Finite Group Databasebeta

← C4×#1901 #354277

Subgroups — C4×#1901 #354277

Order 512.

2064 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
1C22×C4×M16C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#1901C212yesmaximal
4C2×C4×#101C212yesmaximal, complemented
5C4×#407C211yesmaximal, characteristic
6#10963C212yesmaximal, complemented
7#10913C212yesmaximal, complemented
8#10739C212yesmaximal
9C4×#343C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×C22⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#101C2214yes—
15C2×#407C2212yes—
16C2×C4×M16—21no—
17C2×C4×M16—21no—
18C2×C4×M16—28no—
19C2×C4×M16C2212yes—
20C2×C4×M16C2212yes—
21C2×C4×M16C2212yes—
22C22×#97—21no—
23C22×C4×C8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28C4×C22⋊Q8—21no—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C2×#391C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387C412yes—
35C2×#385C412yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C412yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C412yes—
40#1901C4116yesdirect product factor
41C4×#101—28no—
42#1749—21no—
43#1749—21no—
44C4×C22⋊C8—21no—
45C4×C22⋊C8—21no—
46C2×#343C2214yes—
47C2×#343C2214yes—
48C2×#343C2214yes—
49#1728C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—28no—
57C22×M16—28no—
58C22×M16C2×C418yes—
59C22×M16C2314yes—
60C22×M16C2314yes—
61C23×C8—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8C2×C418yes—
68C2×C4×Q8—24no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—24no—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C412yes—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42C2×C412yes—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C412yes—
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×#101—28no—
99C2×#101—28no—
100C2×#101C2×C4116yescomplemented
101#407—24no—
102#407C2×C418yes—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C4×M16—28no—
106C4×M16—28no—
107C4×M16—28no—
108C4×M16—24no—
109C4×M16—24no—
110C4×M16—24no—
111C4×M16D418yes—
112C2×#97—28no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×C4×C8—28no—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120#391—21no—
121#391—21no—
122#391—21no—
123#391—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#387—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#385—22no—
133#385—22no—
134#385—22no—
135#385—22no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C2×#95C2×C412yes—
153C2×#95C2×C412yes—
154C2×#95C2×C412yes—
155C2×#95C2×C412yes—
156C2×#95C2×C412yes—
157#343—28no—
158#343—28no—
159#343C2×C4116yescomplemented
160#336—22no—
161#336—22no—
162C25C22×C411yescharacteristic
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8C22×C414yes—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C412yes—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C414yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C412yes—
194C23×C4C22×C412yes—
195C23×C4C22×C412yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4#4114yes—
202C23×C4C2411yescharacteristic
203C2×M16—232no—
204C2×M16—28no—
205C2×M16—28no—
206C2×M16—28no—
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—232no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—24no—
215C2×M16—24no—
216C2×M16C22×C418yes—
217C2×M16C22×C418yes—
218C2×M16C2×D418yes—
219C2×M16#4118yes—
220C22×C8—28no—
221C22×C8—28no—
222C22×C8—28no—
223C22×C8—24no—
224C22×C8—24no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—44no—
232C4×Q8—44no—
233C4×Q8—42no—
234C4×Q8—42no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C4218yes—
255C2×C4⋊C4C22×C414yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C4218yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—28no—
284C2×C42—44no—
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D414yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4114yes—
303#101—232no—
304C22⋊C8—28no—
305C22⋊C8—28no—
306#97—28no—
307#97—28no—
308#97—44no—
309C4×C8—28no—
310C4×C8—28no—
311C4×C8—44no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—41no—
327C24—41no—
328C24C2×C4212yes—
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24#11712yes—
332C24C23×C411yescharacteristic
333C2×Q8—28no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—28no—
337C2×Q8—41no—
338C2×Q8—42no—
339C2×Q8—41no—
340C2×Q8—44no—
341C2×Q8—42no—
342C2×Q8—44no—
343C2×Q8—42no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—41no—
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—41no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4#10114yescomplemented
412C22×C4C2×C4214yes—
413C22×C4C2×C4212yes—
414C22×C4C2×C4214yes—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4#11712yes—
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4C4×D414yes—
429C22×C4C4×D414yes—
430C22×C4C4×D414yes—
431C22×C4C4×D414yes—
432C22×C4C23×C412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4112yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4112yes—
439M16—232no—
440M16—28no—
441M16—28no—
442M16—28no—
443M16—28no—
444M16—28no—
445M16—44no—
446M16—44no—
447M16C4×D4116yes—
448C2×C8—232no—
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—28no—
453C2×C8—44no—
454C2×C8—44no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—44no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—44no—
487C42—44no—
488C42—44no—
489C42—44no—
490C42#12012yes—
491C42#12012yes—
492C42#12114yes—
493C42#12114yes—
494C42#12312yes—
495C42#12312yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—21no—
512C23—24no—
513C23—21no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23C4×C22⋊C412yes—
521C23C4×C22⋊C412yes—
522C23C2×#10112yes—
523C23#41312yes—
524C23C22×C4211yescharacteristic
525C23C22×C22⋊C411yescharacteristic
526C23C2×#11711yescharacteristic
527C23C2×C4×D412yes—
528C23C4×#4112yes—
529Q8—44no—
530Q8—48no—
531Q8—44no—
532C2×C4—28no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4C4×C22⋊C412yes—
591C2×C4C4×C22⋊C412yes—
592C2×C4#38612yes—
593C2×C4#38612yes—
594C2×C4#38612yes—
595C2×C4#38612yes—
596C2×C4C2×#10112yes—
597C2×C4C2×#10112yes—
598C2×C4#41312yes—
599C2×C4#41312yes—
600C2×C4C2×C4×D414yes—
601C2×C4C2×C4×D412yes—
602C2×C4C2×C4×D412yes—
603C2×C4C4×#4114yes—
604C2×C4C2×#12011yescharacteristic
605C2×C4C2×#12112yes—
606C2×C4C2×#12311yescharacteristic
607C2×C4#52514yes—
608C2×C4#52512yes—
609C2×C4#52914yes—
610C2×C4#53212yes—
611C8—216no—
612C8—48no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—48no—
631C22—48no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22C4×#10114yescomplemented
637C22C2×C4×C22⋊C411yescharacteristic
638C22C42×D412yes—
639C22C2×#38611yescharacteristic
640C22#229012yes—
641C22#229711yescharacteristic
642C22C22×#10111yescharacteristic
643C22C2×#41311yescharacteristic
644C22C2×#41311yescharacteristic
645C22C2×#41311yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—44no—
654C4—44no—
655C4—41no—
656C4—41no—
657C4—82no—
658C4—82no—
659C4#190114yesdirect product factor
660C4C4×#12012yes—
661C4C4×#12114yes—
662C4C4×#12312yes—
663C2—24no—
664C2—22no—
665C2—22no—
666C2—42no—
667C2C4×#38611yescharacteristic
668C2C2×C4×#10111yescharacteristic
669C2C4×#41311yescharacteristic
670C2C2×#190111yescharacteristic
671C2#2098911yescharacteristic
672C2#2142511yescharacteristic
673C2#2142611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×C4×#101C2semidirect
6. #10963C2semidirect
7. #10913C2semidirect
9. C4×#343C2semidirect
40. #1901C4direct (N × H)
100. C2×#101C2×C4semidirect
159. #343C2×C4semidirect
411. C22×C4#101semidirect
636. C22C4×#101semidirect
659. C4#1901direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×C4×C22⋊Q82
C2×#19012
C2×C4×#1012
C4×#4072
#109632
#109132
#107392
C4×#3432