Finite Group Databasebeta

← #369993

Subgroups — #369993

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1852C211yesmaximal, characteristic
2C4×#380C211yesmaximal, characteristic
3C2×C4⋊#95C211yesmaximal, characteristic
4#11377C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#10907C212yesmaximal, complemented
7#10807C212yesmaximal, complemented
8#10730C211yesmaximal, characteristic
9#10730C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4—21no—
23C22×#95—21no—
24C22×#95—21no—
25#1852C418yescomplemented
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C22⋊#95—21no—
43C4×#95—21no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4C2×C414yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—24no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—42no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—42no—
80C2×#107—42no—
81C2×#107C2×C418yescomplemented
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2×C414yes—
88#380—22no—
89#380—22no—
90#380C2×C414yes—
91#380Q814yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—42no—
94C4×C22⋊C4—44no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—44no—
108C2×#95—44no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340C2×C418yescomplemented
116#336—24no—
117#336—42no—
118C25C2×D411yescharacteristic
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4Q1614yes—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×Q812yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C22×C414yes—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—48no—
239C2×C42—48no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42#4114yes—
249#107—48no—
250C22⋊C8—28no—
251C22⋊C8—44no—
252#95—24no—
253#95—24no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—44no—
263#95—42no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24C4×D412yes—
273C24C22⋊Q812yes—
274C24#12312yes—
275C24C2×Q1612yes—
276C24#13512yes—
277C24AGL(1,ℤ/8)12yes—
278C24AGL(1,ℤ/8)12yes—
279C24C22×D411yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—48no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4#11714yes—
389C22×C4C4×D412yes—
390C22×C4C4×Q814yes—
391C22×C4C22⋊Q812yes—
392C22×C4#12312yes—
393C22×C4#12312yes—
394C22×C4#12312yes—
395C22×C4#12514yes—
396C22×C4#12614yes—
397C22×C4C2×Q1612yes—
398C22×C4C2×Q1612yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4#13712yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×Q811yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C42—24no—
446C42—24no—
447C42—44no—
448C42—44no—
449C42—42no—
450C42—42no—
451C42—84no—
452C42—84no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23#38714yes—
482C23C4×Q1614yes—
483C23#44214yes—
484C23#47714yes—
485C23#47814yes—
486C23#48214yes—
487C23#48414yes—
488C23C2×C4×D411yescharacteristic
489C23C2×C22⋊Q811yescharacteristic
490C23C2×#12311yescharacteristic
491C23C2×#12311yescharacteristic
492C23#52511yescharacteristic
493C23#53411yescharacteristic
494C23#53511yescharacteristic
495C23C22×Q1611yescharacteristic
496C23C2×#13511yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57511yescharacteristic
502C23#57511yescharacteristic
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
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—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—416no—
529C2×C4—416no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—816no—
561C2×C4—816no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4#38714yes—
587C2×C4#48414yes—
588C2×C4C2×#11711yescharacteristic
589C2×C4C2×C4×Q811yescharacteristic
590C2×C4C4×#4112yes—
591C2×C4#51812yes—
592C2×C4C2×#12511yescharacteristic
593C2×C4C2×#12611yescharacteristic
594C2×C4#52912yes—
595C2×C4#53812yes—
596C2×C4#55112yes—
597C2×C4#55112yes—
598C2×C4#55112yes—
599C2×C4#55112yes—
600C2×C4#55312yes—
601C2×C4#55312yes—
602C2×C4#55412yes—
603C2×C4#55412yes—
604C8—44no—
605C8—82no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22#192118yes—
621C22C2×#38711yescharacteristic
622C22C4×#12311yescharacteristic
623C22#232111yescharacteristic
624C22#248211yescharacteristic
625C22#248311yescharacteristic
626C22#248711yescharacteristic
627C22#257911yescharacteristic
628C22C2×C4×Q1611yescharacteristic
629C22C2×#44211yescharacteristic
630C22C4×AGL(1,ℤ/8)11yescharacteristic
631C22C4×#13711yescharacteristic
632C22C2×#47811yescharacteristic
633C22C2×#47711yescharacteristic
634C22#307911yescharacteristic
635C22#308011yescharacteristic
636C22C2×#48211yescharacteristic
637C22C2×#48411yescharacteristic
638C22C2×#48411yescharacteristic
639C22#309111yescharacteristic
640C22#309211yescharacteristic
641C22#309411yescharacteristic
642C22#309411yescharacteristic
643C22#309511yescharacteristic
644C22#310411yescharacteristic
645C22#310511yescharacteristic
646C22#311211yescharacteristic
647C22#311411yescharacteristic
648C22#318911yescharacteristic
649C22#319011yescharacteristic
650C22#319211yescharacteristic
651C22#319411yescharacteristic
652C22#320911yescharacteristic
653C22#321111yescharacteristic
654C22#321311yescharacteristic
655C22#321711yescharacteristic
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4#235312yes—
668C4#236812yes—
669C4#250512yes—
670C4#251012yes—
671C2—28no—
672C2#1404011yescharacteristic
673C2C2×#192111yescharacteristic
674C2#2103411yescharacteristic
675C2C4×#48411yescharacteristic
676C2#2157811yescharacteristic
677C2#2204411yescharacteristic
678C2#2204611yescharacteristic
679C2#2758611yescharacteristic
680C2#2758811yescharacteristic
681C2#2760211yescharacteristic
682C2#2760311yescharacteristic
683C2#2768111yescharacteristic
684C2#2768411yescharacteristic
685C2#2923711yescharacteristic
686C2#2924111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11377C2semidirect
5. #11316C2semidirect
6. #10907C2semidirect
7. #10807C2semidirect
25. #1852C4semidirect
81. C2×#107C2×C4semidirect
115. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18522
C4×#3802
C2×C4⋊#952
#113772
#113162
#109072
#108072
#107302
#107122