#19252
C2×#18932
C2×#18742
#141372
C22⋊C4×C4⋊C42
#112562
#111232
#109072
#107302
#107142