Finite Group Databasebeta

← C2×#14300 #515118

Subgroups — C2×#14300 #515118

Order 512.

8010 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic
2C22×#394C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic, complemented
4C22×#385C212yesmaximal, complemented
5#14300C2116yesmaximal, direct product factor
6C2×#1444C211yesmaximal, characteristic, complemented
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1440C212yesmaximal, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1438C211yesmaximal, characteristic
11C2×#1434C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C212yesmaximal, complemented
13C24×D4C2211yescharacteristic
14C25×C4C2212yes—
15C22×C4×D4—28no—
16C22×C4×D4—216no—
17C22×C4×D4C2218yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28C2×#394C4116yescomplemented
29C2×#394C2218yes—
30C2×#387—28no—
31C2×#387C2218yescomplemented
32C2×#385—216no—
33C2×#385C22116yescomplemented
34C22×#95C2211yescharacteristic
35C22×#95C2211yescharacteristic
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2211yescharacteristic
45#1444C2218yescomplemented
46#1441C2218yescomplemented
47#1440C22116yescomplemented
48#1439C22116yescomplemented
49#1438C2218yes—
50#1434C2218yescomplemented
51C22⋊#95C22116yescomplemented
52C26C2312yes—
53C23×D4—28no—
54C23×D4—28no—
55C23×D4—24no—
56C23×D4—24no—
57C23×D4C2×C418yes—
58C23×D4C2314yes—
59C24×C4—28no—
60C24×C4—28no—
61C24×C4—44no—
62C24×C4—44no—
63C24×C4—44no—
64C24×C4—44no—
65C24×C4D4116yes—
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2318yes—
77C2×C4×D4—2128no—
78C2×C4×D4—264no—
79C2×C4×D4—232no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2×C418yescomplemented
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2318yes—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2×C4116yescomplemented
108C22×C22⋊C4C2×C4116yescomplemented
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114C22×C42—28no—
115C22×C42D418yes—
116C22×C42C2314yes—
117#394C2×C4164yescomplemented
118#387—232no—
119#385—264no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95—44no—
128C2×#95—44no—
129C2×#95—44no—
130C2×#95C2×C418yescomplemented
131C2×#95C2314yes—
132C2×#95C2318yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2314yes—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25C22×C418yes—
155C25C2×D418yes—
156C25#4118yes—
157C25C2414yes—
158C25C2411yescharacteristic
159C22×D4—232no—
160C22×D4—232no—
161C22×D4—264no—
162C22×D4—264no—
163C22×D4—216no—
164C22×D4—216no—
165C22×D4—232no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4C22×C4116yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—264no—
201C23×C4—216no—
202C23×C4—216no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—432no—
221C23×C4—432no—
222C23×C4—48no—
223C23×C4—48no—
224C23×C4—48no—
225C23×C4—48no—
226C23×C4—432no—
227C23×C4—432no—
228C23×C4—432no—
229C23×C4—432no—
230C23×C4C22×C414yes—
231C23×C4C22×C414yes—
232C23×C4C22×C418yes—
233C23×C4C22×C414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D418yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D418yes—
239C23×C4C2×D4132yes—
240C23×C4#4114yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C4×D4—2256no—
252C2×C4⋊C4—232no—
253C2×C4⋊C4—232no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—216no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4C22×C4116yescomplemented
266C2×C22⋊C4—264no—
267C2×C22⋊C4—264no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—432no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—432no—
284C2×C22⋊C4C22×C4132yescomplemented
285C2×C22⋊C4C22×C4132yescomplemented
286C2×C42—232no—
287C2×C42—216no—
288C2×C42C2×D4116yes—
289#95—216no—
290#95—216no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—48no—
297#95—48no—
298#95—48no—
299#95C22×C4116yescomplemented
300C24—216no—
301C24—216no—
302C24—216no—
303C24—216no—
304C24—216no—
305C24—216no—
306C24—216no—
307C24—216no—
308C24—216no—
309C24—216no—
310C24—216no—
311C24—216no—
312C24—216no—
313C24—216no—
314C24—216no—
315C24—216no—
316C24—216no—
317C24—216no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24C4×D4132yes—
347C24C23×C418yes—
348C24C23×C412yes—
349C24C22×D418yes—
350C24C22×D414yes—
351C24C22×D412yes—
352C24C2×#4118yes—
353C24C2×#4114yes—
354C24C2×#4112yes—
355C24H(5,𝔽2)12yes—
356C24H(5,𝔽2)12yes—
357C24H(5,𝔽2)14yes—
358C24H(5,𝔽2)12yes—
359C242−1+412yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C2×D4—232no—
363C2×D4—232no—
364C2×D4—264no—
365C2×D4—2256no—
366C2×D4—264no—
367C2×D4—232no—
368C22×C4—216no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—264no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—432no—
440C22×C4—432no—
441C22×C4—4256no—
442C22×C4—432no—
443C22×C4—432no—
444C22×C4—432no—
445C22×C4—432no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4C4×D4116yes—
481C22×C4#120116yes—
482C22×C4C23×C418yes—
483C22×C4C23×C414yes—
484C22×C4C23×C414yes—
485C22×C4C23×C414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D418yes—
488C22×C4C22×D414yes—
489C22×C4C22×D418yes—
490C22×C4C22×D414yes—
491C22×C4C2×#4114yes—
492C4⋊C4—232no—
493C4⋊C4—232no—
494C4⋊C4—432no—
495C4⋊C4—416no—
496C22⋊C4—264no—
497C22⋊C4—264no—
498C22⋊C4—432no—
499C22⋊C4—432no—
500C22⋊C4—432no—
501C42—232no—
502C23—264no—
503C23—264no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—264no—
523C23—432no—
524C23—432no—
525C23—432no—
526C23—432no—
527C23—432no—
528C23—432no—
529C23—432no—
530C23—432no—
531C23—432no—
532C23—432no—
533C23—432no—
534C23—432no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×C4×D4116yes—
552C23C2×C4×D418yes—
553C23C2×C4×D414yes—
554C23#51814yes—
555C23#52014yes—
556C23#52018yes—
557C23C2×#12014yes—
558C23#52518yes—
559C23#52514yes—
560C23#53418yes—
561C23#53418yes—
562C23#53514yes—
563C23#53514yes—
564C23#53618yes—
565C23#53614yes—
566C23#53714yes—
567C23#53718yes—
568C23#53818yes—
569C23#53918yes—
570C23#53918yes—
571C23#53918yes—
572C23#54014yes—
573C23#54214yes—
574C23#54218yes—
575C23C24×C411yescharacteristic
576C23C23×D412yes—
577C23C23×D411yescharacteristic
578C23C22×#4112yes—
579C23C22×#4111yescharacteristic
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+412yes—
586D4—2128no—
587C2×C4—232no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—232no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—4128no—
612C2×C4—416no—
613C2×C4—432no—
614C2×C4—432no—
615C2×C4—416no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—416no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—88no—
652C2×C4—816no—
653C2×C4—88no—
654C2×C4C2×C4×D418yes—
655C2×C4C2×#12018yes—
656C22—216no—
657C22—216no—
658C22—216no—
659C22—48no—
660C22—48no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—4128no—
669C22—48no—
670C22—48no—
671C22—48no—
672C22C4×#12018yes—
673C22#2322116yes—
674C22#232418yes—
675C22#2325116yes—
676C22#232618yes—
677C22#261318yes—
678C22#261618yes—
679C22#2630116yes—
680C22#263318yes—
681C22#263618yes—
682C22#2638116yes—
683C22C22×C4×D412yes—
684C22C22×C4×D411yescharacteristic
685C22C2×#51811yescharacteristic
686C22C2×#52011yescharacteristic
687C22C2×#52012yes—
688C22C22×#12011yescharacteristic
689C22C2×#52512yes—
690C22C2×#52511yescharacteristic
691C22C2×#53412yes—
692C22C2×#53412yes—
693C22C2×#53511yescharacteristic
694C22C2×#53511yescharacteristic
695C22C2×#53612yes—
696C22C2×#53611yescharacteristic
697C22C2×#53711yescharacteristic
698C22C2×#53712yes—
699C22C2×#53812yes—
700C22C2×#53912yes—
701C22C2×#53912yes—
702C22C2×#53912yes—
703C22C2×#54011yescharacteristic
704C22C2×#54211yescharacteristic
705C22C2×#54212yes—
706C4—28no—
707C4—28no—
708C4—44no—
709C4—48no—
710C4—44no—
711C4—44no—
712C4—48no—
713C4—82no—
714C4—82no—
715C4—84no—
716C2—416no—
717C2#14300116yesdirect product factor
718C2C2×C4×#12011yescharacteristic
719C2C2×#232212yes—
720C2C2×#232411yescharacteristic
721C2C2×#232512yes—
722C2C2×#232611yescharacteristic
723C2C2×#261311yescharacteristic
724C2C2×#263012yes—
725C2C2×#263812yes—
726C2C2×#263611yescharacteristic
727C2C2×#263311yescharacteristic
728C2C2×#261611yescharacteristic
subgroupG/H
center, socleC25C24
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#387C2semidirect
4. C22×#385C2semidirect
5. #14300C2direct (N × H)
6. C2×#1444C2semidirect
7. C2×#1441C2semidirect
8. C2×#1440C2semidirect
9. C2×#1439C2semidirect
11. C2×#1434C2semidirect
12. C2×C22⋊#95C2semidirect
28. C2×#394C4semidirect
31. C2×#387C22semidirect
33. C2×#385C22semidirect
45. #1444C22semidirect
46. #1441C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
50. #1434C22semidirect
51. C22⋊#95C22semidirect
89. C22×C4⋊C4C2×C4semidirect
107. C22×C22⋊C4C2×C4semidirect
108. C22×C22⋊C4C2×C4semidirect
117. #394C2×C4semidirect
130. C2×#95C2×C4semidirect
265. C2×C4⋊C4C22×C4semidirect
284. C2×C22⋊C4C22×C4semidirect
285. C2×C22⋊C4C22×C4semidirect
299. #95C22×C4semidirect
717. C2#14300direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×D42
C22×#3942
C22×#3872
C22×#3852
#143002
C2×#14442
C2×#14412
C2×#14402
C2×#14392
C2×#14382
C2×#14342
C2×C22⋊#952