Finite Group Databasebeta

← #361679

Subgroups — #361679

Order 512.

4018 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×#1788C214yesmaximal, complemented
7C2×C4×#102C211yesmaximal, characteristic, complemented
8#13906C211yesmaximal, characteristic, complemented
9#10970C214yesmaximal, complemented
10C23×#41C2212yes—
11C2×#547—24no—
12C2×#547—24no—
13C2×#547—24no—
14C2×#547—24no—
15C2×#547—24no—
16C2×#547C418yescomplemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#123—22no—
19C22×C22⋊Q8C2212yes—
20C22×#121—21no—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2214yes—
26C22×#103C2214yes—
27C22×#102C2212yes—
28C22×#102C2212yes—
29C22×C22⋊C8C2214yes—
30C22×C4×C8C2211yescharacteristic
31C2×#390—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1981—24no—
36#1956—24no—
37#1954—24no—
38#1802—24no—
39#1788—216no—
40C4×#102—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1439—22no—
44C4⋊#95—21no—
45C22×#41—24no—
46C22×#41—24no—
47C22×#41—24no—
48C22×#41—24no—
49C22×#41—24no—
50C22×#41—28no—
51C22×#41—28no—
52C22×#41—44no—
53C22×#41—44no—
54C22×#41C2×C418yes—
55C23×Q8C2312yes—
56C23×D4—22no—
57C23×D4D411yescharacteristic
58C23×D4C2311yescharacteristic
59C24×C4—22no—
60C24×C4—21no—
61C24×C4C2312yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64#547—232no—
65#547—232no—
66#547—232no—
67#547—232no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C414yes—
72C2×C4⋊Q8D414yes—
73C2×C4⋊Q8D414yes—
74C2×#123—216no—
75C2×#123—216no—
76C2×#123—216no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8C2×C418yes—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—28no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4C2×C414yes—
94C2×C4×D4C2×C414yes—
95C2×C4×D4D414yes—
96C2×C4×D4D414yes—
97C2×C4×D4D414yes—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—22no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4D412yes—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×#103—28no—
120C2×#103—28no—
121C2×#103—28no—
122C2×#103—28no—
123C2×#103—44no—
124C2×#103—44no—
125C2×#102—24no—
126C2×#102—24no—
127C2×#102—24no—
128C2×#102—28no—
129C2×#102D418yescomplemented
130C2×C22⋊C8—28no—
131C2×C22⋊C8—28no—
132C2×C22⋊C8—44no—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136#390—44no—
137#384—22no—
138#384—22no—
139#380—28no—
140#380—44no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C2×#95—24no—
144C2×#95—21no—
145#340—24no—
146#340—24no—
147C25—21no—
148C25C2×D412yes—
149C2×#41—216no—
150C2×#41—216no—
151C2×#41—216no—
152C2×#41—216no—
153C2×#41—232no—
154C2×#41—416no—
155C2×#41—416no—
156C2×#41—416no—
157C2×#41—416no—
158C2×#41—416no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—416no—
162C2×#41—48no—
163C2×#41—48no—
164C2×#41—416no—
165C2×#41—48no—
166C2×#41—48no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8C22×C414yes—
175C22×Q8C2×D414yes—
176C22×Q8C2×D414yes—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4C22⋊C414yes—
194C22×D4C22×C412yes—
195C22×D4C2×D412yes—
196C22×D4C2×D412yes—
197C22×D4C2×D414yes—
198C22×D4#4114yes—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—48no—
218C23×C4—48no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C22×C414yes—
224C23×C4C2×D412yes—
225C23×C4C2×D414yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—24no—
238C22×C8—22no—
239C22×C8—24no—
240C22×C8C2×D414yes—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—28no—
244C4⋊Q8C22⋊C4116yescomplemented
245#123—2128no—
246C22⋊Q8—216no—
247C22⋊Q8—216no—
248C22⋊Q8—216no—
249C22⋊Q8—216no—
250#121—264no—
251C4×D4—28no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—28no—
257C4×D4—28no—
258C4×D4C22⋊C4116yescomplemented
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—264no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4C22⋊C414yes—
289C2×C4⋊C4C22×C412yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—84no—
304C2×C22⋊C4C22⋊C418yes—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42C22×C412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C4⋊C8—44no—
315#103—416no—
316#103—416no—
317#102—216no—
318#102—48no—
319C22⋊C8—416no—
320C4×C8—44no—
321#95—48no—
322#95—42no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24—42no—
343C24—42no—
344C24C2×C22⋊C414yes—
345C24C22×D411yescharacteristic
346#41—464no—
347#41—432no—
348#41—464no—
349#41—432no—
350#41—432no—
351#41—432no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8C2×C22⋊C418yes—
370C2×D4—232no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—432no—
388C2×D4—432no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4C2×C22⋊C414yes—
396C2×D4C4×D418yes—
397C2×D4#12118yes—
398C2×D4#12414yes—
399C2×D4#12714yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—232no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—216no—
422C22×C4—24no—
423C22×C4—432no—
424C22×C4—432no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4C2×C22⋊C414yes—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4#12012yes—
464C22×C4#12012yes—
465C22×C4#12012yes—
466C22×C4#12012yes—
467C22×C4#13512yes—
468C22×C4#13512yes—
469C22×C4AGL(1,ℤ/8)12yes—
470C22×C4#13712yes—
471C22×C4C23×C411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—44no—
492C2×C8—44no—
493C4⋊C4—232no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4C2×C22⋊C414yes—
510C22⋊C4—232no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—816no—
514C42—84no—
515C42C2×C22⋊C414yes—
516C42C2×C22⋊C414yes—
517C42#12014yes—
518C42#12014yes—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—24no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23C22×C22⋊C411yescharacteristic
566C23C2×#12011yescharacteristic
567C23C2×#12011yescharacteristic
568C23D4212yes—
569C23#54612yes—
570C23C2×#13511yescharacteristic
571C23C2×#13511yescharacteristic
572C23C2×AGL(1,ℤ/8)11yescharacteristic
573C23C2×#13711yescharacteristic
574C23#57712yes—
575C23#57812yes—
576Q8—216no—
577Q8—44no—
578Q8—48no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582D4—24no—
583D4—28no—
584D4—28no—
585D4—24no—
586D4—24no—
587D4—464no—
588D4—416no—
589D4#39018yescomplemented
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—464no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—432no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4#37914yes—
632C2×C4#37914yes—
633C2×C4#41614yes—
634C2×C4#41714yes—
635C2×C4#44914yes—
636C2×C4#44914yes—
637C2×C4#45214yes—
638C2×C4#45214yes—
639C2×C4C22×C22⋊C411yescharacteristic
640C2×C4C22×C22⋊C411yescharacteristic
641C2×C4C2×C4×D412yes—
642C2×C4#51812yes—
643C2×C4C2×#12011yescharacteristic
644C2×C4C2×#12011yescharacteristic
645C2×C4C2×#12112yes—
646C2×C4C2×#12411yescharacteristic
647C2×C4C2×#12711yescharacteristic
648C2×C4#53512yes—
649C2×C4#53511yescharacteristic
650C2×C4#53511yescharacteristic
651C2×C4D4212yes—
652C2×C4D4212yes—
653C2×C4D4212yes—
654C2×C4#54612yes—
655C2×C4#54612yes—
656C2×C4#54612yes—
657C8—44no—
658C8—82no—
659C22—28no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—48no—
674C22—48no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—84no—
680C22—84no—
681C22C2×#37911yescharacteristic
682C22D4×C22⋊C412yes—
683C22#240812yes—
684C22#243311yescharacteristic
685C22#243511yescharacteristic
686C22C2×#41611yescharacteristic
687C22C2×#41711yescharacteristic
688C22#289812yes—
689C22C2×#44911yescharacteristic
690C22C2×#44911yescharacteristic
691C22C2×#45211yescharacteristic
692C22C2×#45211yescharacteristic
693C22#301011yescharacteristic
694C22#301111yescharacteristic
695C22#301312yes—
696C22#327212yes—
697C22#327612yes—
698C22#328312yes—
699C22#328412yes—
700C22#330212yes—
701C22#330312yes—
702C22#333012yes—
703C22#333112yes—
704C4—24no—
705C4—44no—
706C4—48no—
707C4—44no—
708C4—84no—
709C4—162no—
710C4#178818yescomplemented
711C4C2×#39011yescharacteristic
712C4#230911yescharacteristic
713C4D4×C22⋊C412yes—
714C4#240812yes—
715C4#243311yescharacteristic
716C4#243511yescharacteristic
717C2—48no—
718C2—84no—
719C2#1390211yescharacteristic
720C2C2×#178811yescharacteristic
721C2#2060311yescharacteristic
722C2#2241712yes—
723C2#2243112yes—
724C2#2423712yes—
725C2#2424112yes—
726C2#2575411yescharacteristic
727C2#2575811yescharacteristic
728C2#2588211yescharacteristic
729C2#2588611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1981C2semidirect
3. C2×#1956C2semidirect
4. C2×#1954C2semidirect
5. C2×#1802C2semidirect
6. C2×#1788C2semidirect
7. C2×C4×#102C2semidirect
8. #13906C2semidirect
9. #10970C2semidirect
16. C2×#547C4semidirect
129. C2×#102D4semidirect
244. C4⋊Q8C22⋊C4semidirect
258. C4×D4C22⋊C4semidirect
589. D4#390semidirect
710. C4#1788semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#19812
C2×#19562
C2×#19542
C2×#18022
C2×#17882
C2×C4×#1022
#139062
#109702