Finite Group Databasebeta

← #404048

Subgroups — #404048

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#2016C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5#14714C211yesmaximal, characteristic
6#11207C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11050C211yesmaximal, characteristic
9#11024C212yesmaximal
10#10932C212yesmaximal
11#10795C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2295—24no—
20#2295—24no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376—21no—
33C2×C43C2211yescharacteristic
34#2043—24no—
35#2016—24no—
36#1878—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C23×Q8C2312yes—
49C23×Q8C2311yescharacteristic
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—42no—
56C22×Q16—42no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—42no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8D414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391—24no—
105#391—24no—
106#391—42no—
107#391—42no—
108#391D414yes—
109#389—22no—
110#389—22no—
111#389—22no—
112#389D414yes—
113#387—28no—
114#387—28no—
115#384—42no—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119#376—28no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C43—22no—
132C43—22no—
133#340—24no—
134#340—42no—
135#340—42no—
136#336—24no—
137#336—24no—
138#336—22no—
139#336—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
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×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8C2×D412yes—
161C22×Q8C2×D412yes—
162C22×Q8#4114yes—
163C22×Q8#4112yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C2×Q16—216no—
178C2×Q16—48no—
179C2×Q16—48no—
180C2×Q16—48no—
181C2×Q16—48no—
182C2×Q16—48no—
183C2×Q16—48no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8#4114yes—
198C4⋊Q8—48no—
199C4⋊Q8—48no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—28no—
203C4×Q8—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4#4112yes—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42Q1614yescomplemented
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42#4112yes—
277#103—48no—
278#103—48no—
279#103—48no—
280#103—48no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292C24C22×D411yescharacteristic
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
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×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8#12114yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
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—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4#12012yes—
410C22×C4#12012yes—
411C22×C4#12112yes—
412C22×C4#12112yes—
413C22×C4C2×Q1612yes—
414C22×C4C2×Q1612yes—
415C22×C4C2×Q1612yes—
416C22×C4#13512yes—
417C22×C4#13712yes—
418C22×C4#13712yes—
419C22×C4#13712yes—
420C22×C4#13712yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431Q16—432no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C4⋊C4—216no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4#12114yes—
465C42—216no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—24no—
471C42—24no—
472C42—28no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42—42no—
477C42—42no—
478C42—44no—
479C23C2×#12011yescharacteristic
480C23C2×#12111yescharacteristic
481C23#52511yescharacteristic
482C23#52511yescharacteristic
483C23#53211yescharacteristic
484C23#53511yescharacteristic
485C23#53511yescharacteristic
486C23C22×Q1611yescharacteristic
487C23C2×#13511yescharacteristic
488C23C2×#13711yescharacteristic
489C23C2×#13711yescharacteristic
490C23C2×#13711yescharacteristic
491C23C2×#13711yescharacteristic
492C23#57511yescharacteristic
493C23#57511yescharacteristic
494Q8—28no—
495Q8—44no—
496Q8—44no—
497Q8—44no—
498Q8—44no—
499Q8—44no—
500Q8—44no—
501Q8—44no—
502Q8—82no—
503Q8—82no—
504Q8—82no—
505Q8—82no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#45114yes—
588C2×C4#45214yes—
589C2×C4#46214yes—
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×#12111yescharacteristic
592C2×C4#52512yes—
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52912yes—
596C2×C4#53512yes—
597C2×C4#54012yes—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4#54612yes—
601C2×C4#55012yes—
602C2×C4#55112yes—
603C2×C4#55112yes—
604C2×C4#55312yes—
605C2×C4#55512yes—
606C8—44no—
607C8—82no—
608C22#239211yescharacteristic
609C22#240411yescharacteristic
610C22#243111yescharacteristic
611C22#259811yescharacteristic
612C22#261011yescharacteristic
613C22#261911yescharacteristic
614C22#284711yescharacteristic
615C22C2×#45111yescharacteristic
616C22C2×#45211yescharacteristic
617C22#300711yescharacteristic
618C22#300711yescharacteristic
619C22C2×#46211yescharacteristic
620C22#303511yescharacteristic
621C22#303711yescharacteristic
622C22#303911yescharacteristic
623C22#310411yescharacteristic
624C22#310411yescharacteristic
625C22#310711yescharacteristic
626C22#310811yescharacteristic
627C22#311311yescharacteristic
628C22#311311yescharacteristic
629C22#311611yescharacteristic
630C22#311711yescharacteristic
631C22#317011yescharacteristic
632C22#317211yescharacteristic
633C22#318111yescharacteristic
634C22#318511yescharacteristic
635C22#320911yescharacteristic
636C22#320911yescharacteristic
637C22#321011yescharacteristic
638C22#321111yescharacteristic
639C22#321511yescharacteristic
640C22#321811yescharacteristic
641C22#321811yescharacteristic
642C22#321911yescharacteristic
643C4—24no—
644C4—24no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—44no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—162no—
655C4#241112yes—
656C4#241412yes—
657C4#247112yes—
658C4#247612yes—
659C2#1488611yescharacteristic
660C2#2367911yescharacteristic
661C2#2368511yescharacteristic
662C2#2393411yescharacteristic
663C2#2393711yescharacteristic
664C2#2560011yescharacteristic
665C2#2560411yescharacteristic
666C2#2937011yescharacteristic
667C2#2937111yescharacteristic
668C2#2955311yescharacteristic
669C2#2955911yescharacteristic
670C2#2979911yescharacteristic
671C2#2980111yescharacteristic
672C2#3290611yescharacteristic
673C2#3290711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
272. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
C2×#20432
C2×#20162
C2×#18782
#147142
#112072
#110672
#110502
#110242
#109322
#107952